Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory

Springer Monographs in Mathematics Dietlinde Lau Function Algebras on Finite Sets A Basic Course on Many-Valued Logic...

Author: Dietlinde Lau

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Springer Monographs in Mathematics

Dietlinde Lau

Function Algebras on Finite Sets A Basic Course on Many-Valued Logic and Clone Theory

With 42 Figures and 46 Tables

123

Dietlinde Lau Institute for Mathematics University of Rostock Universitätsplatz 1 18055 Rostock, Germany e-mail: [email protected]

Library of Congress Control Number: 2006929534

Mathematics Subject Classification (2000): 03B50, 08Axx, 08A40, 08A30, 08A05, 06A15 ISSN 1439-7382 ISBN-10 3-540-36022-0 Springer-Verlag Berlin Heidelberg New York ISBN-13 978-3-540-36022-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2006 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the author using a Springer TEX macro package Production: LE-TEX Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: Erich Kirchner, Heidelberg Printed on acid-free paper

44/3100YL - 5 4 3 2 1 0

To my mother, Brigitte Lau

Preface

Functions (or operations), which are defined on finite sets, occur in almost all fields of mathematics. For more than 80 years, algebras (so-called function algebras), whose universes are such functions, have been studied. Particularly in Mathematical Logic, in Universal Algebra (more precise in the Clone Theory), and in parts of Computer Science, certain knowledge about these algebras are subject of the fundamental knowledge. Currently only one book has been published about function algebras, apart from certain monographs or dissertations of specific themes, survey articles and books that contain sections about function algebras or clones. This book has been written by R. P¨ oschel und L. A. Kaluˇznin in the German language and gives a very good overview about the results achieved up to 1979. During the last 26 years, many new results have been obtained; however, a new book about function algebras is overdue. The aim of the present book is to introduce the reader to the theory of function algebras and to give the latest state of research for some selected fields. The author would like to acquaint the reader with proof of the fundamental theorems and the different proof methods, to enable research in the field of the function algebras. This book is self-contained. All necessary fundamental concepts and facts are introduced, but some background knowledge about linear and abstract algebra would be helpful for readers. In the following Introduction, the reader finds short summaries of the 26 chapters of this book. The adjoined section Preliminaries explains abbreviations and some general symbols, which are more or less standard, and gives some facts from basic mathematics. Part I of this book introduces the reader to Universal Algebra to provide almost every knowledge concerning other fields of mathematics. Moreover, this part of Universal Algebra informs the reader that many of the following results

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Preface

of function algebras reply to questions that arise upon studying an algebra. The structure of this book enables the reader to skip the first part and immediately start reading Part II Function Algebras. The author provides the reader with new proofs concerning classic results of the theory of function algebras. The remaining proofs are adapted to the style of the book. The theorems from Sections 14.10, 15.4, 18.2, 18.3, and from Chapter 17 have not yet been published. Small mistakes from the original papers (including the papers of the author) have been corrected in this book without referring to the original mistakes. During the writing process I have tried to solve open mathematics questions and problems, which I recognized during my study of the corresponding literature. In some cases colleagues helped me. In such a case their names are mentioned in the relevant places in the book. I would like to thank my colleagues as well as the authors of the articles I referred to in my book. Especially, I would like to thank my doctoral thesis supervisors, Prof. G. Burosch (Rövershagen), and Prof. V. B. Kudrjavcev (Moscow). I received first information on the complexity of themes in this book from their lectures. I owe Prof. I. G. Rosenberg (Montreal) much gratitude, as I learned many proof methods while studying his papers. I thank my colleagues Prof. Dr. K. Denecke (Potsdam), Prof. Dr. L. Haddad (Kingston), and Prof. Dr. R. P¨ oschel (Dresden) for their fruitful cooperation during many years. In particular, Prof. Haddad was a very great aid during the finishing of the book. I also owe him the organization and the financing (by the Natural Sciences and Engineering Council of Canada) of a linguistic correction of my text by Ms Eryn Kirkwood (from RedInk Editors, Ottawa ON, Canada). Ms Kirkwood deserves my special thanks.

Rostock, June 2006

Dietlinde Lau

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Part I Universal Algebra 1

Basic Concepts of Universal Algebra . . . . . . . . . . . . . . . . . . . . . . . 1.1 Universal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Examples of Universal Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Gruppoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Semirings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.7 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.8 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.9 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.10 Semilattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.11 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.12 Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.13 Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 27 27 28 28 28 28 28 29 29 29 29 30 30 30 31

2

Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Two Definitions of a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Examples for Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Isomorphic Lattices and Sublattices . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Complete Lattices and Equivalence Relations . . . . . . . . . . . . . . .

35 35 39 39 41

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3

Hull Systems and Closure Operators . . . . . . . . . . . . . . . . . . . . . . . 45 3.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Some Properties of Hull Systems and Closure Operators . . . . . . 46

4

Homomorphisms, Congruences, and Galois Connections . . . 4.1 Homomorphisms and Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Congruence Relations and Factor Algebras of Algebras . . . . . . . 4.3 Examples for Congruence Relations and Some Homomorphism Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Congruences on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Congruences on Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Galois Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 52 56 56 58 59

5

Direct and Subdirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.1 Direct Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 5.2 Subdirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6

Varieties, Equational Classes, and Free Algebras . . . . . . . . . . . 6.1 Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Terms, Term Algebras, and Term Functions . . . . . . . . . . . . . . . . . 6.3 Equations and Equational Classes . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Free Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Connections Between Varieties and Equational Defined Classes 6.6 Deductive Closure of Equation Sets and Equational Theory . . . 6.7 Finite Axiomatizability of Algebras . . . . . . . . . . . . . . . . . . . . . . . .

71 71 73 76 78 81 82 84

Part II Function Algebras 1

Basic Concepts, Notations, and First Properties . . . . . . . . . . . 91 1.1 Functions on Finite Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1.2 Operations on PA , Function Algebras . . . . . . . . . . . . . . . . . . . . . . 94 1.3 Superpositions, Subclasses, and Clones . . . . . . . . . . . . . . . . . . . . . 96 1.4 Generating Systems for PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 1.5 Some Applications of the Function Algebras . . . . . . . . . . . . . . . . 104 1.5.1 Classification of Universal Algebras . . . . . . . . . . . . . . . . . . 104 1.5.2 Propositional Logic and First Order Logic . . . . . . . . . . . . 105 1.5.3 Many-Valued Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 1.5.4 Information Transformer . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 1.5.5 Classification of Combinatorial Problems . . . . . . . . . . . . . 118

2

The Galois-Connection Between Function- and Relation-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 2.2 Diagonal Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

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2.3 Elementary Operations on Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.4 Relation Algebras, Co-Clones, and Derivation of Relations . . . . 127 2.5 Some Operations on Rk Derivable from the Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.6 The Preserving of Relations; Pol, Inv . . . . . . . . . . . . . . . . . . . . . . 130 2.7 The Relations χn and Gn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 2.8 The Operator ΓA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 2.9 The Galois Theory for Function- and Relation-Algebras . . . . . . 135 2.10 Some Modifications of the P ol-Inv-Connection . . . . . . . . . . . . . . 137 2.10.1 Galois Theory for Finite Monoids and Finite Groups . . . 137 2.10.2 Galois Theory for Iterative Function Algebras . . . . . . . . . 139 2.11 Some Connections Between the Relation Operations . . . . . . . . . 142 3

The Subclasses of P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 3.1 Definitions of the Subclasses of P2 and Post’s Theorem . . . . . . . 145 3.2 A Proof for Post’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.2.1 The Subclasses A of P2 with A ⊆ L and A ⊆ S . . . . . . . . 149 3.2.2 The Subclasses of L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.2.3 The Subclasses of S, Which Are Not Subsets of L . . . . . 155 3.2.4 A Completeness Criterion for P2 . . . . . . . . . . . . . . . . . . . . 156

4

The Subclasses of Pk Which Contain Pk1 . . . . . . . . . . . . . . . . . . . 159

5

The Maximal Classes of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1 Introduction, a Rough Description of the Maximal Classes . . . . 163 5.2 Definitions of the Maximal Classes of Pk . . . . . . . . . . . . . . . . . . . 165 5.2.1 Maximal Classes of Type M (Maximal Classes of Monotone Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.2.2 Maximal Classes of Type S (Maximal Classes of Autodual Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.2.3 Maximal Classes of Type U (Maximal Classes of Functions, Which Preserve Non-Trivial Equivalence Relations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 5.2.4 Maximal Classes of Type L (Maximal Classes of Quasi-Linear Functions) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 5.2.5 Maximal Classes of Type C (Maximal Classes of Functions, Which Preserve Central Relations) . . . . . . . . . 173 5.2.6 Maximal Classes of Type B (Maximal Classes of Functions, Which Preserve h-Universal Relations) . . . . . 174 5.3 Proof of the Maximality of the Classes Defined in Section 5.2 . 179 5.4 The Number of the Maximal Classes of Pk . . . . . . . . . . . . . . . . . . 183 5.5 Remarks to the Maximal Classes of Pk (l) . . . . . . . . . . . . . . . . . . . 188

6

Rosenberg’s Completeness Criterion for Pk . . . . . . . . . . . . . . . . 191 6.1 Proof of Completeness Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

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7

Further Completeness Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.1 A Criterion for Sheffer-Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2 A Completeness Criterion for Surjective Functions . . . . . . . . . . . 216 7.3 Fundamental Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

8

Some Properties of the Lattice Lk . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.1 Cardinality Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.2 On the Cardinalities of Maximal Sublattices of Lk . . . . . . . . . . . 224 8.3 Some Strategies for the Determination of Sublattices of Lk . . . . 229

9

Congruences and Automorphisms on Function Algebras . . . 233 9.1 Some Basic Concepts and First Properties . . . . . . . . . . . . . . . . . . 234 9.2 Congruences on the Subclasses of P2 . . . . . . . . . . . . . . . . . . . . . . . 235 9.3 Characterization of the Non-Arity Congruences . . . . . . . . . . . . . 238 9.4 About the Number of the Congruences on a Subclass of Pk . . . 243 9.5 A Criterion for the Proof of the Countability of Con A for Certain A ⊆ Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 9.6 Congruences on Some Classes of Linear Functions . . . . . . . . . . . 250 9.7 Congruences on the Maximal Classes of Pk . . . . . . . . . . . . . . . . . 256 9.8 Congruences on Subclasses of [Pk1 ] . . . . . . . . . . . . . . . . . . . . . . . . . 265 9.9 Congruences on Some Subclasses of Pk,l . . . . . . . . . . . . . . . . . . . . 273 9.10 Some Further General Properties of the Congruences and the l-Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.11 The Connection Between Clone Congruences and Fully Invariant Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 9.12 Automorphisms of Function Algebras . . . . . . . . . . . . . . . . . . . . . . 285

10 The Relation Degree and the Dimension of Subclasses of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.1 The Definition of the Relation Degree and of the Dimension of a Subclass of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 10.2 The Dimensions and Relation Degrees of Post’s Classes . . . . . . 293 10.3 Further Examples of the Dimension and Relation Degree of Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 11 On Generating Systems and Orders of the Subclasses of Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 11.1 Some General Properties of Generating Systems and Bases . . . 308 11.2 The Orders and Sheffer-Functions of the Classes of Type C1 , S or U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11.3 Orders of the Classes of Type L, C, B . . . . . . . . . . . . . . . . . . . . . . 314 11.4 The Order of P olk for ∈ Mk and k ≤ 7 . . . . . . . . . . . . . . . . . . 319 11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 11.6 Classifications and Basis Enumerations in Pk . . . . . . . . . . . . . . . 332

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12 Subclasses of Pk,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 12.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 12.2 Some Properties of the Inverse Images . . . . . . . . . . . . . . . . . . . . . 337 12.3 On the Number of the B-projectable Subclasses of Pk,2 , B ⊆ P2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 12.4 The Pl -projectable and the P oll {α}-projectable Subclasses of Pk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 12.5 The Maximal and the Submaximal Classes of Pk,2 . . . . . . . . . . . 354 12.6 The Classes A with M ∩ T0 ∩ T1 ⊆ prA or L ∩ T0 ∩ S ⊆ prA or prA = M ∩ S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 13 Classes of Linear Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 13.1 Some Properties of the Subclasses of Ud That Contain rd . . . . . 384 13.2 The Subclasses of Linear Functions of Pk with k ∈ P . . . . . . . . . 387 13.3 A Survey of Further Results on Linear Functions . . . . . . . . . . . . 390 14 Submaximal Classes of P3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 14.1 A Survey of the Submaximal Classes of P3 . . . . . . . . . . . . . . . . . . 400 14.2 Some Declarations and Lemmas for Sections 14.3–14.9 . . . . . . . 408 14.3 Proof of Theorem 14.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 14.4 Proof of Theorem 14.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 14.5 Proof of Theorem 14.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415 14.6 Proof of Theorem 14.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 14.7 Proof of Theorem 14.1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418 14.8 Proof of Theorem 14.1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 14.9 Proof of Theorem 14.1.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 14.10On the Cardinality of L↓3 (A) for Submaximal Clones A . . . . . . . 425 15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 15.1 The Lattice of Subclasses of P3 of Linear Functions . . . . . . . . . . 433 15.2 The Subsemigroups of (P31 ; ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 15.3 Classes of Quasilinear Functions of P3 . . . . . . . . . . . . . . . . . . . . . . 456 15.3.1 Some Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 15.3.2 Subclasses of L0,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 15.3.3 The Subclasses of L0,1 ∪ L0,2 That Are Not Subclasses of L0,1 or L0,2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 15.3.4 The Remaining Subclasses of L . . . . . . . . . . . . . . . . . . . . . 463 15.4 The Subclasses of [O1 ∪ {max}] . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 15.4.1 Some Descriptions of the Class M . . . . . . . . . . . . . . . . . . . 464 15.4.2 Some Lemmas and a Rough Partition of the Subclasses of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 15.4.3 The Subclasses of [M 1 ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 15.4.4 The Subclasses of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 15.4.5 The Subclasses of M ∩ P ol3 {(0, 2)} . . . . . . . . . . . . . . . . . . 482

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15.4.6 The Remaining Subclasses of M . . . . . . . . . . . . . . . . . . . . . 488 16 The Maximal Classes of a∈Q P olk {a} for Q ⊆ Ek . . . . . . . . . 499 16.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 16.2 Results of Chapter 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 16.3 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502 16.4 Proof of Theorem 16.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 17 Maximal Classes of P olk El for 2 ≤ l < k . . . . . . . . . . . . . . . . . . . 515 17.1 Notations, Definitions, and Some Lemmas . . . . . . . . . . . . . . . . . . 515 17.2 Results of Chapter 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 17.3 Maximality Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520 17.4 Some Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528 17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529 17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk ) . . . . 549 18 Further Submaximal Classes of Pk . . . . . . . . . . . . . . . . . . . . . . . . . 555 18.1 The Maximal Classes of P olk s for s ∈ Sk . . . . . . . . . . . . . . . . . 555 18.2 Some Maximal Classes of a Maximal Class of Type U . . . . . . . . 561 18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)}) . . . . . . . . . . . . 573 18.3.1 Definitions of the U -Maximal Classes . . . . . . . . . . . . . . . . 573 18.3.2 Proof of the U -Maximality of the Classes Defined in 18.3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576 18.3.3 Proof of the Completeness Criterion for U . . . . . . . . . . . . 584 19 Minimal Classes and Minimal Clones of Pk . . . . . . . . . . . . . . . . 589 19.1 Minimal Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589 19.2 The Five Types of Minimal Clones . . . . . . . . . . . . . . . . . . . . . . . . . 590 20 Partial Function Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 20.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 20.2 One-Point Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 20.3 Description of Partial Clones by Relations . . . . . . . . . . . . . . . . . . 604 3 . . . . . . . . . . . . . . . . . . 606 2 and P 20.4 The Maximal Partial Classes of P k . . . . . . . . . . . . . . . . . . . . . . . . 614 20.5 The Completeness Criterion for P k . . . . . . . . 616 20.6 Some Properties of the Maximal Partial Clones of P 20.7 Intervals of Partial Clones That Contain a Maximal Clone . . . . 619 20.8 Intervals of Boolean Partial Classes . . . . . . . . . . . . . . . . . . . . . . . . 627 20.9 On Congruences of Partial Clones . . . . . . . . . . . . . . . . . . . . . . . . . 628 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663

Introduction

The present book deals with a subarea of the Discrete Mathematics. We study functions, which are defined on finite sets, and we study the composition of these functions. Such functions are used, for example, in Computer Science (in particular, in the Switching Theory and in the Theory of Automata), in Mathematical Logic, and in Universal Algebra (in particular, the Clone Theory).1 In other words, we choose an arbitrary finite set A and study an algebra, whose universe is the set PA

(or Pk := P{0,1,2,..,k−1} )

of all n-ary mappings (n ∈ N), which maps the Cartesian power An (of all ordered n-tuples of elements from A) into A, and whose operations are the so-called superposition operations that are described as follows:2 – permutation of variables – identification of variables – adding of fictitious variables and – substitution of variables of a function by functions Denote F (n) the set of all n-ary functions of F ⊆ PA . Moreover, let Ω be the set of all superposition operations described above. Then, PA := (PA ; Ω) is called (full) function algebra (on A). The universe of a subalgebra of PA is called closed set or a subclass (or briefly a class) of PA . If F is a subclass of Pk then F is also called a subclass of the k-valued logic. The set of all closed sets of Pk together with the set inclusion forms a lattice Lk . 1 2

See, for this purpose, also Section 1.5 of Part II. One finds an exact definition of these operations in Section 1.2 of Part II.

2

Introduction

The closed sets of P2 were already determined in the papers [Pos 20] and [Pos 41] by E. L. Post. For over 50 years, many papers have dealt with function algebras or closed sets of Pk for arbitrary k. One finds a survey of the essential articles, which were published up to the year 1978, on the topic in [P¨ os-K 79] with 730 references and in [Ros 77] with 464 references. To give the reader a first impression of the problems handled in the theory of the function algebras, some explanations are subsequently given to the completeness problem3 , which stood in the center of a line of investigations in many articles: One finds a criterion, to decide if a set of functions that belong to Pk is sufficient for the construction of any arbitrary other function of Pk (by means of the superposition operations). A general answer to that is given by the following criterion, which was formulated by E. L. Post in 1921, first, for Boolean functions, i.e., for functions of P2 : A set F ⊆ Pk is complete in Pk if and only if F is a subset of no maximal class of Pk . A closed set M ⊂ Pk is said to be maximal in Pk , if M can not be properly extended to a closed proper subset of Pk . With regard to such and similar question formulations, one naturally deals with the structure of the subclasses of Pk or with the lattice of the subclasses of Pk . The first and most important result in this direction is Post’s pioneering description of L2 ([Pos 41], see Figure 1), now known as Post lattice. One can describe the many Post’s classes with the aid of the classes P2 , M , S, L, Ta (1) Ta,µ , D ∪ C, K ∪ C, [P2 ], C, (a ∈ {0, 1}, µ ∈ N \ {1}) and through formation of certain intersections of these classes, where T0 , T1 , M , S, and L are exactly the maximal classes of P2 .4 For some time, it was thought that L3 is equally simple. Then, Ju. I. Janov and A. A. Muˇcnik showed in the year 1959 the existence of subclasses of Pk for k ≥ 3 with infinite and without basis, respectively, and this implies the existence of a non-countable infinite set of subclasses in Lk (see also [Ehr 55]). Thus, there are as many closed sets in Pk as there are subsets of Pk for k ≥ 3. In contrast to k = 2, therefore, it seems hard to give an effective description of Lk for k ≥ 3. Nevertheless, one could determine the maximal classes of Pk for arbitrary k. S. V. Jablonskij determined all 18 maximal classes of P3 in [Jab 54] and [Jab 58]. Moreover, he extended his results by describing several types of maximal classes of Pk . This work was continued by V. V. Martynyuk [Mar 60], E. Ju. Zacharova [Zac 67], R. A. Bairamov [Bai 67] ˇ ˇ 67]. In [Zac-K-J (some results incorrect) and V. L. Rvaˇcev/ L. I. Sljarov [Rva-S 71] it has been reported that late A. I. Mal’tsev had all 82 maximal sets of P4 . I. G. Rosenberg published the missing maximal classes for Pk in [Ros 65] and 3 4

In Section 1.5 of Part II, it is shown that this problem is a mathematical way to describe problems that occur during the construction of electronic circuits. See Chapter 3 of Part II for details.

Introduction

3

he proved in the book [Ros 70a] that there can not be any further maximal classes for arbitrary k, whereby the general completeness problem for Pk was solved. Rosenberg’s description of the maximal classes is based on the idea of a function preserving a relation; i.e., he could prove that every maximal class has the form P olk , where is a certain h-ary relation on Ek , 1 ≤ h ≤ k, k ≥ 3 and P olk is the set of all functions of Pk , which preserve the relation . To find the maximal ones among classes of the form P olk , he designed a sieve method, which eliminates at each step every relation for which a relation σ is found in the list such that P olk ⊆ P olk σ. The process terminates when the candidate list contains only relations such that P olk can be proved to be maximal. This approach was twofold in scope. To discover the maximal classes and, at the same time, to prove that one has a full list of them. The idea to describe classes through relations kept on being developed to a Galois-theory for function algebras and relation algebra by V. G. Bodnarˇcuk, L. A. Kaluˇznin, V.N. Kotov, B. A. Romov (in [Bod-K-K-R 69]). The interesting subclasses of Pk are not only the maximal ones, and so the Galois theory was and is an important aid for the study of subclasses of Pk . Namely, one can classify finite universal algebras and also combinatorial problems with the aid of the elements of Lk (see Section 1.5 of Part II). These and many other applications presuppose, however, precise knowledge about the subclasses of Pk . Therefore, the object of this book is to explain some methods to the finding of subclasses of Pk and to give a survey of subclasses and their properties, which were determined in the last years. To familiarize the reader from the beginning also with the algebraic side of the function algebras and to show that the concepts introduced for function algebras are only special cases of more general concepts mostly from the Universal Algebra, we begin with an introduction to the Universal Algebra in the first part of this book. One finds supplements to this short introduction to the Universal Algebra in the books [Bur-S 81], [Coh 65], [Gr¨ a 68], [Ihr 93] (or [Ihr 2003]), [McK-M-T 87], [Wer 78] and [Lau 2004], volume 2. In Part II, we deal only with the function algebras. The construction of the book is chosen in a way that one can immediately begin reading Part II and, if one needs concepts and facts from Part I, one can reference these. Unlike Part I, which is strongly linearly structured in that each chapter is based on the preceding chapter, after studying the first two chapters of Part II, all successive studies can also be studied. We concentrate in Part II on the following topics: • • • • •

Basic concepts and notations Galois-connection between function algebras and relation algebras Post’s results on the subclasses of P2 the maximal classes of Pk completeness criterions for Pk (in particular, the Rosenberg’s completeness criterion)

4

Introduction

• congruences on subclasses of Pk • complexity measures for generating systems (for example as order or relation degree and dimension) of certain subclasses of Pk • subclasses of linear functions of Pp (p prime number) • a survey on subclasses of Pk , whose functions have at most two different values • submaximal classes of P3 • the description of all finite and countably infinite sublattices of the depth 1 or 2 of the lattice of all subclasses of P3 • submaximal classes of Pk , which are subsets of maximal classes described by unary relations • a survey of results to maximal classes of Pk for arbitrary k • minimal clones • partial function algebras. One finds completions to this book and a survey on further topics of the function-algebras-theory in [P¨ os-K 79] and [Ros 84]. Subsequently the content of this text is described in brief without the necessary concepts and notations.

Part I Chapter 1 begins with the definition of a universal algebra (briefly: algebra) as a pair (A;F) consists of a nonempty set A and a set F of certain operations on A and gives numerous examples of algebras (among that also the function algebras). In addition, one finds the very important concept of the subalgebra in this chapter. Chapter 2 compiles needed order concepts within the framework of an introduction to the lattice theory. The usual definitions of a lattice are indicated, and the calculation in the lattice theory illustrates by means of the proof of a few classical theorems of the lattice theory. Chapter 3 generalizes observations, which one can make when one examines more closely such concepts as subalgebra or linear hull of subsets of a vector space (or another closure operators of the classical algebra), to the concepts hull system and closure operator. Then, it is shown that these concepts deliver, roughly, the same. In addition, certain combinations to the lattice theory are given, and the lattices, formed from the subalgebras of an algebra, are uniquely characterized through certain properties. Chapter 4 combines properties of such important concepts as the homomorphic and isomorphic mappings between universal algebras, which the readers surely know from the classical algebra here, and which are defined obviously for universal algebras. It is shown, how homomorphic mappings are

Introduction

5

ultimately determined by congruence relations (i.e., by equivalence relations, which are compatible with the operations of algebra). The (general) homomorphism theorem keeps on being proved and is shown by examples, as this theorem can be improved for concrete algebras. A section of Chapter 4 deals with Galois connections between set systems that are continued later for function algebras in Chapter 2 of the second part. Chapter 5 shows how one can form new algebras from given algebras through direct or subdirect products and how one can recognize whether a given algebra is isomorphic to an algebra formed in this way. More precisely: we deal with the following two questions: Which algebras are smallest constituents of given algebras? How can one reduce a given algebra to its smallest constituents? Answers to these questions are given by theorems found by G. Birkhoff. We prove these theorems in Chapter 5. In particular, we prove the representation theorems for algebras: Each finite algebra is isomorphic to a direct product of directly irreducible algebras. Each algebra is isomorphic to a subdirect product of subdirectly irreducible algebras. Chapter 6 deals with classes of algebras of the same type and some theorems on such classes, which were found also by G. Birkhoff. At first we introduce so-called varieties as classes of algebras, which are closed in respect to the formation of subalgebras, homomorphic images and direct products. We then come to a method for the construction of algebra classes that strongly differs from the first method at first sight: Based upon certain equations from variables and operation symbols of a certain type τ , we form the class of all algebras of type τ that these equations fulfill. The result is so-called equational classes. We will see, however, that there is a close connection between the two methods of the algebra class construction: class of algebras is equational defined if and only if it is a variety. In the section on equational classes, we will also treat such concepts as the conclusion of an equational set. In addition, we treat ways to receive such conclusions. In Chapter 9 of Part II, we show that the results on congruences of a subclass F ⊆ PA imply results on subvarieties of the variety, which is generated by the algebra (A; F ) and vice versa.

6

Introduction

Part II Chapter 1 begins with the precise definition of function algebras (PA ; Ω), where Ω is an infinite set of operations, which describe the above superposition operations exactly, on PA first. It is shown then how one can receive the operations of the set Ω through five elementary operations (so-called “Mal’tsevOperationen”) ζ, τ, ∆, ∇ and by means of composition. The basis of further investigations is, then, the (full iterative) function algebra PA := (PA ; ζ, τ, ∆, ∇, ) bzw. Pk := PEk mit Ek := {0, 1, ..., k − 1}. A universe of a subalgebra of Pk is called subclass (or more briefly, class) in the following. If [T ] denotes the set of all functions of PA , which one can form from the functions of T ⊆ PA by means of superposition operations, then we have [F ] = F for arbitrary subclass F of PA . (n) n A mapping defined by ei : EA −→ A, (x1 , x2 , ..., xn ) → xi , where i ∈ {1, 2, ..., n}, is called a projection. A clone is a subclass of PA , which contains all projections of PA . In Chapter 1 there are notations, concepts . . . that are used in the following chapters repeatedly. In addition, it is shown that the set PA and some subsets of PA can be formed from binary functions of PA by means of superposition operations. At the end of Chapter 1, we briefly show how one can use the results of this book and how certain investigations are motivated in the theory of function algebras. Chapter 2 provides tools (like the the concepts of “a function preserves a relation” and “a relation is an invariant for a function”) by which classes, dealt in later chapters, can be effectively described. The set Rk of all n-ary relations on Ek (n = 1, 2, 3, ...) is defined and operations over the set Rk so that Rk with these operations forms a relation algebra Rk . Each universe of a subalgebra of Rk is called a co-clone of Rk . The main result of Chapter 2 is the proof that the lattice of the clones of Pk is antiisomorphic to the lattice of all co-clones of Rk . The basis of this result is the Galois-connection (P ol, Inv), where P ol : P(Rk ) −→ P(Pk ) and Inv : P(Pk ) −→ P(Rk ) are mappings defined by P olk Q := {f ∈ Pk | f preserves each relation of Q}, Invk F := { ∈ Rk | is an invariant for each function f of F } for arbitrary Q ⊆ Rk and arbitrary F ⊆ Pk . The Galois-connection (P ol, Inv) makes it possible to consider the function algebras instead of relation algebras and vice versa. Moreover, one can handle function algebras and relation algebras with equal significance, as it happens in [P¨ os-K 79], for example. We deal, however, with the clones in this book and proceed only with the co-clones if the proof methods developed for function algebras are not sufficient.

Introduction

T0 r T0,2 r

r T0,3 r

pp pp pp T0,∞ rp

r pp pp p rpp

Pr2

r

r

r

r

rM T r1 r r r

r r pp pp p rpp

r

r pp pp p rpp

r

r r

S∩M

r r

K ∪C

r

rS

r

r

r

K

r C0

r

r

r

rI

rL r [P21 ] r

I r C

r

r

pp pp p rpp

r r D∪C r r

pp pp p rpp

7

rT1,2 r rT1,3 r pp pp p rpp

pp pp p rpp T1,∞

r

D

r C1

∅

Fig. 1. The Post’s lattice

The connection between algebras and relations was introduced by M. Krasner ([Kra 46] and [Kra 68/69]). He has completely developed the special Galois-connection between subsets of the permutation groups Sn := {f ∈ (1) Pk | f is bijective} and subsets of Rk . Later, Krasner also developed a sim(1) ilar theory for subsemigroups of Pk . Krasner’s theory was generalized by V. G. Bodnarˇcuk, L. A. Kaluˇznin, V. N. Kotov, and B. A. Romov in [Bod-K-K-R 69]. Chapter 3 gives all subclasses from P2 (the so-called Post’s classes) and proves the completeness of the list. In addition, smallest generating systems (“bases”) and the orders of the Post’s classes are determined. The results of Chapter 3 were found by E. L. Post (see [Pos 20] and [Pos 41]). Figure 1 gives the Hasse-diagram of the lattice of the subclasses of P2 . A knot of the graph without denotation represents a class that can be described as an intersection of classes that lie above it. Chapter 4 describes all classes that contain all unary functions of Pk . There exist k + 1 of such classes, which form a chain in the lattice of all subclasses of Pk . The results of Chapter 4 were found by G. A. Burle (see [Bur 67]). Chapter 5 begins with a coarse description of the maximal classes of Pk . The idea to describe classes this way was indicated already in [Kuz 59] by

8

Introduction

A. V. Kuznecov. Then classes, from which we show later (in Section 5.3) that they are maximal classes of Pk , are defined. (In Chapter 6, the proof is all maximal classes are determined.) As in [Ros 70] the classes are divided in Section 5.2 into six types and characterized by their invariants (relations). We denote the relation sets needed in this case with with Mk , Sk , Uk , Lk , Ck and Bk . Subsequently a maximal class is called a maximal class of the type X, when it can be described with the aid of a relation from set Xk , where X ∈ {M, S, U, L, C, B}. Next to the definitions of the maximal classes of Pk one finds the proofs of some properties of the maximal classes needed in later chapters, and the derivation of a recursion formula for the number determination of the maximal classes of Pk . Chapter 6 gives the proof found by I. G. Rosenberg and published in [Ros 70a] that the classes defined in Section 5.2 are the only maximal classes of Pk for arbitrary k. In some parts, the original proof can be abbreviated by using proof ideas from some papers (for example from [Qua 82] and [P¨ os-K 79]). As already explained, there is also a solution of the completeness problem for Pk through the description of maximal classes of Pk . Chapter 7 shows how one can derive some further completeness criteria for specific question formulations from the Rosenberg’s completeness criterion 6.1. First of all we will handle a criterion for Sheffer functions which was found by G. Rousseau. 5 Then we will show how one can reduce the conditions from Theorem 6.1 if one considers only surjective functions. Finally, we deal with criteria indicate under which conditions a set (⊆ Pk ) consisting of certain unary functions and a Slupecki-function 6 is complete in Pk . Chapter 8 explains the qualitative differences between the lattice L2 of the subclasses of P2 and the lattice Lk of the subclasses of Pk for k ≥ 3. While L2 is countable-infinite, Lk has the cardinality of the continuum (denoted by c) for k ≥ 3. This results from the fact that, for every k ≥ 3, the set Pk has a subclass with an infinite basis. One finds examples of such classes not only in Chapter 8, but also in Sections 12.3 and 14.10. Moreover, we deal with cardinality statements about chains and antichains (in Lk ) and with the embedding of Lk into Lk in Chapter 8. Then the question is clarified, which cardinalities the lattices L↓k (A) 7 for maximal classes A of Pk have. In the last section of Chapter 8, the reader finds “strategies” for the determination of “manageable” sublattices of Lk . In addition, two examples of theorems prove how one can determine certain sections of Lk . Chapter 9 deals with homomorphic mappings from a subclass of Pk onto a 5 6 7

A Sheffer function is a function that every function of Pk can be formed from by means of the superposition operations. This is a function f ∈ Pk \ [Pk1 ] with Im(f ) = Ek . L↓k (A) denotes the set of all subclasses of A.

Introduction

9

subclass of Pk , As generally accepted,8 one can characterize every homomorphism from a class A through a certain congruence relation9 on A. After some basic concepts are defined, all congruences on the subclasses of P2 are determined in Chapter 9. It is the aim of the sections following then, to specify the general homomorphism theorem for function algebras and to find statements on the number of congruences on a subclass of Pk through determination of some general properties of the congruences on subclasses of Pk . Then, all congruences are determined for selected classes (among other things, these are the maximal classes of Pk and certain classes from linear functions). Criteria with which one can find out whether on a partial class of Pk only trivial congruences exist are in addition derived. A further section deals with the connection between clone congruences and the fully invariant congruences on free algebras. The theorem found in this case has interesting inferences and is a bridge between certain investigations in the Universal Algebra and certain investigations in the theory of the function algebras. At the end of this chapter, one can find some results on automorphisms. It is proven that Pk , the subclasses of P2 , and the maximal classes of Pk have only inner automorphisms. The starting point and the basis of subsequent results are derived from an article by A. I. Mal’tsev (see [Mal 66]). Important contributions to the topic of Chapter 9 are also performed by V. V. Gorlov and I. A. Mal’tsev. Chapter 10 deals with the relation degree and the dimension of subclasses A of Pk . The relation degree of A is the smallest number h so that the class A is unambiguously described by a relation set whose elements have the arity h at most. The dimension of A is the smallest arity of a relation that characterizes the class A unambiguously. Chapter 10 begins by investigating the connections between these two complexity measures of the relational description of classes. After that, the relation degrees and dimensions of the Post’s classes, which were found by G. N. Blochina in [Blo 70], are proven. In Section 10.3, one can find further classes for which one knows the relation degree or the dimension. Chapter 11 deals with a further measure (the so-called order of a class), with whose aid the complexity of a description of a subclass of Pk can be characterized. If the class A ⊆ Pk is finitely generated, we call the smallest number r with [A(n) ] = A the order of A. In the case that the subclass A ⊆ Pk is not finitely generated we write ord A = ∞. Section 11.1 shows that one can determine the order easily for a class, if one knows the subclasses of the treated class. Therefore, order statements are often the “by-products” of considerations, which were actually used for determining certain classes or 8 9

See also Chapter 4 of Part I. Those ones are the equivalence relations, which are compatible with the superposition operations on the considered class.

10

Introduction

sublattices of Pk . So, for example, the orders of the Post’s classes result from the considerations of Chapter 3 for verification of the Post’s lattice. Further order statements are found in Chapters 13 and 14. Therefore, in Chapter 11, only the statements on the maximal classes that are described in Chapter 5 are determined. It is proven that, for k ≥ 3, each maximal class of the type S, U, L, B or C1 has the order 2 and that, for the order of a class of the type Ch , where h ≥ 2, the arity h of the descriptive relation of this class is an upper bound. If M ⊂ Pk is a maximal class of the type M, we can only prove that M has the order 2 for the cases 2 ≤ k ≤ 7 and for case that the descriptive binary relation of M has certain properties. For k ≥ 8 there are maximal classes of the type M that do not have a finite order. To prove this fact, we give an example, published by G. Tardos in [Tar 86]. If one has a finite generating class A, it is an interesting problem to clarify which cardinalities are possible for the bases of this class A. In Section 11.2 we show that the proof for ord P olk ≤ 3, where ∈ C1k ∪ Uk ∪ Sk , implies the existence of functions f with [f ] = P olk . Following corresponding notation for functions from Pk , such function f is called a Sheffer function for P olk . The last section explains shortly, as one can determine the cardinalities of the possible bases of the class A, if ord A < ∞. Furthermore, some basic ideas of [Miy 71], [Miy-S-L-R 86], [Miy 88], and [Sto 87] are explained on basis classifications. For more information on the topic, we direct the reader to [Miy 88] and [Sto 87] of M. Miyakawa and I. Stojmenovic. Chapter 12 summarizes the subclasses of the class Pk,2 := {f ∈ Pk | f has only values of the set {0, 1} }, which were found by G. Burosch, J. Dassow, N. Gr¨ unwald, W. Harnau, and the author. When one restricts the domain of a function f (n) ∈ Pk,2 to the set E2n , a homomorphic mapping pr (“projection”) from Pk,2 onto P2 can be defined. Since the image prA of a subclass A of Pk,2 is a subclass of P2 and the subclasses of P2 are known, one can hope to find of certain properties of the inverse images (⊆ Pk,2 ) through the known properties of the images (⊆ P2 ). This hope confirmed itself in a certain sense (see, e.g., Theorem 12.2.5 and Theorem 9.7.6). Conversely, Pk,2 , k ≥ 3, also reflects the negative properties of Pk because the examples of classes from Section 8.1.1 with infinite and without bases are subclasses of Pk,2 . Further, the functions of Pk,2 are important since they can be interpreted as predicates. Chapter 12 is organized as follows: Section 12.1 contains the basic concepts and notations. Section 12.2 contains results on inverse images of subclasses of P2 (with respect to the above projection). The remaining sections deal with the determination of cardinality and with the determination of the elements

Introduction

r r r r pp pp pp p g

r pp pp pp p g

r r

r

r

r

r

r

r r r

r r pp pp pp p g

g

r pp pp pp p g g

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g g

r

r

w w w g g g g g g

r r w

g

g

g

pp pp pp p g g

pp pp pp p g

11

r r r r pp pp pp p g

pp pp pp p g

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g

Fig. 2. A survey on |Nk (B)| for B ∈ L2

of the set Nk (B) := {A | A is a subclass of Pk,2 with pr A = B} for arbitrary subclass B of P2 . In Section 12.3, one can find some structure statements about the lattice of all subclass of Pk,2 . It is also clarified in this section whether the set Nk (B) is finite or infinite or has the cardinality of continuum. Section 12.4 generalizes some statements from Section 12.3 for Pk,l with 2 ≤ l < k. In Section 12.5, the maximal and the submaximal classes of Pk,2 are determined. Then, the investigations are continued from Section 12.3 for k = 3, i.e., for many classes B ⊆ P2 with |N3 (B)| ≤ ℵ0 the elements of the set N3 (B) are determined. Figure 2 shows a survey of the obtained results. This figure shows the Post’s graph, where the knots of this graph are differently labeled. If a class B of P2 is marked in this graph through g, this means that the set Nk (B) has the cardinality of the continuum. The second marker w means that |Nk (B)| ≥ ℵ0 and |N3 (B)| = ℵ0 are valid. For the remaining classes B whose knots do not have any marker the set Nk (B) is finite for arbitrary k ∈ N. Chapter 13 deals with classes of linear functions of Pk , i.e., with closed subsets of the set

12

Introduction

Lk :=

n≥1

{f (n) ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 +

n

ai · xi (mod k)}.

i=1

The lattice of the subclasses of Lk belongs to the earliest and best investigated sublattices of Lk . For the case that k is a prime number, all subclasses, which are no subsets of [L1k ], were determined by A. A. Salomaa in [Sal 64]. The results of [Sal 64] were proven by J. Bagyinszki and J. Demetrovics in [Bag-D 82] and complemented with the remaining subclasses of [L1k ], p ∈ P. ´ Szendrei. For examMany results about linear functions were obtained by A. ple, she proved in [Sze 78] that Lk has only finitely many subclasses, if k is square-free. In addition, she showed that an arbitrary class has, at most, the order 2 (or 3), if k is quare-free and k is an odd number (or an even number), respectively. For the case that k is not square-free, one easily finds a class that does not have any finite basis, whereby the set Lk has infinitely-many subclasses in this case. Section 13.1 starts with properties of certain subclasses of the set (n) := ∈ Lk | ∃a0 , ..., an ∈ Ek ∃j ∈ {1, ..., n} : Ud n≥1 {f f (x) = a0 + aj · xj + d · i=1,i=j ai · xi }. This set is closed, if d is a divisor of k (notation: d | k). Then, with the aid of the results from the first section new proofs are indicated for the theorems of [Sal 64] and [Bag-D 82] in Section 13.2. Chapter 13 ends with a survey on further results about linear and quasi-linear functions. Chapter 14 is the first chapter in this book of this book that deals only with submaximal classes. A subclass (or a subclone) of Pk is called submaximal if it is covered by a maximal class (clone). The concept submaximal class was introduced by I. G. Rosenberg in [Ros 74]. In [Ros 74] one finds also the first results about submaximal classes of Pk (see Chapter 17 for details). The submaximal classes are interesting not only because of their position in the second layer below Pk , but also because further completeness criteria for Pk result from a list of all submaximal classes. For arbitrary k, the full list of the maximal classes of a maximal class A of Pk is only known if A has the type S ([Ros-S 84], see Section 18.1), C1 (see Chapters 16 and 17) or A = P olk , 2 ∪ {(k − 1, k − 1)} (see Section 18.3). There are, however, where := Ek−1 some papers, in which one finds submaximal classes for specific k or only such submaximal classes that contain certain functions. In Section 14.1, one finds a complete description of all submaximal classes of P3 and some remarks about generalizations of the indicated theorems. The following papers provide a basis for this description: [Mac 79], [Mar-D-H 80], [Sal 64], [Bag-D 82] and [Lau 82a]. The theorems from Section 14.1 are proven then in Sections 14.2 – 14.9. In Section 14.10, we will prove that there are 5 submaximal classes with finitely many subclasses; 7 have countably many

Introduction

13

subclasses, and the remaining 146 submaximal classes have uncountably many subclasses. All elements of the lattices L↓3 (A), where A is a submaximal class with|L↓3 (A)| ≤ ℵ0 are determined in Chapter 15. Chapter 15 gives all finite or countably infinite sublattices of depth 1 or 2 of the lattice of the subclasses of P3 , where the finite cases are easy conclusions from Chapter 13. We say that the lattice L↓k (A) of subclasses of A = [A] ⊆ Pk has the depth t, if t is the least integer for which there are some classes A1 , ..., At−1 ∈ Lk with A ⊂ A1 ⊂ A2 ⊂ ... ⊂ At−1 ⊂ Pk . For k = 3 by the Theorems 13.2.3 and 8.1.6 there exist finite and countably infinite sublattices of the depth 1 or 2. In Chapter 15 these sublattices shall be determined exactly. The finite lattices L↓3 (L3 ) of depth 1 are found in Section 15.1. This lattice is a conclusion from Chapter 13. In addition, by Section 14.10, this lattice is the only finite lattice of depth 1 and, further, this lattice contains all finite sublattices of L3 of depth 2. Because of Theorem 14.10.1, there are exactly 7 submaximal classes with |L↓3 (A)| = ℵ0 . One obtains these classes through formation of isomorphic pictures of the following two classes: (1) L3 = [P3 ] ∪ n≥1 {f (n) ∈ P3 | ∃f0 , f1 , ..., fn ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + f2 (x2 ) + ... + fn (xn ) (mod 2))} and M :=

{f (n) ∈ P3 | ∃f1 , ..., fn ∈ O1 : f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (x)},

n≥1

where O1 is the set of all unary monotonous functions (in respect to the total order 0 < 1 < 2) of P3 and x ∨ y := max{x, y}. Since L3 contains also all subclasses, which are generated from unary functions of P3 . First of all, the 1299 subsemigroups of the semigroup (P31 ; ) are determined in Section 15.2. The list of these semigroups is then an important aid in Section 15.3 during the determination of the remaining subclasses of L3 . In Section 15.4 one finds all subclasses of M. Chapter 16 supplies the description (coming from [Sze 91] or [Lau 82b, 95a]) of all maximal classes of the subclass P olk {a} TQ := a∈Q

of Pk for arbitrary Q with ∅ = Q ⊆ Ek , k ≥ 2. With the aid of these classes, a completeness criterion for TQ can be formulated easily. This criterion implies

14

Introduction

necessary and sufficient conditions for whether a finite algebra is semi-primal and has only trivial subalgebras. If |Q| = 1, then TQ is a maximal class of Pk and the maximal classes of TQ , given in Chapter 15, are submaximal classes of Pk . Moreover, if |Q| ≥ 2, we prove that every maximal class of TQ is an intersection of TQ with certain maximal classes of Pk or P olk {a} (a ∈ Q). Chapter 17 continues the investigations of Chapter 16 and generalizes Theorem 14.1.3. For arbitrary k, l ∈ N with 2 ≤ l ≤ k − 1 all maximal classes of P olk El are determined, where 9 relation sets are needed. It is important to note that, for the description of the maximal classes of P olk {a}, a ∈ Ek , one needs only 6 relation sets. With the help of the maximal classes of P olk El , one can easily give a completeness criterion for P olk El . The proofs given in Chapter 17 resemble those ones from Chapter 6, i.e., the results of this chapter were achieved with the means developed by I. G. Rosenberg in [Ros 70a]. Chapter 18 gives a survey (partial without proof) over further submaximal classes found until now. In supplement to Chapters 16 and 17, all submaximal classes of a maximal class of the type S are described in this chapter. It is shown, then, how one can prove the special case k ∈ P of this general description easily. The rest of this chapter deals with submaximal classes of Pk that lie below a maximal class of the type U. In Section 18.2, one can find some maximal classes of P olk , where ∈ Uk is arbitrary. Then, in Section 18.3, the list is completed from Section 18.2 to the list of all maximal classes 2 ∪ {(k − 1, k − 1)} . of P olk for = Ek−1 Chapter 19 Chapter 19 deals with classes of Lk , which are either direct predecessors of the empty set (so-called minimal classes) or which are direct predecessors of the set of all projections (so-called minimal clones). Consequently, it is not difficult to determine the minimal classes. However, for the minimal clones, only partial results can be given. In Chapter 19, one finds a description of all minimal classes. Moreover, Rosenberg’s classification of minimal clones is proven. At the end of Chapter 19, one finds a survey of further results about minimal clones and the description of all partial minimal clones. Chapter 20 deals with partial function algebras. A partial n-ary function k the set on Ek is a mapping from a subset of Ekn into Ek , n ∈ N. Let P of all such functions with n ∈ N. Then one can introduce certain modified k of all partial functions on Ek . Then, the Mal’tsev-operations over the set P set Pk , together with these operations, forms a so-called (full) partial function k ; τ, ζ, ∆, ∇, ), which can similarly be examined like the function algebra (P algebra (Pk ; τ, ζ, ∆, ∇, ). The choice of results on partial function algebras in this chapter focuses on questions already treated for Pk in previous chapters.

Introduction

15

After a composition of some basic concepts in Section 20.1, Section 20.2 shows k is isomorphic to a certain sublattice that the lattice of all partial clones of P of the lattice of all clones of Pk+1 . Thus, one gets many properties of the partial clones from the properties of the clones, which were already found. However many results on clones that may be helpful to find certain partial clones with the aid of the above-mentioned isomorphism are missing. One k with the help could, for example, not solve the completeness problem for P of an isomorphism. In Section 20.3, we show how to describe partial clones by relations. Sections 20.4 and 20.5 deal with the maximal partial clones, with whose help, as for Pk , the completeness problem of the partial logic is soluble. We will 3 has exactly 58 maximal partial clones. In 2 has exactly 8 and P prove that P k for arbitrary Section 20.5, there is a complete list of all maximal clones of P k ∈ N, found by L. Haddad and I. G. Rosenberg. The list is given without proof. In Section 20.6, we determine those descriptive relations of the maximal clones of Pk , which are also descriptive relations of the maximal partial clones k . In addition, we survey those papers that deal with the determination of of P the orders of maximal partial clones. Section 20.7 deals with the determinak | C = [C] ∧ C ∩ Pk = A}, tion of the cardinality of the set I(A) := {C ⊆ P where A is an arbitrary maximal clone of Pk . We will prove that, if A has the type U, S or C, I(A) is a finite set. On the other hand, the set I(A) has the cardinality of continuum if A has the type L or B. For the type M we can give only partial results. Section 20.8 surveys of the cardinalities of the sets I(A), where A is an arbitrary subclass of P2 . In the last section we determine the congruences on the maximal partial k has exactly 4 congruences, whereas clones. It is proven, particularly, that P a maximal partial clone has exactly 4, 8, or 10 pairwise distinct congruences. Finally, one can find some technical references: As already noted, the chapters of this book are not continuously numbered, except for the chapters of the first and second part. References to parts, chapters, sections, lemmas, theorems, ... are given through their numbers. If a reference of the first or second part is missing, then the part discussed is referred to. About the basics of Universal Algebra there are many publications and books. Therefore, only some references for the theorems are given in Part I of this text. In Part II of the book, references for a theorem are left out only if one can prove the theorem easily and if there are several references for this theorem. Normally, one finds references at the beginning of the chapters and with the theorems, whereby it is also clear that the lemmas appertaining to these theorems are based on considerations from the quoted references. For some few theorems, the proofs are taken over from the quoted references directly.

16

Introduction

However new and shorter proofs are presented particularly for theorems that come from older books. The end of a proof (or of a statement with easy proof) is marked by . Some book parts that can be ignored during the first reading differ from the remaining text through a smaller writing.

Preliminaries

We assume that the reader has some knowledge about the basic concepts from the set theory1 , linear algebra, classical algebraic structures and mathematical logic. The following is a list of terms to which we assume the reader is familiar. 1) Logical Symbols So that we can write down mathematical statements quickly and correctly, we use the following symbols from mathematical logic:

symbol

denotation for

∧

and

∨

or

¬

not

=⇒

implies; if - then

⇐⇒

if and only if; iff

:= :⇐⇒

equal by definition equivalent by definition

∃

there exists; it exists at least

∃!

it exists exactly

∀

for all

In order to save parentheses, we write instead of ∃x ( E ) (read: “There exists an x with the property E”) shortly ∃x : E. We arrange analogous one for 1

A naive theory of sets is sufficient for our purposes.

18

Preliminaries

formulas that contain the sign ∀. In addition, ∀x ∃y ... stands for ∀x (∃y (...)), and so forth. 2) Symbols and Concepts of the Set Theory In dealing with sets, we use the following standard notations: membership (∈), nonmembership (∈), set-builder notations ({.... | ...........} 2 ), the empty set (∅), (⊆), proper inclusion (⊂), intersection (∩ inclusion (⊆), noninclusion and ), union (∪ and ) and difference (\). The power set of a set A is the set {B | B ⊆ A} of all subsets of A. It is denoted by P(A). N, N0 , Z, Q, R, C denote respectively the set {1, 2, 3, ....} of all natural numbers, the set N ∪ {0}, the set of all integers, the set of all rational numbers, the set of all real numbers, and the set of all complex numbers. For n ∈ N, we write the order n-tuples (briefly n-tuples) in the form (x1 , x2 , ..., xn ). For two n-tuples x := (x1 , x2 , ..., xn ) and y := (y1 , y2 , ..., yn ) is x = y iff xi = yi for all i ∈ {1, 2, ..., n}. A × B is the set of all 2-tuples (a, b) with a ∈ A and b ∈ B and is called the (Cartesian or direct) product of the sets A and B. For n ∈ N, Π1n Ai (or Πi∈I Ai , where I := {1, 2, ..., n}) denotes the set of all (a1 , a2 , ..., an ) with ai ∈ Ai for all i ∈ I. 3 Further, let An := {(a1 , ..., an ) | ∀i ∈ {1, 2, ..., n} : ai ∈ A} be the direct n-powers of the set A, n ∈ N. A correspondence F of A into B is a subset of A × B, where D(f ) := {a ∈ A | ∃b ∈ B : (a, b) ∈ F } is the domain of F and Im(f ) := {b ∈ b | ∃a ∈ A : (a, b) ∈ F } is the image (or range) of F . If D(F ) = A, then F is a correspondence from F . If Im(F ) = B, then we say that F is a correspondence onto B. The inverse (or converse) F −1 of a correspondence F ⊆ A × B is given by F −1 := {(b, a) ∈ B × A | (a, b) ∈ F }. If F ⊆ A × B and G ⊆ B × C, then the correspondence product F 2G is defined by F 2G := {(a, c) ∈ A × C | ∃b ∈ B : (a, b) ∈ F ∧ (b, c) ∈ G}. We remark that (F 2G)−1 = G−1 2F −1 and 2 is associative. A mapping (or map) f from A into B is a subset of A × B such that for each a ∈ A there is exactly one b ∈ B with (a, b) ∈ f . If f is a mapping from A into B, then we write f : A −→ B and, if (a, b) ∈ f , f (a) = b, or a → b. A mapping f : A −→ B is called injective (or is an injection) iff f (a) = f (a ) implies a = a for all a, a ∈ A. The mapping f : A −→ B is surjective (or is a surjection) iff Im(f ) = B. The mapping f : A −→ B is bijective (or is a bijection) iff is both injective and surjective. Let A be a set and n ∈ N. Then, an n-ary relation on A (or an n-ary 2 3

For example, we write S := {x | x ∈ A ∧ P (x)} or briefly S := {x ∈ A | P (x)} instead of “S is the set of all x ∈ A with the property P (x)”. For arbitrary I, we will define Πi∈I Ai in Part I, Section 5.

Preliminaries

19

relation over A) is a subset of An . A 2-ary , 3-ary or 4-ary relation on A is called respectively a binary, ternary or quaternary relation on A. Two sets A and B have the same cardinality (in symbol |A| = |B|) iff there exists a bijection from A onto B. A set A is infinite, iff there is a subset B ⊂ A with |A| = |B|. A set A is called finite iff A is not infinite. If A is a finite set, then |A| denotes the number of elements of A, i.e., |∅| = 0 and |x ∪ {x}| = |x| + 1 (or n = |{0, 1, 2, ..., n − 1}| for all n ∈ N0 , where n is called a (finite) ordinal). We put ℵ0 := |N| and c := |R|. The set A is called countable if |A| = ℵ0 . If |A| = c, we say that A has the cardinality of the continuum. It is well-known that ℵ0 is the least infinite ordinal and that N and R do not have ∞ the same cardinality. Further, it holds |Q| = ℵ0 , |C| = c, |P(N)| = c and | n=1 An | = ℵ0 if |An | = ℵ0 for all n ∈ N.

Part I

Universal Algebra

Part I Universal Algebra

23

The Universal Algebra is a mathematical discipline, which examines so-called universal algebras, where the classic algebraic structures – such as groups, rings, and lattices – are special cases of such algebras. Roughly said, a universal algebra is nothing other than a nonempty set together with a family of finite-ary operations over this set. Although the concept universal algebra was used already at the end of the 19th century, the development of the theory Universal Algebra started only in the 1930’s. This development has two historical roots. The first root is the classical algebra and the studies of properties that have all algebraic structures (such as semigroups, groups, rings, etc.). The second root is the Mathematical Logic. In the last decades, the Universal Algebra has continued the classic examinations and has also developed new areas. An introduction to the bases of the Universal Algebra follows. The topic of the following six chapters is based on what is needed later, during the investigation of the function algebras. The first five chapters show that the Universal Algebra is a general theory of the algebraic partial disciplines. Chapter 6 treats problems with an impression of the mode of operation of the algebra and that have partial also contact points to the mathematical logic. 1 Additions to the topics can be found in [Lau 2004] 2 . In detail one can study the Universal Algebra with the aid of the following books [Bur-S 81], [Coh 65], [Den-T 96], [Den 2003], [Gr¨ a 68], [Ihr 2003], [McKM-T 87], [Wer 78]. The book [McK-M-T 87] can particularly be recommended. In the preface, we have already noted that to understand the second part of the text need not be read absolutely. Thus, readers can select for themselves the required terms and theorems from Part I, during the reading of Part II. One night notice, after reading Part II, that some examinations are special cases of general problems.

1 2

To realize this, reads [Bur-S 81] and [Ric 78]. There, one finds a continuation to the chapters of Universal Algebra of this book and additional chapters.

1 Basic Concepts of Universal Algebra

1.1 Universal Algebras First, we define the concept of an n-ary partial operation: Let A be a nonempty set, n ∈ N and ∅ ⊂ ⊆ An . An n-ary partial operation is a mapping f from into A. The number n is called arity of f and the arity of f is also denoted by af . To denote the arity of f we also write f (n) or briefly f n , since for content-related reasons, the interpretation is impossible of f n as Cartesian product in the following. If (a1 , ..., an ) ∈ then let f (a1 , ..., an ) be the image of (a1 , ..., an ) under an n-ary operation f . The set we call domain of f , and we denote the domain of f by D(f, A) or briefly by D(f ). Let Im(f ) be the set {f (a1 , ..., an ) | (a1 , ..., an ) ∈ D(f )}, which is called image or range of f . (n) (n) For n ∈ N let c∞ be the n-ary partial operation with D(c∞ ) = ∅.1 n If = A and n ≥ 1, then f is called an n-ary operation on A. If D(f (n) ) ⊂ An we call f (n) a proper partial operation. A nullary operation f on A is an element f of A with af := 0, D(f ) := ∅ and Im(f ) := {f }. Example The operation ◦ of a group (see also Section 1.2) is a binary operation. One can understand the formation of inverse elements in a group as a unary operation −1 and the identity e (unit element) of a group as a nullary operation. As is generally accepted, one can form compositions of mappings in the following manner: If f : M1 −→ M2 and g : M2 −→ M3 are mappings, then we denote the mapping 1

The necessity of such a definition is seen, if one forms superpositions of partial (1) functions (see Part II, Chapter 1). For example, it holds g2f = c∞ for the unary operations f , g with D(f ) := {a} and a ∈ Im(g).

26


f 2g : M1 −→ M3 , x → g(f (x)) by f 2g. It is well-known (or it is easy to see) that 2 is associative. Thus we can renounce the putting of brackets in the case of more than two compositions. Furthermore, we write g1 g2 ...gr instead of g1 2g2 2...2gr (gi : Mi −→ Mi+1 , i = 1, ..., r), if the operation 2 follows from the context. We notice that the above definition of 2 is a special case of the following definition, if one writes the mappings in the form of subsets of Cartesian products: Let R ⊆ A × B and Q ⊆ B × C. Then let R2Q := {(a, c) | ∃b ∈ B : (a, b) ∈ R ∧ (b, c) ∈ Q}. With the help of the partial operations, we can define the following concepts: A partial algebra is an ordered pair A := (A; F ), where A is an arbitrary nonempty set and F is a set of partial operations on A. The set A is called universe (or underlying set) of A. The elements of F are called (partial) fundamental operations of A. A universal algebra or briefly an algebra is a partial algebra (A; F ), where every element of F is an operation, i.e., F does not have any proper partial operations. Here we usually only deal with algebras. Therefore, we often introduce the following concepts only for algebras. If F = {f1 , ..., fr } holds we write (A; F ) in the form (A; f1 , ..., fr ), where we often choose af1 ≥ af2 ≥ ... ≥ afr . We also use the notation (A; (fi )i∈I ), if F = {fi | i ∈ I} and I is a certain index set. A (partial) algebra (A; F ) is called finite, if A is a finite set, otherwise infinite. In order to receive a first classification of the algebras, it offers itself to carry out a classification according to the arities of the (partial) operations of the algebras. For this purpose, we define the concept type of an algebra (A; (fi )i∈I ) as a sequence

1.2 Examples of Universal Algebras

27

τ := (afi | i ∈ I) of the arities of their fundamental operations, if I is a finite or countable set. In the case that I is uncountable, we define as type of A the mapping τ : I −→ N, i → afi . If I is a finite set, then (A; (fi )i∈I ) is called of finite type. If A = (A; f1 , ..., fr ), af1 ≥ af2 ≥ ... ≥ afr , then let τ = (af1 , af2 , ..., afr ). Two (partial) algebras (A; F ), (B; G) are of same type, iff there is a bijection ϕ : F −→ G, f → g with af = ag. Examples2 1) A semigroup (H, ◦) is an algebra of type (2) (F := {◦}). 2) By the above remark, a group is an algebra of the type (2, 1, 0). 3) A lattice (see 1.2.11) is an algebra of type (2, 2). Because of existence of a bijection between the operations of two algebras A := (A; F ) and B := (B; G) of same type, we often denote the operation sets of A and of B with the same symbol (F or G). In analog mode we deal with the denotation of the elements. If it is necessary to denote the operations of two different algebras A and B of the same type differently, then we write f A or f B (or also fA or fB ), respectively. Before we deal with further basic terms from the theory of the algebras, let’s define some specific algebras. We will study some of these specific algebras in detail later.

1.2 Examples of Universal Algebras We start with some “classical algebras”. In defining these algebras, we try to use no existence quantor. 3 We call the equations, which are given for the definition of an algebra, axioms. As usual, we will try to use only a few (independent) axioms. We differ from this principle in the following only in few places (see e.q. the definition of a lattice). 1.2.1 Gruppoids An algebra (A; ◦) of type (2) is called gruppoid. Thus, a gruppoid is not a different one as a nonempty set together with a binary operation. 2 3

Section 1.2 contains the explanations to the concepts used. In Chapter 6, we see that this manner of description has interesting conclusions. In the following, we use the descriptions of the algebraic structures given in this section.

28


1.2.2 Semigroups A gruppoid (H; ◦) is called semigroup if the algebra (H; ◦) fulfills the following axiom: (A)

∀x, y, z ∈ H : (x ◦ y) ◦ z = x ◦ (y ◦ z)

(associativity).

A semigroup is called commutative, if it satisfies the following condition in addition: (C) ∀x, y ∈ H : x ◦ y = y ◦ x. 1.2.3 Monoids An algebra (M ; ◦, e) of type (2, 0) is called monoid, if (M ; ◦) is a semigroup and (E) ∀x ∈ M : e ◦ x = x ◦ e = x holds. The (nullary) operation e is the the neutral element of M . 1.2.4 Groups A group is an algebra (G; ◦,−1 , e) of type (2, 1, 0), which fulfills the above axioms (A), (E) for x, y, z, e ∈ G and (I)

∀x ∈ G : x ◦ x−1 = x−1 ◦ x = e .

x−1 is called the inverse element of x. A group, which (C) fulfills in addition, is called Abelian (or commutative). It is usual in Abelean groups to use +, −x, 0 instead of ◦, x−1 , e (the additive notation). 1.2.5 Semirings An algebra (R; +, ·) of type (2, 2) is called a semiring if (R; +) is a commutative semigroup, (R; ·) is a semigroup, and if the following distributive laws hold: ∀x, y, z ∈ R : x · (y + z) = (x · y) + (x · z), (D1 ) ∀x, y, z ∈ R : (x + y) · z = (x · z) + (y · z). (D2 ) 1.2.6 Rings An algebra (R; +, ·, −, 0) of type (2, 2, 1, 0) is called ring if (R; +, −, 0) is an Abelean group and (R; +, ·) is a semiring. In order to save parentheses, we use the known rule subsequently “The dot bill is carried out before the stroke bill”. A unitary ring (or a ring with a unit element) is an algebra (R; +, ·, −, 0, 1) of type (2, 2, 1, 0, 0), where (R; +, ·, −, 0) is a ring and (E) holds with e = 1 and ◦ = ·.

1.2 Examples of Universal Algebras

29

1.2.7 Fields A partial algebra (K; +, ·, −,−1 , 0, 1) is called field if the algebra (K; +, ·, −, 0, 1) is an unitary ring and (K\{0}; ·,−1 , 1) is an Abelean group. We remark that the operation −1 on K is only a partial operation, since 0−1 is not defined. 1.2.8 Modules Let R := (R; +, ·, −, 0) be a ring. An algebra (M ; F ) where F := {+, −, 0}∪R, + is binary, 0 is a nullary and − and all r ∈ R are unary operations is called a R-module (or module over the ring R), if (M ; +, −, 0) is an Abelean group and if for all r, s ∈ R the following equations hold: (M1 ) (M2 ) (M3 )

∀x, y ∈ M : r(x + y) = r(x) + r(y), ∀x, y ∈ M : (r + s)(x) = r(x) + s(x), ∀x ∈ M : (r · s)(x) = r(s(x)).

Let M := (M ; +, −, (r)r∈R , 0) be the short notation for the R-module defined above. Further, we say that M has the type (2, 1, (1)r∈R , 0). A module over a unitary ring (R; +, ·, −, 0, 1) is an above module that also satisfies the following condition: (M4 )

∀x ∈ M : 1(x) = x.

Three remarks to the above definitions: – Instead of the unary operations r ∈ R one can also define a mapping : R × M −→ M, (r, x) → r x, which fulfills the axioms. However, a module would then not be a universal algebra. – A module has infinite many operations if the ring R is infinite. – The operation symbols +, −, 0 have two different meanings: on the one hand, they describe the operations of the Abelean group (R; +, −, 0); on the other hand, they describe the operations of the Abelean group (M ; +, −, 0). 1.2.9 Vector Spaces Let K := (K; +, ·, −, 0, 1) be a field. Then every K-module (V ; +, −, (k)k∈K , 0) is called K-vector space (or vector space over the field K). 1.2.10 Semilattices A commutative semigroup (S; ◦), in which ∀x ∈ S : x ◦ x = x holds, is called semilattice.

30


1.2.11 Lattices A lattice is an algebra (L; ∨, ∧) of type (2, 2), in which for arbitrary x, y, z ∈ L it holds: (L1 ) x ∨ y = y ∨ x, x ∧ y = y ∧ x (L2 ) x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∧ (y ∧ z) = (x ∧ y) ∧ z (L3 ) x ∨ x = x, x ∧ x = x (L4 ) x ∨ (x ∧ y) = x, x ∧ (x ∨ y) = x

(commutativity), (associativity), (idempotency), (absorption).

A bounded lattice (or a lattice with 0 and 1) is an algebra (L; ∨, ∧, 0, 1) of type (2, 2, 0, 0) such that (L; ∨, ∧) is a lattice and furthermore the following equations hold for all x ∈ L: x ∧ 0 = 0, x ∨ 1 = 1.

(L5 )

A lattice is called distributive, if the following distributive laws hold: (DL1 ) (DL2 )

∀x, y, z ∈ L : x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), ∀x, y, z ∈ L : x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z).

Remark: One can show (with the help of (L1 )–(L4 )) that (DL1 ) and (DL2 ) are equivalent; that is, it suffices to require either (D1 ) or (D2 ) in the definition of a distributive lattice. 1.2.12 Boolean Algebras An algebra (B; ∨, ∧,− , 0, 1) of type (2, 2, 1, 0, 0) is called Boolean algebra if (B; ∨, ∧, 0, 1) is a bounded distributive lattice and the following equalities hold for all x ∈ B: x ∧ x = 0, x ∨ x = 1 x ∧ 0 = 0, x ∨ 1 = 1,

(B1 ) (B2 ) where x :=

−

(x).

1.2.13 Function Algebras One can choose the set of all operations defined on A as a universe of an algebra and can then define operations on the operations on A. For the purpose of distinction we will subsequently replace the concept “operation (on the set A)” by the concept “function (on the set A)”. In Part II, the set A is always a finite set. Therefore, the following concepts become explained only for a specific k-element set Ek . Put Ek := {0, 1, ..., k − 1},

1.3 Subalgebras

31

k ∈ N\{1}. Let Pkn be the set of all n-ary functions f n , which map the n-fold n n Cartesian product Ek into Ek . Put Pk := n≥1 Pk . Elementary operations (called Mal’tsev-operations) over Pk are ζ, τ, ∆, ∇ (unary operations) and (a binary operation) defined by (ζf )(x1 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) if n ≥ 2, ζf = τ f = ∆f := f if n = 1, (∇f )(x1 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ), (f g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) (f n , g m ∈ Pk ). It holds the following (see Part II, Chapter 1): With the help of the Mal’tsev-operations, one can form the following operations (for arbitrary functions and arbitrary variables of the functions): – permutation of variables – identification of variables – adding of fictitious variables and – substitution of variables of a function by functions The set of all functions that can be obtained by a finite number of applications of the above operations from the functions of F ⊆ Pk is called the closure of F , and is denoted by [F ]. If F = [F ], then we say that F is closed or F is a (sub)class of Pk . The algebra (Pk ; ζ, τ, ∆, ∇, ) of type (1, 1, 1, 1, 2) is called (full) iterative function algebra. If A is a subclass of Pk then (A; ζ, τ, ∆, ∇, ) is called function algebra.

1.3 Subalgebras Let A = (A; F ) be a (partial) algebra and let B be a nonempty subset of A. The (partial) algebra B = (B; F ) of the same type as A is called (partial) subalgebra of A (or A is an extension of B) iff it holds D(fB , B) ⊆ D(fA , A) for arbitrary f ∈ F and fA (x1 , ..., xaf ) = fB (x1 , ..., xaf ) for all (x1 , ..., xaf ) ∈ B af ∩ D(f, A). Then we write

32


B ≤ A. If A has a nullary operation f ∈ A and B is a subalgebra of A, then f also belongs to B. The following lemma summarizes easy consequences from the above definition of a subalgebra. Lemma 1.3.1 (a) A subset B of the universe A of an algebra (A; F ) together with the restrictions f|B of the operations f of A to B forms an algebra if and only if f (b1 , ..., baf ) ∈ B for all b1 , ..., baf , if af ≥ 1, and f ∈ B if af = 0 holds for arbitrary f ∈ F . (b) Let I bean arbitrary index set, Bi ≤ A for every i ∈ I and i∈I Bi = ∅. Then ( i∈I Bi ; F ) is a subalgebra of A. (c) For every nonempty subset T of the universe A of an algebra A = (A; F ) there exists exactly one smallest subalgebra T of A, which contains T and which one can describe as follows: T = (T ; F ) with T := {B | B ≤ A and T ⊆ B}.

The set T from Lemma 1.3.1, (c) is called generating system of the algebra T and the universe T of T is denoted by [T ]A or [T ]F or by [T ]f1 ,...,fr , if F = {f1 , ..., fr }, or simply by [T ]. Example For the algebra A = ({0, 1, 2, ..., k − 1}; f ) of the type (3) with f (x, y, z) = x + y − z (mod k) for arbitrary x, y, z ∈ A, it holds e.g. [{a}] = {a} for every a ∈ A and [{0, 1}] = A. If [T ] = T for a subset T ⊆ A of an algebra A = (A; F ), then we say that T is closed. The following lemma gives another possibility of the description of the set [T ], defined above. Lemma 1.3.2 Let A = (A; F ) be an algebra, T be a subset of A and F 0 be the set of all nullary operations of F . Then, for T we can define recursively the following subsets Tn of A as follows: T0 := T ∪ F 0 Tn+1 := Tn ∪ {f (g1 , ..., gaf ) | f ∈ F \F 0 and {g1 , ..., gaf } ⊆ Tn } (n ∈ N0 ).

1.3 Subalgebras

Then [T ]A =

Tn .

n≥0

Proof. To prove “⊆” in (1.1), we have to show that

33

(1.1)

is the universe of n≥0 Tn a subalgebra of A. This would be shown if we have shown that n≥0 Tn is closed regarding the application of all operations of F \ F 0 . Now let f ∈ F \ F 0 and {g1 , ..., gaf } ⊆ n≥0 Tn be arbitrary. Since af ∈ N, there is an m ∈ N with {g1 , ..., gaf } ⊆ Tm . Consequently, f (g1 , ..., gaf ) belongs to Tm+1 , i.e., the set n≥0 Tn is closed. “⊇” follows from Tn ⊆ [T ]A for all n ≥ 0. This inclusion is easy to prove by induction on n, since [T ]A is the universe of a subalgebra of A.

The set S(A) := {B | B is subalgebra of A} ∪ {∅} we call briefly set of all subalgebras of A. Per definitionem (essentially for technical reasons) is also the empty set a subalgebra of each algebra. Then (as the reader can easily verify) S(A) together with the operations ∧ : S(A) × S(A) −→ S(A), B1 ∧ B2 = (B1 ∩ B2 ; F ) ∨ : S(A) × S(A) −→ S(A), B1 ∨ B2 = ([B1 ∪ B2 ]F ; F ) forms a lattice.4

4

This property would not hold for all algebras, if the empty set was not an algebra.

2 Lattices

Lattices arise often in algebraic investigations. In the following we see that the concept “lattice” is always needed, if the elements of sets are ordered in a certain meaning. In addition, the lattice theory is an interesting branch of the Universal Algebra. For space reasons, only the most important basis concepts and some proof ideas of the lattice theory can be indicated here. For a secondary study of the lattice theory, refer to the books on the Universal Algebra and to [Bir 48], [Ern 82], [Sko 73] and [Dav-P 90].

2.1 Two Definitions of a Lattice There are two standard ways of defining lattices. One of these ways was already given in Section 1.2.11: Definition (First Definition of a Lattice) Let L be a nonempty set on which the binary operations ∨ (called “join”) and ∧ (called “meet”) are defined. (L; ∨, ∧) is called lattice if for arbitrary x, y, z ∈ L the following identities hold: (L1 a) x ∨ y = y ∨ x, (L1 b) x ∧ y = y ∧ x

(commutativity),

(L2 a) x ∨ (y ∨ z) = (x ∨ y) ∨ z, (L2 b) x ∧ (y ∧ z) = (x ∧ y) ∧ z

(associativity),

(L3 a) x ∨ x = x, (L3 b) x ∧ x = x

(idempotency),

(L4 a) x ∨ (x ∧ y) = x, (L4 b) x ∧ (x ∨ y) = x

(absorption).

36

2 Lattices

Since one receives an axiom again by exchanging of ∨ and ∧ in above axioms, results from every equation, which the axioms imply, by exchanging of ∨ and ∧ a further valid equation. One names this procedure of deriving equations in lattices, using the duality principle of the lattice theory. In the following proofs we often use the equivalence x ∨ y = y ⇐⇒ x ∧ y = x,

(2.1)

which is a conclusion from the absorption laws and from the commutative laws. For the second definition of a lattice we need some concepts and notations: Definitions • A binary relation ≤ (⊆ A × A) is called a partial order on the set A, if it fulfills the following conditions: (O1 ) ∀a ∈ A : a ≤ a (reflexivity), (O2 ) ∀a, b ∈ A : (a ≤ b and b ≤ a) =⇒ a = b (antisymmetry), (O3 ) ∀a, b, c ∈ A : (a ≤ b and b ≤ c) =⇒ a ≤ c (transitivity). • The pair (A; ≤), where ≤ satisfies the above conditions, we call partially ordered set or, briefly, poset. In examples, we often declare the posets through order diagrams1 (or Hasse diagrams). • A poset P := (P ; ≤) with P ⊆ A × A, which besides fulfills the condition (O4 ) ∀a, b ∈ A : a ≤ b or b ≤ a , is called totally ordered set or linearly ordered set or, briefly, chain. We also use the notation a < b if a ≤ b and a = b is valid, where (L; ≤) is a poset and a, b ∈ L. We also write b ≥ a instead of a ≤ b. Definition Let Q be a subset of P , where P = (P ; ≤) is a poset. The element s ∈ P is said to be a supremum of Q (denoted by sup Q) iff s has the following properties: (S1 ) ∀q ∈ Q : q ≤ s; (S2 ) ∀p ∈ P ((∀q ∈ Q : q ≤ p) =⇒ s ≤ p). Remark The supremum of Q does not exist in general for every subset Q of a poset P . Let e.g. P = {0, 1, 2, 3, 4} and let ≤ defined by the following order diagram: 1

Here, the elements of the poset are represented as points in the plane and, if x < y and no z exists with x < z < y, we draw y higher up than x and connect x and y with a line segment.

2.1 Two Definitions of a Lattice

37

4r @ @

@r 3 HH H 0 r Hr 1 2 rH

Fig. 2.1

Then, sup{0, 1} does not exist, since for p ∈ {2, 3} 0 ≤ p and 1 ≤ p are valid, the elements 2 and 3 are not comparable in respect to ≤. Definition Let Q be a subset of P , where P = (P ; ≤) is a poset. The element i ∈ P is said to be an infimum of Q (denoted by inf Q) iff i has the following properties: (I1 ) ∀q ∈ Q : i ≤ q; (I2 ) ∀p ∈ P ((∀q ∈ Q : p ≤ q) =⇒ p ≤ i). Definition (Second Definition of a Lattice) A poset L := (L; ≤) is called lattice iff for arbitrary a, b ∈ L both sup{a, b} and inf{a, b} in L exist. Some elementary properties of sup and inf in a lattice are summarized in the following lemma: Lemma 2.1.1 Let (P ; ≤) be a lattice by the second definition. Then, for arbitrary x, y, z, u, v ∈ P it holds: (a) x ≤ y =⇒ sup{x, z} ≤ sup{y, z}, (b) x ≤ y =⇒ inf{x, z} ≤ inf{y, z}, (c) (x ≤ y and u ≤ v) =⇒ sup{x, u} ≤ sup{y, v}, (d) (x ≤ y and u ≤ v) =⇒ inf{x, u} ≤ inf{y, v}. Proof. (a) and (b) are easy to check. (c): Let x ≤ y and u ≤ v. Then by (a) it holds sup{x, u} ≤ sup{y, u} and sup{y, u} ≤ sup{y, v}. Since ≤ is transitive, by this (c) follows. One can prove (d) analogously to (c). The next theorem shows how the two lattice definitions are associated. Theorem 2.1.2 It holds: (a) If (L; ∨, ∧) is a lattice by the first definition, then one can define by a ≤ b :⇐⇒ a = a ∧ b

(2.2)

a partial order ≤, which together with L forms a lattice by the second definition.2

38

2 Lattices

(b) Conversely, if (L; ≤) is a lattice by the second definition, then one can define two binary operations ∨, ∧ by a ∨ b := sup{a, b} a ∧ b := inf{a, b}

and

and (L; ∨, ∧) is a lattice by the first definition. Proof. (a): Let (L; ∨, ∧) be a lattice and ≤ is defined by (2.2). Then a ∧ a = a

holds and thus a ≤ a for every a ∈ A. Hence ≤ is reflexive. If a ≤ b and b ≤ a, we have a = a ∧ b and b = b ∧ a. This and (L1 b) imply a = b. Thus ≤ is antisymmetric. Let now a ≤ b and b ≤ c. Then it holds a = a ∧ b and b = b ∧ c by definition of ≤. By this and by (L2 b) we have a = a ∧ (b ∧ c) = (a ∧ b) ∧ c = a ∧ c. Hence a ≤ c and thus ≤ is transitive. It remains to show that there exist sup{a, b} and inf{a, b} for all a, b ∈ L. For these let a, b ∈ L be arbitrary. By (L4 a) and (L4 b) then we have a = a ∧ (a ∨ b) and b = b ∧ (a ∨ b). Thus a ≤ a ∨ b and b ≤ a ∨ b. Let now a ≤ u and b ≤ u for a certain u ∈ L. Then the following equations hold: a = a ∧ u, b = b ∧ u, a ∨ u = (a ∧ u) ∨ u = u, b ∨ u = (b ∧ u) ∨ u = u, and (a ∨ b) ∨ u = (a ∨ u) ∨ (b ∨ u) = u ∨ u = u. If one uses the equation (a ∨ b) ∨ u = u, which just was proven, then one gets: (a ∨ b) ∧ u = (a ∨ b) ∧ ((a ∨ b) ∨ u) = a ∨ b. Therefore, we have a ∨ b ≤ u and sup{a, b} = a ∨ b holds by definition of sup. Analogously, one can show a ∧ b = inf{a, b}. For this, one mixed up ∧ in above considerations by ∨ and one replaced ≤ by ≥. Thus (L; ≤) is a lattice. (b): Let (L; ≤) be a lattice, a ∨ b := sup{a, b} and a ∧ b := inf{a, b}. Obviously, the so defined operations ∨ and ∧ are commutatively and idempotent. To show the validity of the associative law sup{x, sup{y, z}} = sup{sup{x, y}, z} one can prove (under use of Lemma 2.1.1) sup{x, sup{y, z}} ≤ sup{sup{x, y}, z} and sup{x, sup{y, z}} ≥ sup{sup{x, y}, z}. Analogously, one can show the associativity of inf. Because of sup{x, inf{x, y}} ≥ x and sup{x, inf{x, y}} ≤ x it holds obvious sup{x, inf{x, y}} = x. Analogously, one can show another absorption law. Consequently, (L; ∨, ∧) is a lattice by the first definition. 2

Because of (2.1) one also could have defined the following: a ≤ b :⇐⇒ a ∨ b = b.

2.3 Isomorphic Lattices and Sublattices

39

Because of Theorem 2.1.2 we can use the first or the second definition of an lattice subsequently according to requirement, where the statements of the Theorem 2.1.2 are used where appropriate. If one wants the above-mentioned duality principle on inequalities of the form ... ≤ ... enlarge, one has to define the exchanging of ≤ through ≥ as an additional replacement rule. In addition the implication (x ≤ y and u ≤ v) =⇒ (x ∨ u ≤ y ∨ v and x ∧ u ≤ y ∧ v),

(2.3)

resulting from Lemma 2.1.1 and from Theorem 2.1.2, will be an important aid in the below-given proofs.

2.2 Examples for Lattices 2.2.1 Let L := {0, 1}. Further, let ∨ be the conjunction on L and let ∧ be the disjunction on L. Obviously, (L; ∨, ∧) is a lattice. 2.2.2 Let L := N0 , let a ∨ b be the least common multiple and let a ∧ b be greatest common divisor of the integers a, b ∈ N0 . In this case, it is also easy to check that (L; ∨, ∧) is a lattice. 2.2.3 Examples for lattices by the second definition are: (P(A); ⊆), where A is an arbitrary nonempty set; (R; ≤), where ≤ is the usual order on R. One finds further examples in Section 2.4 and in the following chapters.

2.3 Isomorphic Lattices and Sublattices Definition Two lattices L1 , L2 are called isomorphic, iff there exists a bijective mapping α from L1 onto L2 such that for arbitrary a, b ∈ L1 the following two equations hold: α(a ∨ b) = α(a) ∨ α(b)

and

α(a ∧ b) = α(a) ∧ α(b). The mapping α is also called an isomorphism. Definition Let (P1 , ≤) and (P2 ; ≤) be posets. A mapping α from P1 onto P2 is called order-preserving, if the following holds: ∀a, b ∈ P1 : a ≤ b =⇒ α(a) ≤ α(b). Figure 2.2 gives an example of an order-preserving mapping between the lattices L1 and L2 :

40

2 Lattices

r A

r A A A

- r A A

- r AAr

- r

AAr

- r L2

L1 Fig. 2.2

Theorem 2.3.1 Two lattices L1 and L2 are isomorphic iff there is a bijective mapping α from L1 onto L2 such that both α and α−1 are order-preserving. Proof. “=⇒”: Let α be an isomorphism from L1 onto L2 . Then for all a, b ∈ L1 it holds: a ≤ b =⇒ a = a ∧ b =⇒ α(a) = α(a ∧ b) = α(a) ∧ α(b). Consequently, we have α(a) ≤ α(b), i.e., α is order-preserving. Let now c, d ∈ L2 arbitrary with c ≤ d. Then there exist a, b ∈ L1 with α(a) = c and α(b) = d. That α−1 is also order-preserving follows then from c ≤ d =⇒ α(a) ≤ α(b) =⇒ α(a) ∧ α(b) = α(a) =⇒ α(a ∧ b) = α(a) =⇒ a ∧ b = a =⇒ a ≤ b. “⇐=”: Let now α be a bijective mapping from L1 onto L2 with the property that both α and α−1 are order-preserving. Then for arbitrary a, b ∈ L1 we have a ≤ a ∨ b and b ≤ a ∨ b. Thus, α(a) ≤ α(a ∨ b) and α(b) ≤ α(a ∨ b). From this, it follows α(a) ∨ α(b) ≤ α(a ∨ b). Further, for arbitrary u ∈ L2 it holds α(a) ∨ α(b) ≤ u =⇒ α(a) ≤ u and α(b) ≤ u =⇒ a ≤ α−1 (u) and b ≤ α−1 (u) =⇒ a ∨ b ≤ α−1 (u) =⇒ α(a ∨ b) ≤ u. Consequently, α(a ∨ b) = sup{α(a), α(b)} and therefore α(a) ∨ α(b) = α(a ∨ b). Analogously, one can prove α(a) ∧ α(b) = α(a ∧ b).

Remark Figure 2.2 shows that one cannot leave the condition “α−1 orderpreserving” from Theorem 2.3.1.

2.4 Complete Lattices and Equivalence Relations

41

Theorem 2.3.2 For every poset (P ; ≤) there is a set system MP such that (P ; ≤) and (MP ; ⊆) are isomorphic; i.e., there exists a bijective mapping α from P onto MP with the property that α and α−1 are order-preserving. Proof. For every p ∈ P let M (p) := {x ∈ P | x ≤ p}. Then the set MP := {M (p) | p ∈ P } is a set system with the properties claimed in the theorem. To show this, we study the mapping α : P −→ MP . α is surjective by definition. From M (a) = M (b) it follows a ∈ M (b) and b ∈ M (a). Hence a ≤ b and b ≤ a, i.e., a = b. Thus α is injective. α is order-preserving, since a ≤ b and x ∈ M (a) imply x ≤ a ≤ b. Therefore, a ≤ b implies M (a) ⊆ M (b). Furthermore, M (a) ⊆ M (b) implies a ≤ b, i.e., α−1 is also order-preserving and thus α is an isomorphism. The following definition is a special case of a definition from Section 1.3: Definition

If L is a lattice and L is a subset of L with the property ∀x, y ∈ L : x ∨ y ∈ L and x ∧ y ∈ L ,

then (L ; ∨, ∧) is called a sublattice of (L; ∨, ∧). Remark Let (P ; ≤) and (Q; ≤) be posets with Q ⊆ P . Then, it is not valid in general that (Q; ∨, ∧) is a sublattice of the lattice (P ; ∨, ∧), where ∨, ∧ are defined in Theorem 2.1.2. An example is the poset ({0, 1, 2, 3, 4}; ≤) defined by the following order diagram: 4 r r3 @ @r2

1 r @ @r 0

Fig. 2.3

Obviously, ({0, 1, 2, 4}; ≤) is a poset, but ({0, 1, 2, 4}; ∨, ∧) is not a sublattice of ({0, 1, 2, 3, 4}; ∨, ∧), since 1 ∨ 2 = 3 ∈ {0, 1, 2, 4}. Definition A lattice L1 can be embedded into a lattice L2 , if there is a sublattice of L2 isomorphic to L1 .

2.4 Complete Lattices and Equivalence Relations Definition A poset is called complete iff for every subset A of P both sup A (denoted by A) and inf A (denoted by A) exist.

42

2 Lattices

A lattice which is complete as poset is a complete lattice. Examples 1) Obviously, each complete poset is also a lattice. 2) Every finite lattice is complete. 3) (R; ≤) is not complete. One can find further examples in Theorems 2.4.3 and 2.4.5. Theorem 2.4.1 For an arbitrary poset (L; ≤) with a greatest and a least element it holds:

∀A ⊆ L ∃ A ⇐⇒ ∀A ⊆ L ∃ A. Proof. “=⇒”: Let Ao := {x ∈ L | ∀a ∈ A : a ≤ x}. Since L has a greatest element, we have Ao = ∅. It is easy to check that then can prove “⇐=”.

A=

Ao holds. Analogously, one

Theorem 2.4.2 Let A be an algebra. Then the lattice (S(A); ⊆) of all subalgebras of A is a complete lattice. Proof. The proof follows from Theorem 2.4.1 and the fact that every intersection of subalgebras of A is also a subalgebra of A (see Lemma 1.3.1, (b)). The following is a short summary of the equivalence relations: Definitions A binary relation is called an equivalence relation on a (nonempty) set A, if fulfills the following conditions: (E1 ) {(a, a) | a ∈ A} ⊆ (reflexivity); (E2 ) ∀a, b ∈ A : (a, b) ∈ =⇒ (b, a) ∈ (symmetry); (E3 ) ∀a, b, c ∈ A : ((a, b) ∈ and (b, c) ∈ ) =⇒ (a, c) ∈ (transitivity). Let Eq(A) be the set of all equivalence relations on A. By defining of the operation

−1

: P(A × A) −→ P(A × A), → −1 by

−1 := {(b, a) | (a, b) ∈ } and of the binary operation 2 : P(A × A) × P(A × A) −→ P(A × A), (, ) → ◦ by

2 := {(a, c) | ∃b ∈ A : (a, b) ∈ and (b, c) ∈ },

one can also write down the above conditions (E2 ) and (E3 ) as follows: (E2 ) −1 = , (E3 ) 2 ⊆ .

2.4 Complete Lattices and Equivalence Relations

43

Instead of (a, b) ∈ we often write ab or (a, b) or a = b (mod ) or a ∼ b (mod ) (one say: a is equals (or equivalent) to b modulo ). For every set A there are two trivial equivalence relations: ∇A (:= κ1 ) := A2 (all relation)

and

∆A (:= κ0 ) := {(a, a) | a ∈ A} (identity or diagonale). Theorem 2.4.3 (Eq(A); ⊆) is a complete lattice for every nonempty set A. One can determine the infimum and the supremum of an arbitrary subset T := {i | i ∈ I} of Eq(A) as follows: T = i∈I i , T = i0 ,...,it ∈I;t 0, and ϕ(fi ) = gi for all fi ∈ F with afi = 0.

52

4 Homomorphisms, Congruences, and Galois Connections

If ϕ in addition is bijective from A onto B, then ϕ is called an isomorphism or an isomorphic mapping from A onto B. A homomorphism from an algebra A into A is called an endomorphism of A. An isomorphism from A onto A is an automorphism of A. Obviously, ϕ−1 is an isomorphism from B onto A, if ϕ is an isomorphism from A onto B. Therefore, we can say: “A and B are isomorphic”, we write in this case A∼ =B (read: “A isomorphic B”) for the identification of this fact. It is easy to see that the composition of homomorphic mappings ϕ1 2ϕ2 (ϕ1 : A1 −→ A2 , ϕ2 : A2 −→ A3 ) is a homomorphism from A1 = (A1 ; F1 ) into A3 = (A3 ; F3 ). With the help of this property, one can show that the relation “∼ =” is an equivalence relation on sets of algebras. The following lemma summarizes further elementary properties of homomorphic mappings. Lemma 4.1.1 Let ϕ be a homomorphic mapping from A = (A; F ) into B = (B; G). Then it holds: (a) ϕ(A) is closed in respect to B; i.e., (ϕ(A); G) is a subalgebra of B, which is called homomorphic image of A by ϕ. (b) If A is a subalgebra of A, then (ϕ(A ); G) is a subalgebra of (ϕ(A); G). (c) If T ⊆ A is a generating system of A, then ϕ(T ) is a generating system of ϕ(A). (d) If (B ; G) is a subalgebra of (ϕ(A); G), then the so-called inverse image (ϕ−1 (B ); F ) with ϕ−1 (B ) := {a ∈ A | ϕ(a) ∈ B } is a subalgebra of A. (e) Let ψ be a homomorphism from A into B, which is identical with ϕ on a generating system of A. Then, ϕ(a) = ψ(a) for each a ∈ A.

4.2 Congruence Relations and Factor Algebras of Algebras Definitions A congruence relation (briefly: congruence) or a kernel of a homomorphic mapping ϕ of an algebra A = (A; F ) is an equivalence relation κϕ on A that is induced by the homomorphism ϕ from A into B, that is, for all a, a ∈ A it holds: (a, a ) ∈ κϕ :⇐⇒ ϕ(a) = ϕ(a ). In the following, we also denote the relation κϕ with Ker ϕ. Let Con(A) be the set of all congruences of A. Examples The identity mapping idA with idA : A −→ A, a → a

4.2 Congruence Relations and Factor Algebras of Algebras

53

and the mapping ϕC with ϕC : A −→ {c}, a → c from A onto the 1-element algebra of the same type ({c}; G), where g(c, c, ..., c) = c for all g ∈ G, if ag > 0, and g = c, if ag = 0 holds, induce two so-called trivial congruences, the zero-congruence κ0 and the all-congruence (or one-congruence) κ1 : κ0 := {(a, a) | a ∈ A}, κ1 := A × A. Definition An algebra, which has only both congruences κ0 and κ1 , is called simple. Examples for simple algebras are groups of prime order and the fields. As the following lemma demonstrates, it is possible to describe the concept “congruence” without use of the concept “homomorphism”. Lemma 4.2.1 An equivalence relation κ on A is a congruence on the algebra A = (A; F ) if and only if it is compatible with all not nullary operations of F ; i.e., if for all f n ∈ F with n = af > 0 and arbitrary a1 , ..., an , a1 , ..., an ∈ A it holds: {(a1 , a1 ), ..., (an , an )} ⊆ κ =⇒ (f (a1 , ..., an ), f (a1 , ..., an )) ∈ κ. Proof. Let F := {fini | i ∈ I}, where I denotes a certain index set. “=⇒”: Let κ be a congruence on A. Then, by the definition of a congruence, there exists an algebra B := (B; G), where G := {gini | i ∈ I}, and a homomorphic mapping ϕ : A −→ B with the property

κ = {(a, a ) | ϕ(a) = ϕ(a )}.

We have to show that κ is compatible with all f ∈ F \ F 0 . To show this let f := fini ∈ F \F 0 be arbitrary. For the purpose of simplification, we put n := ni and g := gini . Furthermore, let (a1 , a1 ), ..., (an , an ) ∈ κ be arbitrary. Since ϕ is a homomorphism, then we have

54

4 Homomorphisms, Congruences, and Galois Connections ϕ(f (a1 , ..., an )) = g(ϕ(a1 ), ..., ϕ(an )) = g(ϕ(a1 ), ..., ϕ(an )) = ϕ(f (a1 , ..., an )).

Consequently, (f (a1 , ..., an ), f (a1 , ..., an )) ∈ κ. “⇐=”: Conversely, let the equivalence relation κ on A be compatible with all operations of the algebra A = (A; F ). We have to show: There is an algebra B = (B; G) of the same type as A and a homomorphic mapping ϕ : A −→ B with the property {(x, y) | ϕ(x) = ϕ(y)} = κ. To construct the algebra B, we define: a/κ := {x ∈ A | (x, a) ∈ κ}

(a ∈ A)

(i.e., a/κ is the notation of the equivalence class in which lies a) and B := A/κ := {a/κ | a ∈ A} (i.e., B is the set of all equivalence classes of κ). Now, we are able to define the operations gini on the set B as follows: gini (a1 /κ, ..., ani /κ) := fini (a1 , ..., ani )/κ

(4.1)

(i.e., gini (a1 /κ, ..., ani /κ) is exactly the equivalence class in which fini (a1 , ..., ani ) lies). The above definition is possible, since fi is compatible with κ. Detailed: Choosing (a1 , a1 ), ..., (ani , ani ) ∈ κ, then it holds (a1 /κ, ..., ani /κ) = (a1 /κ, ..., ani /κ) and fini (a1 , ..., ani )/κ = fini (a1 , ..., ani )/κ. We put G := {gini | i ∈ I}. Then the mapping ϕ : A −→ B, a → a/κ is a homomorphic mapping from A onto B := (B; G), since by definition of κ ϕ(fini (a1 , ..., ani )) = fini (a1 , ..., ani )/κ holds and (by the compatibility of κ with fini and by the definition (4.1) of gi ∈ G further fini (a1 , ..., ani )/κ = gini (a1 /κ, ..., ani /κ) = gini (ϕ(a1 ), ..., ϕ(ani )) holds, which implies ϕ(fini (a1 , ..., ani )) = gini (ϕ(a1 ), ..., ϕ(ani )). Furthermore, by definition of ϕ and by the properties of an equivalence relation, it holds: {(x, y) ∈ A × A | ϕ(x) = ϕ(y)} = {(x, y) ∈ A × A | x/κ = y/κ} = κ.

4.2 Congruence Relations and Factor Algebras of Algebras

Definitions Lemma 4.2.1,

55

The above algebra, which was designed in the proof of the (A/κ; G)

of the so-called congruence classes of A modulo κ is called factor algebra of (A; F ) (or quotient algebra). The homomorphic mapping ϕ : A −→ A/κ, a → a/κ, defined in the proof of Lemma 4.2.1, is called natural homomorphism (or quotient homomorphism) from A onto A/κ. We make following use of our agreement from the first chapter according to which the operations of algebras of the same type are described equal. Only then, if distinctions are necessary at the denotation, do we indicate the operations. After these preparations, we can prove the following theorem (in generalization of analogous theorems over groups, rings, ...): Theorem 4.2.2 (General Homomorphism Theorem) For each homomorphism ϕ from an algebra A := (A; F ) into an algebra of the same type B := (B; F ) the algebra ϕ(A) := (ϕ(A); F ) is isomorphic to the factor algebra A/κϕ := (A/κϕ ; F ), where κϕ = {(a, a ) ∈ A × A | ϕ(a) = ϕ(a )}. Proof. It suffices to check that α : ϕ(A) −→ A/κϕ , ϕ(a) → a/κϕ is an isomorphism. Since a/κϕ = a /κϕ =⇒ (a, a ) ∈ κϕ =⇒ ϕ(a) = ϕ(a ), α is a mapping. Obviously, by definition, α is a surjection. The injectivity of α follows from the following: α(ϕ(a)) = α(ϕ(a )) =⇒ a/κϕ = a /κϕ =⇒ (a, a ) ∈ κϕ =⇒ ϕ(a) = ϕ(a ) (a, a ∈ A). Thus, α is bijective. To prove that α is a homomorphism, let f n ∈ F and let ϕ(a1 ), ..., ϕ(an ) ∈ ϕ(A) be arbitrary. Then it holds:

56

4 Homomorphisms, Congruences, and Galois Connections α(fϕ(A) (ϕ(a1 ), ..., ϕ(an ))) = α(ϕ(fA (a1 , ..., an )))

(since ϕ is a homomorphism)

= f (a1 , ..., an )/κϕ

(by definition of α)

= fA/κϕ (a1 /κϕ , ..., an /κϕ )

(since κϕ is compatible with f )

= fA/κϕ (α(ϕ(a1 )), ..., α(ϕ(an ))) (by definition of α), Thus our bijective mapping α is an isomorphism. The next lemma summarizes some elementary properties of congruences. The proof for this lemma is left to the reader. Lemma 4.2.3 It holds: (a) The intersection of arbitrary many congruences of an algebra A is also a congruence on A. (b) If κ is a congruence on (A; F ) and ϕ is a homomorphism defined over A with κϕ ⊆ κ, then ϕ(κ) := {(ϕ(a), ϕ(a )) | (a, a ) ∈ κ} is a congruence on (ϕ(A); F ). (c) If π is a congruence on (ϕ(A); F ) and ϕ is a homomorphism defined on (A; F ), then ϕ−1 (π) := {(a, a ) ∈ A × A | (ϕ(a), ϕ(a )) ∈ π} is a congruence on (A; F ), which κϕ includes. (d) If κ is a congruence on A = (A; F ) and B = (B; F ) is a subalgebra of A, then κ|B := {(b, b ) ∈ κ | b, b ∈ B} is a congruence on B. (e) The set ConA of all congruence relations of an algebra A is a complete lattice in respect to the inclusion ⊆ with inf{κj | j ∈ J} = j∈J κj , sup{κj | j ∈ J} =< j∈J κj >ConA , where {κj | j ∈ J} ⊆ ConA and < j∈J κj >ConA is the intersection of all congruences which contain j∈J κj .

4.3 Examples for Congruence Relations and Some Homomorphism Theorems In the following, we characterize the congruence relations by groups and rings more closely. In addition, we give the special homomorphism theorems that follow from these characterizations. 4.3.1 Congruences on Groups Let G = (G; ◦,−1 , e) be a group. A subgroup N of G is called a normal subgroup iff

4.3 Examples for Congruence Relations and Some Homomorphism Theorems

57

∀x ∈ G : x ◦ N = N ◦ x holds. Let NG be the set of all normal subgroups of G. If N is a normal subgroup of G, one also writes N G. The following connections between normal subgroups and congruences on G can easily be checked: (a) For every κ ∈ ConG, e/κ is a normal subgroup of G, and for arbitrary a, b ∈ G it holds: (a, b) ∈ κ ⇐⇒ a ◦ b−1 ∈ e/κ. (b) If N is a normal subgroup then one can obtain by κN := {(a, b) | a ◦ b−1 ∈ N } a congruence on G with e/κ = N . Consequently, the mapping α : ConG −→ NG, κ → e/κ is an order-preserving bijection. Since for every homomorphism ϕ from G into a group G the neutral element e is mapped to the neutral element of G , the properties (a), (b), and Theorem 4.2.2 imply the following Theorem 4.3.1.1 (Homomorphism Theorem for Groups) For every homomorphism ϕ from a group G into a group G there exists as so-called kernel (notation: ker ϕ) a normal subgroup K of G, which consists of all the elements of G which are mapped to the neutral element of G . The group G is isomorphic to the factor group G/K = ({x ◦ K | x ∈ G}; ◦,−1 , K)), where (x ◦ K) ◦ (y ◦ K) := (x ◦ y) ◦ K for arbitrary x, y ∈ G. Conversely, if K is a normal subgroup of G, then K is the kernel of the natural homomorphism from G onto the factor group G/K (∀x ∈ G : x → x ◦ K). We notice that the concept kernel of a group homomorphism differs from the general term kernel of a homomorphism. The corresponding is valid for the concept kernel from Theorem 4.3.2.1.

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4 Homomorphisms, Congruences, and Galois Connections

4.3.2 Congruences on Rings Let R = (R; +, −, 0, ·) be a ring. A subgroup I of (R; +, −, 0) is called an ideal of R iff ∀x ∈ R : x · I ⊆ I ∧ I · x ⊆ I holds. Let IR be the set of all ideals of R. Connections between congruences on R and ideals of R are the following: For every κ ∈ ConR, 0/κ is an ideal of R and we have for arbitrary a, b ∈ R: (a, b) ∈ κ ⇐⇒ a − b ∈ 0/κ. If I is an ideal of R then one can obtain by κI := {(a, b) | a − b ∈ I} a congruence on R with 0/κI = I. Consequently, the mapping ConR −→ IR, κ → 0/κ is an order-preserving bijection. Further it holds: Theorem 4.3.2.1 (Homomorphism Theorem for Rings) For every homomorphism ϕ from a ring R into a ring R belongs as a kernel (notation: ker ϕ) an ideal I of R, which consists of all the elements of R which are mapped to the zero element of R . The ring R is isomorphic to the residue class ring R/I = ({x + I | x ∈ R}; +, −, I, ·), where (x + I) + (y + I) := (x + y) + I and (x + I) · (y + I) := (x · y) + I for arbitrary x, y ∈ R. Conversely, if I is an ideal of R then I is the kernel of the natural homomorphism from R onto R/I (∀x ∈ R : x → x + I). It is obvious idea, according to a homomorphism theorem for rings, to find also a homomorphism theorem for fields. The next theorem shows, however, that such a theorem is only a trivial and special case of the general homomorphism theorem. Theorem 4.3.2.2 A ring R, which is also a field, has only the two trivial ideals {0} and R. In other words: A field has only the trivial congruences κ0 and κ1 . Proof. Let R be a field and let κ be a congruence of R, which is different from κ0 . Denote I the ideal belonging to κ. Since κ = κ0 , there is certain a, b ∈ R with a = b and (a, b) ∈ κ. Consequently, we have (c, 0) := (a − b, b − b) ∈ κ. Hence, I contains the invertable element c. Since I also contains all elements r · c for every r ∈ R (specially, r = r · c−1 with arbitrary r ∈ R) we have I = R. Thus, κ = κ1 .

4.4 Galois Connections

59

4.4 Galois Connections Definitions A Galois connection (or a Galois correspondence) between the sets A and B is a pair (σ, τ ) of mappings σ : P(A) −→ P(B) and τ : P(B) −→ P(A), such that for all X, X ⊆ A and all Y, Y ⊆ B the following conditions are fulfilled:

X ⊆ X =⇒ σ(X) ⊇ σ(X ) (antitony) (GC1) Y ⊆ Y =⇒ τ (Y ) ⊇ τ (Y ) X ⊆ τ (σ((X))) (GC2) (extensivity). Y ⊆ σ(τ ((Y ))) Let (P ; ≤) be a poset. The dual order ≤δ to the order ≤ is defined by x ≤δ y :⇐⇒ y ≤ x. (P ; ≤) is called dual isomorphic (or antiisomorphic) to (Q; ≤), iff (P ; ≤) is isomorphic to (Q; ≤δ ). The bijective mapping appertaining to this fact is called dual isomorphism (or antiisomorphism) . Theorem 4.4.1 Let the pair (σ, τ ) of mappings σ: P(A) −→ P(B) and τ : P(B) −→ P(A) be a Galois connection between A and B. Then it holds: (a) The mappings στ := σ2τ : P(A) −→ P(A) and τ σ := τ 2σ : P(B) −→ P(B) are hull operators on A or B, respectively. (b) The στ -closed sets are exactly the sets of the form τ (Y ), Y ⊆ B. The τ σ-closed sets are exactly the sets of the form σ(X), X ⊆ A. (c) Let Hστ and Hτ σ be the hull systems that are assigned to στ and τ σ, respectively. Then the lattices (Hστ ; ⊆) and (Hτ σ ; ⊆) are dual isomorphic, and σ and τ are inversely dual isomorphisms to each other of these lattices. Proof. (a): The extensivity and the monotony of στ and τ σ immediately follow from (GC1) and (GC2), respectively. Thus, for all X ⊆ A we have X ⊆ τ (σ(X)) and therefore σ(X) ⊇ σ(τ (σ(X))). On the other hand, σ(X) ⊆ (τ σ)(σ(X)) = σ(τ (σ(X))) follows from (GC2). Consequently, we have

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4 Homomorphisms, Congruences, and Galois Connections σ(X) = σ(τ (σ(X)))

(4.2)

τ (Y ) = τ (σ(τ (Y ))).

(4.3)

and analogously Then the equations τ (σ(X)) = τ (σ(τ (σ(X)))) and σ(τ (Y )) = σ(τ (σ(τ (Y )))) follow from these. Thus, στ and τ σ are idempotent. (b): For a στ -closed set X we have X = τ (σ(X)); that is, X has the form X = τ (Y ) with Y := σ(X) ⊆ B. Conversely, a set of the form X := τ (Y ), Y ⊆ B, is στ -closed by (4.3). Analogously, one can prove the assertion for τ σ-closed sets. (c): Because of (b) and Theorem 3.2.1 it holds Hστ = {τ (Y ) | Y ⊆ B} and Hτ σ = {σ(X) | X ⊆ A}. Thus, we have σ(Hστ ) := {σ(τ (Y )) | Y ⊆ B} = Hτ σ and τ (Hτ σ ) := {τ (σ(X)) | X ⊆ A} = Hστ . By (GC1) σ and τ are antitone, and therefore the restrictions of these mappings to Hστ and Hτ σ have also this property. It follows from the idempotence of στ that στ on Hστ is the identical mapping. Analogously, one can see that τ σ is also the identical mapping on Hτ σ . Therefore, the mappings σ : Hστ −→ Hτ σ and τ : Hτ σ −→ Hστ are bijective mappings and it holds σ −1 = τ . Consequently, σ, τ are isomorphisms of the lattices (Hστ ; ⊆) and (Hτ σ ; ⊆δ )

An example of a Galois connection concludes this section. Further examples can be found in [Den-E-W 2004]. Theorem 4.4.2 Let A, B be nonempty sets and let R ⊆ A × B with R = ∅. The mappings σ : P(A) −→ P(B), τ : P(B) −→ P(A) are defined by σ(X) := {y ∈ B | ∀x ∈ X : (x, y) ∈ R}, τ (Y ) := {x ∈ A | ∀y ∈ Y : (x, y) ∈ R}. Then the pair (σ, τ ) is a Galois connection between A and B. Proof. Because of symmetry of the assumptions, it suffices to show that σ is an antitone and τ σ is an extensive mapping. Let X ⊆ X ⊆ A. Then for every y ∈ σ(X ) we have: (x, y) ∈ R for all x ∈ X . Thus (by X ⊆ X ) we have also (x, y) ∈ R for all x ∈ X. Therefore σ(X ) ⊆ σ(X) holds. The inclusion X ⊆ τ (σ(X)) follows directly from τ (σ(X)) = {x ∈ A | ∀y ∈ σ(X) : (x, y) ∈ R} and from the definition of σ(X).

5 Direct and Subdirect Products

With the help of direct and subdirect products, it is possible to form new algebras with larger universes from given algebras. Of these constructions one immediately asks the following questions: Which algebras are smallest “constituents” of given algebras? How can one reduce a given algebra to its smallest “constituents”? First, we will show that every finite algebra is isomorphic to a direct product of directly irreducible algebras. Then, we prove that every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras.

5.1 Direct Products First, we consider direct products of two algebras. Definitions Let B = (B; F ) and C = (C; F ) be algebras of the same type τ. The algebra A := B × C of type τ is called direct product of the algebras B and C, iff B × C is the universe of the algebra A and the operations of A are defined as follows: If f ∈ F is nullary, then let fA := (fB , fC ). If af ≥ 1, let fA defined as follows: fB×C ((b1 , c1 ), (b2 , c2 ), ..., (baf , caf )) := (fB (b1 , ..., baf ), fC (c1 , ..., caf )). Obviously, B and C are homomorphic images of the algebra B × C, since the (so-called projection-)mappings pr1 : B × C −→ B, (b, c) → pr1 (b, c) := b and pr2 : B × C −→ C, (b, c) → pr2 (b, c) := c are homomorphisms from B × C onto B or C, respectively. The kernels of these projection mappings (congruences on B × C)

62


Ker pri := {((b, c), (b , c )) ∈ (B × C)2 | pri (b, c) = pri (b , c )} (i = 1, 2) are distinguished from other congruences by certain properties (see Theorem 5.1.2). We can describe these properties with the help of the following Definition Two equivalence relations κ, µ ∈ Eq(A) are called permutable, if κ2µ = µ2κ holds; i.e., ∀x, z ∈ A : (∃y ∈ A : (x, y) ∈ κ ∧ (y, z) ∈ µ) ⇐⇒ (∃y ∈ A : (x, y ) ∈ µ ∧ (y , z) ∈ κ). The following lemma gives some other definition possibilities for the above concept. In this lemma and in the following theorems, (Eq; ⊆), by Theorem 2.4.3, is a complete lattice and operations ∧ = ∩ (“infimum”) and ∨ (“supremum”) are defined on Eq (see Chapter 2). Lemma 5.1.1 For κ, µ ∈ Eq(A) the following statements are equivalent: (a) (b) (c) (d) (e)

κ and µ are permutable κ2µ ⊆ µ2κ µ2κ ⊆ κ2µ κ ∨ µ = κ2µ µ ∨ κ = κ2µ

Proof. Because of the commutativity of ∨, we have (d)⇐⇒(e). The equivalence (b)⇐⇒(c) is a conclusion from (a)⇐⇒(b). Thus, we have to show that (a)⇐⇒(b) and (a)⇐⇒(e) holds. (a)=⇒(b) is trivial. (b)=⇒(a): Let κ2µ ⊆ µ2κ. Then (by (κ2µ)−1 = µ−1 2κ−1 and by the symmetry of κ and µ): (κ2µ)−1 ⊆ (µ2κ)−1 =⇒ µ−1 2κ−1 ⊆ κ−1 2µ−1 . =µ2κ

=κ2µ

Hence, κ2µ = µ2κ. (a)=⇒(e): Let κ2µ = µ2κ. (e) is shown, if it is proven that κ2µ is the smallest equivalence relation on A, which contains κ ∪ µ. Because of ∀(x, y) ∈ κ (( (x, y) ∈ κ ∧ (y, y) ∈ µ) =⇒ (x, y) ∈ κ2µ), ∀(x, y) ∈ µ (( (x, x) ∈ κ ∧ (x, y) ∈ µ) =⇒ (x, y) ∈ κ2µ) we have κ ∪ µ ⊆ κ2µ. κ2µ is reflexive, since κ (or µ) is reflexive. The symmetry of κ2µ follows from (κ2µ)−1 = µ−1 2κ−1 = µ2κ = κ2µ. Because of (κ2µ)2(κ2µ) = κ2 (µ2κ) 2µ = (κ2κ) 2 (µ2µ) ⊆ κ2µ =κ2µ

⊆κ

⊆µ

κ2µ is also transitive. Thus, κ2µ is an equivalence relation on A. κ2µ is the smallest equivalence relation of Eq(A), which contains κ ∪ µ, since every

5.1 Direct Products

63

other equivalence relation, which contains κ ∪ µ, also contains the transitive closure of κ ∪ µ and therefore contains κ2µ. (e)=⇒(a): Let κ ∨ µ = κ2µ. Since κ ∨ µ is an equivalence relation, the compatibility of κ and µ follows from κ2µ = κ ∨ µ = (κ2µ)−1 = µ−1 2κ−1 = µ2κ.

Theorem 5.1.2 For algebras B and C of the same type and the projection mappings pr1 : B × C −→ B and pr2 : B × C −→ C it holds: (a) Ker pr1 ∧ Ker pr2 = κ0 (b) Ker pr1 ∨ Ker pr2 = κ1 (c) Ker pr1 and Ker pr2 are permutable Proof. Let ((b, c), (b , c )) ∈ Ker pr1 ∩ Ker pr2 be arbitrary. Then, pri (b, c) =

pri (b , c ) for i = 1, 2. This means, however, that b = b and c = c . Therefore, (a) is shown. For arbitrary b, b ∈ B, c, c ∈ C we have ( (b, c), (b, c ) ) ∈ Ker pr1 ∧ ( (b, c ), (b , c ) ) ∈ Ker pr1 . Hence, ( (b, c), (b , c ) ) ∈ κ2µ for all b, b ∈ B and c, c ∈ C and thus (Ker pr1 )2(Ker pr2 ) = κ1 . With the help of Lemma 5.1.1, the assertions (b) and (c) follow from this.

Now we will examine the conditions under which an algebra is a direct product of two other smaller algebras. Theorem 5.1.2 provides instructions on how to proceed. Theorem 5.1.3 Let A = (A; F ) be an algebra and let κ, µ ∈ ConA be two congruence relations with the following three properties: (a) κ ∧ µ = κ0 (b) κ ∨ µ = κ1 (c) κ and µ are permutable Then A is isomorphic to the direct product of A/κ and A/µ. An isomorphism ϕ: A −→ A/κ × A/µ is defined by ∀a ∈ A : ϕ(a) := (a/κ, a/µ). Proof. ϕ is injective: Let ϕ(a) = ϕ(b). Then, a/κ = b/κ and a/µ = b/µ. From these (a, b) ∈ κ ∧ µ follows. Thus a = b by (a). ϕ is surjective: By (b) and (c) for every pair a, b there is a c ∈ A with (a, c) ∈ κ and (c, b) ∈ µ. Consequently, (a/κ, b/µ) = (c/κ, c/µ) = ϕ(c). ϕ is an isomorphism: For all fA ∈ F (afA =: n) and arbitrary a1 , ..., an ∈ A it holds ϕ(fA (a1 , ..., an )) = (fA (a1 , ..., an )/κ, fA (a1 , .., an )/µ) = (fA/κ (a1 /κ, ..., an /κ), fA/µ (a1 /µ, ..., an /µ)) = fA/κ×A/µ (ϕ(a1 ), ..., ϕ(an )).

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Definition An algebra A is called directly irreducible, iff A ∼ = B×C implies |B| = 1 or |C| = 1 for all algebras B and C. Example Obviously, every finite algebra A with |A| ∈ P is directly irreducible. One finds further examples after Theorem 5.1.4. Theorem 5.1.4 An algebra A is directly irreducible iff κ0 and κ1 are the only one pair of congruences of ConA, which satisfy the conditions (a)–(c) of Theorem 5.1.3. Proof. Let A be directly irreducible. Further, κ, µ ∈ ConA fulfill the conditions (a)–(c) of Theorem 5.1.3. Then, by Theorem 5.1.3, we have A ∼ = A/κ × A/µ. Thus, w.l.o.g., |A/κ| = 1. From this, κ = κ1 follows, and then µ = κ0 by (a). Conversely, Let κ0 and κ1 be the only one pair of congruences of A with the properties (a)–(c) of Theorem 5.1.3, and let A ∼ = B × C. Obviously, then κ0 and κ1 are the only one pair of congruence relations on B × C with (a)–(c). By Theorem 5.1.2 the kernels of the projection mappings pr1 and pr2 fulfill (a)–(c). Thus Ker pr1 = κ0 or Ker pr2 = κ0 holds and we have |C| = 1 or |B| = 1, respectively.

With the help of Theorem 5.1.4, one can prove the following statements:1 (a) Every simple algebra A; i.e., every algebra A with Con A = {κ0 , κ1 }, is directly irreducible. (b) If A is a Boolean algebra then A is directly irreducible ⇐⇒ |A| ≤ 2. (c) The residue class group (Zn ; +, −, 0) is directly irreducible iff n is a prime number power. (d) A vector space V := (V ; +, −, K, 0) over the field K is directly irreducible iff |V | = 1 or V is 1-dimensional. One can generalize our definition of the direct product of two algebras in an obvious way for finite many algebras of the same type. As the next definitions demonstrate, it is also possible to form direct products of arbitrarily many algebras of the same type. Definitions The Cartesian product Πj∈J Aj of the sets Aj (j ∈ J) is the set of all mappings α from J into j∈J Aj with α(j) ∈ Aj for all j ∈ J. We write down the elements of Πj∈J Aj in the form (xj | j ∈ J). In analog mode too, one can define then the algebra (Πj∈J Aj ; (fi )i∈I ) as a direct product Πj∈J Aj of the algebras Aj (j ∈ J), where 1

The proof for the statements can be found in [Lau 2004], volume 2.

5.1 Direct Products

65

fi ((aj1 |j ∈ J), (aj2 |j ∈ J), ..., (aj,afi |j ∈ J)) := (fji (aj1 , aj2 , ..., aj,afi )|j ∈ J) for arbitrary ((aj1 |j ∈ J), ..., (aj,afi |j ∈ J) ∈ Πj∈J Aj , if afi > 0, and fi = (fji |j ∈ J), if afi = 0. If J = ∅, then let Πj∈J Aj be the 1-element algebra of the corresponding type. If Aj = A for all j ∈ J, we write AJ instead of Πj∈J Aj . If J = {1, 2, .., n}, one also uses the denotation A1 × ... × An for the direct product of the algebras Aj . Theorem 5.1.5 Every finite algebra is isomorphic to a direct product of direct irreducible algebras. Proof. Induction over the cardinality of the universes of the algebras: Obviously, every algebra A with |A| = 1 is directly irreducible. Now, let A be a finite algebra with |A| ≥ 2, and assume that the assertion is proven for all algebras A with |A | < |A|. If A is directly irreducible, we have to show nothing more. If, however, A ∼ = B × C with |B| > 1, |C| > 1 is valid, then we have |B| < |A| and |C| < |A|; i.e., one can find direct irreducible algebras B1 , ..., Bm , C1 , ..., Cn with B∼ = B1 × ... × Bm , C∼ = C1 × ... × Cn . Thus A ∼ = B1 × ... × Bm × C1 × ... × Cn .

The direct products of more than two algebras have similar properties, like the direct products of only two algebras. For example, the j0 -th projection prj0 from Πj∈J Aj onto Aj with (aj | j ∈ J) → aj0 is a homomorphism for every j0 ∈ J. One can easily check the following property of direct products. Theorem 5.1.6 For every family2 ϕi : B −→ Ai , i ∈ I, of homomorphisms one can form a homomorphism ϕ: B −→ Πi∈I Ai by (ϕ(b))i := ϕi (b).

2

A family (ai | i ∈ I) of elements of a set A is a mapping ϕ : I → A, i → ai . This denotation is used around a certain selection of (not necessarily different) elements from A to characterize. Then, I is called index set of the family (ai | i ∈ I).

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5.2 Subdirect Products Unlike finite algebras, infinite algebras cannot always be represented by direct products of directly irreducible algebras. Example As we already noticed above, a Boolean algebra A is directly irreducible iff |A| ≤ 2. Furthermore, one can easily prove that every 2-element Boolean algebra is isomorphic to the algebra B = ({0, 1}; ∨, ∧,− , 0, 1). An infinite direct product of B is not countable. Consequently, the countable Boolean algebra C = (C; ∨ , ∧ , ¬) with C := {(a1 , a2 , ...) ∈ {0, 1}N | |{i ∈ N | ai = 0}| < ℵ0 or |{i ∈ N | ai = 1}| < ℵ0 }, (a1 , a2 , ...) ◦ (b1 , b2 , ...) := (a1 ◦ b1 , a2 ◦ b2 , ...) for ◦ ∈ {∨, ∧} and ¬(a1 , a2 , ...) := (a1 , a2 , ...) is not isomorphic to a direct product of directly irreducible algebras, however, it is a subalgebra of the direct product BN . In generalizing this example, we get the following new product concept: Definition Let the algebras Ai , i ∈ I, be of the same type. A subalgebra B of Πi∈I Ai is called a subdirect product of the Ai iff prj (B) = Aj holds for all j ∈ I. Example

Obviously, every direct product is also a subdirect product.

Theorem 5.2.1 For a subdirect product B of the algebras Ai , i ∈ I, and the projection mappings prj : Πi∈I Ai −→ Aj it holds Ker(prj )|B = κ0 . j∈I

Proof. a = b.

If (a, b) ∈

j∈I

Ker(prj )|B , then aj = bj follows for all j ∈ I and therefore

By the above theorem and by the fact that all prj|B are surjective, the subdirect products are already characterized: Theorem 5.2.2 Let A be an algebra. For certain congruences κi ∈ ConA, i ∈ I, let κi = κ0 . i∈I

Then A is isomorphic to a subdirect product of the algebras A/κi , i ∈ I. By ϕ(a) := (a/κi | i ∈ I) an injective homomorphism ϕ: A −→ Πi∈I (A/κi ) is defined, and ϕ(A) is a subdirect product of the algebras A/κi .

5.2 Subdirect Products

67

Proof. By Theorem 5.1.6 ϕ is a homomorphism. ϕ is also injective: Let ϕ(a) = ϕ(b). This implies a/κi = b/κi and thus (a, b) ∈ κi for all i ∈ I. Therefore, (a, b) ∈ κ i∈I i = κ0 ; i.e., a = b. Consequently, A and ϕ(A) are isomorphic. Further, by definition of ϕ we have prj (ϕ(A)) = A/κj for all j ∈ I. Hence, ϕ(A) is a subdirect product of the algebras A/κi . Definition An injective homomorphism (a so-called embedding) ϕ: A −→ Πi∈I Ai is called a subdirect representation of A, if ϕ(A) is a subdirect product of the algebras Ai . Example The mapping ϕ of Theorem 5.2.2 is a subdirect representation. Definition An algebra A is called subdirectly irreducible, if for every subdirect representation ϕ : A −→ Πi∈I Ai there exists a j ∈ I such that the mapping ϕ2prj : A −→ Aj is an isomorphism. Thus, an algebra is subdirectly irreducible if and only if one gets by with a single component in every subdirect representation. Theorem 5.2.3 An algebra A is subdirectly irreducible, if and only if the universe of A contains at most an element, or if (ConA\{κ0 }) = κ0 holds. The latter holds obviously iff κ0 exactly has an upper neighbor in ConA: κ1

r @

ConA :

@

@ @

@ @

@

@r

(ConA\{κ0 })

r

κ0 Proof. W.l.o.g. let |A| ∈ {0, 1} in the following. Suppose, (ConA\{κ0 }) = κ0 . Put I := ConA\{κ0 }. Then, with the help of Theorem 5.2.2, one obtains a subdirect representation ϕ: A −→ Πκ∈I (A/κ). For every mapping prκ (κ ∈ I) and all a ∈ A it holds (ϕ2prκ )(a) = a/κ. By κ0 ∈ I, therefore ϕ2prκ : A −→ A/κ is not injective (i.e., is not an isomorphism). Thus, A is not subdirectly irreducible. Let now µ := (ConA\{κ0 }) = κ0 ; i.e., there exists (a, b) ∈ µ \ κ0 , and denote ϕ: A −→ Πi∈I Ai a subdirect representation of A. We have to show that there exists an

68


i ∈ I so that ϕ2pri is an isomorphism from A onto Ai . Since ϕ is injective and a = b, there exists a j ∈ I with prj (ϕ(a)) = prj (ϕ(b)), whereby (a, b) ∈ Ker(ϕ2prj ). Thus, by (a, b) ∈ µ, we have µ ⊆ Ker(ϕ2prj ). Then, by definition of µ, Ker(ϕ2prj ) = κ0 holds; i.e., ϕ2prj is injective. Since ϕ(A) is a subdirect product, ϕ2prj is surjective. Therefore, ϕ2prj is an isomorphism. Consequently, A is subdirectly irreducible. With the help of Theorems 5.2.3 and 5.1.4, one can easily prove the following connection between the direct and the subdirectly irreducible algebras: A is subdirectly irreducible =⇒ A is directly irreducible.

(5.1)

The reversal of the statement (5.1) is not valid, because one can prove with the help of a 3-element lattice which is directly irreducible but which is not subdirectly irreducible. An essential aid for proving the following Theorem is Zorn’s Lemma which is indicated without proof here. This lemma is equivalent to the axiom of choice (see e.g. [Her 55]). Lemma 5.2.4 (Zorn’s Lemma) In every set system M with the property ∀T ⊆ M ((∀X, Y ∈ T ∃Z ∈ T : X ∪ Y ⊆ Z) =⇒

X ∈ M)

X∈T

(i.e., M is an inductively set system) there is a maximal element3 ; i.e., an element M ∈ M that is not contained in any proper subset of M. After these preparations we can prove the following theorem, published by G. Birkhoff in 1944. Theorem 5.2.5 Every algebra is isomorphic to a subdirect product of subdirectly irreducible algebras. Proof. Let A be an algebra. One can easily see that, for every pair a, b ∈ A with a = b, the set Ma,b := {κ ∈ ConA | (a, b) ∈ κ} is an inductive set system. By Lemma 5.2.4 Ma,b has a maximal element Φ(a, b). In the lattice ConA the element Φ(a, b) has exactly an upper neighbor, namely Φ(a, b) ∨ Ω(a, b), where Ω(a, b) is the congruence relation generated by (a, b). It is easy to check that the factor algebra A/Φ(a, b) is isomorphic to an interval [Φ(a, b), κ1 ] := {κ ∈ ConA | Φ(a, b) ⊆ κ ⊆ κ1 } 3

Let (B; ≤) be a poset. A maximal element of the set A ⊆ B is then an element a ∈ A with a < x =⇒ x ∈ A for all x ∈ B.

5.2 Subdirect Products

69

of ConA. By Theorem 5.2.3 the algebra A/Φ(a, b) is subdirectly irreducible. From {Φ(a, b) | a, b ∈ A ∧ a = b} = κ0 and from Theorem 5.2.2, it follows that A is isomorphic to a subdirect product of subdirectly irreducible algebras (namely of the algebras A/Φ(a, b)).

6 Varieties, Equational Classes, and Free Algebras

In this Chapter, only certain classes1 of algebras of the same type shall be regarded. First we introduce so-called varieties as classes of algebras, which are closed in respect to formation of subalgebras, homomorphic images, and direct products. We then come to a method for constructing algebra classes that strongly differs from the first method at first sight: Starting from certain equations from variables and operation symbols of a certain type τ , we form the class of all algebras of type τ that fulfill these equations. The result is an equational class. We will see, however, that there is a close connection between the two methods of algebra class construction: A class of algebras is equationally definable if and only if it is a variety. Free algebras are “the most general” algebras within a variety or an equational class (or an equationally definable class). In the section on equational classes, we will also address such concepts such as conclusion of an equational set. In addition, we investigate methods to receive such conclusions.

6.1 Varieties The following operators S, H, P , I map a class K of algebras of type τ to a class of algebras of the same type. Let S(K) be the class of all subalgebras of algebras aus K, H(K) be the class of all homomorphic images of algebras of K, P (K) be the class of all direct products of families of algebras of K, I(K) be the class of all algebras which are isomorphic to algebras of K. 1

The concept “class” is a generalization of the concept “set”. Informally speaking, a class is a collection so large that subjecting it to the operations admissible for sets would lead to logical contradictions.

72


We denote the composition of the operators Y, X ∈ {H, S, P, I} by XY ; i.e., it holds XY (K) := X(Y (K)). It is easy to check that S, H and IP are hull operators; i.e., for all classes K and L of algebras of the same type we have: ∀X ∈ {S, H, IP } : K ⊆ X(K) ∧ (K ⊆ L =⇒ X(K) ⊆ X(L)) ∧ X(K) = X(X(K)). The operator P is not idempotent: For all A, B, C ∈ K it holds (A×B)×C ∈ P (P (K)), but (A×B)×C ∈ P (K) generally does not hold. However, (A × B) × C ∼ = A × B × C ∈ P (K) is right; i.e., (A × B) × C ∈ IP (K). Definitions • A class K of algebras of the same type is called under X ∈ {S, H, P } closed, if X(K) ⊆ K holds. • If the class K of algebras of the same type is closed under the three operators S, H and P , then the class K is called a variety. Examples (1) It is easy to see that the class of all groups is a variety. (2) Since the direct product of the field Z2 with the field Z2 because of ∀x, y ∈ Z2 : (0, 1) · (x, y) = (0 · x, 1 · y) = (0, y) = (1, 1) is not a field, the class of all fields is not a variety. Lemma 6.1.1 For every class K of algebras of the same type it holds: (a) SH(K) ⊆ HS(K), (b) P S(K) ⊆ SP (K), (c) P H(K) ⊆ HP (K). Proof. (a): Let A ∈ SH(K); i.e., there is an algebra B ∈ H(K) with A ≤ B and B is homomorphic image of an algebra C ∈ K. Let ϕ: C −→ B be a surjective homomorphism. Then, for the subalgebra ϕ−1 (A) of C it holds ϕ(ϕ−1 (A)) = A. Consequently, we have A ∈ HS(K). (b): Let A ∈ P S(K). Then it holds A = Πi∈I Bi with Bi ≤ Ci ∈ K for all i ∈ I. Since obviously Πi∈I Bi is a subalgebra of Πi∈I Ci , we have A ∈ SP (K). (c): Let A ∈ P H(K). Then, A = Πi∈I Bi , where for every i ∈ I there are an algebra Ci and a surjective homomorphism ϕi : Ci −→ Bi . If prj : Πi∈I Ci −→ Cj is the projection mapping, then prj 2ϕj : Πi∈I Ci −→ Bj is a surjective homomorphism. By Theorem 5.1.6 we get by ϕ(c)j := (prj 2ϕj )(c) a homomorphism ϕ: Πi∈I Ci −→ Πi∈I Bi , which is also surjective. Consequently, we have A ∈ HP (K).

Theorem 6.1.2 A class K of algebras of the same type is a variety if and only if HSP (K) = K holds.

6.2 Terms, Term Algebras, and Term Functions

73

Proof. If K is a variety, then obviously HSP (K) = K. Let now HSP (K) = K. We have to show H(K) ⊆ K, S(K) ⊆ K and P (K) ⊆ K. It holds: H(K) = H(HSP (K)) = HSP (K), since H is idempotent. Thus, H(K) = K by assumption. Further we have: S(K) = S(HSP (K)) = SH(SP (K)) ⊆ HS(SP (K)) ⊆ HSP (K) by Lemma 6.1.1, (a) and since S is idempotent. Therefore, S(K) ⊆ K holds. Furthermore, P (K) = P HSP (K) ⊆ HP SP (K) ⊆ HSP P (K) ⊆ HSIP IP (K) = HSIP (K) ⊆ HSHP (K) ⊆ HHSP (K) = HSP (K) by Lemma 6.1.1 and since the operators IP and H are idempotent. Thus, P (K) ⊆ K holds. Consequently, K is a variety.

Every variety is exactly determined by its elements, which are subdirect irreducible algebras: Theorem 6.1.3 Every algebra of a variety K is isomorphic to a subdirect product of subdirect irreducible algebras of K. Proof. By Theorem 5.2.5, every algebra A is isomorphic to a subdirect product of subdirect irreducible algebras Ai , where every Ai is isomorphic to a factor algebra of A; i.e., it holds Ai ∈ H(A). If A belongs to a variety K, then Ai ∈ H(K) ⊆ K.

6.2 Terms, Term Algebras, and Term Functions In Section 1.1, we had agreed to describe the operations of algebras of the same type by the same notations, if it is clearly from the context to which algebra a given operation belongs. Since we will often make use of this convention, we generalize the concept “type of an algebra” as follows: Definitions A type of algebras is an ordered pair (F, τ ), where F is a set, whose elements are called operation symbols, and τ : F −→ N0 is a mapping, which assigns the arity af to every operation f ∈ F. Then, the algebra A = (A; F ) with F := {fA | f ∈ F} is called algebra of type (F, τ ). Let Fn be the set of the n-ary operation symbols of F . Now let (F, τ ) be a type of algebras and let X

74


be a finite or countable infinite set, whose elements are called variables, where X ∩ F0 = ∅. T (X) denotes the smallest set with the following two properties: (1) X ∪ F0 ⊆ T (X), (2) (f ∈ Fn ∧ {t1 , ..., tn } ⊆ T (X)) =⇒ f (t1 , ..., tn ) ∈ T (X). One observes that f (t1 , ..., tn ) is a syntactic expression (a symbol sequence) and not a function value. The elements of T (X) are called terms of type (F, τ ) over the alphabet X. Example Let F := {e, f }, τ (e) := 0, τ (f ) := 2 and X := {x, y, z}. Then T (X) = {e, x, y, z, f (e, e), f (e, x), ..., f (z, y), f (f (e, e), e), f (f (x, e), e), ..., f (f (x, z), f (z, y)), ........}. We agree that (u ◦ v) := f (u, v) for arbitrary u, v ∈ T (X) and, furthermore, we do without outer brackets. Then the set T (X) can be also written down as follows: {e, x, y, z, e ◦ e, e ◦ x, ..., z ◦ y, (e ◦ e) ◦ e, (x ◦ e) ◦ e, ..., (x ◦ z) ◦ (z ◦ y), ...}. T (X) is the universe of the so-called term algebra T(X) := (T (X); F ) of type (F, τ ), where for every f ∈ Fn (n ∈ N ∪ {0}) the operations of this algebra are defined as follows: fT(X) := f, if n = 0, and ∀t1 , ..., tn ∈ T (X) : fT(X) (t1 , ..., tn ) := f (t1 , ..., tn ) for n ≥ 1. A part of the operation table of the operation fT(X) from the above example then looks as follows: u e e x ◦ (y ◦ e)

v e e◦x (x ◦ x) ◦ z

fT(X) (u, v) e◦e e ◦ (e ◦ x) (x ◦ (y ◦ e)) ◦ ((x ◦ x) ◦ z)

The following lemma follows directly from the definition of T(X):

6.2 Terms, Term Algebras, and Term Functions

75

Lemma 6.2.1 The term algebra T(X) is generated by X; i.e., it holds [X] = T (X). The following theorem gives an important property of the algebra T(X): Theorem 6.2.2 Let T(X) be the term algebra of type (F, τ ) over X. Then, for every algebra A of type (F, τ ) and for every mapping ϕ: X −→ A there is exactly one homomorphism ϕ : T(X) −→ A, which ϕ continues, i.e., for which ϕ| X = ϕ holds. Proof. For a given algebra A of type (F, τ ) and for a given mapping ϕ: X −→ A let ϕ: T (X) −→ A be a mapping with the properties: ∀x ∈ X : ϕ(x) := ϕ(x) and ∀f (t1 , ..., tn ) ∈ T (X)\X : ϕ(f (t1 , ...tn )) := fA (ϕ(t 1 ), ..., ϕ(t n )). Obviously, ϕ is defined over T (X) by the above conditions, and it is the only possible continuation of ϕ over T (X).

Using the concepts from Part II, Chapter 1, we can briefly define the following Definition The term functions of an algebra A = (A; F ) of type (F, τ ) are operations over A that can be formed by superposition from the fundamental operations of F and from the projections. Without using concepts from Part II, Chapter 1, we can define the term functions an algebra A of type (F, τ ) as follows: Let t be a term of type (F, τ ) over X = {x1 , ..., xn }, and for a1 , ..., an ∈ A let ϕa1 ,...,an : T(X) −→ A be the unique homomorphism with xi → ai , i = 1, 2, ..., n. Then we can define an n-ary operation tA : An −→ A by ∀a1 , ..., an ∈ A : tA (a1 , ..., an ) := ϕa1 ,...,an (t). These operations are identical with the term functions already defined above. If t and TA are defined as above, we say that the term t induces the term function tA . Let T F (A) be the set of all term functions of A. One can prove the next two lemmas with properties of T F (A) easily. Lemma 6.2.3 Let [..] be the subalgebra-hull-operator (see Chapter 3). Then for every algebra A and every subset B of A it holds: [B] = {tA (b1 , ..., bn ) | n ∈ N ∧ t ∈ T ({x1 , ..., xn }) ∧ {b1 , ..., bn } ⊆ B}.

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Lemma 6.2.4 The algebras A, B and the n-ary term t have the same type. Then, for every homomorphism ϕ: A −→ B and all a1 , ..., an ∈ A it holds: ϕ(tA (a1 , ..., an )) = tB (ϕ(a1 ), ..., ϕ(an )); i.e., the term functions react just like the fundamental operations in respect to homomorphisms.

6.3 Equations and Equational Classes In this section, let T (X) be the set of all terms of type (F, τ ). To show that the variables of the term t ∈ T (X) are of the set {x1 , ..., xn } ⊆ X, we write t < x1 , ..., xn > and set t = t < x1 , ..., xn > . The notation s := t < t1 , ..., tn > meant that the term s was formed from the term t by substituting every variable xi (1 ≤ i ≤ n) by ti in every place the variable xi in t appeared. We agree on an analogous notation for term functions. Definitions • The elements of T (X) × T (X) are called equations (or identities) over X and we write s ≈ t :⇐⇒ (s, t) ∈ T (X) × T (X). • An algebra A of type (F, τ ) fulfills the equation s < x1 , ..., xn >≈ t < x1 , ..., xn > (or the equation s ≈ t holds in A), if for all a1 , ..., an ∈ A sA < a1 , ..., an >= tA < a1 , ..., an > is right. In this case, we also write A |= s ≈ t. Further, we set IdX (A) := {(s, t) ∈ T (X) × T (X) | A |= s ≈ t}.

6.3 Equations and Equational Classes

77

• Let be for Σ ⊆ T (X) × T (X) and classes K of algebras of the same type (F, τ ): A |= Σ :⇐⇒ (∀s ≈ t ∈ Σ : A |= s ≈ t). • The class

M od(Σ) := {A | A |= Σ}

is called the set of all models of Σ. • Conversely, for every class K of algebras of type (F, τ ) let IdX (K) := {(s, t) ∈ T (X) × T (X) | ∀A ∈ K : A |= s ≈ t} be the class of all equations over X that hold in all algebras of K. • A class K of algebras is equationally definable, if there exists a Σ ⊆ T (X) × T (X) with M od(Σ) = K. • A set Σ ⊆ T (X) × T (X) is called equational theory over X, if there is a class K of algebras with Σ = IdX (K). • An equation s ≈ t is called a conclusion of Σ ⊆ T (X) × T (X), if A |= s ≈ t holds for all A ∈ M od(Σ). Let ConsX (Σ) be the set of all conclusions of Σ; i.e., it holds ConsX (Σ) := IdX (M od(Σ)). Instead of Y (Z(..)), where Y, Z ∈ {M od, IdX , ConsX }, we write briefly Y Z(..). The following theorem summarizes elementary properties of the sets defined above and connections between the concepts just defined. Theorem 6.3.1 For arbitrary Σ, Σ ⊆ T (X) × T (X) and arbitrary classes K, K of algebras of type F it holds: (1) Σ ⊆ Σ =⇒ M od(Σ ) ⊆ M od(Σ), K ⊆ K =⇒ IdX (K ) ⊆ IdX (K); (2) Σ ⊆ IdX M od(Σ), K ⊆ M odIdX (K); (3) M odIdX M od(Σ) = M od(Σ), IdX M odIdX (K) = IdX (K); (4) Σ ⊆ ConsX (Σ), Σ ⊆ Σ =⇒ ConsX (Σ) ⊆ ConsX (Σ ), ConsX ConsX (Σ) = ConsX (Σ); (5) K ⊆ M odIdX (K), K ⊆ K =⇒ M odIdX (K) ⊆ M odIdX (K ), M odIdX M odIdX (K) = M odIdX (K); (6) Σ is equational theory ⇐⇒ Σ = ConsX (Σ), K is equationally definable ⇐⇒ K = M odIdX (K).

78


Proof. (1) and (2) immediately follow from the definitions of M od and IdX . (3): By (2) Σ ⊆ IdX M od(Σ) =: Σ holds. Thus, by means of (1), we have M odIdX M od(Σ) ⊆ M od(Σ). Conversely, we have also by (2): K := M od(Σ) ⊆ M odIdX M od(Σ). Therefore, M od(Σ) = M odIdX M od(Σ). Analogously, one can show IdX M odIdX (K) = IdX (K). (4) and (5) one can easily prove by means of (1)–(3). (6): Let Σ be an equational theory; i.e., there is a class K of algebras of type (F, τ ) with Σ = IdX (K). Then ConsX (Σ) = IdX M od(Σ)

assumption

=

(3)

IdX M odIdX (K) = IdX (K)

assumption

=

Σ.

Conversely, let Σ = ConsX (Σ). Then we have Σ = IdX M od(Σ); i.e., Σ is an equational theory. The statement over equational definable classes can be proven analogously.

If we neglect that the objects formed by classes need an exact definition, 2 then the following theorem is an immediate conclusion of the above theorem and of Theorem 4.4.1. Theorem 6.3.2 Let X be a countable infinite set, Alg(F, τ ) the class of all algebras of type (F, τ ) over X and let T (X) be the set of all terms of type (F, τ ). Then the pair (IdX , M od) forms a Galois connection between P(T (X) × T (X)) and P(Alg(F, τ )). Furthermore, the lattice of all equational classes of Alg(F, τ ) is antiisomorphic to the lattice of all equational theories of type (F, τ ).

6.4 Free Algebras To define a free algebra, we need the following properties of T(X): Theorem 6.4.1 Let K be a class of algebras of type (F, τ ), and let T(X) be the term algebra of the same type over the alphabet X. Then (a) IdX (K) = {Ker ϕ | ∃A ∈ K : ϕ : T(X) −→ A is a homomorphic mapping}, (b) IdX (K) ∈ ConT(X). Proof. (a): Let s, t ∈ T (X ) with X := {x1 , ..., xn } ⊆ X. For every algebra A ∈ K and all a1 , ..., an ∈ A there is by Theorem 6.2.2 a homomorphism ϕ: T(X) −→ A with ϕ(xi ) = ai , i = 1, ..., n. For every ϕ it holds ϕ(s) = sA (a1 , ..., an ) and ϕ(t) = tA (a1 , ..., an ). Therefore we have (s, t) ∈ Ker ϕ for all ϕ : T(X) −→ A with A ∈ K if and only if for all A ∈ K and all a1 , .., an ∈ A the equation sA (a1 , ..., an ) = tA (a1 , ..., an ) holds. But this is equivalently with A |= s ≈ t for all A ∈ K. (b) follows directly from (a), since the intersection of congruences of an algebra is a congruence of the algebra, as is well-known. 2

See, for example, [Sch 74], Chapter II.

6.4 Free Algebras

79

By the above theorem, we can form the factor algebra: T(X)/IdX (K)

(6.1)

for an arbitrary class K of algebras of the same type and of a set X of variables. Definitions If the factor algebra (6.1)belongs to K, then T(X)/IdX (K) is called the free algebra of K with free generating set X. If (6.1) belongs to K, we describe (6.1) with FK (X). In case X = {x1 , ..., xn } we also write FK (x1 , ..., xn ) or briefly FK (n), and, if X = {xi | i ∈ N}, we write FK (x1 , x2 , ...) or FK (ℵ0 ) (or FK (ω)). We notice, that, strictly speaking, the free algebra FK (X) is not generated by the set X but by the congruence classes x/IdX (K), x ∈ X. Nevertheless, one often writes x instead of x/IdX (K), since in a nontrivial class K of algebras (i.e., K contains not only 0- or 1-element algebras), the equation x/IdX (K) = y/IdX (K) implies x = y. The importance of the free algebras results from the following theorems with whose aid the main theorems to the equational theory are proven in Section 6.5. The factor algebra T(X)/IdX (K), in respect to the class K, has the same property as T(X) in respect to the class of all algebras of type (F, τ ) (see Theorem 6.2.2): Theorem 6.4.2 Let K be a class of algebras of type (F, τ ) and T(X) be the term algebra of the same type. Furthermore, let x := x/IdX (K) and X := {x | x ∈ X}. Then there is for every algebra A ∈ K and every mapping ϕ : X −→ A exactly one homomorphism ϕ : T(X)/IdX (K) −→ A, which continues ϕ; i.e., for which ϕ|X = ϕ holds. Proof. Let α : X −→ A be a mapping defined by α(x) := ϕ(x). Then, by Theorem 6.2.2 there is a homomorphism α : T(X) −→ A, which continues α. For the homomorphism π : T(X) −→ T(X)/IdX (K) with Ker π = IdX (K) we have by Theorem 6.4.1, (a) that Ker π ⊆ Ker α holds; i.e., it holds: π(s) = π(t) =⇒ α(s) = α(t). By ϕ(π(t)) := α(t) one receives a well-defined mapping ϕ : T (X)/IdX (K) −→ A. It is easy to see that ϕ is a homomorphism and that ϕ(x) = ϕ(x) for all x ∈ X holds. Because of [X] = [π(X)] = π[X] = π(T (X)) = T (X)/IdX (K) the mapping ϕ is uniquely defined through the definition over X. Theorem 6.4.3 For every class K of algebras of the same type and every variable set X it holds T(X)/IdX (K) ∈ ISP (K). Proof. Let T := T (X) and κ := IdX (K). By Theorem 6.4.1 (a) and with the help of Lemma 17.4.1 from [Lau 2004], volume 2 one can prove {(Ker ϕ)/κ | ∃A ∈ K : ϕ : T −→ A is a homomorphic mapping} = ∆T /κ (= κ0 on T /κ). Because of Theorem 5.2.2, the algebra T/κ is isomorphic to a subdirect product of the algebras (T/κ)/((Ker ϕ)/κ)

80


with ϕ : T −→ A, A ∈ K. For every such ϕ it holds (by the First Isomorphism Theorem 3 ) (T/κ)/((Ker ϕ)/κ) ∼ = T/(Ker ϕ) ∼ = ϕ(T) ∈ S(K). Therefore one gets altogether T/κ ∈ ISP (IS(K)) ⊆ ISP (S(K)) ⊆ ISP (K), where, obviously, the first inclusion is valid and the second inclusion follows from Lemma 6.1.1, (b). An immediate conclusion from Theorem 6.4.3 is as follows: Theorem 6.4.4 For every class K of algebras of the same type (in particular for a variety K) which is closed in respect to the operators I, S and P , it holds T(X)/IdX (K) ∈ K; i.e., K contains a free algebra FK (X). Lemma 6.4.5 Every free algebra FK (X) of a variety is isomorphic to a subdirect product of certain algebras FK (E), where E ⊆ X is finite and E = ∅. Proof. For x ∈ X let x := x/IdX (K). Further, for every E ⊆ X let E := {e ∈ FK (X) | e ∈ E}. Denote U(E) the subalgebra of FK (X) which is generated by E. It is easy to check that U(E) and FK (E) are isomorphic. Therefore, it is sufficient to show that FK (X) is isomorphic to a subdirect product of the algebras U(E), where E ⊆ X is nonempty and finite. For every such E, one chooses a surjective mapping ϕE : X −→ E with (ϕE )|E = idE . Then, the homomorphic continuation ϕE is surjective, and it holds (ϕE )|U (E) = idU (E) . Every term has only finite many variables. Thus for every pair s, t ∈ FK (X), there is a finite subset E ⊆ X with = t we have even (s, t) ∈ Ker (ϕE ) because of ϕE (s) = s s, t ∈ U (E). In the case s and ϕE (t) = t. Therefore {Kern(ϕE ) | ∅ ⊂ E ⊆ X ∧ E is finite} = κ0 . By Theorem 5.2.2 FK (X) is also isomorphic to an subdirect product of FK (x)/Kern(ϕE ). Because of FK (X)/Kern(ϕE ) ∼ = U(E), the assertion follows. Theorem 6.4.6 For every variety K it holds K = HSP ({FK (n) | n ∈ N}) = HSP ({FK (ω)}). Proof. Every algebra A ∈ K is a homomorphic image of FK (X), if |X| ≥ |A| (one chooses a mapping ϕ : X −→ A and then one uses Theorem 6.4.2). Thus, the first equality sign in our theorem follows from Lemma 6.4.5, and the second equality sign follows from the fact that FK (n) is isomorphic to a subalgebra of FK (ω) for all n ∈ N.

3

See for example [Wec 92], p. 140 or [Den-W 2002], Theorem 3.2.2 or Theorem 17.4.2 from [Lau 2004], volume 2.

6.5 Connections Between Varieties and Equational Defined Classes

81

6.5 Connections Between Varieties and Equational Defined Classes We need the following statement. Lemma 6.5.1 Let K be a class of algebras of the same type. Then it holds for an arbitrary alphabet X that: (a) ∀Op ∈ {H, S, P } : Op(K) ⊆ M odIdX (K); (b) M odIdX (K) is a variety. Proof. (a): Let Op = H. First, we will show that IdX (K) ⊆ IdX (H(K)) is right. Let s < x1 , ..., xn >≈ t < x1 , ..., xn > be an equation of IdX (K) with {x1 , ..., xn } ⊆ X. Then, this equation also holds in an arbitrary algebra B ∈ H(K): If namely ϕ(A) = B for a certain algebra A ∈ K and a surjective homomorphism ϕ, then for arbitrary b1 , ..., bn ∈ B there are a1 , ..., an ∈ A with the property sB < b1 , ..., bn > = sB < ϕ(a1 ), ..., ϕ(an ) > = ϕ(sA < a1 , ..., an >) = ϕ(tA < a1 , ..., an >) = tB < ϕ(a1 ), ..., ϕ(an ) > = tB < b1 , ..., bn >, i.e., s ≈ t ∈ IdX ({B}) holds, and thus we have IdX (K) ⊆ IdX (H(K)). Now, if one uses Theorem 6.3.1, (1), then one gets M odIdX (H(K)) ⊆ M odIdX (K). Furthermore, by Theorem 6.3.1, (2), we have H(K) ⊆ M odIdX (H(K)). Thus H(K) ⊆ M odIdX (K). Analogously one can prove (a) for Op ∈ {S, P }. (b): Let K ∗ = M odIdX (K). Then, by (a) and with the help of Theorem 6.3.1, (3) it holds for every Op ∈ {H, S, P }: Op(K ∗ ) ⊆ M odIdX (K ∗ ) = M od(IdX M odIdX (K)) = M odIdX (K) = K ∗ . Therefore, K ∗ is a variety.

Theorem 6.5.2 (First Main Theorem of the Equational Theory; [Bir 35]) A class K of algebras of the same type is a variety iff it is equationally definable; i.e., it holds (by Theorem 6.3.1, (6) and Theorem 6.1.2): K = HSP (K) ⇐⇒ ∃X : K = M odIdX (K). Proof. “⇐=”: By Lemma 6.5.1, (a) it holds Op(K) ⊆ M odIdX (K) for every Op ∈ {H, S, P }. If now K = M odIdX (K), then this implies Op(K) ⊆ K. Thus K is a variety. “=⇒”: Let K be a variety. By Lemma 6.5.1, (b) the class K ∗ := M odIdX (K) is also a variety and we have for an arbitrary alphabet X:

82

6 Varieties, Equational Classes, and Free Algebras FK∗ (X) = T(X)/IdX (K∗ ) (by definition) = T(X)/IdX (K)

(since by Theorem 6.3.1, (3) : IdX (K ∗ ) = IdX M odIdX (K) = IdX (K))

= FK (X)

(by definition).

From that (with the aid of Theorem 6.4.6 and X := {x1 , x2 , ...}) we get the equations K = HSP ({FK (ω)}) = HSP ({FK∗ (ω)}) = K ∗ Therefore, K is equationally definable.

6.6 Deductive Closure of Equation Sets and Equational Theory With the following definition of the deductive closure, we generalize the usual procedure of deriving equations from already proven equations. At the end of this section, we will be able to prove that the deductive closure of an equation sets Σ is identical with the set of all conclusions of Σ. Definitions Let (F, τ ) be a type of algebras, T (X) is defined as in Section 6.2 and Σ ⊆ T (X) × T (X). Then, the deductive closure D(Σ) of Σ is the smallest subset of T (X) × T (X) containing Σ such that the following five conditions hold: (R1) ∀p ∈ T (X) : p ≈ p ∈ D(Σ); (R2) ∀p, q ∈ T (X) : p ≈ q ∈ D(Σ) =⇒ q ≈ p ∈ D(Σ); (R3) ∀p, q, r ∈ T (X) : (p ≈ q ∈ D(Σ) ∧ q ≈ r ∈ D(Σ) =⇒ p ≈ r ∈ D(Σ)); (Rep) ∀f n ∈ F ∀{s1 ≈ t1 , ...., sn ≈ tn } ⊆ D(Σ) : f (s1 , ..., sn ) ≈ f (t1 , ..., tn ) ∈ D(Σ); (“replacement rule”); (Sub) ∀s < x1 , ..., xn >≈ t < x1 , ..., xn >∈ D(Σ) ∀t1 , ..., tn ∈ T (X) : s < t1 , ..., tn >≈ t < t1 , ..., tn >∈ D(Σ) (“substitution rule”). Σ ⊆ T (X) × T (X) is called deductively closed if D(Σ) = Σ holds. Obviously, every set Σ of equations with Σ = IdX (K), where K denotes a class of algebras of type (F, τ ), is deductively closed. In other words, if Σ is an equational theory of a class of algebras, then Σ is deductively closed. Furthermore, it holds that D(Σ) ⊆ ConsX (Σ). The aim of the following considerations is the proof that the reversals of the above two statements are also right. Exacter: It shall be shown that every deductively closed set of equations is

6.6 Deductive Closure of Equation Sets and Equational Theory

83

the equational theory of a certain class of algebras and that for every set Σ of equations, it holds ConsX (Σ) ⊆ D(Σ). Obviously, a deductively closed set Σ ⊆ T (X) × T (X) can also be characterized as follows: Because of (R1)–(R3) Σ is an equivalence relation, because of (Rep) Σ is a congruence on T (X), and because of (Sub) Σ is compatible with every endomorphism of T(X) (this is a homomorphism from T(X) into T(X)) (Proof: Let t1 , ..., tn ∈ T (X) be arbitrary. Then there exists an endomorphism ϕ of T(X) with ϕ(x1 ) = t1 , ϕ(x2 ) = t2 , ..., ϕ(xn ) = tn , and for every such endomorphism, it holds that ϕ(s) = s < t1 , ..., tn > and ϕ(t) = t < t1 , ..., tn >.). Definition A congruence relation κ of an algebra A is called fully invariant, if it is compatible with all endomorphisms of A; i.e., if for every endomorphism ϕ of A, (a, b) ∈ κ implies (ϕ(a), ϕ(b)) ∈ κ. The following two lemmas follow immediately from this definition and the above considerations. Lemma 6.6.1 A set Σ ⊆ T (X) × T (X) is deductively closed iff Σ is a fully invariant congruence on T(X). Lemma 6.6.2 For every class K of algebras of the same type and every variable set X, IdX (K) is a fully invariant congruence on T(X). The reversal of Lemma 6.6.2 is also valid: Lemma 6.6.3 For every fully invariant congruence κ over T(X) it holds that IdX ({T(X)/κ}) = κ, i.e., for arbitrary s, t ∈ T (X) we have: (s, t) ∈ κ ⇐⇒ T(X)/κ |= s ≈ t. In other words, an arbitrary fully invariant congruence κ on T(X) is an equational theory of the algebra T(X)/κ. Proof. “=⇒”: Let s = s < x1 , ..., xn >, t = t < x1 , ..., xn > and (s, t) ∈ κ. For arbitrary t1 , ..., tn ∈ T (X) it holds because of full invariance of κ: (s < t1 , ..., tn >, t < t1 , ..., tn >) ∈ κ. Consequently, we have: sT(X)/κ < t1 /κ, ..., tn /κ >= tT(X)/κ < t1 /κ, ..., tn /κ >, i.e., the equation s ≈ t holds in T(X)/κ. “⇐=”: Let s ≈ t ∈ IdX (T(X)/κ). Then it holds sT(X)/κ < x1 /κ, ..., xn /κ >= tT(X)/κ < x1 /κ, ..., xn /κ > . Thus (sT(X) , tT(X) ) ∈ κ and (s, t) ∈ κ.

The following theorem is a conclusion from the Lemmas 6.6.2 and 6.6.3:

84


Theorem 6.6.4 (Second Main Theorem of the Equational Theory; [Bir 35]) A set Σ ⊆ T (X) × T (X) is an equational theory iff Σ is a fully invariant congruence on T (X). Because of Theorem 6.3.1, (6) and Lemma 6.6.1, one can also write Theorem 6.6.4 as follows: Theorem 6.6.5 (Completeness Theorem for the Equational Logic; [Bir 35]) For an arbitrary alphabet X and an arbitrary Σ ⊆ T (X) × T (X) it holds: (a) Σ = ConsX (Σ) ⇐⇒ D(Σ) = Σ; (b) D(Σ) = ConsX (Σ). Proof. (a): “=⇒”: Let Σ = ConsX (Σ). Then, we have D(Σ) ⊆ ConsX (Σ) = Σ and Σ ⊆ D(Σ). Thus D(Σ) = Σ. “⇐=”: Let D(Σ) = Σ. Then, by Theorem 6.6.1, Σ is a fully invariant congruence on T (X). With the help of Theorem 6.6.4 it follows from this that Σ is an equational theory. Therefore, by Theorem 6.3.1, (6) Σ = ConsX (Σ). (b): Let Σ1 := D(Σ). Then we have D(Σ1 ) = Σ1 , Σ ⊆ Σ1 and D(Σ) ⊆ ConsX (Σ) ⊆ ConsX (Σ1 ).

(6.2)

By D(Σ1 ) = Σ1 it follows from (a): ConsX (Σ1 ) = Σ1 . Then, because of the idempotency of D, we have: D(Σ) = D(D(Σ)) = D(Σ1 ) = ConsX (Σ1 ). This and (6.2) imply D(Σ) = ConsX (Σ).

6.7 Finite Axiomatizability of Algebras The reader needs knowledge of the other sections of this chapter, as well as some knowledge of Part II for this section. An old question in universal algebra is whether, for given algebra A of finite type, there is a finite set Σ ⊆ IdX (A) with D(Σ) = IdX (A), where X := {x1 , x2 , x3 , ...}. For the case that D(Σ) = IdX (A) holds for a finite set Σ, we say that A is finitely axiomatizable or finitely based. The following theorem is easy to prove.

6.7 Finite Axiomatizability of Algebras

85

Theorem 6.7.1 ([Lyn 51]) Let A := (A; F ), B := (B; G) be finite and equivalent algebras of finite types; i.e., |A| < ℵ0 , A = B, |F | < ℵ0 , |G| < ℵ0 and T F (A) = T F (B) 4 . Then A is finitely axiomatizable if and only if B is.

The next theorem was founded by G. Birkhoff. Theorem 6.7.2 Let A be a finite algebra of finite type (F, τ ) and let Xn := {x1 , x2 , ..., xn } be a finite set of variables. Then IdXn (A) is finitely axiomatizable. Proof. Let κ := {(s, t) ∈ T (Xn )×T (Xn ) | A |= s ≈ t}. Then κ ∈ ConT(Xn ) by Theorem 6.4.2, (b). The congruence κ has only a finitely many equivalence classes ε1 , ..., εq , since A, F and Xn are finite, and since (s, t) ∈ κ iff the induced term functions sA , tA satisfy sA = tA . For t ∈ T (X) we denote by #t the number of operation symbols (∈ F) occurring in t. In particular, if t ∈ X then #t = 0. Now, from each equivalence class εi (i ∈ {1, ..., q}) of κ, we choose one representative ri with #ri ≤ #t for all terms t ∈ εi . Set M := {r1 , ..., rq }∪{f (ri1 , ri2 , ..., riaf ) | f ∈ F\F0 , {ri1 , ..., riaf } ⊆ {r1 , ..., rq }}, m := max{#ϕ | ϕ ∈ M } and Σ := {s ≈ t ∈ IdXn (A) | #s ≤ m ∧ # t ≤ m}. By induction on α we prove that ∀α ∈ N0 : (s ≈ t ∈ IdXn (A) ∧ #s ≤ α ∧ #t ≤ α) =⇒ s ≈ t ∈ D(Σ) . =:S(α)

(6.3) (I) If α ≤ m, then the statement S(α) is obviously valid. (II) Assume, S(β) holds for certain β ≥ m. Let s ≈ t ∈ IdXn (A) be arbitrary with #s ≤ β + 1 and #t ≤ β + 1. First, we consider the case s := f (s1 , s2 , ..., saf ), t := g(t1 , t2 , ..., tag ),

(6.4)

where f, g ∈ F and s1 , ..., saf , t1 , ...tag ∈ T (Xn ). Then #s1 +...+#saf ≤ β and #t1 + ... + #tag ≤ β. By definition of r1 , ..., rq there exist u1 , u2 , ...., uaf , v1 , v2 , ..., vag , w ∈ {1, ..., q} with [s1 ]κ = [ru1 ]κ , ..., [saf ]κ = [ruaf ]κ , [t1 ]κ = 4

In other words, the operations of algebra A can be represented as superpositions over the operations of algebra B and vice versa (see Part II, Section 1.5.1).

86


[rv1 ]κ , ...., [tag ]κ = [rvag ]κ , [f (s1 , ..., saf )]κ = [rw ]κ = [g(t1 , ..., tag )]κ . Then, by assumption, we have si ≈ rui , tj ≈ rvj ∈ D(Σ) and (by definition of Σ) f (ru1 , ..., ruaf ) ≈ rw , g(rv1 , ..., rvag ) ≈ rw ∈ Σ. With the aid of these equations and the rules (R1)–(R3), (Rep) and (Sub) we get Σ f (s1 , ..., saf ) ≈ f (s1 , ..., sn ) =⇒ Σ f (s1 , ..., saf ) ≈ f (ru1 , ..., ruaf ) =⇒ Σ f (s1 , ..., saf ) ≈ rw =⇒ Σ f (s1 , ..., saf ) ≈ g(rv1 , ..., rvag ) =⇒ Σ f (s1 , ..., saf ) ≈ g(t1 , ..., tag ), i.e., (s, t) ∈ D(Σ) in case (6.4). In the remaining cases (i.e., {s, t}∩Xn = ∅), one can prove the above in analog mode as well. Thus, (6.3) is right, whereby IdXn (A) is finitely axiomatizable. Remark: The above proof shows that one can choose, instead of Σ, the following finite set Σ : Σ := {x ≈ y | x, y ∈ Xm ∧ (x, y) ∈ κ}∪ {x ≈ r | x ∈ Xn ∧ r ∈ {r1 , ..., rq } ∧ (x, r) ∈ κ}∪ {f (g1 , ..., gaf ) ≈ g | f ∈ F ∧ {g1 , ..., gaf , g} ⊆ {r1 , ..., rq } ∧ (f (g1 , ..., gaf ), g) ∈ κ} As a conclusion of Theorems 6.7.1 and 6.7.2 we get: Theorem 6.7.3 Let A := (A; F ) be a finite algebra of finite type with [A] ⊆ (1) [PA ], i.e, the operations of F have at most an essential variable (see Part II, Chapter 1). Then A is finitely axiomatizable. The equational theory and parts of the mathematical logic deal with similar problems. Therefore, one can use sometimes results of the one theory for the other and vice versa. For this purpose, the next theorem provides an example. Because of the better survey, we agree with the following notation: If A = ({0, 1}; F ), f ∈ F defined by f (x) := ¬x or f (x, y) := x ◦ y

(6.5)

(◦ ∈ {∨, ∧, ⇒}) and t is a term, whose operation symbols belong to F , then tˆ denotes a formula of P rop (see Part II, Section 1.5.2), which one obtains by (6.5).

6.7 Finite Axiomatizability of Algebras

87

Theorem 6.7.4 ([Lyn 51]) Let A = ({0, 1}; 0, 1, f1 , f2 , f3 , g) be an algebra of the type (0, 0, 1, 2, 2, 2) with f1 (x) := ¬x, f2 (x, y) := x ∧ y, f3 (x, y) := x ∨ y and g(x, y) := x ⇒ y (see Table 1.2 of Part II). Further, let T be a set of tautologies (⊆ P rop) with the property that every tautology has a derivation from T with the help of sub and modus ponens. 5 Then, the equations (1) x ⇒ x ≈ 1 ( g(x, x) ≈ 1 ), (2) 1 ⇒ x ≈ x ( g(1, x) ≈ x ), (3) (x ⇒ y) ⇒ y ≈ (y ⇒ x) ⇒ x ( g(g(x, y), y) ≈ g(g(y, x), x) ) and the equations of the type (4) t ≈ 1 f¨ ur every t ∈ T form a system Σ of equations with D(Σ) = IdX (A) (more precise: D(Σ) = {ˆ α | α ∈ IdX (A)}). Proof. First, we show that (ϕ ∈ P rop is a tautology) implies ( ϕ ≈ 1 ∈ D(Σ) ).

(6.6)

By assumption, we can derive a tautology ϕ from T with the aid of sub and modus ponens. Consequently, we have to prove that it is possible to copy the modus ponens through the rules (R1)–(R3), (Rep) and (Sub). Let σ ≈ 1, σ ⇒ τ ≈ 1 ∈ D(Σ). Then τ ≈ 1 ∈ D(Σ) follows from (2)

Sub

σ ≈ 1, σ ⇒ τ ≈ 1 1 ⇒ τ ≈ 1 τ ≈ 1. Thus (6.6) holds. Now, let α ≈ β ∈ IdX (A) be arbitrary; i.e., α ⇒ β and β ⇒ α are tautologies. We show α ≈ β ∈ D(Σ). By (6.6) we have that α ⇒ β ≈ 1 and β⇒α≈1

(6.7)

belong to D(Σ). Furthermore, by (3) and (Rep): (β ⇒ α) ⇒ α ≈ (α ⇒ β) ⇒ β ∈ D(Σ). Thus (β ⇒ α) ⇒ α ≈ (α ⇒ β) ⇒ β

(Sub),(6.7)

1⇒α≈1⇒β

(2),(Sub)

α ≈ β,

whereby α ≈ β ∈ D(Σ) is shown. R. L. Lyndon proven the following basic result with the aid of Theorem 6.7.1 and Post’s theorem (see Part II, Theorem 3.1.1): 5

see Part II, Section 1.5.2.

88


Theorem 6.7.5 ([Lyn 51], without proof ) Every two-element algebra of finite type is finitely axiomatizable. Notice that J. Berman gave a short proof for the above theorem in [Ber 80] with the aid of theorems by Baker ([Bak 77]) and McKenzie ([McK 78]). R. C. Lyndon constructed a 7-element algebra of type (0, 2) whose equations are not finitely based (see [Lyn 54]). The smallest such example was found by V. L. Murskij: Theorem 6.7.6 ([Mur 65]; without proof ) The algebra ({0, 1, 2}; ◦), where ◦ is defined by ◦ 0 1 2 0 0 0 0 , 1 0 0 1 2 0 2 2 is not finitely axiomatizable. Notice that all finite groups, rings and lattices are finitely axiomatizable (see [Oat-P 65], [Kru 73], and [McK 70], respectively). The result for lattices was considerably generalized by K. Baker ([Bak 77]), who proven that every finite algebra whose generated variety is congruence distributive is finitely axiomatizable. In [Per 69] was proven that the multiplicative semigroup of all 2 × 2matrices over a 2-element field has a 6-element subsemigroup with no finite basis. One can find further important results on the topic in [McK 78] and [Wil 2001].

Part II

Function Algebras

1 Basic Concepts, Notations, and First Properties

In this chapter, we begin by investigating multi-digit operations, which are defined on a finite set A. We define some operations on the set of these operations. For the purpose of distinction, we subsequently replace the concept “operation (on the set A)” by the concept “function (on the set A)”.

1.1 Functions on Finite Sets Let A be a finite set with at least two elements. Often we choose the set Ek := {0, 1, 2, ..., k − 1}, k ≥ 2 instead of A in the following. We say that f is an n-ary (n-digit) function on A (or an n-ary function of the |A|-valued logic1 ), if f a mapping from the n-fold Cartesian product An into A, n ≥ 1. For technical reasons (see Section 1.3), we renounce that, in this section, we consider also nullary functions. Otherwise, we use the concepts introduced in Chapter 1 of Part I for operations: af , f n , D(f ), ... . Let PAn be the set of all n-ary functions on A 2 , n ≥ 1. Instead of PEnk we write also Pkn . Further let

1 2

One finds an explanation for this notation in Section 1.5. A confusion with the direct product is not possible for content-related reasons.

92


PA :=

n≥1

PAn ,

F n := F ∩ PAn for every subset F of PA , Pk := n≥1 Pkn , PA,B := {f ∈ PA | Im(f ) ⊆ B}, Pk,l := PEk ,El , PA (l) := {f ∈ PA | |Im(f )| ≤ l}, Pk (l) := PEk (l), PA [l] := {f ∈ PA | |Im(f )| = l} and Pk [l] := PEk [l] (2 ≤ l ≤ k). With (x1 , ..., xn ) (briefly x(n) ) or x we denote an arbitrary n-tuple of An or Ekn and usually we say that the xi (i = 1, 2, ..., n) are variables. If n = 2 or n = 3 we also write (x, y) or (x, y, z) instead of (x1 ..., xn ), respectively. We will define, subsequently, specific functions f n from Pk either through a table of the form Table 1.1 x1 x2 0 0 0 0 . . a1 a2 . . k−1 k−1

... xn ... 0 ... 1 ... . ... an ... . ... k − 1

f (x1 , x2 , ..., xn ) f (0, 0, ..., 0) f (0, 0, ..., 1) . f (a1 , a2 , ..., an ) . f (k − 1, k − 1, ..., k − 1)

or through formulas, for example, of the form ∀x ∈ Ekn : f (x1 , ..., xn ) := x1 + ... + xn (mod k),

(1.1)

on the (variable-) alphabet {x, y, z, x1 , x2 , ...}. We write often instead of (1.1) briefly (1.2) f (x1 , ..., xn ) := x1 + ... + xn (mod k) or (if the arity of f is clear from the context or is without importance) we write still more briefly “f is defined by x1 + ... + xn (mod k) ”;

(1.3)

i.e., we do not distinguish between a function and the formula (or term) defining it.3 3

One finds the concept “term” explained in Part I, Section 6.2.

1.1 Functions on Finite Sets

93

Two functions f n , g m ∈ PA are identical (we write f n = g m ) iff n = m and f (x) = g(x) for all x ∈ An hold. Let f n ∈ PA and i ∈ {1, 2, ..., n}. Then we say that the i-th variable (or the i-th place) of the function f ∈ PA is essential, iff there are n-tuples a = (a1 , ..., ai−1 , b, ai+1 , ..., an ) and a = (a1 , ..., ai−1 , c, ai+1 , ..., an ) such that b = c and f (a) = f (a ) hold. In the opposite case, one calls the i-th variable (or i-th place) of f fictitious (or non-essential). If the i-th variable of f is not fictitious, we say that f depends on the i-th variable. The function eni defined by eni (x1 , ..., xn ) := xi (i ∈ {1, ..., n}) is called projection or also selector. Let JA (or Jk ) be the set of all projections of PA (or Pk ), respectively. A constant function (briefly, constant) is a function cna defined by cna (x1 , ..., xn ) := a, where a ∈ A. Notations for certain functions of P2 , the Boolean functions, are given in the following table, where it is defined, as usual ◦(x, y) := x ◦ y if ◦ ∈ {∧, ∨, +, ⇒, ⇐⇒} and

−

(x) := x.

Table 1.2

x x x y x ∧ y x ∨ y x + y x ⇒ y x ⇐⇒ y 0 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 1 0 1 0 0 1 1 0 0 1 1 1 1 0 1 1 Instead of x ∧ y we also write x · y or we write xy briefly. It can easily be shown that the above functions have the following properties: Theorem 1.1.1 It holds: (a) ∀◦ ∈ {∨, ∧, ⇐⇒, +} : x ◦ (y ◦ z) = (x ◦ y) ◦ z; (b) x ∨ x = x, x ∧ x = x, x ⇐⇒ x = 1, x ⇒ x = 1, x + x = 0, x ∨ 0 = x, x ∧ 1 = x, x ∨ 1 = 1, x ∧ 0 = 0;

94


(c) ∀◦ ∈ {∨, ∧, +, ⇐⇒} : x ◦ y = y ◦ x; (d) x ∧ x = 0, x ∨ x = 1, x = x, x ∨ y = x ∧ y, x ∧ y = x ∨ y (“de Morgan’s laws”); (e) x ⇒ y = x ∨ y, x ⇒ y = x ∧ y; (f ) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z), x ∨ (y ∧ z) = (x ∨ y) ∧ (x ∨ z); (g) x ∧ (x ∨ y) = x, x ∨ (x ∧ y) = x.

1.2 Operations on PA, Function Algebras The “formula notation” of our functions from the first section motivates the determination of the following operations on PA : – permutation of variables, – identification of variables, – adding of fictitious variables and – substitution of variables of a function by functions; which are called superposition operations on PA and which one can describe in different way exactly. We give only two possibilities here. First we want describe the above operations through the following (infinite many) partial operations: πs : PAn −→ PAn ∆t : PAn −→ PAr (r < n) ∇q : PAn −→ PAu (u > n) i : PAn × PAm −→ PAn+m−1 Let f n , g m be functions of PA , let s be a permutation on the set {1, 2, ..., n}, let t be a mapping from {1, 2, ..., n} onto {1, 2, ..., r} (r < n), let q be an injective mapping from {1, 2, ..., n} into {1, 2, ..., u} (u > n) and let i ∈ {1, 2, ..., n}. Then, πs f ∈ PAn , ∆t f ∈ PAr , ∇q f ∈ PAu , f i g ∈ PAm+n−1 are defined by (πs f )(x1 , ..., xn ) := f (xs(1) , xs(2) , ..., xs(n) ) (“permutation of variables of f ”), (∆t f )(x1 , ..., xr ) := f (xt(1) , xt(2) , ..., xt(n) ) (“identification of certain variables of f ”), (∇q f )(x1 , x2 , ..., xu ) := f (xq(1) , xq(2) , ..., xq(n) ) (“adding of certain fictitious variables”)

1.2 Operations on PA , Function Algebras

95

and (f i g)(x1 , ..., xm+n−1 ) := f (x1 , ..., xi−1 , g(xi , ..., xi+m−1 ), xi+m , ..., xm+n−1 ) (“the replacement of the i-th variable of f through the function g and the changing of the denotation of variables of f ”). For partial operations α ∈ {πs , ∆t , ∇q , i } defined above, one can continue to certain operations α on PA . For later investigations, however, it is better that we choose the minimal number of the operations on PA . Therefore, next we consider that there are five elementary operations (or Mal’tsevoperations) ζ, τ, ∆, ∇, on PA , with which we can form certain continuations of the partial operations πs , ∆t , ∇q , i for arbitrary s, t, q, i, n, m through composition. These operations were published by A. I. Mal’tsev in [Mal 66], and one can define these operations as follows for arbitrary f n , g m ∈ PA : max{1,n−1}

ζf n ∈ PAn , τ f n ∈ PAn , ∆f n ∈ PA

, ∇f n ∈ PAn+1 , f n g m ∈ PAm+n−1

and (ζf )(x1 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) if n ≥ 2, ζf = τ f = ∆f = f if n = 1, (∇f )(x1 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ), (f g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ). To prove that one can describe the operation πs (over PAn ) by means of operations ζ, τ (over PA ), it suffices to show that the set Sn of all permutations on the set {1, 2, ..., n} is generating by the permutations (12...n) and (12) (given in cyclic description); that is, that [{(12...n), (12)}]2 = Sn holds, where (s2s )(x) := s (s(x)) for all s, s ∈ Sn . But, this follows from the fact that every permutation s ∈ Sn is a product of pairwise disjunct cycles: s = (i1 ...ip )(j1 ...jq )..., that every cycle is a product of transpositions (i1 ...ip ) = (i1 i2 )(i1 i3 )...(i1 ip ), and that (ij) = (1i)(1j)(i1) for arbitrary i > j, i > 1, (1i) = (12)(23)...(i − 1i)(i − 1, i − 2)...(21) and (i, i + 1) = (12...n)n−i+1 (12)(12...n)i−1

96


are valid. Evidently, then, the operations ∆t or ∇q or i can be created (in respect to 2) through the operations πs (s ∈ Sn ), ∆ or πs (s ∈ Sn ), ∇ or πs (s ∈ Sn ), respectively. Thus we proven the following lemma. Lemma 1.2.1 It holds: (a) For every permutation s ∈ Sn there exists an operation π ∈ [{ζ, τ }]2 with πs f = π s f for all f ∈ PAn . (b) For every mapping t from {1, 2, ..., n} onto {1, 2, ..., r} (r < n) there ex ∈ [{ζ, τ, ∆}]2 with ∆t f = ∆ t f for arbitrary f ∈ P n . ists an operation ∆ A (c) For every injective mapping q from {1, 2, ..., n} into {1, 2, ..., u} there ex q f for arbitrary f ∈ P n . q ∈ [{ζ, τ, ∇}]2 with ∇q f = ∇ ists an operation ∇ A (d) For every i ∈ {1, 2, ..., n}, there exists an operation i ∈ [{ζ, τ, }]2 with i g for arbitrary f ∈ PAn and arbitrary g ∈ PAm . f i g = f By means of the operations ζ, τ, ∆, ∇, we can describe the subject of investigation of this chapter. PA together with the operations e21 , ζ, τ, ∆, forms an algebra (PA ; e21 , ζ, τ, ∆, ) of the type (0, 1, 1, 1, 2), which is called (full) function algebra on A. A little changed form of the full function algebra is the so-called iterative (full) function algebra (PA ; ζ, τ, ∆, ∇, ) of the type (1, 1, 1, 1, 2). Since ∇f = f (τ e21 ) is valid, however, both algebras can be regarded as equivalent in a certain sense: If (S; e21 , ζ, τ, ∆, ) is a subalgebra of (PA ; e21 , ζ, τ, ∆, ), then (S; ζ, τ, ∆, ∇, ) is also a subalgebra of (PA ; ζ, τ, ∆, ∇, ). Conversely, if (T ; ζ, τ, ∆, ∇, ) is a subalgebra of (PA ; ζ, τ, ∆, ∇, ), then (T ∪JA ; e21 , ζ, τ, ∆, ) is a subalgebra of (PA ; e21 , ζ, τ, ∆, ). Therefore, we often deal only with the algebra PA = (PA ; ζ, τ, ∆, ∇, ).

1.3 Superpositions, Subclasses, and Clones A function f ∈ PA is called a superposition over F (⊆ PA ), if f can be obtained by a finite number of applications of the operations ζ, τ, ∆, ∇, from the functions of F . We describe a superposition f over F in the rarest cases f through a term over certain function symbols, ζ, τ, ∆, ∇, and parentheses. We use the variable alphabet {x, y, z, x1 , x2 , ...},

1.3 Superpositions, Subclasses, and Clones

97

certain function symbols, commas, and parenthesis instead of this. In some cases, where an equation for the precise definition of the function would be necessary formally, we are satisfied with the the right side of the defining equation if the remaining information on the function results from the context. Further, if f ∈ Pkn , g1 , ...gn ∈ Pkm and the m-ary function h ∈ Pk is defined by h(x1 , ..., xm ) := f (g1 (x1 , ..., xm ), g2 (x1 , ..., xm ), ..., gn (x1 , ..., xm )), then, we write briefly h := f (g1 , ..., gn ). The set of all superpositions over F (⊆ PA ) is called hull or closure of F and it is denoted by [F ]. Obviously, [..] is a hull operator on the set PA . A set F ⊆ PA satisfying [F ] = F is called a closed set or a subclass or briefly class of PA . We define that the empty set is also a closed set, i.e., ∅ = [∅]. One can form many examples of closed sets with the aid of the following concept: Let T ⊆ Pkm and f ∈ Pkn . Then we say that f preserves the set T iff ∀g1 , ..., gn ∈ T : f (g1 , ..., gn ) ∈ T. It is easy to see that a projection preserves every set T ⊆ Pkm and that the set of all functions, which preserve the set T , is closed. The set F ⊆ PA is called a clone of PA , if F is closed and JA ⊆ F holds.4 Obviously, the subclasses of PA are exactly the universes of subalgebras of (PA ; ζ, τ, ∆, ∇, ) and clones are exactly the universes of subalgebras of (PA ; e21 , ζ, τ, ∆, ). Let LA be the set of all closed subsets of PA . Put Lk := LEk . (LA ; ⊆) is a lattice (see Part I, Chapter 2). Further, let L↓A (F ) := {F ∈ LA | F ⊆ F } and

L↑A (F ) := {F ∈ LA | F ⊆ F }.

Analogously, one can define L↑k (F ) and L↓k (F ) for A = Ek . Further, let LA (F ; G) := L↑A (F ) ∩ L↓A (G), where F, G ∈ LA and F ⊂ G. 4

If F is a class of PA , then JA ⊆ F iff e11 ∈ F .

98


If [G] = F (⊆ PA ), then G is called complete in F . In particular, if F = PA , we say, G is complete or G is a complete set. A closed set F is called a maximal subclass of the closed set F , if F ⊂ F and [F ∪ {f }] = F for every f ∈ F \F . If F = PA then, we say briefly, F is a maximal class. The maximal classes of the maximal classes of PA are called submaximal classes. As usual, we call a subset F of F a generating system of F , if [F ] = F . A generating system F of F is called basis of the closed set F , if every proper subset of F is not a generating system of F . If a subclass F of PA has a finite generating system, then the order of F we denote with ord F . We understand from that, the smallest number with [F r ] = F . If F does not have any finite generating system, we write ord F = ∞.

1.4 Generating Systems for PA For the purpose of determining of generating systems for the set PA , we consider some descriptions (so-called “normal forms”) for an arbitrary function f n ∈ PA . These descriptions are superpositions over certain functions of PA , which are to be described easily and which have small arities. We use the following notations: 1 if x = a, ja (x) := 0 otherwise (a ∈ A) and

ja (x1 , ..., xn ) :=

1 if (x1 , ..., xn ) = a, 0 otherwise

(a ∈ An , n ∈ N). Theorem 1.4.1 (Representation Theorem for Functions of PA ) Let 0, 1 ∈ A and let ∧, ∨ be two binary associative5 operations on A with a ∧ 1 = a, 0 ∨ a = a ∨ 0 = a and a ∧ 0 = 0

(1.4)

for each a ∈ A. Then, for every function f n ∈ PA it holds: f (x) = (

m

fai (x) := )fa1 (x) ∨ fa2 (x) ∨ ... ∨ fam (x),

i=1

where An := {a1 , ..., am }, m := |A|n and

(1.5)

1.4 Generating Systems for PA

99

fai (x) := cf (ai ) (x1 ) ∧ jai (x) (i = 1, ..., m). Furthermore: jai (x) = jai1 (x1 ) ∧ jai2 (x2 ) ∧ ... ∧ jain (xn ), where ai := (ai1 , ..., ain ). Proof. The correctness of the equation (1.5) can be assured by checking that on both the left side and the right side of the formula, the same value stands for every x. If A = {0, 1}, the functions ∨, ∧ (= ·) are defined as in Table 1.2, j0 (x) = x and j1 (x) = x, then one receives the disjunctive normal form (or DNF) of an arbitrary Boolean function f n ∈ P2 as an conclusion from (1.5): f (x1 , ..., xn ) = f (a1 , ..., an ) · xa1 1 · xa2 2 · ... · xann , (1.6) a∈E2n

where α

x :=

x if α = 0, x if α = 1

(α ∈ E2 ). If f = cn0 , then we can write f (x1 , ..., xn ) =

xa1 1 · xa2 2 · ... · xann

(1.7)

a∈E2n ,f (x)=1

instead of (1.6). For example, f (x, y, z) = x · y · z ∨ x · y · z ∨ x · y · z is the DNF for the ternary function f defined by Table 1.3. Table 1.3

x 0 0 0 0 1 1 1 1 5

y 0 0 1 1 0 0 1 1

z f (x, y, z) 0 0 1 0 0 1 1 0 0 1 1 0 0 0 1 1

The associativity can be renounced if the necessary parentheses are put in the following formulas.

100


If A is an arbitrary finite set, then one can choose ∧ and ∨ as lattice operations with 0 = A and 1 = A. Then, by Theorem 2.1.2 of Part I, we have ∨(x, y) = sup {x, y} and ∧(x, y) = inf {x, y}, where is the partial order (appertaining to the lattice) over A with the greatest element 1 and the least element 0. If A = Ek , then the functions ∨ := + (mod k) and ∧ := · (mod k) also fulfill (1.4) and we get the following normal form for an arbitrary function f n ∈ Pk : f (a1 , ..., an ) · ja1 (x1 ) · ... · jan (xn ) (mod k) (1.8) f (x) = a∈Ekn

The following theorem results from the above considerations immediately: Theorem 1.4.2 It holds: (a) Let ∨ and ∧ be binary operations on A, which (1.4) fulfill. Then, {∨, ∧}∪ {c1a , ja1 | a ∈ A} is a generating system for PA . In particular, if A = E2 , then [{∨, ∧,− }] = P2 and (because of Theorem 1.1.1, (d)) [{∨,− }] = [{∧,− }] = P2 . (b) ord PA = 2.

Theorem 1.4.3 Let A = Ek and let k = pm be a prime number power. Then one can define operations + and · on Ek so that (Ek ; +, ·) is a field with the neutral element o in respect to + and the neutral element e of the group (Ek \{o}; ·). Then, one can represent an arbitrary function f n ∈ Pk with the aid of these field operations as follows: ai1 i2 ...in · xi11 · xi22 · ... · xinn (1.9) f (x) = (i1 ,...,in )∈Ekn

(x0 := e; ai1 i2 ...in ∈ Ek ). This representation is unique except the order of addends, i.e., the equality of the corresponding coefficients results from the equality of two functions ∈ Pkn . Proof. The existence of a field (Ek ; +, ·) for k = pm , p ∈ P and m ∈ N is well-known. (see for example [Lid-N 87] or [Lau 2004], volume 2).


101

Every polynomial of the form (1.9) is uniquely represented through the sen quence of coefficients. Thus there are k (k ) different formulas of the form (1.9). n (kn ) , our theorem is proven if Since |Pk | = k f (x) = (i1 ,...,in )∈E n ai1 i2 ...in xi11 xi22 ...xinn and k f (x) = (i1 ,...,in )∈E n bi1 i2 ...in xi11 xi22 ...xinn k

implies ai1 ...in = bi1 ...in for all (i1 , ..., in ) ∈ Ekn . This is clear for (i1 , ..., in ) = (o, o, ..., o) (one forms f (o, ..., o)!). Let I := {xij | ij = o ∧ j ∈ {1, ..., n}} for the proof of ai1 ...in = bi1 ...in in the case (i1 , ..., in ) ∈ Ekn \{o}. If one identifies now the variables in f from I with x and one replaces the remaining variables through c0 (x), then one receives a unary function f that can be represented as follows: (1.10) f (x) = a0 + a1 · x + a2 · x2 + ... + ar−1 · xr−1 or

f (x) = b0 + b1 · x + b2 · x2 + ... + br−1 · xr−1

(1.11)

with r−1 := |I|, ar−1 = ai1 ...in and br−1 = bi1 ...in for certain a0 , ..., ar−2 , b0 , ..., br−2 . If one forms now in (1.10) and (1.11) f (α1 ), f (α2 ), ..., f (αr ) for pairwise distinct α1 , α2 , ..., αr ∈ Ek , then one sees that both (a0 , ..., ar−1 )T as also (b0 , ..., br−1 )T is a solution of the matrix equation A · x = (f (α1 ), ..., f (αr ))T with ⎛ ⎞ 1 α1 α12 ... α1r−1 ⎜ 1 α2 α2 ... αr−1 ⎟ 2 2 ⎟ A := ⎜ ⎝ ................. ⎠. 1 αr αr2 ... αrr−1 But, because of det A = o,6 this is only possible for a0 = b0 , ..., ar−1 = br−1 . The next property of functions of Pk for k ≥ 3 is not only useful while determining from generating systems for Pk ; it also has interesting consequences, which we will deal with later. We need the following denotation:7 ιhk := {(a1 , ..., ah ) ∈ Ekh | |{a1 , ..., ah } | ≤ h − 1} (h ≥ 2), 3 δ{α,β} := {(a1 , a2 , a3 ) ∈ Ek3 | aα = aβ } (α, β ∈ {1, 2, 3}) and 3 δ{1,2,3} := {(x, x, x) | x ∈ Ek }. 6 7

This follows from the fact that A is a Vandermonde matrix (see for example [Lau 2004], volume 1). See also Chapter 2.

102


Furthermore, for arbitrary ri := (r1i , r2i , ..., rhi ) ∈ Ekh , i = 1, 2, .., n, and f ∈ Pkn we put: f (r1 , ..., rn ) := (f (r11 , r12 , ..., r1n ), f (r21 , r22 , ..., r2n ), .., f (rh1 , rh2 , ..., rhn )). Theorem 1.4.4 Let f be an n-ary function of Pk , which is essentially dependent of at least two variables (w.l.o.g. of x1 and x2 ) and which has q pairwise distinct values. Then: 3 3 (a) q ≥ 3 =⇒ ∃r1 , ..., rn ∈ δ{1,2} ∪ δ{2,3} : f (r1 , ..., rn ) ∈ Ek3 \ι3k

(“Fundamental Lemma of Jablonskij”); (Jab 58]) (b) q ≥ 3 =⇒ ∃r1 , ..., rn ∈ ιqk : f (r1 , ..., rn ) ∈ Ekq \ιqk ; 3 3 3 3 ∪ δ{2,3} : f (r1 , ..., rn ) ∈ δ{1,3} \δ{1,2,3} . (c) q = 2 =⇒ ∃r1 , ..., rn ∈ δ{1,2}

Proof. (a), (c): Since f depends on the variable x1 essentially, there is an a := (a2 , ..., an ) ∈ Ekn−1 , so that Ta := {f (x, a2 , ..., an ) | x ∈ Ek } has at least two different elements. We distinguish two cases: Case 1: |Ta | < q. In this case, one can find a tuple c = (c1 , ..., cn ) with γ := f (c) ∈ Ta . Consequently, we have ⎞ ⎛ ⎞ ⎞ ⎛ ⎛ α f (a1 , a2 , ..., an ) a1 a2 ... an f ⎝ c1 a2 ... an ⎠ := ⎝ f (c1 , a2 , ..., an ) ⎠ = ⎝ β ⎠ c1 c2 ... cn γ f (c1 , c2 , ..., cn ) and |{α, β, γ}| = 3 for certain a1 ∈ Ek . Case 2: |Ta | = q. Since f also depends on x2 essentially, the function f1 (x1 , ..., xn−1 ) := f (d, x1 , ..., xn−1 ) is not a constant function for certain d ∈ Ek . Let now β := f (d, a2 , ..., an ). Because of f1 = cβ there are certain c2 , ..., cn and a γ with γ := f (d, c2 , ..., cn ) = β . Since |Ta | = q, then one can find an a1 ∈ Ek with γ if q = 2, α := f (a1 , a2 , ..., an ) = α ∈ {β , γ } if q ≥ 3. Consequently, we have ⎞ ⎛ ⎞ α a1 a2 ... an f ⎝ d a2 ... an ⎠ = ⎝ β ⎠ . d c2 ... cn γ ⎛


103

(b) follows easy from (a). From the many conclusions from this theorem, we provide only the following, first. Lemma 1.4.5 Let f n be a function of Pk , which is essentially dependent on two variables and which has q ≥ 3 distinguish values. Then: Pk,Im(f ) ⊆ [{f } ∪ Pk (q − 1)]. Proof. W.l.o.g. let Im(f ) = Eq . By Theorem 1.4.4, (b) there exist r1 , ..., rn ∈ ιqk with f (r1 , ..., rn ) = (0, 1, ..., q − 1)T and ⎞ ⎛ a02 ... a0n a01 ⎜ a11 a12 ... a1n ⎟ ⎟ (r1T , ..., rnT ) = ⎜ ⎝ ....................... ⎠. aq−1,1 aq−1,2 ... aq−1,n For an arbitrary function g m ∈ Pk,Im(f ) let gj (x1 , ..., xm ) = aij :⇐⇒ ∃i : g(x1 , ..., xm ) = i, (j = 1, 2, ..., n). Obviously, the functions g1 , ..., gn belong to Pk (q − 1) and it holds: g(x1 , ..., xm ) = f (g1 (x1 , ..., xm ), ..., gn (x1 , ..., xm )). Thus g ∈ [{f } ∪ Pk (q − 1)]. Subsequently, we declare another possibility of the characterization of functions of Pk that we need later during the description of a certain type of maximal classes of Pk . Lemma 1.4.6 Let Ai (i ∈ Ek ) be a partition of the set Ek and ai ∈ Ai (i ∈ Ek ). Furthermore, let y if ∃i ∈ Ek : x = ai and y ∈ Ai , x y := x otherwise. Then, an arbitrary function f n of Pk is a superposition over the functions z, gf , fi (i ∈ Ek ) defined by z(x, y) := x y, ai ⇐⇒ f (x1 , ..., xn ) ∈ Ai , gf (x1 , ..., xn ) := f (x1 , ..., xn ) if f (x1 , ..., xn ) ∈ Ai , fi (x1 , ..., xn ) := otherwise ai (i ∈ Ek ), and f (x) = ((...((gf (x) f0 (x)) f1 (x)) ...) fk −1 (x)) n

is a representation of f .

(1.12)

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1.5 Some Applications of the Function Algebras It is the aim of this section to show by examples how results in function algebras can be used in other mathematical disciplines. In addition, some problems that motivate certain investigations with function algebras are explained. 1.5.1 Classification of Universal Algebras In Part I, Chapter 1, we introduced the concept of the type of an algebra, and in Chapter 6 combinations (classes) of algebras of the same type, which fulfill certain equations. Such a decomposition of the algebras is, however, very coarse. The set MA of all finite algebras on the same universal A can be more finely decomposed with the aid of the following equivalence relation RA : The algebras (A; F ) and (A; G) with F, G ⊆ PA are called equivalent in respect to RA , iff [F ] = [G] holds, i.e., iff the operations of the one algebra can be represented as superpositions over the operations of the other algebra and vice versa. The equivalence classes (blocks) of the relation RA form a partition of the set MA , where the set of all algebras (A; F ) with F ∈ LA is a representative system of these equivalence classes. Reducts of (A; F ) are defined to be algebras of the form (A; G) with G ⊆ [F ]. An algebra (A; F ) with [F ] = PA is called primal ([Fos 59]). For example, by Theorem 1.4.2, the two-element Boolean algebra is primal. The Rosenberg’s completeness criterion from Chapter 6 can be used to obtain further examples of primal algebras (see also Chapter 7). An algebra (A; F ) is called preprimal iff [F ] is a maximal clone of PA ([Den 82], [Kno 85]). For a description of further generalizations of “primal algebras”, we need some concepts and results of Chapter 2 and the notations S(A) (the set of all subalgebras of the algebra A), Aut(A) (the set of all automorphisms on A), SubIso(A) (the set of all isomorphisms between subalgebras of A) and Con(A) (the set of all congruences of A). Let A := (A; F ) be an algebra, for which there is a relation set Q on A with [F ] = P olA Q. Then A is called • semiprimal iff Q ⊆ Rk1 , i.e., iff Q = S(A) (or with other words: iff every operation on A which preserves all subalgebras of A belongs to [F ]) [Fos-P 64]; • demiprimal iff Q = Aut(A) ([Qua 71]); • infraprimal (or demisemiprimal) iff Q = S(A) ∪ Aut(A) [Qua 71]; • quasiprimal iff Q = SubIso(A) ([Pix 71]); • hemiprimal iff Q = Con(A) ([Fos 70]);

1.5 Some Applications of the Function Algebras

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For further concepts and properties of the above-defined algebras, refer to [Den 82], [Den-W 2002] and [Sze 86]. Since the present book describes many clones, one finds properties of the above-defined algebras, in many places in this book. 1.5.2 Propositional Logic and First Order Logic A proposition is a “sentence” (of a natural or artificial language) for which it makes sense to ask whether it false (notation: 0) or true (notation: 1). At the basis of the concept “proposition” we have the two-value principle (also called principle of the excluded middle). This means that each proposition must be either false or true, there is no other possibility “in between”. The following are propositions: – Rostock is a city in Germany. – There are infinite many prime numbers. – 2 · 3 = 5. – There exists extra-terrestrial life. The following are not propositions: – Two chickens on the way after the day before yesterday. – Everything that I say is false. – It is a beautiful day. – Be quiet! Propositions will be denoted by capital letters A, B, .... Instead of “A is an arbitrary proposition” we say “A is a proportional variable”. Therefore, a propositional variable takes the values 0 and 1. One can associate propositions (“sentences”) in the informal language in multiple ways with each other (for example through such conjunctions as “and”, “or”, “if – then”, ...). The result of this connection is normally, again, a proposition, whose value (0 or 1) is dependent from the values of associated single propositions. In the propositional logic, a part of the colloquial connections is modelled and defined exactly (unlike the informal language). Since we abstract from the content of a proposition during the consideration of a proposition (i.e., we have interest only in the so-called truth value 0 or 1 of the proposition), proposition combinations are multi-digit functions on {0, 1}, i.e., Boolean functions (or function of the 2-valued logic). Interpretations of the Boolean functions that are defined in Table 1.2 are, for example, • The negation of the proposition A: A (“not A”). A is true if and only if A is false. • The conjunction of the propositions A, B: A ∧ B (“A and B”). A ∧ B is true if and only if A as well as B are true. • The disjunction of the propositions A, B: A ∨ B (“A or B”). A ∨ B is true if and only if either A or B or both are true.

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• contravalence: A + B (“either A or B”). A + B is true if and only if either A or B is true. • equivalence: A ⇐⇒ B (“A if and only if B”). A ⇐⇒ B is true if and only if A and B have the same truth value. • implication: A ⇒ B (“A implies B”; “If A, then B”). A ⇒ B is false if and only if A has the value 1 and B has the value 0. To provide mathematics with a precise language, the mathematical logic creates an artificial, formal language. Next, we give a short introduction on proportional logic; that is, the logic that deals only with propositions. Later, we extend our treatment to the first order logic, which also takes properties of individuals into account. The process of formalization of proportional logic consists of two stages: (i) present a formal language (ii) specify a procedure for obtaining valid or true propositions. The language of propositional logic has an alphabet consisting of • the set of proposition symbols At := {A, B, C, ..., A1 , B1 , C1 , ..., An , Bn , Cn , ...} (the elements of At are called atoms or atomic propositions) • the set of connectives J0 := {∧, ∨, ¬, ⇒, ⇔} • the set of auxilliary symbols {(, )} The set P rop is the smallest set X with the following three properties: • At ∪ {0, 1} ⊆ X • if α, β ∈ X, then (α ◦ β) ∈ X for all ◦ ∈ J0 • if α ∈ X, then (¬α) ∈ X Notice that P rop = T (At) (see Part I, Section 6.2). A mapping v : P rop −→ {0, 1} is called a valuation if v(0) = 0, v(1) = 1, v(¬α) = ¬v(α) and v(α ◦ β) = v(α) ◦ v(β) for all α, β ∈ P rop and all ◦ ∈ J0 . 8 If v(α) = 1 (or v(α) = 0) then α is true (or false) under v, respectively. If α = (β) ∈ P rop, then we only write β instead of α in the following. Obviously, we have: If v0 : At −→ {0, 1}, then there exists a unique valuation v such that v(α) = v0 (α) for all α ∈ At. α ∈ P rop is a tautology if v(α) = 1 for all valuations v. We write |= α (or ∅ |= α) for “α is a tautology”. If Σ ⊆ P rop and α ∈ P rop, then 8

Notice that the right sides of the equations are determined by Table 1.2.


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Σ |= α iff for all valuations v: v(σ) = 1 for all σ ∈ Σ implies v(α) = 1. If Σ |= α then α is called a consequence of Σ. Further, set Cons(Σ) := {α ∈ P rop | Σ |= α}. It is a classical problem of the propositional logic to find a system of axioms and rules with which one can determine all tautologies and all consequences from a set of propositions. In books about mathematical logic, one finds many solutions for this purpose. We declare a solution with proof here. The proof is chosen so that it can be used as proof for a corresponding theorem of the predicate logic as a basic idea. It is normal to write down the rule “R: if α1 , ..., αr ∈ P rop are derivable, then αr+1 is derivable” in the following form: α1 , ..., αr R αr+1 As an example, we give the substitution rule sub: Let α ∈ P rop, which contains x ∈ At (in symbol: α(..., x, ...) ). If α is a tautology, then one can form a tautology by replacing every occurrence of x by β ∈ P rop in α: α(..., x, ...); β ∈ F orm sub . α(..., β, ...) The Hilbert-type-calculus for the classical proportional logic is defined by the following 13 axioms and a rule, where α, β, γ, σ, τ are arbitrary elements of P rop: (I) Axioms are all formulas of the form (A1) α ⇒ α (A2) α ⇒ (β ⇒ α) (A3) (α ⇒ β) ⇒ ((β ⇒ γ) ⇒ (α ⇒ γ)) (A4) (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)) (A5) α ⇒ (α ∨ β), β ⇒ (α ∨ β) (A6) (α ⇒ γ) ⇒ ((β ⇒ γ) ⇒ ((α ∨ β) ⇒ γ)) (A7) (α ∧ β) ⇒ α, (α ∧ β) ⇒ β (A8) (γ ⇒ α) ⇒ ((γ ⇒ β) ⇒ (γ ⇒ (α ∧ β))) (A9) ((α ∧ β) ∨ γ) ⇒ ((α ∨ γ) ∧ (β ∨ γ)), ((α ∨ γ) ∧ (β ∨ γ)) ⇒ ((α ∧ β) ∨ γ) (A10) ((α ∨ β) ∧ γ) ⇒ ((α ∧ γ) ∨ (β ∧ γ)), ((α ∧ γ) ∨ (β ∧ γ)) ⇒ ((α ∨ β) ∧ γ) (A11) (α ⇒ β) ⇒ (¬β ⇒ ¬α) (A12) (α ∧ ¬α) ⇒ β (A13) β ⇒ (α ∨ ¬α). 9 (II) Rules: There is only one rule: “from σ and σ ⇒ τ conclude τ ”, written as 9

One can also choose α, β, γ as three different variables and the substitution rule sub as an additional rule.

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σ, σ ⇒ τ (“modus ponens”) τ A derivation of ϕ from Σ (⊆ P rop) is a finite sequence (ϕ1 , ϕ2 , ..., ϕn ) of formulas with ϕn = ϕ, where each ϕi (1 ≤ i ≤ n) is an axiom or is an element of Σ or is the result of the application of modus ponens on ϕu , ϕv (u, v < i). Let be the derivation operator defined in this way, i.e., we write Σϕ iff there is a derivation of ϕ from Σ. Put cons(Σ) := {ϕ | Σ ϕ} (the set of all derivation consequences from Σ). It is easy to check that cons(Σ) ⊆ Cons(Σ), since the axioms are tautologies and since, for each valuation v, v(σ) = 1 and v(σ ⇒ τ ) = 1 imply v(τ ) = 1. For proof that cons(Σ) ⊂ Cons(Σ) is false, we need the following notations and facts: Since P rop is a set of terms, one can form the term algebra Prop := (P rop; ∨, ∧, ⇒, ¬, 0, 1) of the type (2, 2, 2, 1, 0, 0). For arbitrary α, β ∈ P rop let α ≈Σ β iff (Σ α ⇒ β and Σ β ⇒ α). Theorem 1.5.2.1 The relation ≈Σ has the following properties for arbitrary Σ: (a) (b) (c) (d)

≈Σ is a congruence of the algebra Prop. The factor algebra (P rop/≈Σ ; ∨, ∧, ¬, 0, 1) is a Boolean algebra. The set cons(Σ) is the 1 of the Boolean algebra (P rop/≈Σ ; ∨, ∧, ¬, 0, 1). There exists a homomorphic mapping ν from Prop onto Prop/≈Σ with ν(ϕ) = 1 iff Σ ϕ.

Proof. (a): First we show that ≈Σ is an equivalence relation. By Σ α ⇒ α (see (A1)) is ≈Σ reflexive. The symmetry of ≈Σ follows from the definition of ≈Σ . One can prove the transitivity of ≈Σ with the aid of (A3) and the modus ponens as follows: Let α ≈Σ β and β ≈Σ γ be arbitrary, i.e., it holds: Σ α ⇒ β, Σ β ⇒ α, Σ β ⇒ γ, Σ γ ⇒ β. By means of the modus pones ( σ := α ⇒ β, σ ⇒ τ := (A3) ) and (A3) one obtains Σ (β ⇒ γ) ⇒ (α ⇒ γ). When one uses the modus pones (σ := β ⇒ γ) again, one receives Σ α ⇒ γ. Analogously, one can show Σ γ ⇒ α. Thus, ≈Σ is an equivalence relation. To prove the compatibility of ≈Σ with ⇒ we consider


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ϕ ≈Σ ϕ , ψ ≈Σ ψ , ( ϕ, ϕ , ψ, ψ ∈ P rop), i.e., it holds Σ ϕ ⇒ ϕ , Σ ϕ ⇒ ϕ, Σ ψ ⇒ ψ , Σ ψ ⇒ ψ. (ψ ⇒ ψ ) ⇒ (ϕ ⇒ (ψ ⇒ ψ )) (α = ψ ⇒ ψ and τ = β in (A2)) and the modus ponens imply Σ ϕ ⇒ (ψ ⇒ ψ ). Furthermore, by (A4), we have ((ϕ ⇒ (ψ ⇒ ψ )) ⇒ ((ϕ ⇒ ψ) ⇒ (ϕ ⇒ ψ )). Hence we get Σ ((ϕ ⇒ ψ) ⇒ (ϕ ⇒ ψ )) with the aid of modus ponens. Analogously, one can show Σ ((ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ)). Consequently,

ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ .

(1.13)

By (A3) we have Σ (ϕ ⇒ ϕ) ⇒ ((ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ )), whereby Analogously, and therefore, Then

Σ (ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ ). Σ (ϕ ⇒ ψ ) ⇒ (ϕ ⇒ ψ ), ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ .

(1.14)

ϕ ⇒ ψ ≈Σ ϕ ⇒ ψ ,

since ≈Σ is transitive and (1.13) and (1.14) are valid. Hence, ≈Σ is compatible with ⇒. To prove that ≈Σ is compatible with ¬ we assume α ≈Σ β; i.e., we have Σ α ⇒ β and Σ β ⇒ α. Then, by (A11) and the modus ponens, we get Σ ¬β ⇒ ¬α and Σ ¬α ⇒ ¬β, whereby ¬α ≈Σ ¬β. Because of this property of ≈Σ , we can define a partial order on Prop/≈Σ as follows: [ϕ]≈Σ ≤ [ψ]≈Σ iff Σ ϕ ⇒ ψ, where [ϕ]≈Σ = [ψ]≈Σ means that ϕ ≈Σ ψ (i.e., Σ ϕ ⇒ ψ and Σ ψ ⇒ ϕ) holds. (By (A1) is ≤ reflexive; (A3) implies the transitivity of ≤. The antisymmetry follows from the definition of ≈Σ : I If Σ ϕ ⇒ ψ and Σ ψ ⇒ ϕ, then we have ϕ ≈Σ ψ.) Now (A5) shows that [α]≈Σ ≤ [α ∨ β]≈Σ and [β]≈Σ ≤ [α ∨ β]≈Σ .

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Assume there is a γ with [α]≈Σ ≤ [γ]≈Σ and [β]≈Σ ≤ [γ]≈Σ . By (A6) and the modus ponens, this implies [α ∨ β]≈Σ ≤ [γ]≈Σ , whereby sup([α]≈Σ , [β]≈Σ ) = [α]≈Σ ∨ [β]≈Σ = [α ∨ β]≈Σ holds. Analogously, inf ([α]≈Σ , [β]≈Σ ) = [α]≈Σ ∧ [β]≈Σ = [α ∧ β]≈Σ follows from (A7), (A8) and the modus ponens. Thus ≈Σ is also compatible with the operations ∨ and ∧. (b): (a) implies that Prop/≈Σ is a lattice. Because of the axioms (A9) and (A10) this lattice is distributive. Assume Σ ϕ and Σ ψ. Then [ϕ]≈Σ = [ψ]≈Σ , since Σ ψ ⇒ ϕ and Σ ϕ ⇒ ψ follows from this with the aid of (A2) and the modus ponens. Because of (A2), we have Σ (β ⇒ α) for all β ∈ P rop and all α ∈ P rop with Σ α, i.e., β ≤ α. Therefore, {α | Σ α} is the greatest element of the lattice Prop/≈Σ . By (A12) the smallest element is the equivalence class [α ∧ ¬α]≈Σ . Consequently, the algebra Prop/≈Σ also fulfills the axioms (B1 ) (see Section 1.2.12). (B2 ) follows from the fact that ≈Σ is compatible with ∧, ∨ and ¬ with the aid of (A13) as follows: [α]≈Σ ∧ (¬[α]≈Σ ) = [α ∧ (¬α)]≈Σ = 0 (see above), [α]≈Σ ∨ (¬[α]≈Σ ) = [α ∨ (¬α)]≈Σ = 1 (because of [β]≈Σ ≤ [α ∨ ¬α]≈Σ for each β ∈ P rop by (A13)). Hence, Prop/≈Σ is a Boolean algebra. (c): We have shown above that the 1 of P rop/≈Σ contains all formulas derivable from Σ. Further, if α ∈ P rop belongs to the equivalence class 1 of P rop/≈Σ , then we get Σ ϕ ⇒ α for each ϕ with Σ ϕ by the above shown. Thus (by modus ponens) we have Σ α. Hence, (c) is proven. (d) By Part I, Section 4.1, the natural homomorphism ϕ : P rop −→ P rop/≈Σ , α → [α]/≈Σ fulfills (d). Theorem 1.5.2.2 (Completeness Theorem of Proportional Logic) Let Σ ⊆ P rop. Then cons(Σ) = Cons(Σ).


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Proof. Obviously, cons(Σ) ⊆ Cons(Σ). Assume there exists an α ∈ Cons(Σ) \ cons(Σ). Then [α]/≈Σ = [1]/≈Σ by Theorem 1.5.2.1, whereby α is an element of a certain prime ideal (maximal ideal) of the Boolean algebra Prop/≈Σ .10 Consequently, there exists a homomorphism µ from Prop/≈Σ onto the two-element Boolean algebra {0, 1} with µ([α]/≈Σ ) = 0. Then, with the aid of Theorem 1.5.2.1, (d), it is easy to see that v := ν2µ is a valuation with v(α) = 0 and v(σ) = 1 for all σ ∈ Σ. But, this is a contradiction to α ∈ Cons(Σ). Next is a short introduction to predicate logic (or first order logic). Predicate logic is a language to describe statements about algebraic structures (to given signature). A signature is a pair δ := ((ni )i∈I , (mj )j∈J ) with ni ∈ N0 , mj ∈ N for all i ∈ I and j ∈ J. A := (A; (fiA )i∈I , (RjA )j∈J ) is called a structure of signature δ, if (A; (fiA )i∈I ) an algebra of the type (ni )i∈I and RjA ⊆ Amj holds for all j ∈ J. We say “fiA is the interpretation of fi ” and “RjA is the interpretation of Rj ”. With the aid of a mapping P from Ah into {0, 1}, one can describe an h-ary relation R on a set A (i.e., R ⊆ Ah ) as follows: P (a1 , .., ah ) = 1 iff (a1 , ..., ah ) ∈ R. Such a mapping P is called an h-ary predicate. Let P, Q be predicates on A. Then one can understand the predicates as propositions and form (¬P ) and (P ◦ Q) for each ◦ ∈ J0 . m Therefore, let Pj j be the mj -ary predicate with mj

Pj (a1 , .., amj ) = 1 iff (a1 , ..., amj ) ∈ Rj

in the following. The alphabet of the first order logic consists of the following symbols: • • • •

the set V ar := {x0 , x1 , x2 , ...} of variables m the set {Pj j | j ∈ J} of predicate symbols the set {fini | i ∈ I} of operation symbols the set of connectives J := {∧, ∨, ¬, ⇒, ⇔, ∃, ∀} (∃ and ∀ are called the existential and universal quantifier) • the set of auxilliary symbols {(, )} As described in Part I, Section 6.2, we form the set of all terms T erm := T (V ar) and the term algebra Term = (T erm; (fi )i∈I ). Then, F ORM is the smallest set X with the following four properties:

10

See books on Boolean algebras or universal algebras.

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• Pj (t1 , ..., tmj ) ∈ X for all j ∈ J and all t1 , ..., tmj ∈ T erm (Pj (t1 , ..., tmj ) is called an atom); • if α, β ∈ X , then (α ◦ β) ∈ X for all ◦ ∈ J0 • if α ∈ X then (¬α) ∈ X • if α ∈ X then (∃xk α) ∈ X and (∀xk α) ∈ X for all k ∈ N0 If t ∈ T erm, then V ar(t) denotes the set of all elements of X, which occur in T . For ϕ ∈ F ORM the set V ar(ϕ) of all variables of ϕ is defined by • V ar(Pj (t1 , ..., tmj )) := V ar(t1 ) ∪ ... ∪ V ar(tmj ) • V ar(¬α) := V ar(α) and V ar(α ◦ β) := V ar(α) ◦ V ar(β) for all ◦ ∈ J0 • V ar(∃xk α) = V ar(∀xk α) := V ar(α) ∪ {xk } (α, β ∈ F ORM ). The set f r(ϕ) of free variables of ϕ is defined by • f r(ϕ) := V ar(ϕ) if ϕ is an atom • f r(¬α) := f r(α) and f r(α ◦ β) := f r(α) ◦ f r(β) for all ◦ ∈ J0 • f r(∀xk α) = f r(∃xk α) := f r(α)\{xk } (α, β ∈ F ORM ). The set bd(ϕ) of bound variables of ϕ is defined by • bd(ϕ) := ∅ if ϕ is an atom • bd(¬α) := bd(α) and bd(α ◦ β) := bd(α) ◦ bd(β) for all ◦ ∈ J0 • bd(∀xk α) = bd(∃xk α) := bd(α) ∪ {xk } (α, β ∈ F ORM ). – ϕ is called open formula if bd(ϕ) = ∅. ϕ is a sentence or is closed, if f r(ϕ) = ∅. Obviously, V ar(ϕ) = f r(ϕ) ∪ bd(ϕ), but f r(α) ∩ bd(ϕ) need not be empty. Let u, u be mappings from V ar into A. Then we write u =xk u iff u(xj ) = u (xj ) for all j = k. Let A := (A; (fi )i∈I , (Rj )j∈J ) be a structure of signature δ and let u : V ar −→ A be a mapping. For u, there is exactly one homomorphism u : T(Var) −→ (A; (fi )i∈I ), which u continues (see Part I, Theorem 6.2.2). In the following manner, one can interpret the elements of F ORM with the aid of A, u and u : An interpretation function (or valuation) is a mapping vA,u from F ORM into {0, 1} defined by • vA,u (Pj (t1 , ..., tmj )) = 1 iff ( u(t1 ), ..., u (tmj )) ∈ RjA • vA,u (¬α) := ¬(vA,u (α)) and vA,u (α ◦ β) := vA,u (α) ◦ vA,u (β) for all ◦ ∈ J0 • vA,u (∀xk ϕ) = 1 iff v A,u (ϕ) = 1 for all u with u =xk u (i.e., vA,u (∀xk ϕ) = u , u=x u vA,u (ϕ)) k • vA,u (∃xk ϕ) = 1 iff there is a u with u =xk u and vA,u (ϕ) = 1 (i.e., vA,u (∃xk ϕ) = u , u=x u vA,u (ϕ) ) k


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ϕ ∈ F ORM is satisfied by u : V ar −→ A iff vA,u (ϕ) = 1. ϕ is true (or valid) in A, iff vA,u (ϕ) = 1 for all u : V ar −→ A. A |= ϕ stands for “ϕ is true in A”. A |= ϕ stands for “ϕ is not true (or false) in A”. If A |= ϕ, we call A a model of ϕ. In general, if A |= σ for all σ ∈ Σ ⊆ F ORM , we call A a model of Σ. Notice that a closed formula of F ORM is either true or false in A. Formulas α, β ∈ F ORM are equivalent (in symbol α ≡ β) iff vA,u (α) = vA,u (β) for every structure A and for every u : V ar −→ A. The following theorem is easy to prove: Theorem 1.5.2.3 Let ϕ, ψ ∈ F ORM . Then: (1) ¬∀xk ϕ ≡ ∃xk ¬ϕ, ¬∃xk ϕ ≡ ∀xk ¬ϕ; (2) if xk ∈ f r(ψ) then (Qxk ϕ ◦ ψ) ≡ Qxk (ϕ ◦ ψ) for all Q ∈ {∃, ∀} and ◦ ∈ {∧, ∨}; (3) (∀xk ϕ ∧ ∀xk ψ) ≡ ∀xk (ϕ ∧ ψ), (∃xk ϕ ∨ ∃xk ψ) ≡ ∃xk (ϕ ∨ ψ); (4) ∀xk ∀xl ϕ ≡ ∀xl ∀xk ϕ, ∃xk ∃xl ϕ ≡ ∃xl ∃xk ϕ. Next, we define the mappings repτ , tutτ and subτ from F ORM into F ORM , where τ is a mapping from V ar into T erm. With repτ , one can describe the replacement of variables by some terms in a formula from F ORM . The mapping tutτ renames the bound variables of a formula so that there are no variables of the formula anymore, which is free and bound. subτ is a combination of repτ and tutτ . Let τ : V ar −→ T erm be an arbitrary mapping and τ : T erm −→ T erm the unique homomorphic continuation of τ . Then the mapping repτ : F ORM −→ F ORM is defined by • repτ (Pj (t1 , ..., tmj )) := Pj ( τ (t1 ), ..., τ(tmj )) • repτ (¬α) := ¬repτ (α) and repτ (α ◦ β) := repτ (α) ◦ repτ (β) for all ◦ ∈ J0 • if ϕ = Qxk α with Q ∈ {∀, ∃}, then repτ (ϕ) := Qxk repτ (α), where xk if xi = xk , τ (xi ) := τ (xi ) if xi = xk . For the mapping τ : V ar −→ T erm and ϕ ∈ F ORM let n(τ, ϕ) be the smallest number j0 ∈ N0 with xj ∈ f r(ϕ) ∪ V ar(τ (x)) for all x ∈ f r(ϕ). Let tutτ : F ORM −→ F ORM be defined by • tutτ (ϕ) = ϕ if ϕ is an atom • tutτ (¬ϕ) = ¬(tutτ (ϕ)) and tutτ (ϕ ◦ ψ) = tutτ (ϕ) ◦ tutτ (ψ) for all ◦ ∈ J0 ; • for Q ∈ {∀, ∃} let tutτ (Qxk ϕ) = Qxj tutτ (repτk,j (ϕ)) with j := n(τ, ϕ) and xj if k = l, τk,j (xl ) := xl otherwise.

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For τ : V ar −→ T erm and arbitrary ϕ ∈ F ORM set subτ (ϕ) := repτ (tutτ (ϕ)). The following lemma gives some properties of the above mappings: Lemma 1.5.2.4 (without proof ) Let A be a structure of signature δ, u : V ar −→ A a mapping with the homomorphic continuation u : T erm −→ A. Then: (a) vA,u (ϕ) = vA,u (tutτ (ϕ)); (b) f r(ϕ) = f r(tutτ (ϕ)); (c) vA,τ 2u (ϕ) = vA,u (subτ (ϕ)). With the aid of the above mapping, we can describe the Hilbert-typecalculus for the classical predicative logic, which consists of following axioms and rules: (I) axioms are: (a) the axioms (A1) - (A13) of the classical propositional logic; (b) all formulas of the form (A14) (∀xk ϕ) ⇒ subτ (ϕ), where τ =xk id (i.e, τ replaces something at most for the variable xk ); (A15) ∃xk ϕ ⇒ ¬∀xk ¬ϕ, ¬∀xk ¬ϕ ⇒ ∃xk ϕ; (A16) subid (ϕ) ⇒ ϕ. (II) rules are: (a) the modus ponens; (b) ϕ ⇒ subτk,l (ψ) ϕ ⇒ ∀xk ψ,

where τk,l (xj ) :=

xl if j = k, xj otherwise

and xl ∈ V ar(ψ) ∪ f r(ϕ) (“∀-introductory-rule”); (c) ϕ subτ (ϕ) for all τ . Analogously to the proportional logic, one can define and cons(Σ) for Σ ∈ F ORM in respect to the above calculus. The completeness theorem of the first order logic says that = |= holds, i.e., cons(Σ) = Cons(Σ) is valid for all Σ ⊆ F ORM and that there are axioms and rules, so that = |= was proven for the first time by K. G¨ odel in his dissertation 1929. For the above calculus one can prove the completeness theorem for first order


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logic similar to Theorem 1.5.2.2, where the Rasiowa-Sikorski-Tarski-Lemma is needed (see [Ric 78] for details). The famous incompleteness theorems of Gödel say that second order logic 11 does not have a completeness theorem in general.

1.5.3 Many-Valued Logics The above considerations can be generalized when one assigns the “sentences” certain values from the set Ek (k ≥ 3). We receive then a so-called k-valued logic or many-valued logic. One finds a full introduction to the logic for example in [Got 89] or [Kre-G-S 88]. From a mathematical perspective it doesn’t matter which interpretations have the elements of Ek . Nevertheless, it is shown which interpretations of the elements of Ek , for example, are possible. According to these interpretations, one can select certain functions of Pk , with which one can form many-valued logics. For reasons of space limitations, we have excluded information. These functions can be found in [Men 85], where literature on the topic is also provided. 1.5.2.1 The three values of a three-valued logic can be interpreted as follows: “false”, “indefinite”, “true”; “false”, “possible”, “true”; “false”, “undecidable”, “true” or “invalid”, “in part valid”, “full-valid”. Legally relevant actions can be divided in “punishable”, “prohibited but not punishable”, “allowed”. 1.5.2.2 The 4-valued logic seems particularly suitable for analyzing problem areas logically, in which there are two different kinds of truth (or validity) and two different kinds of falsehood (or nullity). Examples: “fact falsehood”, “legal nullity”, “legal validity”, “fact truth” ; “knowledge falsehood”, “belief falsehood”, “belief truth”, “knowledge truth”. 1.5.2.3 A legal interpretation of the 6-valued logic is, for example, 0: “logically false”, 1: “legally invalid”, 2: “false according to facts”, 3: “true according to facts”, 4: “legally valid”, 5: “logically true”. 11

In this logic, quantifications over elements of a structure as well at quantifications over partial sets and relations of the structure are allowed.

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1.5.2.4 Instead of Ek , one can also choose a finite subset of the real numbers x with 0 ≤ x ≤ 1. Then, with the aid of such a set, a probability logic can be formed: The value 1 corresponds to the certainty of the truth; the value 0.5 corresponds to the uncertainty, whether true or false; the value 0 corresponds to the certainty of the falsehood (or the impossibility); values between 1 and 0.5 correspond to degrees of higher probability; values between 0.5 and 0 correspond to degrees of low probability. We notice that one could also have chosen the interpretation of the Fuzzy Logic instead of the above-mentioned probability logic (see for example [BanG 90] or [Til 92]). The problems of all the above-mentioned logics correspond basically to those of the propositional logic. 1.5.4 Information Transformer The functions f n ∈ PA can be understood simply as mathematical models of objects that process information (see Figure 1.1). The “object” from Figure 1.1 receives the information x1 , ..., xn ∈ A at the entries, processed to the information f (x1 ..., xn ) ∈ A. In this case, we neglect the time which is needed for the workmanship. x1 x2 ... xn ??

? f

?x ) f (x1 , ..., n Fig. 1.1

Superpositions over functions of PA correspond with this model to the “assembling” of such objects. For example, one can describe the diagram x1

x2

x3

? ? f A U A ?

? g

? g(f (x1 , x2 ), x3 , x3 ) Fig. 1.2


117

through the formula g(f (x1 , x2 ), x3 , x3 ). In particular, Boolean functions are used for the mathematical description of electrical circuits or components in computers. This mathematical description is independent of the concrete technical realization (as, for example, relay contact circuits or transistors). Naturally, one receives the so-called completeness problem from these interpretations of the functions from PA : A necessary and sufficient criterion is searched in order to be able to decide whether a system of certain selected functions (which correspond to certain elementary elements) produces all functions from PA by means of superposition. One way of finding this criterion indicates the following theorem: Theorem 1.5.4.1 Let A be a subclass of Pk with the property that to every proper subclass A of A there exists a certain maximal class M of A with A ⊆ M . Furthermore, denote M the set of all maximal classes of A. Then for an arbitrary subset T of A it holds: [T ] = A ⇐⇒ ∀M ∈ M : T ⊆ M.

(1.15)

Proof. “=⇒”: Let [T ] = A. Suppose there is an M ∈ M with T ⊆ M . Then, a contradiction results; however, from that, immediately, A = [T ] ⊆ [M ] = M ⊂ A. “⇐=”: Let T ⊆ Pk be no subset of a maximal class of A. Suppose, [T ] ⊂ A. Then, by assumption, one can find a certain maximal class M ∈ M with [T ] ⊆ M , a contradiction to the supposition. By Theorem 11.1.1, every finitely generated class fulfills the conditions of Lemma 1.5.3.1 and has only finitely many maximal classes, whereby (1.15) supplies a criterion of the wanted kind if one knows the maximal classes of A. Further problems that result from interpreting the functions of PA as information transformers are the following: • Find minimal generating systems, where “minimal” is related to the number of the generating system’s functions or to the arities of the generating system’s functions. • Find algorithms to construct a minimal realization of a given function by certain elementary functions. One can find further problems and their solutions in [Rin 84] and [P¨ os-K 79].

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1.5.5 Classification of Combinatorial Problems In this section, we need some notations (Rk , P olk , Invk , ...) of Chapter 2. Further, we need some concepts of the algorithm theory:12 Algorithms are techniques for solving problems, wherein the word “problem” is used in a very general sense: A problem class consists of infinitely many instances having a common structure. A problem, whose solution is either “yes” or “no”, is called a decision problem. Often, finding a solution algorithm for the solution of a mathematical problem does not suffice. Statements about the complexity of the algorithms are important for the applicability of algorithms in particular. The complexity of an algorithm is a function f (n), which gives the number of the arithmetic steps (e.g. value assignments, elementary arithmetic operations, comparison operations, ...) which are necessary to the estimating of a solution of a given problem with n master data. For g : N −→ R+ let O(g(n)) := {f (n) | (f : N −→ R+ ) ∧ (∃c > 0 ∀n ∈ N : f (n) ≤ c · g(n))} We say that the algorithm has the complexity O(g(n)), if the complexity f (n) of the algorithm belongs to O(g(n)). Generally one holds algorithms with a complexity O(nt ) for “good” and these algorithms are called polynomial algorithms. A decision problem with the complexity O(nt ), where n, t ∈ N, is called tractable. Let P be the class of all tractable problems. L denotes the class of all problems with the complexity O(log n). The class of decision problems for which a positive answer can be verified in polynomial time is denoted by NP (for “non-deterministic polynomial”). That is, we do not only require the answer “yes” or “no”, but the explicit specification of a certificate which allows to verify the correctness of a positive answer. LP denotes the class of decision problems for which a positive answer can be verified in logarithmic time. Obviously, P ⊆ NP. Further, we know that L ⊆ NL ⊆ P. It is the greatest problem of the algorithm theory to decipher whether P = NP is valid. Most people believe, however, that P = NP. A problem is called NP-complete if is in NP and if the polynomial solvability of this problem would imply that all problems in NP are solvable in polynomial time as well. In other words: Each problem in NP can be “transformed” (in polynomial time) to the given NP-complete problem, such that 12

See e.g. [Pap 94] and [Jun 2005].


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a solution of the NP-complete problem also gives a solution to that other problem in NP. Therefore, if one finds a polynomial algorithm for an NPcomplete problem, this would imply that P = NP. Next, is a short introduction to some papers that deal with classifications of combinatorial problems. Definition A CSP (or a “constraint satisfication problem”) is a triple (V, D, C), where • V is a set of variables • D is a set of values which can take the variables • C := {C1 , C2 , ..., Cq } with Ci := (si , τi ) for i = 1, ..., q, where si is an mi -ary tuple of variables and τi is an mi -ary relation on D (Ci is called constraint). A solution of the CSP is a mapping f : V −→ D with the property ∀ Ci := ((xi1 , xi2 , ..., ximi ), τi ) ∈ C : (f (xi1 ), f (xi2 ), ..., f (ximi )) ∈ τi . Many combinatorial problems can be described as CSP (see [Jea 97] 13 ) Example The problem SAT as CSP An instance of the standard propositional satisfiability problem is specified by giving a formula on propositional logic, and asking whether there are values for the variables that make the formula true. For example, consider the formula t(x1 , x2 , x3 , x4 ) := (x1 ∨ x2 ∨ x3 ∨ x4 ) ∧ (x1 ∨ x2 ∨ x3 )∧ (x3 ∨ x4 ∨ x1 ) ∧ (x3 ∨ x2 ∨ x4 ) ∧ (x1 ∨ x3 ). The question is whether there are a1 , a2 , a3 , a4 ∈ {0, 1} with t(a1 , a2 , a3 , a4 ) = 1. This problem can be expressed as the CSP as follows: Put V := {x1 , x2 , x3 , x4 }, D := {0, 1}, C1 C2 C3 C4 C5 C

:= := := := := :=

((x1 , x2 , x3 , x4 ), D4 \ {(0, 0, 0, 0)}), ((x1 , x2 , x3 ), D3 \ {(1, 1, 0)}), ((x3 , x4 , x1 ), D3 \ {(1, 1, 0)}), ((x3 , x2 , x4 ), D3 \ {(1, 1, 0)}), ((x1 , x3 ), D2 \ {(1, 1)}), {C1 , c2 , C3 , C4 }.

The solutions of the above problem are 13

In this paper are 20 combinatorial problems expressed as CSP.

120


x1 x2 x3 x4 f1 f2 f3 f4 f5 f6

: : : : : :

0 0 0 0 1 1

0 0 1 1 0 0

0 1 0 0 0 0

1 0 0 1 0 1

It is also possible to describe a CSP with the aid of the following notations: Definitions Let V be a nonempty set and let τ1 , τ2 , ..., τt be relations on V . Then, (V ; τ1 , τ2 , ..., τt ) is called relational structure. The mapping : {1, 2, ..., t} −→ N is the rank functions of the relational structure (V ; τ1 , τ2 , ..., τt ) if (i) is the arity of the relation τi (i = 1, 2, ..., t). A relational structure Σ is similar to a relational structure Σ if they have the same rank function. Let Σ := (V ; τ1 , ..., τt ) and Σ := (V ; τ1 , ..., τt ) be two similar relational structure, and let be their common rank function. A homomorphism from Σ into Σ is a mapping h : V −→ V such that ∀i ∈ {1, ..., t} : (a1 , .., ahi ) ∈ τi =⇒ (h(a1 ), ..., h(ahi )) ∈ τi . Let Hom(Σ, Σ ) be the set of all homomorphisms from Σ into Σ . Obviously, it holds that Theorem 1.5.5.1 ([Jea-C-P 98]) (a) Let P := (V, D, C) be a CSP with C := {(s1 , τ1 ), (s2 , τ2 ), ..., (sq , τq )}. Then, the set of all solutions of P is the set Hom(Σ, Σ ), where Σ := (V ; {s1 }, {s2 }..., {sq }) and Σ := (D; τ1 , τ2 , ..., τq ). Conversely: Let Σ := (V ; τ1 , ..., τt ) and Σ := (D; τ1 , ..., τt ) be two similar relational structures. Then, Hom(Σ, Σ ) is the set of all solutions of the CSP (V, D, S) with C :=

t

{(s, τi ) | s ∈ τi }.

i=1

The following theorem is our first example of an application of P ol and Inv (see Chapter 2) in algorithm theory.


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Theorem 1.5.5.2 Let D := Ek , Γ := {1 , ..., q } ⊆ Rk , V := {x1 , ..., xn } and let M be a set of mappings from V into Ek . Then it holds: There exists a CSP (V , D, C), where V ⊆ V and C = {(c1 , 1 ), ..., (cq , q )}, with the solution set S and S|V = M if and only if := {(f (x1 ), ..., f (xn )) | f ∈ M } ∈ Invk (P olk Γ ). The proof of the above theorem is explained by subsequent examples: We choose: D := E2 , 1 := {(0, 1, 1), (1, 0, 0), (1, 1, 1)}, 2 := {(0, 0), (1, 0)}, 3 := {(1, 1)}, Γ := {1 , 2 , 3 }, V := {x1 , x2 , x3 , x4 } and M := {f1 , f2 }, where x f1 (x) f2 (x) x1 0 1 x2 1 0 x3 1 1 x4 1 1 “⇐=”: Assume, := {(f (x1 ), ..., f (xn )) | f ∈ M } ∈ Invk (P olk Γ ). Then can be obtained by a finite number of applications of the elementary operations 3 (see Section 2.3). For example: ζ, τ, pr, ∧ and × from and δk;{1,2} = {(x1 , x2 , x3 , x4 ) ∈ E24 | ∃x5 ∈ E2 : (x1 , x5 , x3 ) ∈ 1 ∧ (x5 , x2 ) ∈ 2 ∧ (x5 , x4 ) ∈ 3 }. Therefore, V = {x1 , x2 , x3 , x4 , x5 } and (V , D, C), where C := {((x1 , x5 , x3 ), 1 ), ((x5 , x2 ), 2 ), ((x5 , x4 ), 3 )}, is a CSP with the solution set S = {f1 , f2 }, where x f1 (x) f2 (x) x1 0 1 x2 1 0 x3 1 1 x4 1 1 x5 1 1 and S|V = M = {f1 , f2 }. “=⇒”: Choose V := {x1 , x2 } and M := {g1 , g2 }, where x g1 (x) g2 (x) x1 0 1 x2 0 0 ({x1 , x2 , x3 , x4 , x5 }, E2 , C) with C := {((x1 , x5 , x3 ), 1 ), ((x5 , x2 ), 2 ), ((x5 , x4 ), 3 )} is a CSP and it holds S|V = M = {g1 , g2 }. Then

122


= {(0, 0), (1, 0)} = {(x1 , x2 ) ∈ E22 | ∃x3 ∈ E2 ∃x4 ∈ E2 ∃x5 ∈ E2 : (x1 , x5 , x3 ) ∈ 1 ∧ (x5 , x2 ) ∈ 2 ∧ (x5 , x4 ) ∈ 3 } Thus ∈ Invk P olk Γ (see Chapter 2). Next we show how one can use the above theorem for solving the following task: Put D := E2 , Γ

:= { ⊆ E22 | = ∅} =: {1 , ..., 15 },

x f1 (x) f2 (x) f3 (x) x1 0 0 0 M := {f1 , f2 , f3 }, where x2 0 0 1 x3 0 1 0 x4 1 0 0 Is there a CSP (V , E2 , C) with V ⊆ V , C = {((c1 , 1 )), ..., (c1515 )} and the solution set S which ones S|V = M is valid for? Because of the above theorem, this question is equivalent for the following question: := {(f (x1 , ..., f (x4 ))) | f ∈ M } = {(0, 0, 0, 1), (0, 0, 1, 0), (0, 1, 0, 0)} ∈ Inv2 P ol2 Γ ? With the help of the results of Chapter 3, it is easy to prove that P ol2 Γ =

15

P ol2 i = [h2 ],

i=1

where h2 (x, y, z) := (x ∧ y) ∨ (x ∧ z) ∧ (y ∨ z). Then, because of ⎛ ⎞ ⎛ ⎞ 0 0 0 0 ⎜0 0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ h2 ⎜ ⎝0 1 0⎠=⎝0⎠ 1 0 0 0 we have ∈ Inv2 P ol2 ; i.e., there is no CSP with the above properties. Definitions The general combinatorial problem (GCP) is the decision problem with: Instance: A pair (Σ1 , Σ2 ) of similar finite relational structures Σ1 and Σ2 . Question: Is there a homomorphism from Σ1 to Σ2 ? For any GCP instance P := (Σ1 , Σ2 ) a homomorphism from Σ1 to Σ2 will be called a solution of P .


123

The GCP (or CSP) is known to be NP-complete. However, certain restrictions may affect the complexity of GCP (or CSP). One of the natural possibilities for restricting CSP is by limiting the relations that can appear in constraints (or relational structures). Definition Let Γ be a set of relations on the set D. Denote by CSP(Γ ) the subclass of CSP defined by the following property: any constraint relation in any instance must belong to Γ . The next theorem is the basis for the following theorems. Theorem 1.5.5.3 ([Jea 98]; without proof ) Let Γ and Ψ be finite sets of relation on D (w.l.o.g. we put D := Ek ). If Ψ ⊆ Invk (P olk Γ ) then GCP(Ψ ) can be reduced in polynomial time to GCP(Γ ). In other words: For any finite set Γ ⊆ Rk , the complexity of GCP(Γ ) is determined, up to polynomial-time reductions, by P olk Γ . Theorem 1.5.5.4 ([Jea 98]; without proof ) Let D := Ek , k ≥ 2 and Γ ⊆ Rk . Then (1) GCP(Γ ) ⊆ P, if one of the following conditions is valid: (a) P olk Γ contains a constant function. (b) P olk Γ contains a binary function which is associative, commutative and idempotent. (c) r ∈ P olk Γ , where r(x, y, z) := x + y + z and (Ek ; +) is Abelian 2group. (2) If P olk Γ contains a ternary function ϕ defined by y if y = z, ϕ(x, y, z) := x otherwise, then GCP(Γ ) ⊆ NL. (3) If each function of P olk Γ is either a projection or a semiprojection, then GCP(Γ ) is NP-complete. With the aid of the above theorem and Post’s description of all closed sets of Boolean functions, one can prove the following: Theorem 1.5.5.5 ([Sch 78]) For arbitrary Γ ⊆ R2 it holds: GCP(Γ ) is NP-complete, if P ol2 Γ ⊆ [P21 ]; in all other cases we have GCP(Γ ) ⊆ P. More can be found on the topic in [Jea 98], [Jea-C-P 98], [Bul-K-J 2000] and [Coh-J-G 2003].

2 The Galois-Connection Between Function- and Relation-Algebras

This section aims to develop a “suitable” means to describe function algebras or clones. I mean, suitable in the sense that “big” function algebras (or clones) can be described with an “expenditure” as small as possible. Analogously to other fields of algebra we introduce invariants for our function algebras. The two papers [Bod-K-K-R 68/69] of V. G. Bodnarˇcuk, L. A. Kaluˇznin, V. N. Kotov and B. A. Romov are basis of this so-called Pol-Inv theory (or Galois theory for function- and relation-algebras). These articles generalize results by M. Krasner on a Galois theory for groups (see [Kra 45], [Kra 68/69]). A full representation of the Pol-Inv theory with many applications can be found in the monograph [P¨ os-K 79] by R. P¨ oschel and L. A. Kaluˇznin.

2.1 Relations An h-ary relation on Ek is a subset of the h-fold Cartesian product Ekh of the set Ek , h ∈ N. The elements (a1 , a2 , ..., ah ) of (we say “h-tuple”) are written as columns ⎞ ⎛ a1 ⎜ a2 ⎟ ⎟ ⎜ ⎝ ... ⎠ , ah and then we also write

⎛

⎞ a1 ⎜ a2 ⎟ ⎜ ⎟ ⎝ ... ⎠ ∈ . ah

The relation is written often as a matrix whose columns are the elements of the relation. For example,

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⎞ a1 a1 a1 := ⎝ ... ... ... ⎠ ah ah ah ⎛

instead of := {(a1 , ..., ah ), (a1 , ..., ah ), (a1 , ..., ah )}. We think of this matrix representation of if we subsequently talk about the length h and the width || of the relation as well as about rows from . Denote Rkh the set of all h-ary relations on Ek and let Rk := Rkh . h≥1

We remark that the empty set is an element of Rk . If Q ⊆ Rk then let Qh := Q ∩ Rkh .

2.2 Diagonal Relations The simplest relations, in a certain sense1 , are the diagonal relations (or diagonals) defined as follows: For an arbitrary equivalence relation ε on {1, 2, ..., h} let h δk,ε := {(a1 , ..., ah ) ∈ Ekh | (i, j) ∈ ε =⇒ ai = aj }.

If h or k follows from the context, then we write only δε or δεh or δk,ε . Every element of the set h Dkh := { δk,ε | ε is an equivalence relation on {1, 2, ..., h} }.

is called a diagonal h-ary relation. Let Dk := {∅} ∪

Dkh

h≥1

be the set of all diagonal relations. h , we often declare this For the purpose of a more simple description of δk,ε relation in the form h δk;ε 1 ,...,εr or, briefly, by δε1 ,...,εr , where ε1 , ..., εr are exactly the equivalence classes of ε, which have at least two elements. In particular, we have h δk; = Ekh

and h = {(x, x, ..., x) ∈ Ekh | x ∈ Ek }. δk;E k 1

See Theorem 2.5.1.

2.4 Relation Algebras, Co-Clones, and Derivation of Relations

127

2.3 Elementary Operations on Rk We define in this section some operation ζ, τ, pr, ∧ and × on Rk with whose aid we can form later more “complex” operations on Rk . We call, therefore, the operations ζ, τ, pr, ∧ and × the elementary operations on Rk .

Let ∈ Rkh and ∈ Rkh , where h, h ∈ N. Then, if = ∅ and = ∅, let ζ ∈ Rkh , τ ∈ Rkh , pr ∈ Rkh−1 for h ≥ 2 and pr = ∅ for h = 1, × ∈ Rkh+h and ∧ ∈ Rkh (only for h = h ) defined by: ζ := {(a2 , a3 , ..., ah , a1 ) | (a1 , a2 , ..., ah ) ∈ } (cyclical exchanging of the rows), τ := {(a2 , a1 , a3 , ..., ah ) | (a1 , a2 , ..., ah ) ∈ } (exchange of the first two rows) for h ≥ 2 and ζ = τ = for h = 1 or = ∅; pr := {(a2 , ..., ah ) | ∃a1 ∈ Ek : (a1 , a2 , ..., ah ) ∈ } (projection onto the 2th,..., h-th coordinate or the strike of the first row) for h ≥ 2, × := {(a1 , ..., ah , b1 , ..., bh ) | (a1 , a2 , ..., ah ) ∈ ∧ (b1 , b2 , ..., bh ) ∈ } (Cartesian product of and ) and ∧ := {(a1 , ..., ah ) | (a1 , ..., ah ) ∈ ∩ } (intersection of the relations and ).

2.4 Relation Algebras, Co-Clones, and Derivation of Relations The algebra

3 , ζ, τ, pr, ∧, ×) Rk := (Rk ; δk;{1,2}

of type (0, 1, 1, 1, 2, 2) is called full relation algebra on Ek . Every subalgebra Q of Rk (in symbol Q ≤ Rk ) is a relation algebra on Ek . Let Q ⊆ Rk . Then, [Q] denotes the set of all relations of Rk that can be obtained by a finite number of applications of the elementary operations 3 ,i.e., [Q] is the universe of ζ, τ, pr, ∧ and × from the relations of Q and δk;{1,2} the smallest relation algebra, which contains Q. If [Q] = Q (⊆ Rk ) we say that Q is closed or Q is a co-clone of Rk . Further, we say a relation can be derived from the relation (or is -derivable), if ∈ [{}]. In this case we also write: .

128


2.5 Some Operations on Rk Derivable from the Elementary Operations We say that an operation on Rk is derivable from the elementary operations ζ, τ, pr, ∧ and × (or {ζ, τ, pr, ∧, ×}-derivable), if their effect on an arbitrary relation ∈ Rk can also be described by the effect of a finite com3 on . position of the elementary operations and δk;{1,2} The following is a small list of such derivable operations. In this case, always describes an h-ary and an h -ary relation of Rk . (O1 ): Permutation of coordinates (or permutations of rows). As is known, the permutations 1 2 ... h − 1 h 1 2 3 ... h and 2 3 ... h 1 2 1 3 ... h form a generating system for the symmetric group Sh (i.e., for the set of all permutations on {1, 2, ..., h} with the operation 2). Consequently one can realize all rearrangements of the rows of with the aid of ζ and τ . Thus, for every s ∈ Sh , σs () := {(as(1) , ..., as(h) ) | (a1 , ..., ah ) ∈ } is a {ζ, τ, pr, ∧, ×}-derivable operation. (O2 ): Projection onto the α1 -th, ..., αt -th coordinates (or deleting of rows). For {α1 , ..., αt } ⊆ {1, 2, .., h} it holds: prα1 ,...,αt () := {(aα1 , ..., aαt ) | (a1 , .., ah ) ∈ } = pr(pr(...(pr (σs ()))...)), h−t times where s ∈ Sh and s(α1 ) = h − t + 1, s(α2 ) = h − t + 2, ..., s(αt ) = h. In particular, we have prs(1),...,s(n) () = σs (). We remark that, in the case where is given in the form {(a0 , a1 , ..., ah−1 ) | ....}, we choose {α1 , ..., αt } ⊆ {0, 1, 2, .., h − 1} and define prα1 ,...,αt () in analog manner, as above.

2.5 Some Operations on Rk Derivable from the Elementary Operations

129

(O3 ): Identification of coordinates. For i, j ∈ {1, 2, ..., h} and i = j let ∆i,j () := {(a1 , ..., aj−1 , aj+1 , ..., ah ) | (a1 , ..., aj−1 , ai , aj+1 , ..., ah ) ∈ } and ∆ := ∆1,2 . ∆i,j can be formed as follows: 3 ) = Ek and It is easy to prove that pr1 (δk;{1,2} h δk;{i,j} = pr1,...,i−1,h+1,i+1,...,j−1,h+2,j+1,...,k (1 ), 3 and i < j. Consequently, ∆i,j is where 1 := Ek × ... × Ek ×pr1,2 δk;{1,2} h−2 times derivable because of h ∆i,j = pr1,...,j−1,j+1,...,h ( ∧ δk,{i,j} ).

(O4 ): Doubling of coordinates (rows). One receives a doubling of the i-th row of as follows: νi () := {(a1 , ..., ai−1 , ai , ai , ai+1 , ..., ah ) | (a1 , ..., ah ) ∈ } 3 = pr1,...,i−1,h,h+1,,i,...,h−1 (∆i,h+1 ( × pr(δk;{2,3} )).

(O5 ): Adding of fictitious coordinates. Let ∇ := {(a1 , ..., ah+1 )|a1 ∈ Ek ∧ (a2 , ..., ah+1 ) ∈ }. The first coordinate of ∇ is a so-called fictitious coordinate. Then, with the aid of (O1 ) one can derive the relation ∇i := {(a1 , ..., ai−1 , ai , ai+1 , ..., ah+1 ) ∈ Ekh+1 | (a1 , ..., ai−1 , ai+1 , ..., ah+1 ) ∈ } for i ∈ {1, ..., h + 1}. (O6 ): General composition (relation product). Let ◦t := {(a1 , ..., ah−t , bt+1 , ..., bh ) | ∃u1 , ..., ut ∈ Ek : (a1 , ..., ah−t , u1 , ..., ut ) ∈ ∧ (u1 , ..., ut , bt+1 , ..., bh ) ∈ } for t ∈ N with t ≤ h and t ≤ h . In particular, o := o1 . For h = h = 2, ◦ is the well-known relation product 2. The following theorem results from the above considerations.

130


Theorem 2.5.1 Every co-clone Q of Rk contains all diagonal relations and is closed in respect to – permutation of coordinates (rows), – projection onto coordinates (or deleting of rows), – identification of coordinates (rows), – doubling of coordinates (rows), – adding of fictitious coordinates, – (finite) intersection formation, – Cartesian products and – general composition.

2.6 The Preserving of Relations; Pol, Inv We say that a function f n ∈ Pk preserves the relation ∈ Rkh (or is invariant for f or is a invariant for f ), if ⎛ ⎛ ⎞ ⎞ f (a11 , a12 , ..., a1n ) a11 a12 ... a1n ⎜ f (a21 , a22 , ..., a2n ) ⎟ ⎜ a21 a22 ... a2n ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . . . . . . . . . . . . . . . ⎠ := ⎝ . . . . . . . . . . . . . . . . . . ⎠ ∈ ah1 ah2 ... ahn f (ah1 , ah2 , ..., ahn ) for all

⎛

⎞ ⎛ a11 a12 ⎜ a21 ⎟ ⎜ a22 ⎜ ⎟ ⎜ ⎝ ... ⎠ , ⎝ ... ah1 ah2

⎞

⎛

⎞ a1n ⎟ ⎜ ⎟ ⎟ , ..., ⎜ a2n ⎟ ∈ . ⎠ ⎝ ... ⎠ ahn

The empty set ∅ is preserved by every function f ∈ PA . Note that f preserves iff is the universe of a subalgebra (A; f )h . By P olk or, briefly, P ol we denote the set of all functions f ∈ Pk that preserve the relation . For Q ⊆ Rk , we put P olk . P olk Q := ∈Q

P olk or P olk Q is a short-cut of polymorphisms of or Q, respectively. The set of all relations ∈ Rk that are preserved from the function f ∈ Pk is Invk f. For A ⊆ Pk let

2.6 The Preserving of Relations; Pol, Inv

Invk A :=

131

Invk f

f ∈A

be the set of all invariants of A and let (Invk A)n := (Invk A) ∩ Rkh be the set of all n-ary invariants of A. If k can be seen from the context, we write Inv instead of Invk . Further notations used by us are P oln Q := (P ol Q)n (Q ⊆ Rk or Q ∈ Rk ) and Inv n A := (Inv A)n (A ⊆ Pk or A ∈ Pk ) for n ∈ N. Elementary connections between P ol and Inv are summarized in the following theorem. Theorem 2.6.1 For arbitrary A, B ⊆ Pk and arbitrary S, T ⊆ Rk , it holds: (a) A ⊆ B =⇒ Inv B ⊆ Inv A, S ⊆ T =⇒ P ol T ⊆ P ol S; (b) A ⊆ P ol Inv A, S ⊆ Inv P ol S; ((a) and (b) mean that the pair (P ol, Inv) of mappings P ol : P(Pk ) −→ P(Rk ), A → P olk A and P ol : P(Rk ) −→ P(Pk ), Q → Invk Q is a Galois connection (see Part I, 4.4) between the sets Pk and Rk .) (c) Inv P ol Inv A = Inv A, P ol Inv P ol S = P ol S; (d) A ⊆ P ol S ⇐⇒ S ⊆ Inv A; (e) P ol (S ∪ T ) = P ol S ∩ P ol T , Inv A ∪ B = Inv A ∩ Inv B. Proof. (a), (b), (d), and (e) are direct conclusions from the definitions of P ol and Inv. (c): Let S := Inv A. By (b) we have S ⊆ Inv P ol S. Conversely, it holds A ⊆ P ol Inv A and thus by (a): Inv P ol Inv A ⊆ Inv A. Therefore, Inv P ol Inv A = Inv A. Analogously, one can show that P ol Inv P ol S = P ol S holds.

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Theorem 2.6.2 For every A ⊆ Pk and every Q ⊆ Rk the sets Inv A and P ol Q are closed (in respect to the operations defined above, respectively); i.e., P ol Q is a clone of Pk and Inv A is a co-clone of Rk . Furthermore, it holds: Inv [A] = Inv A and P ol [Q] = P ol Q. In particular we have: Inv [{e21 }] = Rh and P ol Dk = Pk . Proof. Let , ∈ Inv A and f ∈ A be arbitrary. Then, f preserves the -derivable relations ζ, τ , pr, ∧ and × obviously. Thus {ζ, τ , pr, ∧ , × } ⊆ Inv A, whereby Inv A is closed. Now let f n , g m ∈ P ol Q and ∈ Q be arbitrary. Clear that, then the functions ζf, τ f and ∆f also preserve . Further, the relation is also an invariant of f g, because f (g(r1 , ..., rm ), rm+1 , ..., rm+n−1 ) ∈ holds for all r1 , ..., rm+n−1 ∈ , since g(r1 , .., rm ) ∈ . Inv [A] = Inv A follows from Inv [A] ⊆ Inv A (because of A ⊆ [A] and Theorem 2.6.1, (a)) and Inv A ⊆ Inv [A]. (By Theorem 2.6.1, (a) we have A ⊆ P ol Inv A indeed. Since P ol Inv A is closed, this implies [A] ⊆ P ol Inv A. If one uses Theorem 2.6.1, (a) again and then 2.6.1, (c), one receives Inv P ol Inv A = Inv A ⊆ Inv [A].) Analogously, one can prove P ol [Q] = P ol Q. One can easily checks the remaining statements of the theorem. A conclusion of Theorem 2.6.1, (a) and of the definition of (see Section 2.4) is the following Theorem 2.6.3 For arbitrary relations , ∈ Rk it holds: =⇒ P ol ⊆ P ol .

2.7 The Relations χn and Gn For arbitrary n ∈ N and k ∈ N\{1} denote χk;n or – if k can be seen from the context – χn the k n -ary relation, whose rows are just all (x1 , ..., xn ) ∈ Ekn that are arranged (we say “lexicographical”) unambiguously according to the following regulation:

2.7 The Relations χn and Gn

133

The tuple (x1 , ..., xn ) is before the tuple (y1 , ..., yn ), if the integer x1 · k n−1 + x2 · k n−2 + ... + xn−1 · k + xn is smaller than y1 · k n−1 + y2 · k n−2 + ... + yn−1 · k + yn . For example, the following is valid:

χ2;3

⎛

0 ⎜0 ⎜ ⎜0 ⎜ ⎜0 := ⎜ ⎜1 ⎜ ⎜1 ⎜ ⎝1 1

0 0 1 1 0 0 1 1

⎞ 0 1⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟. 0⎟ ⎟ 1⎟ ⎟ 0⎠ 1

We denote the columns of χn with χ(1), ..., χ(n). Obviously, there is exactly a function fr ∈ Pkn with f (χn ) = r to every column n r ∈ Ekk . The relation n Gn (A) := {r ∈ Ekk |fr ∈ An }, which one can form with the aid of the functions fr , is called n-th graphic of A ⊆ Pk . The following theorem summarizes elementary properties of the relation Gn (A). Theorem 2.7.1 For an arbitrary clone A ⊆ Pk it holds: (a) ∀n ∈ N : Gn (A) ∈ Inv A; (b) f n ∈ An ⇐⇒ f n ∈ P ol Gn (A); (c) A ⊆ ... ⊆ P ol Gn (A) ⊆ P ol Gn−1 (A) ⊆ ... ⊆ P ol G2 (A) ⊆ P ol G1 (A); (d) A = n≥1 P ol Gn (A); (e) ∀ ∈ Inv A : ∈ [ n≥1 {Gn (A)}]; (f ) Inv A = [ n≥1 {Gn (A)}]. Proof. (a): Let g m ∈ A and r1 , ..., rm ∈ Gn (A) be arbitrary. Then g(r1 , ..., rm ) = g(fr1 , ..., frm ) = h(χn ), where

134


h(x1 , ..., xn ) := g(fr1 (x1 , ..., xn ), ..., frm (x1 , ..., xn )). Since h belongs to An , we obtain h(χn ) ∈ Gn (A). (b): If f n ∈ A, we have f n ∈ P ol for every ∈ Inv A. Consequently, f n ∈ P ol Gn (A) because of (a). On the other hand, f n ∈ Gn (A) implies the existence of a certain r ∈ Gn (A) with f n (χn ) = r; thus f = fr ∈ An is valid. (c) and (d) follow from (b) and from the clone properties. (e): Let ∈ Inv h A be arbitrary and t := ||. We show that can be derived from Gt (A) using the operation prα1 ,...,αh . Since Jk ⊆ A, we have χt ⊆ Gt (A). For every j ∈ {1, ..., h}, one can find an αj so that the j-th row of is identical with the αj -th row of χt . Because of ∈ Inv h A, it follows prα1 ,...,αh Gt (A) = ∈ [{Gt (A)}]. (f): By (e) we have Inv A ⊆ [ n≥1 {Gn (A)}]. Further, because of (a) and Theorem 2.6.2, we have [ n≥1 {Gn (A)}] ⊆ Inv A. Hence, (f) is valid.

2.8 The Operator ΓA For arbitrary A ⊆ Pk denote ΓA a mapping from Rk into Rk , which is defined for σ ∈ Rkh as follows: (2.1) ΓA (σ) := { ∈ Rk | ∈ Inv A ∧ σ ⊆ }. In the language of the Universal Algebra, ΓA (σ) is a subalgebra of (Ek ; A)h , which is generated by σ, or ΓA (σ) is the universe of the smallest subalgebra of the algebra (Ek ; A)h , which contains the set σ. Obviously, ΓA is a hull operator. Theorem 2.8.1 For an arbitrary clone A ⊆ Pk and every n ∈ N it holds: (a) ΓA (χn ) ∈ Inv A; (b) ΓA (χn ) = Gn (A); (c) An = {fr |r ∈ ΓA (χn )}. Proof. (a) follows from [Inv A] = Inv A (see Theorem 2.6.2) and the definition of ΓA (χn ). (b): Since every projection eni belongs to An , we have χn ⊆ Gn (A). Denote now an arbitrary k n -ary relation of Rk with χn ⊆ . If ∈ Inv A, then f (χn ) ∈ for every f ∈ An , i.e., Gn (A) ⊆ . Consequently, we have shown that Gn (A) ⊆ ΓA (χn ) holds. ΓA (χn ) ⊆ Gn (A) follows from Gn (A) ∈ Inv A (see Theorem 2.7.1, (a)). (c) is an easy conclusion from (b) and the definition of Gn (A).

2.9 The Galois Theory for Function- and Relation-Algebras

135

2.9 The Galois Theory for Function- and Relation-Algebras Theorem 2.9.1 Let A be a clone of Pk . Then A = P ol Inv A. Proof. By Theorem 2.6.1, (b) we have A ⊆ P ol Inv A. To prove that P ol Inv A ⊆ A let f n ∈ P ol Inv A be arbitrary. Because of Theorem 2.7.1, (a) we have that f ∈ P ol Gn (A) holds and (by Theorem 2.7.1, (b)) f ∈ An . Thus A = P ol Inv A. Theorem 2.9.2 Let Q be a co-clone of Rk . Then Q = Inv P ol Q. Proof. Let A := P ol Q. Because of Theorem 2.6.1, (b) we have Q ⊆ Inv A. To prove that Inv A ⊆ Q it is sufficient to show that Γ (χt ) ∈ Q for arbitrary t ∈ N, since [ t≥1 {ΓA (χt )}] = Inv A (see Theorem 2.7.1, (f) and Theorem 2.8.1, (b)). Let now γ := { ∈ Q|χt ⊆ }. t

Because of Ekk ∈ Q and the fact that Q is closed in respect to ∩, we have t χt ⊆ γ and γ is the smallest relation (∈ Qk ), which contains χt , in respect to cardinality. Further, we have ΓA (χt ) ⊆ γ, since γ ∈ Q ⊆ Inv A (see (2.1)). Consequently, our theorem is proven, if we can show ΓA (χt ) = γ. Suppose, ΓA (χt ) ⊂ γ. Then, there is a column r ∈ γ\ΓA (χt ). Because of At = {fs |s ∈ ΓA (χt )} (see Theorem 2.8.1, (c)) we have fr ∈ At . Consequently, there exists an m-ary relation β ∈ Inv A and certain columns r1 , ..., rm ∈ β with f (r1 , ..., rm ) ∈ β. Every row of the matrix (r1 , ..., rm ) is also a row of the matrix χt . Denote ij the number of a row of χt , which agrees with the j-th row of (r1 , ..., rm ) (j = 1, 2, ..., m). Let now k +m γ := pr1,2,...,kt (γ × β) ∩ δ{i . t t t 1 ,k +1},{i2 ,k +2},...,{im ,k +m} t

Since Q is closed, γ belongs to Q, and by construction of γ we have χt ⊆ γ ⊆ γ. Furthermore, we have r ∈ γ\γ , since r1 , ...., rt ∈ β, fr (r1 , ..., rt ) ∈ β and fr (χt ) = r ∈ γ. With γ we received a contradiction to the choice of γ. Thus, γ = ΓA (χt ). With the aid of Theorems 2.9.1 and 2.9.2, we can prove the important properties of the P ol-Inv-connection:

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Theorem 2.9.3 (Theorem of V. G. Bodnarˇ cuk, L. A. Kaluˇ znin, V. N. Kotov and B. A. Romov; [Bod-K-K-R 68/69]) indextheorem of Bodnarˇcuk, Kaluˇznin,. Kotov and Romov Let L(Pk ) (or L(Rk )) be the set of all clones (or co-clones) of Pk (or Rk ) respectively. Then the mappings Inv : L(Pk ) −→ L(Rk ), A → Inv A and P ol : L(Rk ) −→ L(Pk ), Q → P ol Q are bijective mappings, which reverse the partial order ⊆, i.e., it holds ∀A, B ∈ L(Pk ) : A ⊆ B =⇒ Inv B ⊆ Inv A and ∀S, T ∈ L(Rk ) : S ⊆ T =⇒ P ol T ⊆ P ol S. In other words: The lattices (L(Pk ), ⊆) and (L(Rk ), ⊆) are antiisomorphic. Pk

Rk

@ @ @

@ @

A = P ol @ Inv A

XX

Inv B

@

Inv A

S = Inv P ol S

@

: XXX X X XX X z X P ol T

B

X y XX

XXX XXXX 9 X

P ol S

@

@ @ @

T

@

@

@ @

Jk

Dk

Fig. 2.1

Proof. By Theorem 2.6.2, the mappings Inv and P ol are mappings from L(Pk ) (or L(Rk )) into L(Rk ) (or L(Pk )), respectively. The surjectivity and the injectivity (and then the bijectivity) of these mappings are easy conclusions from Theorem 2.9.1 and Theorem 2.9.2 (with the help of Theorem 2.6.2). The “reversal property” of P ol and Inv (in respect to ⊆) was already given in Theorem 2.6.1, (a).

2.10 Some Modifications of the P ol-Inv-Connection

137

2.10 Some Modifications of the P ol-Inv-Connection 2.10.1 Galois Theory for Finite Monoids and Finite Groups It is well-known that every finite semigroup H = (H; ◦) is isomorphic to a 1 certain subsemigroup of (P|H| ; ) and every finite group G = (G; ◦) is isomorphic to a certain subgroup of the group (S|G| ; ), where Sk := Pk1 [k] for k ∈ N. 2

Furthermore we have Lemma 2.10.1.1 For an arbitrary subset A of [Pk1 ], the set A is a clone of [Pk1 ] if and only if A = [A]∇ holds and (A1 ; ) is a subsemigroup of (Pk1 ; ) with the unit element e. In other words, a clone A ⊆ [Pk1 ] is completely determined by the monoid (A1 ; ). Because of the above properties, it is possible to derive a Galois theory for semigroups and groups from the Galois theory for function algebras (see Section 2.9). For this purpose, we define two new operations ∨ and ¬ on Rk : For arbitrary , ∈ Rkh and h ∈ N we set ∨ := ∪ (union) and ¬ := Ekh \ (negation, complement). The following lemma supplies a reason for these new relation operations. Lemma 2.10.1.2 It holds: (a) ∀H ⊆ Pk1 : , ∈ Inv h H =⇒ ∨ ∈ Inv h H; (b) ∀G ⊆ Sk : ∈ Inv h G =⇒ ¬ ∈ Inv h G. Proof. (a): If f ∈ H ⊆ Pk1 , {, } ⊆ Inv H and r ∈ ∨ , then we have obvious r ∈ or r ∈ , whereby f (r) ∈ ∨ , since f preserves the relations and . Consequently, ∨ is also an invariant of f . (b): Let f ∈ G ⊆ Sk , ∈ Inv G and r ∈ ¬ be arbitrary. Set r := f (r). Then r belongs to ¬, since f −1 (r ) = r, f −1 ∈ [G] and Inv G = Inv [G] (see Theorem 2.6.2). Consequently, f preserves the relation ¬. An algebra of the form 3 (Q; δk;{1,2} , ζ, τ, pr, ×, ∧, ∨) 2

For proof, one can assume w.l.o.g. H = G = {0, 1, ..., k − 1}. Let now fa (x) := a ◦ x ∈ Pk1 for arbitrary a ∈ Ek . Then, the mapping α : Ek −→ Pk1 , a → fa is an isomorphic mapping from H (or G) onto the semigroup (or group) (A)

({f0 , f1 , ..., fk−1 }; ), since α(a ◦ b)(x) = fa◦b (x) = (a ◦ b) ◦ x = a ◦ (b ◦ x) = fa (fb (x)) = (fa fb )(x).

138


with Q ⊆ Rk is called Krasner-algebra of first kind and an algebra of the form 3 , ζ, τ, pr, ×, ∧, ∨, ¬) (Q; δk;{1,2} with Q ⊆ Rk Krasner-algebra of second kind. In Theorem 2.10.1.4, we will see that the algebras just defined are the right partners for semigroups (or groups) for a Galois-correspondence. Following our terminology from Section 2.4, we call this a clone, which is also closed, concerning the operation ∨, a co-monoid. Moreover, a co-monoid, which is closed concerning the operation ¬, is a co-group. The following lemma summarizes all auxiliary statements that are necessary to prove Theorem 2.10.1.4. Lemma 2.10.1.3 Let H, G ⊆ Pk1 and Q, T ⊆ Rk . Further, let H := (H; ) be a monoid, G := (G; ) be a group, Q be a co-monoid, and T be a co-group. Then (a)

H = P ol1 Inv H

(a’)

G = P ol1 Inv G

(b)

P ol Q ⊆ [Pk1 ]

(b’)

P ol T ⊆ [Sk ]

(c)

Q = Inv P ol1 Q

(c’)

T = Inv P ol1 T .

Proof. (a), (a’): Let A be a monoid of Pk1 (or a group of Sk ). Then, by Lemma 2.10.1.1, [A]∇ is a clone of Pk and, because of Theorem 2.6.2, we have [A]∇ = P ol Inv [A]∇ = P ol Inv A. Consequently, A = [A]1 = P ol1 Inv A. (b): Since the diagonal relations belong to Q and since Q is closed in respect to ∨, the relation 3 3 ∪ δk;{2,3} γ := δk;{1,2} belongs to Q. With the help of Theorem 1.4.4, it can easily be shown that the relation is preserved of no function of Pk , which depends on at least two variables essentially. Thus, P ol Q ⊆ [Pk1 ]. (b’): Through (b) we already proven P ol T ⊆ [Pk1 ]. Further, the diagonal relation ι2k := {(x, x)|x ∈ Ek } belongs to Q, whereby also ¬ι2k ∈ Q. Obviously the relation ¬ι2k = {(x, y)|x = y} is preserved, however, only from the permutations of Pk1 . Consequently, we have P ol T ⊆ [Sk ]. (c), (c’): If Q is a co-monoid (or is a co-group) of Rk , then Q is also a co-clone of Rk and it is valid (because of P ol1 Q ⊆ P ol Q, Theorem 2.6.1, (a) and Theorem 2.9.2): Q = Inv P ol Q ⊆ Inv P ol1 Q. By (b) (or (b’)) we have in addition P ol Q ⊆ [P ol1 Q], whereby Inv [P ol1 Q] ⊆ Inv P ol Q and (because of Inv P ol1 Q = Inv [P ol1 Q] by Theorem 2.6.2 and Inv P ol Q = Q by Theorem 2.9.2) Inv P ol1 Q ⊆ Q follow. One obtains the following theorem as an easy conclusion in analogy to the above theorem from Lemmas 2.10.1.2 and 2.10.1.3.


139

Theorem 2.10.1.4 ([Kra 45], [Kra 68/69]) Let M be the set of all monoids of the form (M ; ) with M ⊆ Pk1 and G be the set of all groups of the form (G; ) with G ⊆ Sk . Further, let K1 be the set of all co-monoids of Rk and K2 be the set of all co-groups of Rk . Then, the lattices (M; ⊆) and (K1 ; ⊆) and the lattices (G; ⊆) and (K2 ; ⊆) are antiisomorphic. In particular, for i ∈ {1, 2}, L1 := M and L2 := G are valid: The mappings P ol1 : Ki −→ Li , Q → P ol1 Q and Inv : Li −→ Ki , H → Inv H are bijective mappings, which reverse the partial order ⊆, i.e., it holds ∀H, G ⊆ Li : H ⊆ G =⇒ Inv G ⊆ Inv H and

∀S, T ⊆ Ki : S ⊆ T =⇒ P ol1 T ⊆ P ol1 S.

2.10.2 Galois Theory for Iterative Function Algebras Iterative function algebras are algebras of the form (Pk ; ζ, τ, ∆, ∇; ). The universes of subalgebras of these algebras Pk (k ∈ N\{1}) are called subclasses or, briefly, classes of Pk . Unlike clones of Pk , there are classes of Pk that do not contain the projections. Nevertheless, for the classes of Pk , one can also find certain relation algebras as partners for a Galois-correspondence. Subsequently, two possibilities for this purpose will be presented. For the first possibility, we choose a certain class A of Pk that does not contain the function e11 and which, therefore, is not a clone. Let L↓k (A) be the lattice of all subclasses of A and let Lk (Jk , Jk ∪ A) be the lattice of all subclasses of Jk ∪ A, which have Jk as a subset. Moreover, let Lk (Inv(Jk ∪ A), Rk ) be the set of all co-clones C of Rk with Inv(Jk ∪ A) ⊆ C. Obviously, the mapping α : L↓k (A) −→ Lk (Jk , Jk ∪ A), T → Jk ∪ T is an isomorphism from L↓k onto Lk (Jk , Jk ∪ A). Since the elements of Lk (Jk , Jk ∪ A) are clones of Pk , the lattice Lk (Jk , Jk ∪ A) is antiisomorphic to the lattice Lk (Inv(Jk ∪ A), Rk ) by Theorem 2.9.3. Consequently, we have

140


Theorem 2.10.2.1 Let A be a subclass of Pk . Further, put P olA Q := A ∩ P olk Q for arbitrary Q ⊆ Rk . Then the mappings Inv : L↓k (A) −→ Lk (Inv(Jk ∪ A)), T → Inv T, and

P olA : Lk (Inv(Jk ∪ A), Rk ) −→ L↓k (A), Q → P olA Q

are bijective mappings, and the lattices L↓k (A) and Lk (Inv(Jk ∪ A), Rk ) are antiisomorphic. As W. Harnau in [Har 83] was pointing, another possibility to characterize subclasses of Pk consists that one establishes relation pairs and relation-pairalgebras. Let Rphk := {(, ) ∈ Rkh × Rkh | ⊆ } and Rpk :=

Rphk .

h≥1

The elements of Rpk are called relation pairs. We say that f n ∈ Pk preserves the relation pair (, ) ∈ Rpk , if f (r1 , ..., rn ) ∈ for arbitrary r1 , ..., rn ∈ . By definition, each function of Pk preserves the pair (∅, ∅). If = , then the fact “f preserves the relation pair (, )” is identical with the fact “f preserves ”. Let F ⊆ Pk and Q ⊆ Rpk . Then, denote Invpk F the set of all relation pairs of Rpk , which are preserved from every function of F . Further, let P olpk Q be the set of all functions of Pk , which preserve each relation pair of Q. It can easily be shown that the following lemma holds: Lemma 2.10.2.2 ([Har 83]) (a) ∀Q ⊆ Rpk : P olpk Q = [P olpk Q]ζ,τ,∆,∇, ; (b) ∀(, ) ∈ Rpk : ([P ol(, )]e21 ,ζ,τ,∆, = P ol(, ) ⇐⇒ = ). Analogous to Section 2.3, one can define certain operations on the set Rpk : h+1 (“the doubling For α ∈ {ζ, τ, ∆, pr, ∼} (see Section 2.3, ∼ h := ∇ ∧ δ{1,2} of the first row of ”)) and (, ), (µ, µ ) ∈ Rpk we set: α(, ) := (α, α ), (∇, ∇ ), if = ∅, ∇(, ) := (∇, ∅), if = ∅, and


141

(, ) × (µ, µ ) := ( × µ, × µ ).

Further, put E := h≥1 Ekh in the following. For each a ∈ E one can define two operations on Rpk as follows: (\{a}, ), if ⊆ \{a}, ν1,a (, ) := otherwise, (, ) and ν2,a (, ) :=

(, ∪ {a}), if a ∈ , otherwise (, )

((, ) ∈ Rpk ). 3 The importance of the above-defined operations results from the following lemma, which one can easily check. Lemma 2.10.2.3 Let (, ) be a relation pair which is derivable from Q (⊆ Rpk ) by means of the operations ζ, τ, ∆, ∇, ∼, pr, ×, ν1,a , ν2,a (a ∈ E). Then P olpk Q ⊆ P olpk (, ). The algebra Rpk := (Rpk ; (∅, ∅), ζ, τ, ∆, pr, ∼, ×, (ν1,a )a∈E , (ν2,a )a∈E ) of the type (0,1,1,1,1,1,2,1,1,1,...) is called full relation-pair algebra on Ek . A relation-pair algebras is a subalgebra of this algebra. Further, a universe of a relation-pair algebra is called co-class. Then, the following theorem can be proven with some expenditure (ultimately similar to the proof of Theorem 2.9.3):

Theorem 2.10.2.4 ([Har 83]) Let L(Pk ) (or L(Rpk )) be the set of all classes (or co-classes) of Pk (or Rpk ), respectively. Then the mappings Invk : L(Pk ) −→ L(Rpk ), A → Invk A and P olpk : L(Rpk ) −→ L(Pk ), Q → P olpk Q 3

In [Har 83] so-called multi-operators dh and dv are first defined instead of the operations ν1,a , ν2,a (a ∈ E): dh (, ) := {(, µ) ∈ Rpk | ⊆ µ ⊆ }, dv (, ) := {(µ, ) ∈ Rpk | ⊆ µ ⊆ }. Then, relation-matrix-algebras, in which these multi-operators are describable through two operations, are introduced. It is shown how one can repair the “lack” with the multi-operators by going over to the relation-matrix-algebras with finite many operations.

142


are bijective mappings with the properties ∀A, B ∈ L(Pk ) : A ⊆ B =⇒ Invk B ⊆ Invk A, ∀S, T ∈ L(Rpk ) : S ⊆ T =⇒ P olpk T ⊆ P olpk S; i.e., the lattices (L(Pk ); ⊆) and (L(Rpk ); ⊆) are antiisomorphic.

2.11 Some Connections Between the Relation Operations In this last section of Chapter 2, we want to clarify the order in which one can use the relation operations given in Section 2.3, to obtain all relations from the set [Q] (⊆ Rk ) as effectively as possible. Let ∆ := {(x1 , x2 , ..., xh ) ∈ | x1 = x2 } ( ∈ Rkh ). This relation is obvious derivable from our elementary operations. For arbitrary α, α1 , ..., αt ∈ {ζ, τ, ∆, ∆ , ∇, pr} we use the following notations: α1 := α(), αi := α(αi−1 ) for i ∈ N, α1 α2 ...αt := α1 (...(αt−1 (αt ())...) ( ∈ Rk ). It can easily be shown that the Lemma 2.11.1 holds.

Lemma 2.11.1 For arbitrary relations ∈ Rkh , ∈ Rkh and ∈ Rkh it is valid:

(a) × = (ζ h ∆h ) ∩ (∇h ); 3 (b) ∅ = prh , δk;{1,2} = ∆ ∇3 ∅;

(c) ζ(pr) = pr(ζ(τ ())), τ (pr) = pr(ζ h−1 (τ (ζ()))), ∆ (pr) = pr(ζ h−1 (∆ (ζ()))), ∇(pr) = pr(τ ((∇()))); (d) ζ( ∧ ) = (ζ) ∧ (ζ ), τ ( ∧ ) = (τ ) ∧ (τ ), ∆ ( ∧ ) = (∇) ∧ (∇ ), if h = h ; (e) ( ∨ ) ∧ = ( ∧ ) ∨ ( ∧ ), if h = h = h .

2.11 Some Connections Between the Relation Operations

143

3 Theorem 2.11.2 Let Ω := {δk;{1,2} , ζ, τ, pr, ×, ∧}. Then for all Q ⊆ Rk it is valid:

(a) [Q]Ω = [Q]ζ,τ,∆ ,∇,pr,∧ ; (b) [Q]ζ,τ,∆ ,∇,pr,∧ = [ [[Q]ζ,τ,∆ ,∇ ]∧ ]pr ; (c) [Q]Ω∪{∨} = [Q]ζ,τ,∆ ,∇,pr,∧,∨ ; (d) [Q]ζ,τ,∆ ,∇,pr,∧,∨ = [ [ [ [Q]ζ,τ,∆ ,∇ ]∧ ]∨ ]pr . Proof. (a): Since ∆ and ∇ are Ω-derivable operations, we have [Q]ζ,τ,∆ ,∇,pr,∧ ⊆ [Q]Ω . The reversed inclusion [Q]Ω ⊆ [Q]ζ,τ,∆ ,∇,pr,∧ follows from Lemma 2.11.1, (a), (b). (b) follows from Lemma 2.11.1, (c), (d). One proves the statements (c) and (d) analogously to (a) and (b) using Lemma 2.11.1, (e). A translation of these observations into the language of mathematical logic can be found in [P¨ os-K 79], p. 63–68.

3 The Subclasses of P2

A basic result of many-valued logic is the description of all closed sets of Boolean functions given by E. L. Post in [Pos 20] and [Pos 41]. Since Post’s proof is long and rather complicated, revisions (for instance [Jab-G-K 70] and [Ugo 88]) and new proofs (for instance [Ber 80], [McK-M-T 87] and [Res-D 89] or [Den-W 2002]) have been published. The new proof methods of the last years mainly result from the fact that parts of Post’ results are special cases or conclusions of certain theorems of many-valued logic or universal algebra. In this chapter, we tried to verify Post’s results in an elementary way by working out some essential basic ideas.

3.1 Definitions of the Subclasses of P2 and Post’s Theorem We need some notations introduced in Chapter 1 to define the subclasses of P2 and to describe certain generated sets of this subclasses. We shall define functions of P2 by formulae over the alphabet {x, y, z, x1 , x2 , ...} and we use the usual symbols ∧ (“conjunction” or “multiplication modulo 2”), ∨ (“disjunction”), + (“addition modulo 2”) and − (“negation”). By ◦ ∈ {∧, ∨, +},

−

, cna (a ∈ E2 ), eni (1 ≤ i ≤ n), m3 , t2 , q 3 , r3 , hµ+1 (µ ∈ N) µ

we denote functions of P2 given by

146


◦(x, y) := x ◦ y, −

(x) := x,

cna (x1 , ..., xn ) := a, eni (x1 , ..., xn ) := xi , m(x, y, z) := x ∧ (y ∨ z), t(x, y) := x ∧ y, q(x, y, z) := x ∧ (y ∨ z), r(x, y, z) := x + y + z, µ+1 hµ (x1 , ..., xµ+1 ) := i=1 (x1 ∧ x2 ∧ ... ∧ xi−1 ∧ xi+1 ∧ ... ∧ xµ+1 ) = 1 if ∃i ∈ {1, ..., µ + 1} : x1 = ... = xi−1 = xi+1 = ... = xµ+1 = 1, 0 otherwise (h1 = ∨), respectively. We write x for (x1 , ..., xn ), α for (α, α, ..., α) (α ∈ E2 ) and often xy instead of x ∧ y. Finally, let x, if σ = 0, xσ := x, if σ = 1. The mapping δ : P2 −→ P2 , f n −→ (f δ )n with f δ (x1 , ..., xn ) := f (x1 , x2 , ..., xn ) is known to be an automorphism from P2 , which we shall use to describe isomorphic closed subsets of P2 (see Section 9.11), where Aδ := {f δ | f ∈ A}. We remark that δ is the unique non-trivial isomorphism from a subalgebra of P2 to a subalgebra of P2 (see Theorem 9.12.5). In the following, we define some closed subsets of P2 with the help of which we can describe all subclasses of P2 with the applications of ∩ and ∪: 0 0 1 • M := P ol 0 1 1 = n≥1 {f n ∈ P2 | ∀a, b ∈ E2n : a ≤ b =⇒ f (a) ≤ f (b)} (set of all non-decreasing monotone functions), • S := P ol =

0 1 1 0

n≥1 {f

n

∈ P2 | f (x1 , ..., xn ) = f (x1 , x2 , ..., xn )}

(set of all self-dual functions),

3.1 Definitions of the Subclasses of P2 and Post’s Theorem

• L :=

n≥1 {f

n

147

n ∈ P2 | ∃a0 , ..., an ∈ E2 : f (x) = a0 + Σn=1 ai · xi }

(set of all linear functions), • T0,µ := P olE2µ \{1} if µ ∈ N ( f n ∈ T0,µ ⇐⇒ (∀a1 , ..., aµ ∈ E2µ : (∀i ∈ {1, ..., µ} : f (ai ) = 1 and ai = (ai1 , ..., ain )) =⇒ ∃ j ∈ {1, ..., n} : a1j = a2j = ... = aµj = 1) ⇐⇒ (∀a1 , ..., aµ ∈ E2n ∃ j ∈ {1, ..., n} : ∀x ∈ {a1 , ..., aµ } : f (x) = xj ∧ f (x))), δ T1,µ := T0,µ = P olE2µ \{0},

Ta := Ta,1 , where a ∈ E2 , T0,∞ := µ≥1 T0,µ =

n≥1 {f

n

∈ P2 | ∃j ∈ {1, ..., n}∃f ∈ P2 : f (x) = xj ∧ f (x)},

δ , T1,∞ := T0,∞

• K := [∧] (set of all conjunctions), • D := K δ = [∨] (set of all disjunctions), • C := [c0 , c1 ] (set of all constant functions), • Ca := [ca ], a ∈ E2 , • I := [e11 ] (set of all projections), • I := [− ].

148


Theorem 3.1.1 (Post’s Theorem; [Pos 41]) (1) The set of all subclasses of P2 is countably infinite. (2) The non-empty subclasses of P2 are P2 , S, M, L, Ta,µ , Ta,µ ∩ Ta , Ta,µ ∩ M, Ta,µ ∩ M ∩ Ta , K ∪ C, K ∪ Ca , K, D ∪ C, D ∪ Ca , D, S ∩ T0 , S ∩ M, S ∩ L, S ∩ L ∩ T0 , L ∩ Ta , I ∪ C, I ∪ C, I, I ∪ Ca , I, C, Ca , where a ∈ E2 and µ ∈ {1, 2, ..., ∞}. (The Hasse-diagram of these classes is given in Figure 3.1.) (3) In the set P2 , there exists exactly (a) 9 closed subsets of order 1: [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , I, C, C0 , C1 ; (b) 20 closed subsets of order 2: P2 , T0 , T1 , M, L, M ∩ T0 , M ∩ T1 , L ∩ T0 , L ∩ T1 , M ∩ T0 ∩ T1 , K ∪ C, K ∪ C0 , K ∪ C1 , K, D ∪ C, D ∪ C0 , D ∪ C1 , D, T0,∞ , T1,∞ ; (c) 20 closed subsets of order 3: S, S ∩ T0 , S ∩ M, S ∩ L, S ∩ L ∩ T0 , T0,2 , T1,2 , T0,2 ∩ T1 , T1,2 ∩ T0 , T0,2 ∩ M, T1,2 ∩ M, T0,2 ∩ M ∩ T1 , T1,2 ∩ T0 ∩ M, T0,∞ ∩ T1 , T1,∞ ∩ T0 , T0,∞ ∩ M, T1,∞ ∩ M, T0,∞ ∩ M ∩ T1 , T1,∞ ∩ M ∩ T0 , T0 ∩ T1 ; (d) 8 closed subsets of order µ + 1 (µ ≥ 3): Ta,µ , Ta,µ ∩ Ta , Ta,µ ∩ M, Ta,µ ∩ M ∩ Ta (a ∈ E2 ).

3.2 A Proof for Post’s Theorem

T0 r T0,2 r

r T0,3 r

pp pp pp T0,∞ pr

r pp pp pp pr

Pr2

r

r

r

r

rM T r1 r r r

r r pp pp pp pr

r

r pp pp pp pr

r

r r

S∩M

r r

K ∪C

r

rS

r

r

r

K

r C0

r

r

r

rL

r

r [P21 ] r

rI

I r C

r

pp pp pp pr

r r D∪C r r

pp pp pp pr

149

rT1,2 r rT1,3 r pp pp pp pr

pp pp pp prT1,∞

r

D

r C1

∅

Fig. 3.1. The Post Lattice

3.2 A Proof for Post’s Theorem For any subclass A of P2 , the following three cases are possible: Case 1: A ⊆ L and A ⊆ S, Case 2: A ⊆ L, Case 3: A ⊆ L and A ⊆ S. By following this case distinction, all subclasses of P2 and minimal generating subsets of these classes are determined as follows. The orders of the subclasses given are easy conclusions from the proven statements about generating sets of the subclasses. We start with: 3.2.1 The Subclasses A of P2 with A ⊆ L and A ⊆ S Theorem 3.2.1.1 The following holds: P2 = [◦,− ] for ◦ ∈ {∧, ∨}, M = [∨, ∧, c0 , c1 ], T0,µ = [hµ , t], T0,µ ∩T1 = [hµ , q], T0,µ ∩ M = [hµ , m, c0 ] ( T0 ∩ M = [∨, ∧, c0 ] ), T0,µ ∩ M ∩ T1 = [hµ , m], T0,∞ = [t], T0,∞ ∩ T1 = [q], T0,∞ ∩ M = [m, c0 ] and T0,∞ ∩ M ∩ T1 = [m], where µ ∈ N.

150


Proof. Let be f n ∈ P2 , for which µ + 1 distinct tuples a1 , ..., aµ+1 exist with f (a1 ) = ... = f (aµ+1 ) = 1. Then, we have f (x) = hµ (fa1 (x), fa2 (x), ..., faµ+1 (x)),

where fai (x) :=

(3.1)

0 if x = ai , f (x) otherwise

(i = 1, 2, ..., µ + 1). We call every function f n of a subclass A of P2 with {hµ , fa1 , ..., faµ+1 } ⊆ A and f (a1 ) = ... = f (aµ+1 ) = 1 for certain a1 , ..., aµ+1 ∈ E2n a reducible function. We denote by NA the set of all not reducible functions f of a class A. Obviously, the set NA ∪ {hµ } is a generating set for the class A, if hµ ∈ A and A ∩ {h1 , ..., hµ−1 } = ∅ for certain µ ≥ 1. The Table 3.1 gives an easily verifiable description of the set (NA )n for A ∈ {P2 , T0,m , M, T0,m ∩ M, T0,m ∩ T1 , T0,m ∩ M ∩ T1 }, m ∈ N, and the minimal µ with hµ ∈ A. 1 The functions gJ and mJ for J ⊆ E2n from Table 3.1 are defined by 1 if x ∈ J, gJ (x) := 0 otherwise

and mJ (x) :=

1 if ∃a ∈ J : x ≥ a, 0 otherwise. Table 3.1

A

NAn

P2

{gJ | |J| ≤ 1}

1

M

{mJ | |J| ≤ 1}

1

T0,µ

{gJ | |J| ≤ µ and ∃t : gJ (x) = xt ∧ gJ (x)}

µ

T0,µ ∩ T1

{gJ ∈ NTn0,µ | 1 ∈ J}

µ

T0,µ ∩ M

{mJ | |J| ≤ µ and ∃t : mJ (x) = xt ∧ mJ (x)}

µ

T0,µ ∩ M ∩ T1 {mJ ∈ NTn0,µ ∩M | 1 ∈ J} 1

a minimal µ with hµ ∈ A

µ

One possible proof of Table 3.1 is as follows: We start with a ”partition” of an arbitrary f ∈ A in functions fai by (3.1), then we repeat this construction for the functions fai instead of f , if fai ∈ NA , etc. In case A ⊆ M be let all tuples ai minimal with respect to ≤ by this fai ∈ M .


151

Since g∅n = mn∅ = cn0 , n g{(a (x) = xσ1 1 ∧ xσ2 2 ∧ ... ∧ xσnn , 1 ,...,an )}

mn{0} = cn1 , mn{(a1 ,...,an )} (x) = xi1 ∧ xi2 ∧ ... ∧ xiν if {i1 , ..., iν } = {i | ai = 1} = ∅, we have

NP2 ⊆ [∧,− ],

NT0 ⊆ [c0 , ∧, t], = [t],

NM ⊆ [∧, c0 , c1 ], NM ∩T0 ⊆ [∧, c0 ] and NM ∩T0 ∩T1 ⊆ [∧]. Consequently, P2 = [∨, ∧,− ] (and P2 = [◦,− ] for ◦ ∈ {∨, ∧} by Morgan’s laws), T0 = [∨, t], M = [∨, ∧, c0 , c1 ], M ∩ T0 = [∨, ∧, c0 ] and M ∩ T0 ∩ T1 = [∨, ∧]. Furthermore, Table 3.1 implies the following: if A = T0,µ ∩ B with B ∈ T0,∞ ∩ B = {P 2 , T1 , nM, T1 ∩ M } and µ ≥ 2, then A = [{hnµ } ∪ (T0,∞ ∩ B)] and {f ∈ P | ∃i ∈ {1, 2, ..., n} ∃ f ∈ B : f (x) = x ∧ f (x)}. By this 2 i n≥1 fact, from a generating subset {f n , g m , ...} of B, we get a generating subset of the form {∧ f, ∧ g, ...} for the class T0,∞ ∩ B. Thus, T0,∞ = [t], since t(x, t(x, y)) = x ∧ y and P2 = [∧,− ]. T0,∞ ∩ T1 = [q] follows from T1 = T0δ = [∧, tδ ], tδ (x, y) = x ∨ y and ∧ ∧, ∧ tδ ∈ [q]. By T1 ∩ M = [∧, ∨, c1 ] and {∧ ∧, ∧ ∨, ∧ c11 = e22 } ⊆ [m] is T0,∞ ∩ T1 ∩ M = [m]. Finally, T0,∞ ∩ M = [m, c0 ], since T0,∞ ∩ M = (T0,∞ ∩ M ∩ T1 ) ∪ [c0 ]. Lemma 3.2.1.2 If A = [A] ⊆ P2 , ca ∈ A for certain a ∈ E2 and A ⊆ L, then A contains a binary non-linear function. Proof. By x◦y+x+y = x◦ y for {◦, ◦ } = {∧, ∨}, ∆(+c1 ) =− and Theorem 1.4.2 we have [◦, +, c1 ] = P2 for ◦ ∈ {∧, ∨}, i.e., every function g n ∈ P2 has a description of the form ai1 i2 ...iν ◦ xi1 ◦ xi2 ◦ ... ◦ xiν g(x) = a0 + {i1 ,i2 ,...,iν }⊆{1,2,...,n}

for some a0 , ai1 i2 ...iν ∈ E2 (see Theorem 1.4.3). Therefore, because A ⊆ L, there exists a function f n ∈ A with f (x) = a0 + x1 ◦ x2 ◦ ... ◦ xr +

n i=1

where r ≥ 2 and

ai ◦ xi +

ai1 ...iν ◦ xi1 ◦ ... ◦ xiν ,

i1 , ..., iν ∈ {1, 2, ..., n}, ν ≥ r, {i1 , ..., iν } ⊆ {1, 2, ..., r}

152


◦ :=

∨ if a = 1, ∧ if a = 0.

Our statement follows from f (x, y, y, ..., y , ca , ...., ca ) = b + x ◦ y + c ◦ x + d ◦ y ∈ A\L (r−1) times for some b, c, d ∈ E2 . Lemma 3.2.1.3 Let A be a subclass of P2 , which is not a subset of L and not a subset of S. Then, the function ∧ or the function ∨ belongs to A. Proof. It is easy to verify that ∨ or ∧ is a superposition over a binary nonlinear function of P2 . Thus, we have to show A2 ⊆ L. Since A ⊆ S, there exists an f ∈ A with 0 1 a f = , 1 0 a a ∈ E2 , i.e., f ∈ {f1 , f2 , ..., f8 } (see Table 3.2). Table 3.2

x 0 0 1 1

y f1 0 0 1 0 0 0 1 0

f2 1 1 1 1

f3 1 0 0 1

f4 0 1 1 0

f5 0 0 0 1

f6 1 0 0 0

f7 0 1 1 1

f8 1 1 1 0

The functions f5 , ..., f8 are non-linear. If f ∈ {f1 , ..., f4 } then ∆f is a constant function and A2 ⊆ L follows from Lemma 3.2.1.2. Lemma 3.2.1.4 Let A be a subclass of P2 with ∧ ∈ A. Then the following implications hold: (a) (∃a ∈ E2 : A ⊆ Ta ) =⇒ ca ∈ A, (b) (A ⊆ M and A ⊆ K ∪ C) =⇒ m ∈ A, (c) A ⊆ M =⇒ q ∈ A, (d) (A ⊆ K ∪ C and A ⊆ T0 ) =⇒ ∨ ∈ A, (e) (∃µ ∈ N : A ⊆ T0,µ and A ⊆ T0,µ+1 ) =⇒ hµ ∈ A. Proof. (a): If A ⊆ Ta there exists a g 1 ∈ A with g(a) = a, i.e., g ∈ {ca ,− }. ca ∈ A follows from ∧ ∈ A, x ∧ x = 0 and 0 = 1. (b): f denotes an n-ary function of A\(K ∪ C). Then there exist two tuple a = (a1 , ..., an ) and b = (b1 , ..., bn ) with the following properties: f (a) = f (b) = 1, a ≤ b, b ≤ a and f (c) = 0 for every c with c < a or c < b . Defining functions gi3 ∈ A by


⎧ ⎪ ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ⎨ gi (x, y, z) :=

if if

ai bi ai bi

=

=

1 1 0 1

153

, ,

ai 1 ⎪ ⎪ z if = , ⎪ ⎪ b 0 ⎪ i ⎪ ⎪ ⎪ ⎪ ⎪ ai 0 ⎪ ⎪ = ⎪ ⎩ yz if bi 0

(i = 1, 2, ..., n) we obtain x ∧ f (g1 (x, y, z), ..., gn (x, y, z)) = m(x, y, z) ∈ A, since f ∈ M . (c): By A ⊆ M we have a function h3 in A with 1 0 0 1 h = . 1 0 1 0 Consequently, h (x, y, z) := x ∧ h(x, y, z) ∈ A and h (x, y, z) ∈ {x ∧ y ∧ z, x(y+z+1), x∧z, x(y∨z)}. Since x∧(y∧z) = x(y∨z) and x(yz+z+1) = x(y∨z), it holds q ∈ A. (d): By (a), (b), (c) and m(x, y, z) = q(x, y, q(x, y, z)) it holds {c1 , m} ⊆ A and hence ∆(m c11 ) = ∨ ∈ A. (e): If A ⊆ T0,µ+1 , there is an f ∈ A with f (E2µ+1 \{1}) = 1, i.e., w.l.o.g.: ⎛ ⎞ ⎛ ⎞ 1 0 1 1 ... 1 0 0 0 ... 0 ...... 0 ⎜1⎟ ⎜ 1 0 1 ... 1 0 1 1 ... 0 ...... 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ f⎜ ⎜ 1 1 0 ... 1 1 0 1 ... 0 ...... 0 ⎟ = ⎜ 1 ⎟ . ⎝.⎠ ⎝ ................................ ⎠ 1 1 1 1 ... 0 1 1 1 ... 1 ...... 0 (µ+1) times Since ∧ ∈ A, we have f (x1 , ..., xµ+1 ) := f (x1 , ..., xµ+1 , x1 x2 , x1 x3 , ..., x1 x2 x3 , ..., x1 x2 ...xµ+1 ) ∈ A. By f ∈ T0,µ we have

f (x) =

hµ (x) if x = 1, a if x = 1

for a certain a ∈ E2 . If a = 1 then f = hµ ∈ A. If a = 0 and µ ≥ 2 then f (x1 , ..., xµ−1 , xµ xµ+1 , f (x1 , ..., xµ+1 )) = hµ (x) ∈ A. Finally, if a = 0 and µ = 1, we have f = + and h1 (x, y) = xy + x + z ∈ A.

154


Theorem 3.2.1.5 The subclasses A of P2 with A ⊆ S and A ⊆ L are P2 , M, Ta,µ , Ta,µ ∩ B, K, K ∪ Ca , K ∪ C, D, D ∪ Ca , D ∪ C, where a ∈ E2 , µ ∈ {∞, 1, 2, ...} and B ∈ {Ta , M, M ∩ Ta }. Proof. Denote A a subclass of P2 with A ⊆ S and A ⊆ L. By Lemma 3.2.1.3, we have that A obtains K or D. W.l.o.g. [∧] = K ⊆ A. If A ∈ {K, K ∪ C0 , K ∪ C1 , K ∪ C} then we can distinguish 8 cases, which are given in Table 3.3. Using Lemma 3.2.1.4 and Theorem 3.2.1.1, we see that A is a certain class, which is given in the last column of Table 3.3. Table 3.3

∃µ ∈ A ⊆ T1 A ⊆ M {∞, 1, 2, ...} : A ⊆ T0,µ ∧ A ⊆ T0,µ+1 − − − − − + − + − − + + + − − + − + + + − + + +

conclusions from the assumptions

A

{c1 , c0 , q} ⊆ A {c1 , c0 , m} ⊆ A {c1 , q} ⊆ A ⊆ T1 {c1 , m} ⊆ A ⊆ T1 ∩ M {hµ , c0 , q} ⊆ A ⊆ T0,µ {hµ , c0 , m} ⊆ A ⊆ T0,µ ∩ M {hµ , q} ⊆ A ⊆ T0,µ ∩ T1 {hµ , m} ⊆ A ⊆ T0,µ ∩ T1 ∩ M

P2 M T1 T1 ∩ M T0,µ T0,µ ∩ M T0,µ ∩ T1 T0,µ ∩ T1 ∩ M

(+ stands for the truth of the assertion in the first row, − for the truth of the negated assertion. Furthermore, let T0,∞+1 := ∅ and h∞ := e11 ) 3.2.2 The Subclasses of L Obviously, all subclasses of [P21 ] [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , C, C0 , C1 , I, ∅ are also subclasses of L. With the help of these sets, we can determine all subclasses of L with the following. Lemma 3.2.2.1 Let L be subclass of L with L ⊆ [P21 ]. Then L = [L1 ∪ {r}] (r(x, y, z) = x + y + z). Proof. Let L = [L] ⊆ L and L ⊆ [P21 ]. Then, there is a function g ∈ L with g(x, y, z) = a + x + y + bz for some a, b ∈ E2 . Thus g(g(x, y, z), z, z) = r(x, y, z) ∈ L. By forming of the functions of the type r r ... r and by n identifying of variables in these functions, we get that every function i=1 bi xi


155

with b1 + ... + bn = 1 belongs to [r] ⊆ A. Now, denote f an n-ary function of n L and let f (x) = a0 + i=1 ai xi . Then, we have f (x, x1 , ..., xn ) := x + (a2 + ... + an )x1 +

n

ai xi ∈ [r]

i=2

and

f (x) = f (f (x1 , ..., x1 ), x1 , ..., xn ).

Therefore f ∈ [{r} ∪ L1 ] and [{r} ∪ L1 ] = L is proven. Looking for subsemigroups of (P21 ; ), which have the property to be preserved by r, we get the following Theorem 3.2.2.2 All subclasses of L are L, L ∩ T0 = [c0 , +], L ∩ T1 , L ∩ S = [− , r], L ∩ T0 ∩ S = [r], [P21 ], I ∪ C, I, I ∪ C0 , I ∪ C1 , I, C, C0 , C1 , ∅. 3.2.3 The Subclasses of S, Which Are Not Subsets of L Obviously, a function f n ∈ P2 (n ≥ 2) belongs to S iff there exists a function F n−1 ∈ P2 with the property f (x1 , ..., xn ) = x1 F (x2 , ..., xn ) ∨ x1 F (x2 , ..., xn ),

(3.2)

where F (x2 , ..., xn ) := f (0, x2 , ..., xn ). Consequently, we can define a bijective mapping α of S := S\S 1 onto P2 as follows: α : f −→ F. Lemma 3.2.3.1 The mapping α has the following properties: τ, ∆, ∇ and (a) For the operations ζ, , defined by )(x1 , ..., xn ) := f (x1 , x3 , x4 , ..., xn , x2 ), (ζf ( τ f )(x1 , ..., xn ) := f (x1 , x3 , x2 , x4 , ..., xn ), )(x1 , ..., xn−1 ) := f (x1 , x2 , x2 , x3 , ..., xn−1 ), (∆f (∇f )(x1 , ..., xn+1 ) := f (x1 , x3 , x4 , ..., xn+1 ) and (f g)(x1 , ..., xm+n−2 ) := f (x1 , g(x1 , ..., xm ), xm+1 , ..., xm+n−2 ) (n, m ≥ 2), it holds α( γ f ) = γ(α(f )) for every γ ∈ {ζ, τ, ∆, ∇} and α(f g) = α(f ) α(g), i.e., the algebra (S ; ζ, τ, ∆, ∇, ) is isomorphic to the algebra (P2 ; ζ, τ, ∆, ∇, ). (b) For every subclass A (= ∅) of S, α(A) is a subclass of P2 , and it holds α(A) ⊆ S, A ⊆ α(A) and α(A) ∩ S = A.

156


Proof. (a) is easy to check. (b): Let A be a subclass of S. By (a) we have that α(A) is also a closed set. Assume α(A) ⊆ S. Then F (x2 , ..., xn ) = F (x2 , ..., xn ) for every f n ∈ A. Thus by (3.2) we get that the variable x1 is fictitious for every function f n ∈ A. However, this is not possible. Hence α(A) ⊆ S holds. Let f n ∈ A. Then ∇f ∈ A and therefore α(∇f ) = f ∈ α(A), i.e., A ⊆ α(A). If f n ∈ S ∩ α(A), we have ∆(α−1 f ) = f ∈ A and thus S ∩ α(A) ⊆ A. From this, it follows that A = S ∩ α(A), since A ⊆ α(A) and A ⊆ S. With the help of Lemma 3.2.3.1 and Theorem 3.2.1.5, it is not difficult to determine the missing subclasses of S. It holds Theorem 3.2.3.2 (a) The sets S ∩ M and S ∩ T0 are the only proper subclasses of S, which are not subsets of L. (b) It holds that S = [h2 ,− ], S ∩ T0 = [h2 , r], S ∩ M = [h2 ]. Proof. (a): Let A be a subclass of S with A ⊆ L. Then it holds α(A) ⊆ L. By Lemma 3.2.1.2 and Lemma 3.2.1.3, we have {∨, ∧} ∩ α(A) = ∅. Consequently, the function h2 (= xyz ∨ x(y ∨ z)) or the function g(x, y, z) := x(y ∨ z) ∨ xyz belong to A. Since it holds g(g(x, y, z), y, z) = h2 (x, y, z), h2 ∈ A. Further, the functions α(h2 ) = ∧, x ∧ h2 (x, y, z) = x(y ∨ z) and c10 = α(e21 ) are elements of α(A). Thus, T0,2 ∩M ⊆ α(A) and by Lemma 3.2.3.1, (b) we have T0,2 ∩M ∩S ⊆ A. By Theorem 3.2.1.5, there exists only the following possibilities for A: S ∩ T0,2 , S ∩ M = S ∩ M ∩ T0 , S ∩ T0 and (a) follows from S ∩ T0,2 = S ∩ M . (This fact is easy to prove, for example, with the help of the relation product 2 and theproperty P ol ∩ P ol ⊆P ol2 : 1 0 0 0 1 0 0 1 Let 1 := , 2 := and 3 := , Then 2 21 = 3 0 1 0 1 0 0 1 1 and thus S ∩ T0,2 ⊆ S ∩ M . Conversely, S ∩ M ⊆ S ∩ T0,2 holds, since 3 21 = 2 .) (b) follows from Theorem 3.2.1.1 and Lemma 3.2.3.1. 3.2.4 A Completeness Criterion for P2 The following is a conclusion from Theorem 3.1.1: Theorem 3.2.4.1 (Completeness Criterion for P2 ) Let A ⊆ P2 . Then [A] = P2 ⇐⇒ ∀X ∈ {T0 , T1 , M, S, L} : A ⊆ X.


157

Without using of Theorem 3.1.1, one can prove Theorem 3.2.4.1 with the help of Theorem 1.5.4.1 and the following: Theorem 3.2.4.2 P2 has exactly five maximal classes: T0 , T1 , M, S, and L. Proof. It is easy to see that the sets T0 , T1 , M, S, L are pairwise distinct proper subclasses of P2 . Therefore, proof of the following suffices for the proof of our theorem: ∀A ⊆ P2 : ((∀K ∈ {T0 , T1 , M, S, L} : A ⊆ K) =⇒ [A] = P2 ) Let now A ⊆ P2 with {f0 , f1 , fM , fS , fL } ⊆ A, where f0 ∈ T0 , f1 ∈ T1 , fM ∈ M, fS ∈ S and fL ∈ L. If one identifies all variables of f0 with each other, then one gets an unary function f0 ∈ [A] with f0 (0) = 1, i.e., f0 ∈ {c1 , e11 }. Case 1: f0 = c1 . In this case, it holds that f1 (c1 (x), ..., c1 (x)) = c0 (x) ∈ [A]. Since fM ∈ A\M , there are some (ai , bi ) ∈ {(0, 0), (0, 1), (1, 1)} (i = 1, 2, ..., n) with f (a1 , ..., an ) > f (b1 , ..., bn ). Consequently, the function e11 is a superposition over {fM , c0 , c1 } ⊆ [A]. Thus P21 belongs to [A]. By the proof of Lemma 3.2.1.2 (see also Theorem 1.4.3), we can describe the function fLn with the help of a so-called Shegalkin polynom. Since fL does not belong to L, we can assume w.l.o.g. that fL (x) = a0 + x1 · x2 · ... · xr +

n i=1

ai · xi +

ai1 ...iν · xi1 · ... · xiν ,

i1 , ..., iν ∈ {1, ..., n}, ν ≥ r, {i1 , ..., iν } ⊆ {1, ..., r}

holds for r ≥ 2. Now, we consider the function fL (x, y) := fL (x,

y, ..., y , c0 (x), ..., c0 (x)) (r−1) times

which has the form fL (x, y) = a + b · x + c · y + x · y for some a, b, c ∈ {0, 1}. It is easy to check that x · y is a superposition over {fL } ∪ P21 (⊆ A). Consequently, by Theorem 1.4.2, we have [A] = P2 . Case 2: f0 = e11 . Since fSn ∈ S, there exist some a1 , ..., an ∈ {0, 1} with fS (a1 , ..., an ) = fS (a1 , ..., an ). Thus, c1 is a superposition over {fS , e11 } and the Case 2 is put down to the Case 1.

4 The Subclasses of Pk Which Contain Pk1

We start with the definitions of some subclasses of Pk . We show later that these classes are all classes A of Pk with Pk1 ⊆ A. Let Ut := Pk (t) ∪ [Pk1 ] for t = 2, 3, ..., k. In particular, Uk = Pk . Further, let Lk be the set 1 {f n ∈ Pk | ∃a ∈ E2 ∃f0 ∈ Pk1 ∃f1 , ..., fn ∈ Pk,2 : [Pk1 ]∪ n≥1

f (x) = f0 (a + f1 (x1 ) + f2 (x2 ) + ... + fn (xn ) (mod 2))}.

For k = 2 Lk is the set L, already defined in Chapter 3. The sets Ut and Lk can be described with the help of relations: Lemma 4.1 Let ιhk := {(a0 , ..., ah−1 ) ∈ Ekh | ∃i, j ∈ Ek : i = j ∧ ai = aj } and λk := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek }. Then (a) Ut = P ol ιt+1 for each t ∈ {2, 3, ..., k − 1} k and (b) Lk = P ol λk . t+1 Proof. (a): Obviously, Ut is a subset of P ol ιt+1 k . The inclusion Ut ⊂ P ol ιk t+1 (i.e., there is a function (∈ P ol ιk ) that essentially depends on at least two variables and which has at least t + 1 different values) is false because of

160


Theorem 1.4.4, (b). Thus, Ut = P ol ιt+1 k . (b): It is easy to check that Lk ⊆ P ol λk . For the proof of P ol λk ⊆ Lk , denote f n an arbitrary function from Pk,2 ∩ (P ol λk ). First, we will show that then f (x1 , 0, ..., 0) + f (0, x2 , ..., xn ) + f (0, 0, ..., 0) = f (x1 , ..., xn ) (mod 2)

(4.1)

holds. Suppose, for some x1 = a1 , ..., xn = an is this false. Then, we have ⎞ ⎛ ⎛ ⎞ a a a a1 0 ... 0 ⎜ 0 a2 ... an ⎟ ⎜ a b b ⎟ ⎟ ⎜ ⎟ f⎜ ⎝ 0 0 ... 0 ⎠ ∈ ⎝ b a b ⎠ a1 a2 ... an b b a for arbitrary a, b ∈ E2 . However, this is a contradiction to f ∈ P ol λk . Therefore, (4.1) holds and this implies: f (x1 , ..., xn ) = a + f1 (x1 ) + ... + fn (xn ) (mod 2), where a := (n − 1) · f (0, 0, ..., 0) (mod 2) and fi (x) := f (0, ..., 0,

x , 0, ..., 0). i-th place

Because of pr1,2,3 λk = ι3k , an arbitrary function g from (P ol λk )\[Pk1 ] has exactly only two different values. Consequently, there exists a certain permutation s ∈ Sk with s(Im(g)) = {0, 1} such that g has the description s−1 (s g) and s g belongs to Pk,2 ∩ P ol λk . It is obvious then that g ∈ Lk . Definition Let E ⊆ Ek with |E| ≥ 2. Then we can define a mapping prE from Pk,E into PE as follows: prE f n = g m :⇐⇒ (n = m ∧ ∀a ∈ E n : f (a) = g(a)) (f n ∈ Pk,E , g m ∈ PE ). Lemma 4.2 Let f 2 be a function from Lk with f (x, y) = x + y (mod 2) for all x, y ∈ E2 , let g m be a function from Pk,2 with prE2 g ∈ L2 and let h ∈ Ut \Ut−1 . Then, (a) Lk = [Pk1 ∪ {f }], (b) U2 = [Pk1 ∪ {g}] and (c) ∀t ∈ {3, 4, ..., k} : Ut = [Ut−1 ∪ {h}]. Proof. (a) follows directly from the definition of Lk . 1 ] = P2 . Therefore, some binary (b): By Theorem 3.2.4.1, [{prE2 g} ∪ prE2 Pk,2


161

functions ∧ , + with prE2 ∧ = ∧ and prE2 + = + belong to [Pk1 ∪ {g}] and an arbitrary function ut of Pk,2 is a superposition over Pk1 ∪ {∧ , + }: u(a1 , ..., at ) · ja1 (x1 ) · ... · jat (xt ) (mod 2) u(x1 , ..., xt ) = (a1 , ..., at ) ∈ Ekt

(see (1.8) from Section 1.4). This implies (b). (c) is a consequence from Lemma 1.4.5.

Theorem 4.3 (Burle’s Theorem, [Bur 67]) The classes with

[Pk1 ], Lk , U2 , U3 , ..., Uk−1 , Pk [Pk1 ] ⊂ Lk ⊂ U2 ⊂ U3 ⊂ ... ⊂ Uk−1 ⊂ Pk

are the only subclasses of Pk which contain Pk1 . Proof. Suppose there exists a class A of Pk with Pk1 ⊂ A, which is different from the classes of the above theorem. Then A contains a certain function f n , which has at least two essentially variables and at least l ≥ 2 different values. Consequently, by Theorem 1.4.4, (a), (c), there exists some a1 , ..., an , b1 , ..., bn , α, β, γ ∈ Ek with ⎛ ⎞ ⎛ ⎞ a1 a2 a3 ...an α f ⎝ a1 b2 b3 ...bn ⎠ = ⎝ β ⎠ , b1 b2 b3 ...bn γ where |{α, β, γ}| = 3 for l ≥ 3 and α = γ and α = β for l = 2. Then, a binary function f with f (x, y) := g0 (f (g1 (x), g2 (y), ..., gn (y))) and ⎛ ⎞ ⎛ ⎞ 0 0 0 f ⎝ 0 1 ⎠ = ⎝ 1 ⎠ 1 1 0 is a superposition over f and some g0 , ..., gn ∈ Pk1 . We distinguish two cases: Case 1: f (1, 0) = 0. In this case, the function prE2 f is nonlinear. Thus, by Lemma 4.2, (b) it holds U2 ⊆ A. Since we have assumed, however, A = Ut for each t ∈ {2, ..., k}, we obtain a contradiction with the aid of Lemma 4.2, (c). Case 2: f (1, 0) = 1. In this case, by Lemma 4.2, (a) Lk is a subset of A and, because of A = Lk , there is a function g ∈ A, which does not preserve λk . Then, one can form a function g ∈ A ∩ Pk,2 with prE2 g ∈ L2 as a superposition over g and some functions of Lk . Thus, by Case 1, we get a contradiction.

162


Consequently, the classes given in our theorem are the only subclasses of Pk that contain Pk1 . The claimed chain property is an immediate conclusion from the definitions of these classes. The claimed chain property is a direct conclusion from the definitions of these classes.

5 The Maximal Classes of Pk

5.1 Introduction, a Rough Description of the Maximal Classes A subclass A of Pk is maximal in Pk (or A is called maximal class)1 if and only if no further classes of Pk exist between A and Pk . In other words: A = [A] ⊆ Pk is maximal in Pk if and only if A = Pk and [A ∪ {f }] = Pk for each f ∈ Pk \A. One is interested in these classes not only for structural reasons, but particularly because one can solve a central problem of the Many-Valued Logic (the so-called Completeness Problem) with the aid of these classes (see Theorem 1.5.4.1). Inter alia, the following papers dealt with the determination and description of maximal classes: [Pos 41], [Jab 54], [Jab 58], [Ros 70;a], [Mart 60], [Lo 63;a– ˇ 70;b]. c], [Lo 64], [Zac 67], [Zac-K-J 69], [Bai 67] and [Sai One knows the maximal classes T0 , T1 , M, S, and L of P2 (see Theorem 3.2.4.2) through the papers [Pos 20], [Pos 41] by E. L. Post. Efforts to determine all maximal classes of Pk for k ≥ 3 began more than 50 years ago. S. V. Jablonskij determined all 18 maximal classes of P3 in [Jab 54]. A. I. Mal’tsev proved how in the paper [Zac-K-J 69] was mentioned that P4 has exactly 82 maximal classes. I. G. Rosenberg was the first that succeeded in the description of all maximal classes of Pk for each k ∈ N \ {1} (see [Ros 65]), and he proved in [Ros 70a] that the list of the given classes is complete. Rosenberg defined six relation sets, which are subsequently denoted by us with Uk , Mk , Sk , Lk , Ck and Bk , and he proved that the set {P olk | ∈ Uk ∪ Mk ∪ Sk ∪ Lk ∪ Ck ∪ Bk } 1

One uses instead of “maximal class” the concept “precomplete class” in older papers.

164

5 The Maximal Classes of Pk

is exactly the set of all maximal classes of Pk . He could use that some other authors had shown already the maximality of some classes P ol . For example, for classes of the form P ol 1 with 1 ∈ Uk ∪ Sk ∪ C1k and ∈ Lk , where k is a prime number, it was proven in [Jab 58] that these classes are maximal classes. In [Mart 60], one finds maximal classes of the form P ol 2 with 2 ∈ Mk . For further details, we refer the reader to [Ros 70;a]. In Section 5.2, we will describe the maximal classes of Pk in the manner found from Rosenberg.2 Further, we give first properties of the maximal classes, where these properties are consequences from the definitions more or less. Then, in Section 5.3, the maximality of the classes described in the theorem is proven. It turns out, in this case, that one manages with three basic ideas in the proof. Chapter 6 is dedicated to the proof of the completeness of the given set of maximal classes then. This most difficult part of the determination of maximal classes orientates itself strongly onto the proof given in [Ros 70;a]. One could abbreviate, however, the proof through transfer and modification of some ideas from the papers [Qua 82], [P¨ os- K 79] (p. 126-129) and [Lau 92a]. A first coarse description of the maximal classes supplies the following theorem basically already proven by A. V. Kuznezov in 1959. Theorem 5.1.1 ([Kuz 59], [Ros 70;a] (3.2.5), [But 60]) The class Lk for k = 2 or Uk−1 (= P olk ιkk ) for k ≥ 3 is the only maximal class of Pk , which contains Pk1 . For every maximal class A of Pk , which is different from L2 and Uk−1 , is valid: A = P olk G1 (A). Proof. Let A be an arbitrary maximal class of Pk . Then, the following two cases are possible: Case 1: A1 = Pk1 . Then, by Theorem 4.3, A is the set L2 for k = 2 or the set Uk−1 for k ≥ 3. Case 2: A1 ⊂ Pk1 . In this case, (A1 ; ) is a proper subsemigroup of (Pk1 ; ) which e := e11 contains, what one can prove as follows: Suppose, e ∈ A1 . Then A1 ∩ [Pk1 [k]] = ∅, since sk = e for all s ∈ Pk1 [k]. Consequently, we have A ⊂ Jk ∪ A = [Jk ∪ A] ⊂ Pk , which contradicts the presupposed maximality of A. Thus A is a clone, for which, by Theorem 2.7.1, (c) A ⊆ P olk G1 (A) ⊆ Pk 2

This is not the only means to describe maximal classes. In [Den-P 88] one can find descriptions of maximal classes of Pk through hyperidentities. One finds more about hyperidentities in the book [Den-W 2000] of K. Denecke and S. L. Wismath.

5.2 Definitions of the Maximal Classes of Pk

165

holds. Because of A1 = Pk1 and the maximality of A, this is possible only for A = P olk G1 (A).

5.2 Definitions of the Maximal Classes of Pk The maximal classes are defined with the aid of the relation sets Mk , Sk , Uk , Lk , Ck and Bk . Indeed, by Theorem 5.1.1, one can describe every maximal class of Pk for k ≥ 3 with the aid of a certain k-ary relation in the form P ol . If P ol is a maximal class, then, P ol = P ol isvalid for all -derivable non-diagonal k relation . Therefore, the elements of h=1 Rkh \Dkh are possible descriptive relations for a maximal class P olk . The subsequently defined relations of Mk , Sk , Uk , Lk , Ck and Bk are (with few exceptions), with respect to the arity, minimally chosen relations, which one can use to describe the maximal classes (see Chapter 10). We say that a maximal class A is a class of type X, if there exist an X ∈ {M, S, U, L, C, B} and a ∈ Xk with A = P olk . 5.2.1 Maximal Classes of Type M (Maximal Classes of Monotone Functions) Let Mk be the set of all partial orders on Ek with a greatest and a least element. More exactly, a binary relation ∈ Rk belongs to Mk if and only if has the following four properties: 1) is reflexive (i.e., ι2k ⊆ ); 2) is antisymmetric (i.e., ∩ −1 = ι2k ); 3) is transitive (i.e., ◦ = ) and 4) there exist elements o (”least element”) and e (”greatest element”) in Ek with {(o , x), (x, e ) | x ∈ Ek } ⊆ . It can easily be shown (see proof of Lemma 6.1.6) that the elements o and e are uniquely determined. We write a ≤ b instead of (a, b) ∈ and a h and (a0 , ..., ah−1 ) ∈ γ =⇒ ((a0 , ..., ah−1 ) ∈ ιhk ∨ (∃i ∈ Eh : ai ∈ C)), then ord P olk γ = # h2 $ + 1.

2

(11.5)

316

11 On Generating Systems and Orders of the Subclasses of Pk

Proof. Below is a procedure comprising three steps. This procedure shows that an arbitrary function, which takes q ≥ 2 different values, is a superposition over the set (P olk γ)h ∪ (P olk γ ∩ (Pk (q − 1))). The iteration of this procedure takes back the construction of an arbitrary function to the construction of functions that take at most h − 1 different values. Since all such functions preserve the relation γ, we have ord P olk γ ≤ h by Theorem 11.1.6, (c). Let f n be an arbitrary function of P olk γ with n > 2 and let c be any fixed element of the set C, i.e., c is a central element of the relation γ. 1.) To dismantle o f , we need the n-ary functions f1 , f2 and the binary function g of P olk γ: f (x1 , ..., xn ) if f (x1 , ..., xn ) ∈ C, f1 (x1 , ..., xn ) := c otherwise, c if f (x1 , ..., xn ) ∈ C, f2 (x1 , ..., xn ) := f (x1 , ..., xn ) otherwise, ⎧ x1 ∈ C ∧ x2 = c, ⎨ x1 if x1 = c ∧ x2 ∈ C, x2 if g(x1 , x2 ) := ⎩ c otherwise. We get: f (x) = g(f1 (x), f2 (x)), where f2 ∈ Pk,C ⊂ [(P olk γ)2 ] by Theorem 11.1.6, (a). 2.) In the second step the function f1 is dismantled, where we use functions ha ∈ (P olk γ)2 , a ∈ Ek \C, defined as follows: ⎧ (x1 = c ∧ x2 = a) ∨ (x1 = a ∧ x2 = c), ⎨ a if x1 = x2 , ha (x1 , x2 ) := x1 if ⎩ c otherwise. Let Ea := {a ∈ Ekn | f1 (a) = a}. Further, for a ∈ Ea we put: ⎧ if x = a, ⎨a if x ∈ Ea \{a}, fa (x1 , ..., xn ) := c ⎩ f1 (x) otherwise. If Ea = {a1 , ..., as }, then there exists the following representation for f1 : f1 (x) = ha (...ha (ha (fa1 (x), fa2 (x)), fa2 (x))..., fas (x)). Similar to f1 , one can dismantle the functions fa . In this case, we choose a b ∈ Ek \(C ∪ {a}), put Eb := {b1 , ..., bt } and for b ∈ Eb we define ⎧ if x = b, ⎨b if x ∈ Eb \{b}, fa,b (x1 , ..., xn ) := c ⎩ fa (x) otherwise. 2

x denotes the greatest integer z with z ≤ x.

11.3 Orders of the Classes of Type L, C, B

317

Then the function fa has the representation fa (x) = hb (...hb (hb (fa,b1 (x), fa,b2 (x)), fa,b3 (x))...fa,bt (x)). Then, one dismantles the functions in analogous manner fa,b , and so forth. As a result, the function f1 is dismantled in certain functions ha;α ∈ P olk γ defined by α if x = a, ha;α (x1 , ..., xn ) := c otherwise, where α ∈ (Ek \C) ∪ {c} and a ∈ Ekn . 3.) In this step, we construct representations for the n-ary functions u ∈ P olk γ, which are defined by di if x = di , i = 1, 2, ..., q, u(x1 , ..., xn ) := c otherwise, where {d1 , ..., dq } ⊆ Ek \C, di = dj for i = j and di = (di1 , ..., dim ); i, j = 1, 2, ..., q. If q ≤ h − 1 and for the case that q = h and (d1 , ..., dq ) ∈ γ, we have u ∈ [(P olk γ)2 ]. Consequently, the following three cases must still be examined: Case 1: q = h and (d1 , ..., dq ) ∈ γ. In this case, there exists a j (1 ≤ j ≤ n) with (d1j , d2j , ..., dhj ) ∈ γ, since u ∈ P olk γ. Put d1 if x ∈ {d1 , d2 , ..., dh }, u1 (x1 , .., xn ) := c otherwise, and

u2 (x1 , x2 ) :=

di if x1 = dij ∧ x2 = d1 , i = 1, 2, ..., h, c otherwise.

Because of u1 , u2 ∈ P olk kγ and u(x) = u2 (xj , u1 (x)) we have u ∈ [(P olk γ)2 ]. Case 2: There is a set {e1 , e2 , ..., eh } ⊆ {d1 , ..., dq } with (e1 , ..., eh ) ∈ γ\ιhk . The functions ⎧ ei if x1 = ... = xi−1 = xi+1 = ... = xh = ei , ⎪ ⎪ ⎨ xi = c, i = 1, 2, ..., h, v(x1 , ..., xh ) := if x x ⎪ 1 1 = ... = xh ∈ {d1 , .., dq }\{e1 , ..., eh } ⎪ ⎩ c otherwise, and

vi (x1 , ..., xn ) :=

c if u(x1 , ..., xn ) = ei , u(x1 , ..., xn ) otherwise

(i = 1, 2, ..., h) belong to P olk γ. It is easy to check that u(x) = v(v1 (x), v2 (x), ..., vh (x)) holds, where |Im(vi )| < |Im(u)|.

318


Case 3: q > h and for an arbitrary set {e1 , e2 , ..., eh } ⊆ {d1 , ..., dq } it holds: (e1 , e2 , ..., eh ) ∈ γ \ ιhk . We need the functions ⎧ d2i−1 if x1 = ... = x[ h +1 = d2i−1 , i = 1, 2, .., # h2 $, ⎪ ⎪ 2 ⎪ ⎪ ⎪ d2i if x1 = ...xi−1 = xi+1 = ... = x h +1 = d2i , ⎪ 2 ⎪ ⎪ ⎪ xi = d2i−1 , i = 1, 2, .., # h2 $, ⎨ if h odd ∧ x1 = ... = x h +1 = dh , w(x1 , ..., x h +1 ) := dh 2 2 ⎪ ⎪ ⎪ d h+1 if x1 = ... = x h = dh+1 ∧ x h +1 = dh , ⎪ 2 2 ⎪ ⎪ ⎪ if x1 = ... = x h +1 ∈ {dh+2 , ..., dq }, x1 ⎪ ⎪ 2 ⎩ c otherwise, d2i−1 if u(x) ∈ {d2i−1 , d2i }, wi (x1 , ..., xn ) := u(x) otherwise (i = 1, 2, ..., # h2 $) and w h +1 (x1 , ..., xn ) := 2

dh if u(x) ∈ {dh , dh+1 }, u(x) otherwise,

of P olk γ for the dismantling of the function u. It holds: u(x) = w(w1 (x), w2 (x), ..., w h +1 (x)), 2

where |Im(wi )| < |Im(u)|. The iteration of the above procedure proves the contention 2 if k − |C| ≤ k, ord P olk γ = ≤ h otherwise. If k − |C| > h and the second case is not possible for any function f ∈ P olk γ,i.e., γ fulfills the condition (11.1), it results from the above procedure then that ord P olk γ ≤ # h2 $ + 1 is valid. We still have to show that w ∈ h [(P olk γ) 2 holds, if γ fulfills the condition (11.1). h +1

Denote A a matrix whose rows are just the tuples of Ek 2 on which the function w takes the values d1 , d2 , ..., dh+1 . It is easy to check that certain rows a1 , ..., ah (ai := (ai1 , ..., air ), i = 1, 2, ..., h, r ≤ # h2 $) are found in every matrix, which can be formed from the matrix A by deleting at least a column, so that is valid for every i ∈ {1, 2, ..., r}: (ai1 , a2i , ..., air ) ∈ ιhk . Consequently, an arbitrary ( h2 + 1)-ary function of P olk γ, which depends on at most # h2 $ places essentially, takes either a value from the set C on at least a tuple of A or two different rows of A, on which the function takes the very same value, exist. Further, every # h2 $-ary function of P olk γ can take only then h + 1 different values from the set Ek \C on different tuples b1 , ...,bh+1 , if there is at least a column that contains h+1 different values of the set Ek \C among the columns of the matrix

11.4 The Order of P olk for ∈ Mk and k ≤ 7

⎛

319

⎞

b1 ⎜ b2 ⎟ ⎜ ⎟. ⎝ ... ⎠ bh+1 Consequently, we have w ∈ (P olk γ) 2 and ord P olk γ = # h2 $ + 1, if γ fulfills the condition (11.1). h

11.4 The Order of P olk for ∈ Mk and k ≤ 7 In this section let be a relation of Mk . The following lemma generalizes a theorem from [Jab 58], p. 83. Lemma 11.4.1 Let E ⊆ Ek with |E| ≥ 2. Further, let be a partial order relation on E with the following properties: ⊆ and (E; ) is a lattice. Then {f ∈ P olk | Im(f ) ⊆ E} ⊆ [(P olk )2 ]. Proof. Obviously, has a least element o and has a greatest element e. It is easy to check that there exists a unary function i with Im(i) = E and i(a) = a for all a ∈ E. We show that an arbitrary n-ary function f ∈ Pk,E ∩ P olk is a superposition over (sup (i(x1 ), i(x2 ))), (inf (i(x1 ), i(x2 ))) and the functions a if x ≥ b, mb,a (x) := o otherwise (a ∈ E, b ∈ Ek ). For all a := (a1 , a2 , ..., an ) ∈ Ekn is valid: f (a) if x ≥ a, fa (x1 , ..., xn ) := o otherwise, = inf (ma1 ,f (a) (x), ma2 ,f (a) (x), ..., man ,f (a) (x)) ∈ [(P olk )2 ]. Then one can represent the function f n as follows: f (x) = sup (fa1 (x), fa2 (x), ..., fakn (x)), where {a1 , ..., akn } = Ekn . Consequently, f ∈ [(P olk )2 ]. A consequence of Lemma 11.4.1 is Theorem 11.4.2 If (E; ) a lattice then ord P olk = 2.

Theorem 11.4.3 If k ≤ 7 then ord P olk = 2.

320


Proof. Let k = 6 and the partial order relation defined by Figure 11.1. We will prove the theorem only for this relation. In the remaining cases, the relation fulfills the conditions of Theorem 11.4.2 or one can lead the proof similarly to the proof which is subsequently given. 5 s @ @

3 s @s 4 HH H H 1 s Hs 2 @ @ @s 0 Fig. 11.1

We start with the definitions of some functions of (P olk )2 : 0 if x ∈ E3 , x if x ∈ E3 , i1 (x) := i2 (x) := x otherwise, 5 otherwise, ⎧ x = 1, ⎨ 2 if x = 2, e1 (x) := 1 if ⎩ otherwise,

⎧ x = 3, ⎨ 4 if x = 4, e2 (x) := 3 if ⎩ x otherwise,

supi (x1 , x2 ), i = 0, 1, 2 and infj (x1 , x2 ), j = 3, 4, 5, where the relations i and j are defined in Figure 11.2. 0 :

5 s @ 3 s @s 4 @ 0 @s @ @s 2 1 s

3 :

1 :

5 s @ 3 s @s 4 @ 1 @s

5 s @ 3 s @s 4 @ 2 @s

s0

s0

s2

s1

3 s

5 :

s5

s5

s 3@ @s 2

s 4@ @s 2

3 s @

4 s

1 s @ @s 0

4 :

2 :

1 s @ @s 0

Fig. 11.2

s4

@s 5@ @s 2 1 s @ @s 0

11.4 The Order of P olk for ∈ Mk and k ≤ 7

321

Let f n ∈ P ol6 be arbitrary. We prove by induction on n that f n ∈ [(P ol6 )2 ] holds. For n = 1, 2 our assertion is trivial. Assume the assertion is valid for n − 1 ≥ 2 and we show that it is valid for n. If |Im(f )| ≤ 5, then, by Lemma 11.4.1, f ∈ [(P ol6 )2 ]. Let |Im(f )| = 6 in the following. We show through a dismantling procedure that f n is a superposition over certain binary functions and certain functions g m with |Im(g)| ≤ 5 or m ≤ n − 1. Because of induction assumption and Lemma 11.4.1, our theorem would then be proven. 1.) During the first dismantling of function f , we use, the following functions of P ol6 : sup (x1 , i1 (x2 )), = f (x) if f (x) ∈ E3 , f1n (x) ≤ f (x) otherwise, where we assume the validity of ¬∃f1 ∈ P ol6 : = f (x) if f (x) ∈ E3 , ∃a : f1 (a) x2i+1 ∧ f1 (y) =

(g) ∀ 2i ∈ {m+1, ..., m −1} ∃z, z ∈ Q : z, z > x2i ∧ f1 (z) = γ ∧ f1 (z ) = γ . Lemma 11.5.1 Let ∅ = Q ⊆ E8n and let f1 : Q −→ E8 be a mapping. Then there is a monotone function f : E8n −→ E8 with f|Q = f1 if and only if the following two conditions hold: (1) f1 is monotone; (2) there is no zigzag in Q for f1 . Proof. “=⇒”: The condition (1) is trivial. To prove (2) we assume by way of contradiction that xm , ..., xm ∈ Q is a zigzag for f1 according to the above definition and that f : E8n −→ E8 is a monotone extension of f1 . First, let m = 0 and m = 2q + 1, q ∈ N in the zigzag (see Figure 11.6). ; γ

;γ

s A

s

;γ

;α

; α

s A

; γ

s

A A x0 ; β AAsx2q ; s s x2 ; AA @ @ @ @ . @ @ . @ . @ @ @ @ @ @s x2q+1 ; β x1 @ ; s x3@ ; s A A A A A A AAs AAs s s ;α

A A

; α

Fig. 11.6

Since x1 , ..., x2q ∈ E8n and f1 preserves on these tuples, we have f1 (x1 ) = β. Continuation of this consideration supplies f1 (x2q ) = β, which is not possible because of x2q+1 < x2q and f1 (x2q+1 ) = β . For the other possibilities for m and m one shows in analog mode that the assumption of the existence of a zigzag for f1 supplies a contradiction. “⇐=”: Assume that the conditions (1) and (2) are satisfied. We construct a monotone function f : E8n −→ E8 extending the function f1 by defining of the following sets Hx := f −1 (x) for each x ∈ E8 . For q ∈ E8n let f (q) := {f1 (y) | y ∈ Q ∧ y ≤ q}, f (q) := {f1 (z) | z ∈ Q ∧ z ≥ q}.

11.5 A Maximal Clone of Monotone Functions That Is Not Finitely Generated

327

Furthermore, let H0 := {q ∈ E8n | f (q) ⊆ {0}}, H1 := {q ∈ E8n | f (q) ⊆ {1}}\H0 , Hα := {q ∈ E8n | f (q) ⊆ {0, α}}\(H0 ∪ H1 ), Hα := {q ∈ E8n | f (q) ⊆ {0, α }}\(H0 ∪ H1 ∪ Hα ), Hγ := {q ∈ E8n | f (q) ⊆ {1, γ}}\(H0 ∪ H1 ∪ Hα ∪ Hα ), Hγ := {q ∈ E8n | f (q) ⊆ {1, γ }}\(H0 ∪ H1 ∪ Hα ∪ Hα ∪ Hγ ) and H := E8n \(H0 ∪ H1 ∪ Hα ∪ Hα ∪ Hγ ∪ Hγ ). To partition the set H, we consider the elements of H as vertices of a nondirectional graph in which an edge joins the vertices x, y ∈ H if and only if x < y or y < x in E8n is valid. Then, graph G consists of certain connected maximal subgraphs (the so-called components) K1 ..., Kt , with which we define the sets Hβ and Hβ as follows: Hβ := {x ∈ H | ∃i ∈ {1, ..., t} : x ∈ Ki ∧ β ∈ {f1 (a) | a ∈ Ki }}, Hβ := H\Hβ . The subsets Hx for x ∈ E8 now form a partition of E8n , so they really determine a function f : E8n −→ E8 satisfying ∀x ∈ E8n ∀e ∈ E8 : f (x) = e :⇐⇒ x ∈ He . We prove that this function preserves the relation and that it agrees with the function f1 on E8n . One can check the monotony from f easily, using the following implications, which result from the definitions of the sets He , e ∈ E8 and which are valid for every a ∈ E8n : ∀e ∈ {α, α } : f (a) = e =⇒ e ∈ f (a) ∀e ∈ {γ, γ } : f (a) = e =⇒ e ∈ f (a). Finally we show f|Q = f1 . Obviously, for arbitrary z ∈ Q with f1 (z) = β we have f (z) = f1 (z). To show that f (z) = f1 (z) holds for elements z ∈ Q with f1 (z) = β as well, we must prove that no component G contains elements from both f1−1 (β) and f1−1 (β ). We use here the complicated condition (2). By the definition of H for every z ∈ H, we have f (z) ⊆ {0, α}, f (z) ⊆ {0, α }, f (z) ⊆ {1, γ}, f (z) ⊆ {1, γ }. By condition (1) these conditions are equivalent to

328

and


β ∈ f (z) or β ∈ f (z) or {α, α } ⊆ f (z)

(11.6)

β ∈ f (z) or β ∈ f (z) or {γ, γ } ⊆ f (z).

(11.7)

Suppose a component K of the graph G contains a xm ∈ f1−1 (β) and a xm ∈ f −1 (β ). We choose xm and xm so that are the least distance among the possible vertices. So, there is a path between xm and xm . Take the shortest path. If x, y and z are three consecutive elements of this path then neither x < y < z nor x > y > z can hold since y would then be redundant. Thus, we can choose the indices so that the path be xm , ..., xm satisfies the conditions (a), (c), (d), and (e) in the definition of a zigzag. If m < 2i < m then there is no z ∈ Q with f1 (z) = β or f1 (z) = β and z ≥ x2i , since otherwise either z, x2i+1 , ..., xm or xm , ..., x2i−1 , z would be a shorter path in K from a element of f −1 (β) to an element f −1 (β ). Consequently, (b) holds. Condition (f) also holds because for x2i only {γ, γ } ⊆ f (x2i ) can occur among the possibilities in (11.3). Similarly, condition (g) holds. Hence xm , ..., xm is a zigzag for f1 contrary to (2). Next, for n ≥ 3, we shall define (n + 5)-ary relations µ0 , µ1 , ..., µn and µ on E8 , which we need to prove the following lemmas. C1

r

C2

Cn−2

r

r

C

rP P PP PP PP B r D2 r D4 r r B Pr D2n−4 @ @ @ @ @ @ @ @ @ @ @ @r D2n−3 D@ D D ..... D r @ 1 r 3 r 5 r 2n−5 aa Q !! AQ Q A ! Q aa Q aa A Q A !!! Q aAa Q ! A Q Q !! a Q A a Q!A! Q A Q A a a! a ! Q Q aa QAr r ! QA! A

A

Fig. 11.7. Poset Q0

Definitions Let n ≥ 3. Let Q0 (⊆ E8t ) Further, we take

3

the poset defined by Figure 11.7.

Q0 := {A, A , B, B , C1 , C2 , ..., Cn−2 , C }, Q1 := Q0 \{B} and Q2 := Q0 \{B }. Let 3

Later we choose t = 2 · n.


329

T := (a, a , b, b , c1 , c2 , ..., cn , c ) ∈ E8n+5 . The index of c is always understood modulo n. Let fi : Q0 −→ E8 (i = 1, ..., n) be the following mapping: fi (A) := a,

fi (A ) := a ,

fi (B) := b,

fi (B ) := b ,

fi (Cj ) := ci+j (j = 1, ..., n − 2) and fi (C ) := c . (Clearly, the mapping fi depends on the choice of the tuple T .) We declare T ∈ µ0 iff for each i ∈ {1, ..., n} both (fi )|Q1 and (fi )|Q2 can be extended monotonously to Q0 . Furthermore, for j ∈ {1, ..., n} let T ∈ µj iff T ∈ µ0 and fj can be extended monotonously to a monotone mapping on Q0 . Finally, let n µj . (11.8) µ := j=1

Lemma 11.5.2 If T ∈ µ then T belongs to at least n − 2 of the relations µj with 1 ≤ j ≤ n. Proof. Let T := (a, a , b, b , c1 , ..., cn , c ) and l := |{i ∈ {1, ..., n} | T ∈ µi }|. T ∈ µ implies T ∈ µ0 . If T ∈ µi for all i ∈ {1, ..., n} then l = n. Thus, it is enough to consider the case T ∈ µ0 but T ∈ µi for some i ∈ {1, ..., n}. By the cyclical nature of the relations µi , we may assume T ∈ µ1 . It means that f1 cannot be extended monotonously to Q0 . Because of T ∈ µ0 both (f1 )|Q1 and (f1 )|Q2 have monotone extensions to Q0 . Therefore, f1 is monotone. So by Lemma 11.5.1, there is a zigzag xm , ..., xm in Q0 for f1 . Let H be the set of all elements of Q0 comparable with some xi (m < i < m ). Since xm , ...xm is a zigzag for (f1 )|(Q0 ∩H) as well, it follows that (f1 )|(Q0 ∩H) cannot be extended monotonously to Q0 . This implies that both B and B are in H. But xm+1 , ..., xm −1 is a series of distinct elements in Q0 \ Q0 such that every two consecutive elements are comparable. These two properties imply that {xi | m < i < m } = {Dj | 1 ≤ j ≤ 2n − 3}, where the correspondence is either in direct or reverse order depending on whether f1 (B) = β or f1 (B) = β . So, by condition (g) in the definition of a zigzag {a, a } = {fi (A), fi (A )} = {α, α }, and by condition (f) we have {ci , c } = {fi (Ci−1 ), fi (C )} = {γ, γ } for i = 2, ..., n − 1. W.l.o.g. we may assume that c = γ and ci = γ for all i = 2, ..., n − 1. Because of condition (c) of the definition of a zigzag we have

330


{b, b } = {fi (B), fi (B )} = {β, β }. Using the fact that T ∈ µ0 , one can show that {c1 , cn } ⊆ {γ, γ , 1}. So, we have determined all tuples T ∈ µ0 \µ1 . By symmetry, we get similar characterizations for µ0 \µi for i = 2, ..., n. Applying them, we obtain: c1 = c2 = γ =⇒ l = 0 =⇒ T ∈ µ, ((c1 = γ ∧ cn = γ) ∨ (c1 = γ ∧ cn = γ)) =⇒ l = n − 2, (c1 = γ ∧ cn = γ) =⇒ l = n − 1. Lemma 11.5.3 Let n ≥ 3. For every l < E8 preserve the relation µ.

n 2

the l-ary monotone functions on

Proof. Let g be an l-ary function of P ol8 and let Ti ∈ µ for i = 1, 2, ..., l. Then by Lemma 11.5.2 we have ∀i ∈ {1, 2, ..., l} : |{j ∈ {1, ..., n} | Ti ∈ µj }| ≤ 2. Thus there is an index j ∈ {1, ..., n} with Ti ∈ µj for all i = 1, 2, ..., l. Since one can derive the relations µi (1 ≤ i ≤ n) from the t-th graphic Gt (P ol8 ) by means of the operation pr (projection), the relations mi belong to Inv8 (P ol8 ) (see Chapter 2). Therefore, function g preserves the relation µj , from which we receive g(T1 ..., Tl ) ∈ µj ⊆ µ. Lemma 11.5.4 For n ≥ 3 there exist 2n-ary monotone functions on E8 , which do not preserve the relation µ. Proof. We consider a matrix A of the type (n + 5, 2n), whose construction is given in the following table, where a, a , b, b , c1 , ..., cn , c are the rows of A and T1 , T2 , ..., T2n are the columns of A. Furthermore, let T := (α, α , β, β , γ, γ, ..., γ, γ )T be a column matrix of length n + 5, which is also defined in the following table. a a b b c1 c2 c3 c4 . . . cn c

T1 α α β β 1 γ γ γ . . . γ γ

T2 α α β β γ 1 γ γ . . . γ γ

T3 α α β β γ γ 1 γ . . . γ γ

T4 α α β β γ γ γ 1 . . . γ γ

... ... ... ... ... ... ... ... ... ... ... ... ... ...

Tn Tn+1 Tn+2 Tn+3 Tn+4 α α α α α α α α α α β 1 1 1 1 β 1 1 1 1 γ 1 β γ γ γ β 1 β γ γ γ β 1 β γ γ γ β 1 . . . . . . . . . . . . . . . 1 β γ γ γ γ γ γ γ γ

... ... ... ... ... ... ... ... ... ... ... ... ... ...

T2n α α 1 1 β γ γ γ . . . 1 γ

T α α β β γ γ γ γ . . . γ γ


Let

331

Q := {a, a , b, b , c1 , ..., cn , c } ⊆ E82n

and f1 : Q −→ E8 be a function defined by f1 (A) = f1 (T1 , T2 , ..., T2n ) := T. Clearly, f1 is monotone. Suppose that there is a zigzag xm , ..., xm ∈ E82n for f1 . Thus we have xm = b, xm = b , m < 2i + 1 < m =⇒ a < x2i+1 ∧ a < x2n+1 m < 2i < m =⇒ c > x2i ∧ (∃qi ∈ {1, 2, ..., n} : cqi > x2i ). Let p := % 12 (m − 1)&.4 Each j ∈ {1, ..., n} appears among the numbers q1 , ..., qp , since we could not otherwise choose the j-th coordinates of xm , ..., xm to satisfy the above conditions. Let l (1 ≤ l ≤ n) be the number that appears last from the beginning in the sequence q1 , ..., qp . Then there is an interval in this sequence with first last element l − 1 and l + 1 or the reverse and containing no other element equal to l − 1, l or l + 1 considered again modulo n. However, in this case, we cannot choose the (n + l)-th coordinates of xm+1 , ..., xm −1 to satisfy the above conditions. The contradiction proves that there is no zigzag in E82n for f1 . So, by Lemma 11.5.1, we can extend f1 to a 2n-ary monotone function h with h(T1 , ..., Tn ) = T. Using that {T1 , ..., Tn , T } ⊆ µ0 and the characterization of µ0 \µi (1 ≤ i ≤ n) one can easily check that T1 , ..., Tn belong to µ but T ∈ µ. Thus the 2n-ary monotone function h does not preserve µ. Theorem 11.5.5 ([Tar 86]) does not have a finite order.

For k ≥ 8 there is a ∈ Mk , so that P olk

Proof. Let ∈ Mk defined by the Hasse-diagram of Figure 11.4. Choosing a finite set A ⊂ P ol8 we have a number l, such that every element of A is at most l-ary. Taking n ≥ 2 · l and defining µ corresponding to this n, we get that A ⊆ P ol8 {µ, } ⊆ P ol8 by Lemma 11.5.3, but (P ol8 )n ⊆ P ol8 {µ, } by Lemma 11.5.4. So the finite set A does not generate the whole clone. Presumably ord P olk = ∞ is valid for every relation , in whose diagram one can embed the diagram given in Figure 11.4. 4

x denotes the floor of x ∈ R, i.e., the largest integer which is ≤ x.

332


11.6 Classifications and Basis Enumerations in Pk Let A be a class of Pk that is finitely generating. Then, A has only finite many A-maximal classes M1 , M2 , ..., Mt and it is valid for all B ⊆ A: [B] = A ⇐⇒ (∀i ∈ {1, ..., t} : B ⊆ Mi ). A generating system B := {f1 , ..., fr } of A with the following property is especially interesting: [B \ {fi }] = A for every i ∈ {1, ..., r}. Such a set B is called basis of A and the number r is the rank of B. It is briefly explained in this section, as one can classify the bases of A and one can also determine, then, the ranks of the bases. With all maximal classes M1 , ..., Mt of the class A, can be classified the functions, by their membership in the A-maximal sets as follows: For f ∈ A we put 0 if f ∈ Mi , χi (f ) := 1 if f ∈ Mi (i = 1, 2, ..., t) and χ(f ) := (χ1 (f ), χ2 (f ), ..., χt (f )). We call χ(f ) the characteristic vector of f . We put f ≡ g iff f, g ∈ A and χ(f ) = χ(g). Obviously, ≡ is an equivalence relation on A, and so it partitions A into pairwise disjoint nonempty sets (called equivalence classes or blocks). Let f ∈ A with χ(f ) = (a1 , ..., at ). Then, it is easy to see that the block [f ]≡ of ≡ has the form Ta1 ∩ Ta2 ∩ .... ∩ Tat , where, for i ∈ {1, ..., t}, Tai :=

Mi if ai = 0, A \ Mi for ai = 1.

If f ∈ B ⊆ A and χ(f ) = χ(g), then we have [B] = A ⇐⇒ [{g} ∪ (B \ {f })] = A. In other words, it suffices to study the completeness in A up to the equivalence relation ≡. Further, it is easy to see that ∀B ⊆ A : ([B] = A ⇐⇒ ( χ(f )) = (1, 1, 1, ..., 1)), f ∈B

where is the usual componentwise logical ∨ of the tuples (∈ E2t ). A set B := {f1 , ..., fr } ⊆ A is a basis of A iff [B] = A and

11.6 Classifications and Basis Enumerations in Pk

∀j ∈ {1, ..., r} : (

χ(f )) = (1, 1, 1, ..., 1))

333

(11.9)

f ∈B\{fj }

Once we know all the characteristic vectors, we can find all complete sets in A and all bases by a direct combinatorial check (which may be done by a simple computer program, provided t is not large (see [Sto 87])). If to α := (α1 , ..., αt ) ∈ E2t we associate the set Iα := {i | αi = 1} and if I1 , ..., Im are the subsets of {1, 2, ..., t} corresponding to the characteristic vectors, the completeness problem is reduced to listing irredundant coverings (i.e., no proper subset covers {1, 2, ..., t}). The study of classes also provides information on the classes of Pk , which are the intersections of families of A-maximal sets, which is of independent interest.5 The characteristic vectors can also be applied to seek the set of classes of functions that make an incomplete set complete. One finds many results above characteristic vectors and basis enumerations in the books [Mas 88] and [Sto 87]; e.g., classification of A ∈ {P2 , P3 , Pk,2 , Lp }, p ∈ P, and classifications of the maximal classes of P3 and of Pk,2 . Subsequently, the concepts and ideas introduced above are only explained as an example. Let A = P2 . Then, by Theorem 3.2.4.2, we can say M1 := T0 , M2 := T1 , M3 := S, M4 := L, M5 := M. Then there are exactly 15 characteristic vectors (a1 , ..., a5 ) for P2 ([Jab 52], [Ibu-N-N 63], [Kri 65]): number of (a1 , ..., a5 ) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5

a1 a2 a3 a4 a5 example for f with χ(f ) = (a1 , ..., a5 ) 1 1 1 1 1 x∧y 1 1 0 1 1 (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z) 0 1 1 1 1 x∧y 1 0 1 1 1 x∨y 1 1 0 0 1 x 1 0 1 0 1 x+y+1 0 1 1 0 1 x+y 0 0 1 1 1 x ∧ (y ∨ z) 1 0 1 0 0 c1 0 1 1 0 0 c0 0 0 1 1 0 x∨y 0 0 0 1 1 (x ∧ (y ∨ z)) ∨ (x ∧ y ∧ z) 0 0 0 1 0 (x ∧ y) ∨ (x ∧ z) ∨ (y ∧ z) 0 0 0 0 1 x+y+z 0 0 0 0 0 e11

E.g., for A = P3 with one exception the least nontrivial intersections are all minimal clones.

334


With the aid of the above, one can classify the bases of P2 as follows: Theorem 11.6.1 ( [Ibu-N-N 63], [Kri 65]; without proof ) Let B := {f1 , ..., fr } be an arbitrary basis of P2 . Then r ∈ {1, 2, 3, 4} and for χ(B) := {χ(f1 ), ..., χ(fr )} is valid: |B| the sets of the numbers of the lements of χ(B) 1 {1} 2 {2, x}, where x ∈ {3, 4, 6, 7, 8, 9, 10, 11} {3, x}, where x ∈ {4, 5, 6, 9} {4, x}, where x ∈ {5, 7, 10} {5, x}, where x ∈ {8, 11} 3 {5, x, y}, where x ∈ {6, 7, 9, 10} and y ∈ {12, 13} {6, 7, x}, where x ∈ {8, 11, 12, 13} {6, 6, 10} {6, 10, x}, where x ∈ {11, 12, 13} {7, 8, 9} {7, 9, x}, where x ∈ {11, 12, 13} {9, 10, x}, where x ∈ {8, 12} 4 {9, 10, 11, 14}, {9, 10, 13, 14} In other words: There are exactly 42 aggregates6 for P2 . 1 aggregate has the rank 1, 17 aggregates have the rank 2, 22 aggregates have the rank 3 and 2 aggregates have the rank 4.

6

An aggregate is the set of all bases having the the same set of characteristic vectors.

12 Subclasses of Pk,2

In this chapter, we will to deal with subclasses of the class Pk,2 := {f ∈ Pk | W (f ) ⊆ {0, 1}}. When one restricts the domain of a function f n ∈ Pk,2 to the set E2n , a homomorphic mapping pr (”projection”) from Pk,2 onto P2 can be defined. Since the image prA of a subclass A of Pk,2 is a subclass of P2 and the subclasses of P2 are known, one can hope +to find certain properties of the inverse images (⊆ Pk,2 ) through the known properties of the images (⊆ P2 ). This hope confirmed itself in a certain sense (see e.g. Theorem 12.2.5 and Theorem 9.7.6). On the other hand, Pk,2 , k ≥ 3 also reflects the negative properties of Pk because the examples of classes from Section 8.1.1 with infinite and without bases are subclasses of Pk,2 . Further, the functions of Pk,2 are important since they can be interpreted as predicates. Some further applications are subsequently mentioned: Functions of P3,2 permit the description of a decision (values 0, 1) with abstention from voting (value 2). Special functions of P3,2 are of interest in the theory of noncorrect algorithms (see e.g. [Sch 78]). In [Eps-F-R 74] G. Epstein, G. Frieder and D. C. Rine mention that functions of Pk,2 are useful for describing logico-arithmetical branchings in programs where the arithmetical constants (mostly k > 2) are arguments and the two logical constants form the range. They also give a function of P3,2 which is used in control of real-time processes and in aeronautics. In 1973/1974 G. Burosch dealt with the set Pk,2 . Some years later he, J. Dassow, W. Harnau and the authoress continued the study of the subclasses of Pk,2 . The investigations of Pk,2 concerned the following problems:

336


(1) For a given subclass B of P2 determine the set of all subclasses of Pk,2 , which can be projected to B, i.e., the restrictions of the functions to arguments of E2 gives a function of B. (2) As completely as possible, construct the lattice of the subclasses of P3,2 . (3) For a given subclass of Pk,2 , decide whether it is finitely generated and construct a system of generators (if possible). A summary of the achieved results on the closed subsets of Pk,2 was published in [Bur-D-H-L 85]. Unfortunately, a proof could not be given in [Bur-D-H-L 85] for every theorem, since the extent of this proof was too great. After B. Csák´ any showed the author how the Theorem of Baker and Pixley can be generalized (see Theorems 8.3.1 and 12.3.1), she found new and shorter proofs. The subsequently proofs to the theorems and lemmas without reference come mostly from the papers [Lau 88;a;b] (or [Lau 84c]) but also from [Bur-D-H-L 85] and [Lau 77;a;b]. In addition, the results of N. Gr¨ unwald from [Gr¨ u 83;a;b]which continued the investigations of the team Burosch-Dassow-Harnau-Lau. The chapter is organized as follows: Section 12.1 contains the basic concepts and notations. In Section 12.2, one can find results on inverse images of subclasses of P2 (with respect to the above projection). The remaining sections deal with the determination of the cardinality and with the determination of the elements of the set Nk (B) := {A ∈ L↓k (Pk,2 ) | pr A = B} for subclasses B of P2 . In Section 12.3, one can find some structure statements about L↓k (Pk,2 ). In addition, this section clarifies whether the set Nk (B) is finite or infinite or has the cardinality of continuum. In Section 12.4, one can find all subclasses A of Pk,l whose projection is the class Pl or P oll {α} (α ∈ El , 2 ≤ l < k). After that, in Section 12.5, the maximal and the submaximal classes of Pk,2 are determined. Then, the investigations are continued from Section 12.3 for k = 3, i.e., for many classes B ⊆ P2 with |N3 (B)| ≤ ℵ0 the elements of the set N3 (B) are determined.

12.1 Notations In this chapter, let 2 ≤ l < k and Pk,l as defined in Section 1.1. In general, we define functions of Pk,l by formulae over an alphabet X := {x, x1 , x2 , ...}. In contrast, we define the Boolean functions over the alphabet Y := {y, y1 , y2 , ...}, and we use the notations for functions and closed sets of P2 from Chapter 3. By k1 and ja , where a ∈ Ek , we denote the unary functions of Pk,l given by

12.2 Some Properties of the Inverse Images

k1 (x) :=

1 if x ∈ E2 , 0 otherwise,

ja (x) :=

337

1 if x = a, 0 otherwise,

respectively. To characterize the subclasses of Pk,l we need the following homomorphism of Pk,l onto Pl , which we denote by prl or only with pr and which we call “projection” 1 ): For f n ∈ Pk,l and g m ∈ Pl let pr f n = g m if and only if n = m and f (a) = g(a) for all a ∈ Eln . If B ⊆ Pl , we call the subset pr−1 B := {f ∈ Pk,l | pr f ∈ B} of Pk,l inverse image of B. We say a subclass A of Pk,l is B-projectable iff prl A = B. Denote Nk (B) the set of all B-projectable subclasses of Pk,l . By P olPk,l := Pk,2 ∩ P olk one can describe subclasses of Pk,l . If the index Pk,l can be seen from the context, we write only P ol instead of P olPk,l . . If B = P oll (⊆ Pl ), ⊆ Elh and a1 , ..., ar are some tuples of Ekh \Elh , then we also denote the closed set P olPk,l ( ∪ {a1 , ..., ar )} with B a1 ,...,ar . In particular, these notations will be used in Section 12.6 for l = 2 and B ∈ {T0 , T1 , M }. Furthermore, let 0 1 ... l − 1 a {a, b} ⊆ Ek . Za,b := P olPk,l 0 1 ... l − 1 b

12.2 Some Properties of the Inverse Images With the mapping pr, we can prove that some properties of the subclasses of Pl are transmitted to their inverse images in Pk,l . 1

n To avoid confusion with the notation for functions en i , the functions ei are called selectors at some places of this chapter.

338


Lemma 12.2.1 (a) Let A ⊆ Pk,l , [pr A] = Pl and b ∈ Ek . Then, the set A ∪ {ja | a ∈ Ek \{b}} is a generating system for Pk,l . (b) ord Pk,l = 2. Proof. (a): Since pr A is a generating system for Pl , there are binary functions h and g of [A] with (pr h)(y1 , y2 ) = y1 + y2 (mod l) and (pr g)(y1 , y2 ) = y1 · y2 (mod l). Furthermore, the constant functions c0 , c1 , ..., cl−1 and the function k−1 jb (x) = 1 + (l − 1) · ja (x) (mod l) a=0, a=b

belong to [A]. Further, any function f n of Pk,l has a representation of the form f (x1 , ..., xn ) = f (a1 , ..., an ) · ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ) (mod l), (a1 ,...,an )∈Ekn

(12.1) i.e., f belongs to [{g, h, c0 , ..., cl−1 , j0 , ..., jk−1 }] ⊆ [A]. (b) follows from (a) and Theorem 1.4.2, (b). Lemma 12.2.2 Let B be a subclass of Pl , which contains the set Jl of all projections (selectors) of Pl . Then, for every set A ⊆ Pk,l with [pr A] = pr B, the set A ∪ pr−1 Jl is a generating system for pr−1 B. Proof. Let f n ∈ pr−1 B. Since [pr A] = B, there exists a function f1n ∈ [A] with pr f1 = pr f . In pr−1 Jl one can find the (n + 1)-ary function xn+1 if x ∈ Eln+1 , h(x) := f (x1 , ..., xn ) otherwise. Consequently, f (x1 , ..., xn ) = h(x1 , ..., xn , f1 (x1 , ..., xn )) is a superposition over A ∪ pr−1 Jl . By the above lemma, we obtain the following theorem. Theorem 12.2.3 If Jl ⊆ B ⊂ B ⊆ Pl and B is a maximal class of the class B then is also pr−1 B maximal in pr−1 B. It is easy to see that the above theorem does not hold if B does not contain Jl (see also Theorem 12.2.6). Lemma 12.2.4 The order of pr−1 Jl is 3.


Proof. Let

i(x) :=

339

x if x ∈ El , 0 otherwise.

Every n-ary function f of pr−1 Jl with pr f = eni can be represented in the following manner: f (x1 , ..., xn ) = i(xi ) · k1 (x1 ) · ... · k1 (xn )+ a∈Ekn \Eln f (a1 , ..., an ) · ja1 (x1 ) · ... · jan (xn ) (mod l). (12.2) We prove that (12.2) is a superposition over i(x1 ) · k1 (x2 ) (mod l), i(x1 ) + a · jq (x2 ) (mod l), i(x1 ) + i(x2 ) · jq (x3 ) (mod l),

(12.3)

i(x3 ) + i(x1 ) · jp (x2 ) · jq (x3 ) (mod l), (p, q ∈ Ek ; q > 1, a ∈ El ). First, we construct the function i(x) + a · ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ) (mod l)

(12.4)

for (a1 , ..., an ) ∈ Ekn \Eln , a ∈ El . W.l.o.g. let a1 ∈ Ek \El . The function i(x)+(i(x1 )+i(x3 ))ja2 (x2 )ja1 (x1 )ja1 (x1 ) = i(x) + i(x3 )ja1 (x1 )ja2 (x2 ) (mod l) is obtained from i(x) + i(x2 )ja1 (x1 ) if we substitute the variable x2 by i(x1 )+i(x3 )ja2 (x2 )ja1 (x1 ). By substitution of x3 by i(x1 ) + i(x4 )ja3 (x3 )ja1 (x1 ) we generate the function i(x) + i(x4 )ja1 (x1 )ja2 (x2 )ja3 (x3 ) (mod l). By iterated application of these constructions we obtain the function i(x) + i(xn+1 )ja1 (x1 )ja2 (x2 )...jan (xn ) (mod l).

(12.5)

The function (12.4) is generated from (12.5) by substituting xn+1 by i(x1 ) + a · ja1 (x1 ). Obviously, i(x) + f (a1 , ..., an )ja1 (x1 )ja2 (x2 )...jan (xn ) (mod l) (12.6) a∈Ekn \Eln

is a superposition of functions of type (12.4). If we substitute x in (12.6) by i(xi ) · k1 (x1 ) · ... · k1 (xn ) ∈ [i(x) · k1 (x1 )] we obtain the function (12.2); i.e., (12.3) is a generating system for pr−1 Jl and ord pr−1 Jl ≤ 3. Since the functions of (pr−1 Jl )2 preserve the relation

340


λl ∪ {(2, 2, 2, 2)} (see Lemma 4.1), the function i(x1 ) + j1 (x1 )j1 (x2 )j2 (x3 ) ∈ pr−1 Jl does not preserve this relation, however, we have ord pr−1 Jl = 3. Theorem 12.2.5 Let B ⊆ Pl be a clone. Then ord B ≤ ord pr−1 B ≤ max(3, ord B). Every inverse image (⊆ Pk,2 ) of a clone B ⊆ P2 is finitely generated. Proof. The statements of the theorem are consequences of Theorems 12.3.2 and 12.3.4 and of Chapter 3. The following theorem gives information on the order of the remaining inverse images (⊆ Pk,2 ) of subclasses (⊆ P2 ). Theorem 12.2.6 The inverse images pr−1 C0 , pr−1 C1 and pr−1 C have no finite basis. Proof. Let A be an inverse image of Pk,2 whose projection is generated by {ca , cb }, {a, b} ⊆ E2 . Then A contains the functions ⎧ ∃i, α : xi = 2 ∧ ⎨ α if x1 = ... = xi−1 = xi+1 = ... = xn = α ∈ E2 , f n (x1 , ..., xn ) := ⎩ a otherwise for all n ∈ N. A has no finite basis if we can prove that, for each n, the function f n is not a superposition of the (n − 1)-ary functions of A. It suffices to prove that f n has no representation by a formula h(x1 , ..., xn ) := g0 (g1 (x1 , ..., xn ), ..., gn−1 (x1 , ..., xn )),

(12.7)

where gi (0 ≤ i ≤ n) is a function of A or gi (1 ≤ i ≤ n) is defined by gi (x1 , ..., xn ) = xj

(12.8)

for some j ∈ {1, 2, ..., n}. W.l.o.g. we can assume that g1 is not a function of form (12.8). Then, for α ∈ E2 βα := (g1 (2, α, ..., α), ..., gn−1 (2, α, ..., α)) is a tuple of E2n and thus g0 (β0 ) = g0 (β1 ). However by definition f n (2, α, ..., α) = α for every α ∈ E2 . Therefore, formula (12.7) does not define the function f n.


341

Theorem 12.2.7 (a) Let A ⊆ Pk,l be an inverse image whose projection prl A is a clone. Then A has a basis with exactly r elements if and only if pr A has such a basis. (b) ∃f ∈ Pk,l : [f ] = Pk,l . Proof. (a): If [{f1 , ..., fr }] = A then [{pr f1 , pr f2 , ..., pr fr }] = pr A holds. Let pr A = [{g1 , ..., gr }], where g1 denotes a function that stands in the construction formula of e11 “outside”; i.e., we have e11 (x) = g1 (t1 (x), ..., tn (x)) for certain t1 , ..., tn of [{g1 , ..., gr }]. Further, denote f1 , ..., fr certain functions of A with pr fi = gi , i = 1, ..., r and let IV f (x1 , ..., xn , x, ..., xp,q , ..., xp,q , ..., xp,q , ..., x p,q , ..., xp,q , ...) i(xp,q ) · jp (xp,q ) · jq (xp,q ) := f1 (x1 , ..., xn ) · k1 (x) + p, q

q≥l 0 ≤ p < q ≤ k−1

+

IV jp (x p,q ) · jq (xp,q ) (mod l). p, q q≥l 0 ≤ p < q ≤ k−1

We prove that {f, f2 , ..., fr } is a generating system for A. By Lemmas 12.2.2 and 12.2.4, it is sufficient to show that the function system (12.3) belongs to [{f, f2 , ..., fr }]. Due to the choice of the function f1 and by e11 ∈ prl A, the function IV (i(xp,q )jp (xp,q )jq (xp,q ) + jp (x i (x1 )k1 (x) + p,q )jq (xp,q )) p, q q≥l 0≤p d, then by definition of fc,d the statement (12.12) holds. If a = a then w = w , since all considered functions belong to pr−1 S. For a = a the two cases are possible:

12.5 The Maximal and the Submaximal Classes of Pk,2

357

a p has a column with a q p 0 0 1 1 ∈ , , , . 1 0 1 q 0 a q Case 2: The matrix has a column with q ∈ E2 . a q In the second case, w = w holds. In the first case, one can w= w easily prove hi,t (i) 1 = . when one one assumes p > q considering the condition hi,t (t) 1 Case 1: The matrix

Let h be a mapping from W onto E2 defined by h(w) = h(w1 , ..., wσ ) := f (a). Because of (12.12) and (12.13), an α-ary function h1 ∈ S can be found, whose restriction onto W is identical with h. Hence there is an inverse image h of h and hi,t . Thus in [A], and, by construction, f is a superposition over h , fi,t −1 f ∈ [A] and [A] = pr S; i.e., the third statement of Theorem 12.5.4 was proven. One checks the remaining statements of the theorem easily. Lemma 12.5.5 Let A ⊆ pr−1 M , [pr M ] = M and {k1 , j0 (x1 ) · ji (x2 ) | i = 2, 3, ..., k − 1} ⊆ [A]. Then [A] = pr−1 M . Proof. Obviously, for every function g ∈ M there exists a function g ∈ [A] with pr g = g. Consequently, certain inverse images of the conjunction, disjunction, n n = c0 (n ∈ [A], pr n = c0 ), i(x) · k1 (x) = j1 (x) (i ∈ [A], pr i = pr j1 ) and j0 (c0 (x)) · jp (x) = jp (x), 2 ≤ p ≤ k − 1 are superpositions over A. One can describe every function f n = cn0 of pr−1 M by f (x1 , .., xn ) = f1 (x1 , ..., xn ) · k1 (x1 ) · ... · k1 (xn ) ∨ (a1 , ..., an ) ja1 (x1 ) · ja2 (x2 ) · ... · jan (xn ),

(12.14)

∈ Ekn \E2n f (a1 , ..., an ) = 1

where f1 ∈ [A] and pr f1 = pr f . (12.14) is obvious a superposition over A and thus [A] = pr−1 M .

Theorem 12.5.6 (1) pr−1 M has exactly the following maximal classes: (a) pr−1 B, where B is maximal in M ; (b) Zi,t ∩ pr−1 M , 2 ≤ t < i < k;

358


(0,r) (c) M , M (r,s) , 1≤ s < k, 2 ≤ r < k; 0 1 0 a (a,b) := P ol . M 0 1 1 b (2) The sets, defined in a)–(c), with different definitions are pairwise incomparable (with respect to inclusion) so that pr−1 M has exactly 4 + 32 (k − 1) · (k − 2) maximal classes. (3) A set A ⊆ pr−1 M is pr−1 M -complete if and only if A ⊆ T holds for every class T which is defined in (a)–(c).

Proof. Obviously, if [A] = pr−1 M , then A is not a subset of the sets (a)–(c). Now let A be subset of pr−1 M with A ⊆ T for every class T defined in (a)–(c). Then we have [pr A] = M and c0 , c1 ∈ [A]. Further, [A] contains functions fi,t , gr,s , qr and pr with the properties: fi,t ∈ Zi,t ,

gr,s ∈ M (r,s) ,

qr ∈ M (r,r) ,

pr ∈ M (0.r) .

, gr,s , qr and pr with Then certain unary or binary functions fi,t (i) = fi,t (t), fi,t 0 r 1 gr,s , = 1 s 0

qr

0 r 1 r

=

1 0

and

pr

0 0 1 r

=

1 0

,

2 ≤ t < i ≤ k − 1, 1 ≤ s ≤ k − 1, 2 ≤ r ≤ k − 1 are obvious superpositions over {c0 , c1 , fi,t , gr,s , qr , pr }. For the function (x), x), if fi,t (i) = 0, gi,t (fi,t fi,t (x) := gt,i (fi,t (x), x), if fi,t (i) = 1, (a) for a ∈ {i, t}. Thus w.l.o.g. we can assume for all (a) = fi,t it holds fi,t (i) = 1 and fi,t (i) = 0. permissible i, t in the following: fi,t Let i ∈ [A], pr i = pr j1 and let E be the set of all A ∈ Ek \E2 with i(a) = 0. (i(x), x) =: gr,1 (x), we have gr,1 (r) = 1 and gr,1 (1) = Then, for r ∈ E and gr,1 0. Consequently, i(x) ∨ gr,1 (x) = j0 (x) r∈E

and r−1

qr (i(x1 ), x2 )·gr,1 (i(x1 ), x2 )·(

fr,t (x2 ))·(

t=2

k−1

fi,r (x2 )) = j0 (x1 )·jr (x2 ) ∈ [A]

i=r+1

for r ≥ 2. Furthermore, it holds k−1

r=2

pr (j0 (c0 (x)) · jr (x), x) = k1 (x) ∈ [A].

12.5 The Maximal and the Submaximal Classes of Pk,2

359

Thus, we showed that the generating system for pr−1 M of Lemma 12.5.5 belongs to [A]. Therefore, the set A is pr−1 M -complete. One checks the remaining statements of the theorem easily. We come now to the maximal classes of pr−1 L. First we give some remarks on a possible description of functions of pr−1 L. Because of j0 = 1 + j1 (x) + ... + jk−1 (x) it follows from Section 12.1 that every function f n ∈ Pk,2 has an unambiguous description (up to the order of the summands) of the form aI1 ,...,Ik−1 · kI1 ,...,Ik−1 (x), (12.15) f (x) = a + I1 , ..., Ik−1 ⊆ {1, 2, ..., n}

where aI1 ,...,Ik−1 ∈ E2 , kI1 ,...,Ik−1 (x1 , ..., xn ) :=

k−1

ji (xq )

(12.16)

i=1 q∈Ii

and the sets I1 , ..., Ik−1 in (12.15) are pairwise disjunct. The functions kI1 ,...,Ik−1 with aI1 ,...,Ik−1 = 1 in (12.15) are called components of f . Let Kf be the set of all components of f . Denote kf,I an n-ary function of the form kI1 ,...,Ik−1 (x), kf,I (x) := I1 , ..., Ik−1 I1 ∪ ... ∪ Ik−1 = I kI1 ,...,Ik−1 ∈ Kf

where I ⊆ {1, 2, ..., n}. Lemma 12.5.7 Let {f, g, h} ⊆ pr−1 L, (pr g)(y1 , y2 ) = y1 + y2 and (pr h)(y) = y. Then every function of the form kf,I is a superposition over {c0 , f, g, h}. Proof. It is easy to see that the function f (x) := f (x) + f (0, 0, ..., 0) is a superposition over the functions f and h. Denote r the smallest number, for which f has a component with r essential variables. W.l.o.g. let f (x) = kf,{1,2,...,r} + f (x). Then it holds f (x1 , ..., xr , c0 , ..., c0 ) = kf,{1,2,...,r} (x) ∈ [{c0 , f, g, h}] and

f (x) = g(f (x), kf,{1,2,...,r} (x)) ∈ [{c0 , f, g, h}],

where Kf ⊂ Kf ⊆ Kf and every component of f has at least r essential variables. Through repetition of this construction, one receives the statement of the lemma.

360


Theorem 12.5.8 (1) pr−1 L has exactly the following maximal classes: (a) pr−1 B, where B is a maximal class of L; (b) Zi,t ∩ pr−1 L, 2 ≤ t < i < k; (c) Lq := P olPk,l {(q, q, q, q), (a, b, c, d) | {a, b, c, d} ⊆ E2 ∧ a + b = c + d (mod 2)}, 2 ≤ q < k. (2) The classes listed in (a)–(c) are pairwise incomparable (with respect to the inclusion) so that there are exactly 4+ 12 (k−1)·(k−2) pr−1 L-maximal classes. (3) A set A ⊆ pr−1 L is pr−1 L-complete if and only if A ⊆ Q holds for all classes Q listed in (a)–(c). Proof. The statements (1) and (2) are consequences from (3). Since “=⇒” of (3) is trivial, we prove only (3), “⇐=”. Let A be a subset of pr−1 L with A ⊆ Q for all classes Q which are defined in (a)–(c). Because of A ⊆ pr−1 B, B maximal in L, pr [A] = L holds. Consequently, c0 , c1 ∈ [A] and one can find functions g, h ∈ [A] with (pr g)(y1 , y2 ) = y1 +y2 and (pr h)(y) = y. Since A is, in addition, not a subset of the sets Zi,t , 2 ≤ t < i ≤ k − 1, there are unary functions gi,t ∈ [A] with gi,t (i) = 1 and gi,t (t) = 0, t = i, 2 ≤ i ≤ k − 1, 2 ≤ t ≤ k − 1. Let now q ∈ {2, 3, ..., k − 1}. Since the functions of A not all preserve the relation ⎛ ⎞ 0 0 0 1 1 0 1 1 q ⎜0 0 1 1 0 1 0 1 q⎟ ⎜ ⎟ ⎝ 0 1 0 0 1 1 0 1 q ⎠, 0 1 1 0 0 0 1 1 q a function f with pr(f (x1 , ..., xn , q)) ∈ L belongs to [A]. Then by the maximality of L in P2 and by pr[A] = L, for every function um ∈ P2 there is a (m + 1)-ary function u ∈ [A] with u (y1 , ..., ym , q) = u(y1 , ..., ym ). Consequently, a function of the form k v(x1 , ..., xk ) = a + i=1 ai ji (xi ) + j1 (x1 ) · ... · j1 (xk−1 ) · jq (xk ) + k−1 p=2,p=q vp (x1 , ..., xk−1 ) · jp (xk ) belongs to [A] for some a, ai ∈ E2 and some functions vp ∈ Pk,2 . By Lemma 12.5.7 the functions of the type kv,I , I ⊆ {1, 2, ..., k}, are superpositions over A. If one adds all functions of the form kv,I with |I| ≤ k − 1 to v, one receives so the function

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S

361

v(x1 , ..., xk ) := j1 (x1 ) · ... · j1 (xk−1 ) · jq (xk ) + aI1 ,...,Ik−1 · kI1 ,...,Ik−1 (x) · jp (xk ), I1 , ..., Ik−1 , p I1 ∪...∪Ik−1 = {1, ..., k−1} p ∈ {2, 3, ..., k − 1}\{q}

which also belongs to [A] and for which v (x1 , x2 , gq,2 (x3 ), gq,3 (x3 ), ..., gq,q−1 (x3 ), gq,q+1 (x3 ), ..., gq,k−1 (x3 ), x3 ) = j1 (x1 ) · j1 (x2 ) · jq (x3 ) ∈ [A] holds. With the help of (12.15) it is easy to see that every function of pr−1 L is a superposition over {c0 , c1 , g, h} ∪ {j1 (x1 )j1 (x2 )jq (x3 ) | q ∈ {2, 3, ..., k − 1}}. Thus, [A] = pr−1 L.

12.6 The Classes A with M ∩ T0 ∩ T1 ⊆ prA or L ∩ T0 ∩ S ⊆ prA or prA = M ∩ S In this and in the following section, we specify the cardinality statements of Section 12.3 about Nk (A) (A ∈ L2 ) for k = 3. In addition, we provide a concrete description of the elements of N3 (A) if |N3 (A)| ≤ ℵ0 is valid. Table 12.1 gives a first survey of these statements. The statements of the last three rows of Table 12.1 were proven already in Theorems 12.3.8, 12.3.10, and 12.3.11. The cardinality statements of the eighth row are a consequence of Theorem 12.3.1. The remaining statements in the table result from the following theorems. Theorem 12.6.1 It holds: N3 (P2 ) = {P3,2 , Z2,0 , Z2,1 }; N3 (S) = {pr−1 S, pr−1 S ∩ Z2,0 , pr−1 S ∩ Z2,1 }; (2) N3 (Ta ) = {pr−1 Ta , Ta , pr−1 Ta ∩ Z2,0 , pr−1 Ta ∩ Z2,1 } (a ∈ E2 ); (2) (2) N3 (T0 ∩ T1 ) = {pr−1 (T0 ∩ T1 ), T0 ∩ pr−1 (T0 ∩ T1 ), T1 ∩ pr−1 (T0 ∩ T1 ), Z2,0 ∩ pr−1 (T0 ∩ T1 ), Z2,1 ∩ pr−1 (T0 ∩ T1 )}; (2) (2) (5) N3 (T0 ∩ S) = {pr−1 (T0 ∩ S), T0 ∩ pr−1 (T0 ∩ S), T1 ∩ pr−1 (T0 ∩ S), −1 −1 Z2,0 ∩ pr (T0 ∩ S), Z2,1 ∩ pr (T0 ∩ S)}.

(1) (2) (3) (4)

(One finds Hasse diagrams of some of the sets listed above in the Figure 12.1.)

362

12 Subclasses of Pk,2 Table 12.1 |N3 (B)| 3

B P2 , S Ta (a ∈ E2 )

4

T0 ∩ T1 , T0 ∩ S

5

M

23

M ∩ Ta (a ∈ E2 )

36

M ∩ T0 ∩ T1

49

Ta,2 (a ∈ E2 )

148

Ta,m , Ta,m ∩ Ta , Ta,m ∩ M, Ta,m ∩ Ta ∩ M, S ∩ M

< ℵ0

(a ∈ E2 , m ∈ {2, 3, ...}) L, L ∩ T0 , L ∩ T1 , L ∩ S, L ∩ T0 ∩ S Ta,∞ , Ta,∞ ∩ Ta , Ta,∞ ∩ M, Ta,∞ ∩ Ta ∩ M (a ∈ E2 )

c

K ∪ C, K ∪ Ca , K, D ∪ C, D ∪ Ca , D (a ∈ E2 )

c

[P21 ],

c

I ∪ C, I, I ∪ Ca , I, C, Ca (a ∈ E2 )

pr −1 r P2

@

r Z2,0

ℵ0

pr −1 r T0

@

@

@ @r Z2,1

r

(2) T0

pr

−1

pr −1 Tr0 ∩ T1

@

@

@ @r

T0 ∩ Z2,1

rpr−1 T0 ∩ Z2,0

@

@(2) r pr −1T0 ∩ T1 ∩ T0(2) @r −1

pr

r

T0 ∩ T1 ∩ T1

pr −1 T0 ∩ T1 ∩ Z2,0

r

pr −1 T0 ∩ T1 ∩ Z2,1

Fig. 12.1

Proof. The intersections of pr−1 B, B ⊆ P2 , with Z2,0 or Z2,1 are always the smallest B-projectable subclasses of P3,2 . Thus, (1) is a consequence from Theorem 12.5.1 and (2) is a consequence from Theorem 12.5.4. By Theorem 12.5.3, the maximal classes of pr−1 Ta belonging to N3 (Ta ) are just the sets (2) Z2,a ∩ pr−1 Ta and Ta = P olP3,2 {a, 2}. With the aid of Theorem 12.4.4, it is easy to see that Z2,a ∩ pr−1 Ta is the only Ta -projectable maximal class of (2) Ta . From that and from the above remarks, (3) follows. By Theorem 12.3.4, the statements (4) and (5) are consequences from (1)–(3).


363

We come now to the M -projectable subclasses A of P3,2 . Because of h2 ∈ M , Theorem 12.3.1, and Lemma 12.3.5 one can find some 1 , ..., m ⊆ E32 \E22 and an A with pr A = P2 for such class A so that & %m 0 0 1 P ol ∪ i ∩ A (12.17) A= 0 1 1 i=1

holds. Because of Theorem 12.3.6, (6) must be valid for the relations 1 , ..., m in addition: 2 1 (12.18) ⊆ 1 ∪ 2 ∪ ... ∪ m . 0 2 We call relations 1 , ..., m with the above property (12.18) M -permissible. The following lemma gives some properties of M -projectable subclasses of P3,2 , which we need to prove Theorem 12.6.3. Lemma 12.6.2 Let be the relations 1 , ..., m ⊆ E32 \E22 M -permissible and let a ∈ E2 . Then: 0 0 1 0 0 1 (1) 1 ⊆ 2 =⇒ P ol ∪ 2 ⊆ P ol ∪ 1 ; 0 1 1 0 1 1 m a 2 0 0 1 (2) ⊆ 1 ∪ ... ∪ m =⇒ i=1 P ol ∪ i ⊆ pr−1 M ∩ Z2,a ; 2 a 0 1 1 0 0 1 1 0 0 1 0 1 (3) P ol = P ol ; 0 1 1 2 2 0 1 1 2 0 0 1 0 2 0 0 1 0 0 0 1 2 (4) P ol = P ol ∩ P ol ; 0 1 1 1 0 1 1 2 1 0 1 1 2 0 0 1 1 2 0 0 1 0 1 2 (5) P ol = P ol ; 0 1 1 2 2 0 1 1 2 2 2 0 0 1 2 2 0 0 1 2 0 0 1 2 (6) P ol = P ol ∩ P ol ; 0 1 1 0 1 0 1 1 0 0 1 1 1 0 0 1 2 2 0 0 1 2 2 2 (7) P ol = P ol ; 0 1 1 0 2 0 1 1 0 1 2 0 0 1 0 2 2 0 0 1 0 2 0 0 1 2 2 (8) P ol = P ol ∩ P ol . 0 1 1 2 1 2 0 1 1 2 2 0 1 1 1 2 Proof. The statement (1) is trivial. Since ⎧ 0 0 1 0 2 ⎪ ⎪ ⎨ 0 1 1 2 0 0 0 1 2 0 0 1 a ◦ =: = 0 1 1 a 0 1 1 2 ⎪ 0 0 1 1 2 ⎪ ⎩ 0 1 1 2 1 0 1 0 1 2 and ( ∩ τ ) ◦ = , we have 0 1 0 1 a 0 0 1 2 0 0 1 a P ol ∩ ⊆ pr−1 M 0 1 1 a 0 1 1 2

2 2 2 2

2 if a = 0, 1 0 if a = 1 2

∩ Z2,a .

364


With the aid of (1), statement (2) results. (3) follows from (1) and 0 0 1 0 0 1 1 0 0 1 0 1 ◦ = . 0 1 1 0 1 1 2 0 1 1 2 2 Because of (1) and 0 0 1 2 0 0 1 0 0 0 1 2 0 ◦ = 0 1 1 1 0 1 1 2 0 1 1 1 2 (4) holds. (5) follows from (1) and 0 0 1 1 2 0 0 1 1 2 0 0 1 0 1 2 ◦ = . 0 1 1 2 2 0 1 1 2 2 0 1 1 2 2 2 The classes from (6) 1, 1 → 0, 2 → 2). The last statement of 0 0 1 0 0 1 1 2

or (7) are isomorphic to those from (3) or (4) (0 → the lemma follows from (1) and 2 0 0 1 2 2 0 0 1 0 2 2 ◦ = . 2 0 1 1 1 2 0 1 1 2 1 2

Theorem 12.6.3 The M -projectable subclasses of P3,2 are exactly the following: (1) pr−1 M , (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

0 0 1 0 M := P ol , 0 1 1 2 0 0 1 2 M (22) := P ol , 0 1 1 2 0 0 1 2 M (21) := P ol , 0 1 1 1 0 0 1 1 M (12) := P ol , 0 1 1 2 M (02) ∩ M (22) , M (02) ∩ M (21) , M (21) ∩ M (22), 0 0 1 2 (20) M := P ol , 0 1 1 0 M (12) ∩ M (22) , 0 0 1 0 2 M (02)(22) := P ol , 0 1 1 2 2 M (02) ∩ M (22) ∩ M (21) , (02)


(13) M (21)(22) := P ol (14) (15) (16) (17) (18) (19) (20) (21) (22) (23)

0 0 1 2 2 0 1 1 1 2

M (20) ∩ M (22) , M (12) ∩ M (02)(22) , M (21) ∩ M (02)(22) , M (02) ∩ M (21)(22) , M (20) ∩ M (21)(22) , 0 M (02)(12)(22) := P ol 0 0 M (02)(21)(22) := P ol 0 0 M (20)(21)(22) := P ol 0 −1 pr M ∩ Z2,1 , pr−1 M ∩ Z2,0

0 1 0 1 0 1

1 1 1 1 1 1

0 2 0 2 2 0

,

1 2 2 1 2 1

2 , 2 2 , 2 2 , 2

(see Figure 12.2).

pr−1 M

r @ @

@ @ M (2,1) M r M r @r @ @ @ @ @ @ @ @ @ (2,0) @ @r M (1,2) r @r r @rM @ @ @ @ @ @ (0,2),(2,2) (2,1),(2,2) @ M M @ @r @r r r r A A A A A A A A A A A A A AAr A r AAr Ar A A A A A r r M (0,2),(2,1),(2,2) r M (2,1),(2,0),(2,2) M (0,2),(1,2),(2,2) S S S S S S SS SS r r (0,2)

(2,2)

pr−1 M ∩ Z2,1

pr−1 M ∩ Z2,0

Fig. 12.2

365

366


Proof. Because of Lemma 12.6.2 (1), 0 0 1 0 0 1 1 0 0 1 0 1 ◦ = 0 1 1 0 1 1 2 0 1 1 2 2

or

0 0 1 2 0 1 1 0

0 0 1 0 0 1 2 2 ◦ = 0 1 1 0 1 1 0 1

we have M (12) ⊆ M (02) or M (20) ⊆ M (21) . With the help of Table 12.2 one can prove the other inclusions (non-inclusions) of the classes (1) - (23). In Table 12.2, the sign + (or −) shows whether a function fi (i = 1, ..., 10) belongs (or does not belong) to a class of the left column of the table, respectively.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23.

pr−1 M M (02) M (22) M (21) M (12) M (02) ∩ M (22) M (02) ∩ M (21) M (21) ∩ M (22) M (20) M (12) ∩ M (22) M (02)(22) M (02) ∩ M (22) ∩ M (21) M (21)(22) M (20) ∩ M (22) M (12) ∩ M (02)(22) M (21) ∩ M (02)(22) M (02) ∩ M (21)(22) M (20) ∩ M (21)(22) M (02)(12)(22) M (02)(21)(22) M (20)(21)(22) pr−1 M ∩ Z21 pr−1 M ∩ Z20

j1 + + + + − + + + + − + + + + − + + + − + + − +

Table 12.2 j0 j2 k1 f1 + + + + + + − + + + + − + − + − + + − + + + − − + − − − + − + − − − + − + + − − + + − − + − − − + − + − − − + − + + − − + − − − + − − − − − + − + + − − + − − − − − + − + − − − − − − −

f2 + − − + − − − − + − − − − − − − − − − − − − −

f3 + + + − + + − − − + − − − − − − − − − − − − −

f4 + + − + − − + − − − − − − − − − − − − − − − −

f5 + + + + − + + + − − + + − − − + − − − − − − −

f6 + + + + + + + + − + − + + − − − + − − − − − −

f7 + + + + − + + + + − − + − + − − − − − − − − −

f8 + + + + − + + + − − − + + − − − + − − − − − −

The functions f1 –f10 from the above table are defined as follows: f1 (x1 , x2 ) := j0 (x1 )j2 (x2 ), f2 (x1 , x2 ) := j1 (x1 )j2 (x2 ), f3 (x1 , x2 ) := k1 (x1 )j2 (x2 ), f4 (x1 , x2 ) := j0 (x1 )j2 (x2 ) ∨ j1 (x2 ),

f9 + + + − + + − − − + + − − − + − − − − − − − −

f10 + − + + − − − + + − − − + + − − − + − − − − −


367

f5 (x1 , x2 ) := j1 (x1 )j1 (x2 ) ∨ j2 (x1 )j2 (x2 ), f6 (x1 , x2 ) := k1 (x1 )j0 (x2 ) ∨ j2 (x1 )j1 (x2 ), f7 (x1 , x2 ) := k1 (x1 )k1 (x2 ) ∨ j2 (x1 )j2 (x2 ), f8 (x1 , x2 ) := k1 (x1 )j2 (x2 ) ∨ j1 (x2 ), f9 (x1 , x2 ) := j1 (x1 )j2 (x2 ), f10 (x1 , x2 ) := j0 (x1 )j2 (x2 ).

Thus it remains to show that no further classes are described by (12.17) than the ones listed above. By Theorem 12.6.1, only the sets P3,2 , Z2,0 , and Z2,1 are possible for A in (12.17). Since pr−1 M ∩ Z2,0 and pr−1 M ∩ Z2,1 are minimal M -projectable classes of P3,2 , the M -projectable classes different from these classes are described as follows m 0 0 1 P ol ∪ i , (12.19) 0 1 1 i=1

where the relations 1 , ..., m are M -permissible subsets of E32 \E22 . With the aid of the statement (2) of Lemma 12.6.2 and considering (12.18), one can be convinced that the relations i , i = 1, ..., m, must be from the set

0 1 2 2 2 0 1 0 2 0 2 1 2 , , , , , , , , , 1 2 21 2 2 2 2 2 2 0 2 2 2 2 2 2 2 2 0 1 2 0 2 2 2 2 2 , , , , , . 0 1 0 2 1 2 2 2 2 2 1 2 0 1 2

Because of Lemma 12.6.2, (3)–(8) we can assume that the relations 1 , ..., m belong to

0 2

1 2 2 2 0 2 2 2 0 1 2 2 2 2 , , , , , , , , . 2 0 1 2 2 2 1 2 2 2 2 0 1 2

Now it is not difficult to find all classes, which are describable through (12.19), in a step-by-step way. Because of Lemma 12.6.2, (1) or Theorem 12.5.6, the largest M -projectable classes lying below pr−1 M are the sets M (02) , M (22) and M (21) . Then (with the help of Lemma 12.6.2, (2)) possible intersections of these sets and the next largest classes of the form (12.19) with m = 1 belong to layers 1–4 in Figure 12.2. As intersections of classes of the fourth layer (i.e., as next classes of the form (3) with m = 1), only the classes of the fifth layer are possible, and so forth. Theorem 12.6.4 Let a ∈ E2 . Then N3 (M ∩ Ta ) = {A ∩ pr−1 M ∩ Ta | A ∈ N3 (M )}∪ (2) ∪{A ∩ Ta | A ∈ N3 (M ) ∧ A ⊆ M (12) ∧ A ⊆ M (20) } and |N3 (M ∩ Ta )| = 36.

368


Proof. W.l.o.g. let a = 0. Put A3 := M ∩ T0 . It is easy to check that the 36 sets given in the theorem are all A3 -projectable and pairwise different (see Figure 12.3). By Theorem 12.3.4, for every class A of N3 (A3 ) there are certain sets A1 , A2 with A = A1 ∩ A2 , A1 ∈ N3 (M ), and A2 ∈ N3 (T0 ). Because of Theorem 12.3.6, (3) the sets of N3 (A3 ) different from Z2,0 ∩ pr−1 A3 and Z2,1 ∩ pr−1 A3 belong to the set (2)

{A ∩ pr−1 A3 , A ∩ T0

| A ∈ N3 (M )\{Z2,0 ∩ pr−1 M, Z2,1 ∩ pr−1 M }}

(2)

(20)

(2)

Since pr(T0 ∩ M (12) ) = M ∩ T0 and pr(M0 ∩ pr−1 (M ∩ T0 )) ⊆ T0 is obviously valid, the set N3 (A3 ) agrees with the set given in Theorem 12.6.4, and it holds |N3 (A3 )| = 36 by Theorem 12.6.3.

Theorem 12.6.5 Exactly 49 subclasses of P3,2 are M ∩ T0 ∩ T1 -projectable, and it holds N3 (M ∩ T0 ∩ T1 ) = {A ∩ pr−1 (M ∩ T0 ∩ T1 ) | A ∈ N3 (M )}∪ (2)

{A ∩ Ta

| a ∈ E2 ∧ A ∈ N3 (M ) ∧ A ⊆ M (12) ∧ A ⊆ M (20) }

(see Figure 12.4, where A4 := M ∩ T0 ∩ T1 ). Proof. Analogous to the considerations in the proof of Theorem 12.6.4 the statements from Theorem 12.6.5 are a consequence of Theorems 12.3.4, 12.6.1, and 12.6.3, where one must consider that (2)

∩ M (12) ∩ pr−1 T1 ) = M ∩ T0 ∩ T1

(2)

∩ M (20) ∩ pr−1 T0 ) = M ∩ T0 ∩ T1 .

pr(T0 and

pr(T1

Fig. 12.3

pr−1 A3 ∩ Z2,0

PP @ PP PP @ PP PP @ (2) PP −1 −1 (0,2) −1 (2,2)@ −1 (2,1) pr A3 ∩ M pr A ∩ M pr A 3 3 ∩M PPrpr A3 ∩ T0 @P r r r P P PP @PPP @PPP @ PP @ PP PP@ PP @ PP PP P PP @ PP @ PP @ PP@ P P −1 (1,2) P P @ @ pr A3 ∩ M PP PPr PPr @P @P @r r rP r r PP PP @ @PPP @ @ P P @ @ PP PP@ PP @ PP PP PP @ PP PP @ PP @ @ −1 (0,2),(2,2) −1 (2,1),(2,2) @ @ P P P pr A3 ∩ M PP @ PP @ pr A3 ∩ M PP Pr @r @P P P @r @rpr−1 A3 ∩ M (2,0) r r r r P P A @ A PPP A PPP PPP @ A PP PP@ A @ A P PPP PP P P PP PP @ PP A A @ A PP @ PP PP A A @ A P PP PPr P @ Ar @r A r r r r Aaa @ @ a A @ @ aa A aa @ @ aa A @ @ a a A rP pr−1 A3 ∩ M (0,2),(2,1),(2,2) @r r @r ar pr−1 A3 ∩ M (0,2),(1,2),(2,2) S @ PPP PP @ S PP S PP @ PP @ S PP @r S r rpr−1 A3 ∩ M (2,0),(2,1),(2,2) @ pr−1 A3 ∩ Z2,1 @ @ @ @r

pr−1r A3

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S 369

−1

pr −1 A4 ∩ Z2,1

Fig. 12.4

pr −1 A4 ∩ Z2,0

pr r A4 P PP @ PP PP @ PP −1 @ −1 (2) (2) (0,2) −1 (2,2) pr −1 A4 ∩ T pr −1 A4 ∩ T0 pr A4 ∩ M pr A4 ∩ M pr A4 ∩ M (2,1) P PP 1 @ r r r r r PP PP PP @ @ @ @ PP PPP PPP @ P@ PP PP PP @ @ @ PPP PPP PPP@ @ @ PP PPr PPr @r r @r @r r r @ r P PP PPP @ P P @ P @ @ @ @ P P P PP P PP @ @ @ @ PP @ PP PPP @ @ @ PP@ PP@ P @ @ −1 (0,2),(2,2) −1 (2,1),(2,2) P ∩M ∩M PPr PP PP @rpr−1 A4 ∩ M (2,0) @rpr A4 P r r @r @r r @ r @ rpr A4 P pr −1 A4 ∩ M (1,2) r P P P P APP @ PP@ APP @ A P @ A PP PPP P@ @ PP A @ A PPP PPP@ PPP @ PPr PP PPr A A @r ra r r @r r @ r r ! ! @ A!a @ @ !a aa @ @ @ !! A aa ! @ A @ @ ! a aar @r A @r!! @r r r r P PP (0,2),(2,1),(2,2) @ @ pr −1 A 4 ∩M P PP @ PP @ @ PP@ PP rpr−1 A4 ∩ M (2,0),(2,1),(2,2) @ @r r pr −1 A4 ∩ M (0,2),(1,2),(2,2) r @ @ @ @ @ @ @r @r

370 12 Subclasses of Pk,2


371

Now we come to the determining of sets N3 (A) with A ∈ {L, L ∩ T0 , L ∩ T1 , L ∩ S, L ∩ T0 ∩ S}. As already shown in Section 12.5 while preparing proof of Theorem 12.5.8, there are some a, ai , aI,J of E2 with f (x) = a +

n i=1

ai j1 (xi ) +

aI,J · kI,J (x)

(12.20)

I, J I ∪ J ⊆ {1, ..., n} I ∩J =∅ J = ∅

for every function f n ∈ pr−1 L. The functions

kI,J (x) := ( j1 (xi )) · ( j2 (xj )) i∈I

j∈J

with aI,J = 1 in (12.20) are called components of f . Let Kf bethe set of all components of f . Further, let K := f ∈pr−1 L Kf and KM := f ∈M Kf , M ⊆ P3,2 . It is easy to check that the subsequently defined subsets of pr−1 L are closed and are finitely generated by the given subsets: ⎛ ⎞ 0 0 0 1 1 0 1 1 2 ⎜0 0 1 1 0 1 0 1 2⎟ ⎟ L2 := P ol ⎜ ⎝0 1 0 0 1 1 0 1 2⎠ 0 1 1 0 0 0 1 1 2 = {f ∈ pr−1 L | Kf ⊆ {kI,J ∈ K | |I| ≤ 1}} = [j1 (x1 ) + j1 (x2 ), c1 , j1 (x1 ) · j2 (x2 )], L2,r := {f ∈ pr−1 L | Kf ⊆ {kI,J ∈ K | I = ∅ ∧ |J| ≤ 4}} if r < ∞, [j1 (x1 ) + j1 (x2 ), c1 , j2 (x1 ) · ... · j2 (xr )] ' ( = {j1 (x1 ) + j1 (x2 ), c1 } ∪ q≥1 j2 (x1 ) · ... · j2 (xq )} if r = ∞ (1 ≤ r ≤ ∞), Z2,0 ∩ pr−1 L n = n≥1 {f n ∈ P3,2 | ∃a, a1 , ..., an ∈ E2 : f (x) = a + i=1 ai · j1 (xi )}, = [j1 (x1 ) + j1 (x2 ), c1 ], Z2,1 ∩ pr−1 L = n≥1 {f n ∈ P3,2 | ∃a, a1 , ..., an ∈ E2 : n f (x) = a + i=1 ai · (j1 (xi ) + j2 (xi ))}, = [j0 (x1 ) + j0 (x2 ), c1 ].

372


Further put K := {kI,J ∈ K | J = ∅ ∧ I ∩ J = ∅}, K1 := {kI,J ∈ K | |I| ≥ 1}, K2 := {kI,J ∈ K | |I| ≤ 1}, K3 := K1 ∩ K2 = {kI,J ∈ K | |I| = 1}, K0,r := {k∅,J ∈ K | |J| ≤ r},

1 ≤ r ≤ ∞.

Lemma 12.6.6 Let f n ∈ P3,2 , pr f n = cn0 , f n = cn0 and Kf ⊆ K0,∞ . Further, let r be the smallest number for which there is a function k∅,J ∈ Kf with |J| = r. Then k∅,{1,...,r} (x) = j2 (x1 ) · ... · j2 (xr ) ∈ [f, j1 ]. Proof. By assumption, in Kf there is a function k∅,J with J ⊆ J for all k∅,J ∈ Kf \{k∅,J }. Consequently, we have f (h1 (x1 ), ..., hn (xn )) = k∅,J (x),

if hi (x) :=

j1 (x) if i ∈ J, x if i ∈ J

and i = 1, ..., n. Lemma 12.6.7 Let f n ∈ P3,2 , pr f n = cn0 and Kf ⊆ K0,∞ . Then there is an a ∈ E2 with (1) j1 (x1 )j2 (x2 ) + a · j2 (x2 ) ∈ [f, j1 ]; (2) j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) ∈ [f, j1 ], if furthermore Kf ∩ (K1 \K2 ) = ∅ holds. Proof. First we prove (1) through induction on the arity n of f . n = 2: Since Kf ⊆ K0,∞ we can assume w.l.o.g. f 2 (x1 , x2 ) = j1 (x1 )j2 (x2 ) + α · j2 (x1 )j1 (x2 ) + β · j2 (x1 ) + γ · j2 (x2 )+ (α, β, γ, δ ∈ E2 ). +δ · j2 (x1 )j2 (x2 ) Thus we have f (j1 (x1 ), x2 ) = j1 (x1 )j2 (x2 ) + γ · j2 (x2 ), i.e., (1) is right for n = 2. n − 1 −→ n: Suppose (1) holds for all (n − 1)-ary functions f with pr f = cn−1 0 and Kf ⊆ K0,∞ , n > 2. Now let f be an arbitrary n-ary function with pr f = cn0 and Kf ⊆ K0,∞ . Then there are (n − 1)-ary functions g, h and q with (w.l.o.g.) f (x) = j1 (x1 ) · g(x2 , ..., xn ) + j2 (x1 ) · h(x2 , ..., xn ) + q(x2 , ..., xn ),


373

where g = cn−1 . Consequently, 0 f (j1 (x1 ), x2 , ..., xn ) = j1 (x1 ) · g(x2 , ..., xn ) + q(x2 , ..., xn ) =: f (x). Case 1: Kg ⊆ K0,∞ . With the aid of the proof of Lemma 12.6.6, it is easy to see that j1 (x1 )j2 (x2 )+ a · j2 (x2 ) is a superposition over {f , j1 } for certain a ∈ E2 . Case 2: Kg ⊆ K0,∞ . By induction assumption, the function j1 (x2 )j2 (x3 ) + b · j2 (x3 ) is a superposition on g and j1 for certain b ∈ E2 . Thus we can construct the function f (x1 , x2 , x3 ) := j1 (x1 )(j1 (x2 )j2 (x3 ) + b · j2 (x3 )) + q (x2 , x3 ), where q denotes a c0 -projectable function, as a superposition over f and j1 (under the given conditions). For b = 1 we have f (x1 , x2 , x2 ) = j1 (x1 )j2 (x2 ) + a · j2 (x2 ),

a := q (2, 2).

If b = 0 then we construct the function f (x1 , x2 , x3 ) := f (x1 , j1 (x2 ), x3 ) = j1 (x1 )j1 (x2 )j2 (x3 ) + α · j1 (x2 )j2 (x3 ) + β · j2 (x3 ) first. Consequently, f (x2 , x1 , x2 ) = j1 (x1 )j2 (x2 ) + β · j2 (x2 ) if α = 1, and

f (x1 , x1 , x2 ) = j1 (x1 )j2 (x2 ) + β · j2 (x2 )

in the case α = 0. Hence (1) holds for each n-ary function f ∈ P3,2 with pr f = cn0 and Kf ⊆ K0,∞ . Proof for (2) is obtained through the transfer of proof for (1). Lemma 12.6.8 Let B ∈ {L, L ∩ T0 , L ∩ S, L ∩ T0 ∩ S}, A ⊆ Z2,0 ∩ pr−1 B and [pr A] = B. Then (1) (2) (3) (4) (5) (6) (7)

pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ), j1 (x) + j2 (x1 )}], L2 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j2 (x2 ), j1 (x) + j2 (x1 )}], (2) T0 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 )}], (2) T0 ∩ L2 ∩ pr−1 B = [A ∪ {j1 (x) + j1 (x1 )j2 (x2 )}], L2,r ∩ pr−1 B = [A ∪ {j 1 (x) + j2 (x1 )j2 (x2 )...j2 (xr )}], r ≥ 1, L2,∞ ∩ pr−1 B = [A ∪ q≥1 {j1 (x) + j2 (x1 )j2 (x2 )...j2 (xq )}], Z2,0 ∩ pr−1 B = [A].

Proof. The proof results from definitions of the considered classes.

374


Theorem 12.6.9 Let B ∈ {L, L ∩ T0 , L ∩ S, L ∩ T0 ∩ S}. Then the following B-projectable classes (⊆ P3,2 ) only exist: (1) (2) (3) (4) (5) (6) (7)

pr−1 B, L2 ∩ pr−1 B, L2,r ∩ pr−1 B, r = 1, 2, ..., L2,∞ ∩ pr−1 B, Z2,a ∩ pr−1 B, a ∈ E2 , (2) Ta ∩ pr−1 B, a ∈ E2 , if B ∩ Ta = B, (2) Ta ∩ L2 ∩ pr−1 B, a ∈ E2 , if B ∩ Ta = B

(see Figure 12.5). q

pr −1 L

qL2 qL2,∞ q q q qL2,r q q q qL2,2

q

pr −1 L ∩ Z2,0

qL2,1 @ @ @q pr −1 L ∩ Z

2,1

Fig. 12.5

Proof. Let A be a B-projectable subclass of P3,2 . Then, j1 or j1 + j2 belongs to A. W.l.o.g. we can assume j1 ∈ A, since s j1 s = j1 + j2 holds for s(x) := 2x+1 (mod 3) and thus all considerations are in the case j1 ∈ A isomorphically to those of the case j1 + j2 ∈ A. Consequently, we have Z2,0 ∩ pr−1 B ⊆ A and j1 (x1 ) + j1 (x 2 ) + j1 (x3 ) ∈ A. For KA (= f ∈A Kf ) the following cases are possible: Case 1: KA ⊆ K0,∞ (i.e., A ⊆ L2,∞ ). Case 1.1: KA = ∅. In this case we have: A = Z2,0 ∩ pr−1 B. Case 1.2: It exists an r ≥ 1 with KA ⊆ K0,r−1 and KA ⊆ K0,r . Then, there is a function f n ∈ A with Kf ∩ K0,r = ∅. Consequently, the function f1 (x, x) := j1 (x) + f (j1 (x1 ), ..., j1 (xn )) + f (x) = j1 (x) + f1 (x) with


f1 (x) :=

375

k∅,J (x).

k∅,J ∈Kf

belongs to A. By Lemma 12.6.6, one can construct the function j2 (x1 )...j2 (xt ), where t := min |J|, k∅,J ∈Kf

f1

and j1 . Consequently, j1 (x) + j2 (x1 )....j2 (xt ) ∈ A as a superposition over for t ≤ t. A superposition over these functions and f1 is also f2 (x, x) := j1 (x) + f1 (x) + k∅,J ∈ Kf k∅,J (x) |J| ≤ t

= j1 (x) +

f2 (x)

with f2 (x) :=

k∅,J ∈ Kf k∅,J (x). |J| > t

If f2 = cn0 , then we have the function j2 (x1 )...j2 (xs ), where s :=

min

k∅,J ∈Kf , |J|>t

|J|,

by substituting certain variables by j1 and by possibly changing the numbering of the variables from f2 . Consequently, the function k∅,J (x) f3 (x, x) := j1 (x) + k∅,J ∈ Kf |J| > s

belongs to A, etc. By iterated application of these constructions we obtain the function j1 (x) + j2 (x1 )...j2 (xr ) ∈ A. Hence, by Lemma 12.6.8, (5), we have A = L2,r ∩ pr−1 B. Case 1.3: KA ∩ K0,r = ∅ for all r ≥ 1. Because of the considerations from Case 1.2 and Lemma 12.6.8, (6) is valid: A = L2,∞ ∩−1 B. A = L2,∞ ∩−1 B. Case 2: KA ⊆ K2,∞ . In this case, there is a function f n ∈ A with Kf ∩ K1 = ∅. Then, the function f1 (x, x) := j1 (x) + f (x) + f (j1 (x1 ), ..., j1 (xn )) = j1 (x) + f1 (x) with

f1 (x) := j1 (x) +

kI,J (x)

kI,J ∈Kf

belongs to A. With the aid of Lemma 12.6.6, it is easy to see that the function

376


f1 (x1 , x2 ) := j1 (x1 )j2 (x2 ) + a · j2 (x2 ), a ∈ E2 is a superposition over {j1 , f1 }. Therefore, the function g(x, x1 , x2 ) := j1 (x) + j1 (x1 )j2 (x2 ) + a · j2 (x2 ) is a superposition on {f1 , j1 }, i.e., g ∈ A. Consequently, g(g(x, x1 , x2 ), x2 , x2 ) = j1 (x) + j1 (x1 )j2 (x2 ) (2)

belongs to A. Then, by Lemma 12.6.8, we have T0 ∩ L2 ∩ pr−1 B ⊆ A. If (2) A = T0 ∩ L2 ∩ pr−1 B then the following three cases are possible: Case 2.1: A ⊆ L2 and pr A ⊆ T0 . In this case, there is a function p ∈ A with p(0) = 1. For (pr p)(y) = y it holds j1 (x) + j1 (x1 )j2 (x1 ) = j1 (x) + j2 (x1 ) ∈ A. If pr p = c1 then we also obtain j1 (x) + j2 (x1 ) = j1 (x) + p(p(x1 ))j2 (x1 ) ∈ A. Consequently, by Lemma 12.6.8, A = L2 ∩ pr−1 B. (2)

Case 2.2: A ⊆ L2 , pr A ⊆ T0 and pr A ⊆ T0 . In this case, there is a function r2 ∈ A with r(0, 0) = 0 and r(0, 2) = 1, and it holds that j1 (x) + r(j1 (x1 ), x1 ) + r(j1 (x1 ), j1 (x1 )) = j1 (x) + j2 (x1 ) ∈ A. Therefore, by Lemma 12.6.8, A = L2 ∩ pr−1 B. Case 2.3: A ⊆ L2 . Denote q an n-ary function of A, which does not belong to L2 . Then, q1 (x, x) := j1 (x) + q(x) + q(j1 (x1 ), ..., j1 (xn )) = j1 (x) + q1 (x), where

q1 (x) :=

kI,j (x),

kI,J ∈Kq

is a function of A. By Lemma 12.6.6, a function of the form j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) is a superposition over {q1 , j1 }. Thus, q2 (x, x1 , x2 , x3 ) := j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ) + a · j2 (x3 ) ∈ A and therefore q2 (q2 (x, x1 , x2 , x3 ), x3 , x3 ) := j1 (x) + j1 (x1 )j1 (x2 )j2 (x3 ) ∈ A.

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S (2)

377

(2)

Thus by Lemma 12.6.8, we have T0 ∩ pr−1 B ⊆ A. If a = T0 ∩ pr−1 B, then there is either a function p ∈ A with p(0) = 1, or a function r ∈ A with r(0, 0) = 0 and r(0, 2) = 1. As already shown in the above Case 2.1 or 2.2, j1 (x) + j2 (x1 ) ∈ A results. Thus by Lemma 12.6.8, (1), A = pr−1 B holds. Next we study the set N3 (T0,m ) with m ∈ N\{1}. By Theorem 12.3.1 (or 12.3.2), for every m ∈ N\{1}, we know that N3 (T0,m ) is a finite set. In the following, we determine the elements of N3 (T0,m ) and we show that |N3 (T0,2 )| = 148 holds. Since the description of the elements of the other sets N3 (A) with S ∩ M ⊆ A or Ta,m ∩ Ta ∩ M ⊆ A requires a lot of place, we must renounce this description here and refer the reader to the dissertation by N. Gr¨ unwald (see [Gr¨ u 84]). In the dissertation by N. Gr¨ unwald, one also finds the proofs left out in the following. Lemma 12.6.10 For every set P olP3,2 (E2m \{1} ∪ j )) T := A ∩ ( j∈J (2)

with A ∈ {T0 , P3,2 }, j ⊆ E3m \E2m , and every set B := {g ∈ K | g has the value 1 on at most m tuples} ∪ {g1m+1 } with pr g1m+1 = hm (∈ P2 ; see Chapter 3) it holds [B] = T . Proof. Let f n ∈ T and let a1 , ..., ar be the tuples of E3n , on which the function f n has the value 1. By induction on r, we prove f n ∈ [B]: For r ≤ m the assertion is obviously right. For r > m we assume that all functions of T , which have the value 1 on less than r tuples, belong to [B]. By definition of T , the functions 0 if x = ai , n fi (x) := f n (x) otherwise (i = 1, ..., m + 1) belong to T . Thus, by assumption, we have f1 , ..., fn ∈ [B]. Then our assertion follows from f n (x) = g1 (f1 (x), f2 (x), ..., fm+1 (x)) ∈ [B]. Lemma 12.6.11 Let W1,m := P olP3,2 (E3m \{1}), W2,m := P olP3,2 (E3m \{1, 2}m ). Then (a) For arbitrary sets Ai ⊆ Wi,m (i ∈ {1, 2}) with [pr Ai ] = T0,m the set B1 := A1 ∪ {j1 (x1 ) · j2 (x2 ), j1 (x), j1 (x1 ) · j0 (x2 )}

378


is a generating system for W1,m and B2 := A2 ∪ {j1 (x1 ) · j2 (x2 ), j1 (x), j1 (x1 ) · j0 (x2 )} is a generating system for W2,m . (b) The set pr−1 T0,2 ∩ Z2,0 is maximal in W1,m . (c) The set pr−1 T0,2 ∩ Z2,1 is maximal in W2,m . (d) For all A ∈ N3 (T0,m )\{pr−1 T0,2 ∩Z2,0 , pr−1 T0,2 ∩Z2,1 } it holds W1,n ⊆ A. Proof. See [Gr¨ u 84]. Lemma 12.6.12 Let A be a set of the following form P ol ((E2m \{1}) ∪ i ) mit i ⊆ (E3m \(E2m ∪ {2})), m ≥ 2.

(12.21)

i∈I

Then (1) A is T0,m -projectable and the only maximal classes of A that belong to N3 (T0,m ) are the following classes: (2) (a) A ∩ T0 ; (b) A ∩ P ol ((E2m \{1} ∪ δ), where δ fulfills the following conditions: δ ⊆ (E3m \(E2m ∪ {2}, ∃i ∈ I : δ ⊆ i , ∀ε ⊂ δ∃i ∈ I : εi . (2) If B ⊆ A, [pr B] = T0,m , B ⊆ Z2,i ∩ pr−1 T0,m , i = 0, 1, and if B is not contained in the sets of the above type (a) or (b), then [B] = A holds. (3) If special A = W2,m , then Z2,1 ∩ pr−1 T0,m is the only T0,m -projectable maximal class of W2,m . Proof. See [Gr¨ u 84]. Lemma 12.6.13 Denote A a set of the form (2) P ol ((E2m \{1}) ∪ i ) mit i ⊆ (E3m \E2m , m ≥ 2. T0 ∩

(12.22)

i∈I

Then: (1) A is T0,m -projectable, and the only maximal classes of A, which belong to N3 (T0,m ), are classes of the following form: A ∩ P ol ((E2m \{1} ∪ δ), where δ fulfills the following conditions: δ ⊆ E3m \E2m , ∃i ∈ I : δ ⊆ i , ∀ε ⊂ δ∃i ∈ I : εi .

(12.23)


379

(2) If B ⊆ A, [pr B] = T0,m , B ⊆ Z2,i ∩ pr−1 T0,m , i = 0, 1, and B ⊆ C for every class C that was described in (1), then it holds [B] = A. (3) If in particular A = W1,m then Z2,0 ∩pr−1 T0,m is the only T0,m -projectable maximal class of W1,m . Proof. See [Gr¨ u 84]. Theorem 12.6.14 ([Gr¨ u 84]) There are only finitely many subclasses of P3,2 , which are T0,m -projectable. One receives a concrete description of these subclasses with the aid of Lemmas 12.6.12 and 12.6.13. Proof. See [Gr¨ u 84]. In the following table are explanations to notations that we need to describe the T0,2 -projectable subclasses of P3,2 . Table 12.3

i 1 2 3 4 5 6 7 8 9

τi 1 2 0 2 2 2 1 2 2 1 1 2 2 2 1 0 2 2 1 2 2 0 2 2 0 2 2 0 0 2

i 10 11 12 13 14 15 16 17 18 19

1 2 1 2 2 1 2 1 2 1 0 2 1 2 2 1 1 2 2 1 2 2 1 0 1 2 0 2 1 2

2 1 2 1 2 0 0 2 2 0 2 0 0 2 0 2 0 2 2 0

τi

0 2 2 2 2 2 2 2 0 2 2 2 2 0 2 2 2 2 2 2

Theorem 12.6.15 ([Gr¨ u 83;a;b]) 0 0 1 ∪ τi for i = 1, 2, ..., 19. Let τ0 := {0, 2} and τi := 0 1 0

380


There are exactly 148 T0,2 -projectable subclasses of P3,2 that can be described as follows: 1): pr−1 T0,2 , 2)–5): P ol τi with i ∈ {0, 1, 2, 3}, 6)–10): P ol {τi , τj } with (i, j) ∈ {(1, 3), (1, 2), (2, 3), (1, 0), (2, 0)}; 11)–12): P ol {τi , τj , τk } with (i, j, k) ∈ {(1, 2, 3), (1, 2, 0)}; 13)–18): P ol τi with i ∈ {4, 5, 6, 7, 8, 9}; 19)–30): P ol {τi , τj } with (i, j) ∈ {(4, 3), (4, 2), (4, 0), (5, 2), (6, 3), (6, 0), (7, 3), (7, 0), (8, 1), (9, 0), (9, 1), (9, 3)}; 31)–34): P ol {τi , τj , τk } with (i, j, k) ∈ {(4, 2, 3), (4, 2, 0), (9, 1, 3), (9, 1, 0)}; 35)–49): P ol {τi , τj } with (i, j) ∈ {(4, 5), ((4, 6), (4, 7), (4, 8), (4, 9), (5, 6), (5, 7), (5, 8), (5, 9), ((6, 7), (6, 8), (6, 9), (7, 8), (7, 9), (8, 9)}; 50)–63): P ol {τi , τj , τk } with (i, j, k) ∈ {(4, 5, 2), (4, 6, 0), (4, 7, 0), (4, 6, 3), (4, 7, 3), (4, 9, 0), (4, 9, 3), (6, 7, 0), (6, 7, 3), (6, 9, 4), (6, 9, 3), (7, 9, 0), (7, 9, 3), (8, 9, 1)}; 64)–69): P ol τi with i ∈ {10, 11, 12, 13, 14, 15}; 70)–75): P ol {τi , τj } with (i, j) ∈ {(10, 0), (10, 3), (11, 2), (14, 0), (14, 3), (15, 1)}; 76)–95): P ol {τi , τj } with (i, j) ∈ {(10, 5), (10, 8), (10, 9), (14, 0), (11, 6), (11, 7), (11, 9), (12, 4), (12, 7), (12, 9), (13, 4), (13, 7), (13, 9), (14, 4), (14, 5), (14, 8), (15, 4), (15, 5), (15, 7)}; 96)–99): P ol {τi , τj , τk } with (i, j, k) ∈ {((10, 9, 0), (10, 9, 3), (14, 4, 0), (14, 4, 3)}; 100)–114): P ol {τi , τj } with (i, j) ∈ {(10, 11), (10, 12), (10, 13), (10, 14), (10, 15), (11, 12), (11, 13), (11, 14), (11, 15), (12, 13), (12, 14), (12, 15), (13, 14), (13, 15), (14, 15)}; 115)–117): P ol τi with i ∈ {16, 17, 18}; 118)–119): P ol {τi , τj } with (i, j) ∈ {(16, 0), (16, 3)}; 120)–123): P ol {τi , τj } with (i, j) ∈ {(16, 8), (16, 5), (17, 9), (18, 4)}; 124): P ol {τ16 , τ8 , τ5 }; 125)–132): P ol {τi , τj } with (i, j) ∈ {(16, 15), (16, 12), (16, 13), (16, 11), (17, 15), (17, 14), (18, 11), (18, 10)}; 133)–134): P ol {τi , τj , τk } with (i, j, k) ∈ {(16, 15, 5), (16, 11, 8)}; ∈ {(16, 15, 12), (16, 15, 14), 135)–142): P ol {τi , τj , τk } with (i, j, k) (16, 15, 11), (16, 12, 13), (16, 12, 11), (16, 13, 11), (17, 15, 14), (18, 11, 10)}; 143)–144): P ol {τi , τj } with (i, j) ∈ {(16, 17), (16, 18), (17, 18)}; 146): P ol τ19 ; 147)–148): Z2,1 ∩ pr−1 T0,2 , Z2,0 ∩ pr−1 T0,2 . Proof. The following five statements are direct consequences from Lemmas 12.6.12 and 12.6.13:

12.6 The Classes A with M ∩T0 ∩T1 ⊆ prA or L∩T0 ∩S ⊆ prA or prA = M ∩S (02)(20)

(1) T0,2

:= P ol

0 1 0 0 2 0 0 1 2 0

381

has exactly the following T0,2 -projectable

maximal classes: (2) (02)(20) a) T0 ∩ T0,2 ; 0 1 0 1 (02)(20) b) T0,2 ∩ P ol ; 0 0 1 2 c) Z2,1 ∩ pr−1 T 0,2 . 2 0 2 1 0 1 0 and A = (2) If A := P ol ∪ i with i ⊆ 0 2 1 2 0 0 1 0 1 0 0 2 P ol , then A has exactly the following T0,2 -projectable 0 0 1 2 0 maximal classes: (2) a) T0 ∩ A; 0 1 0 b) A ∩ P ol ∪ δ , where δ fulfills the conditions 0 0 1 2 0 2 1 δ⊆ 0 2 1 2 ∃i ∈ I : δ ⊆ i ∀ε ⊂ δ ∃i ∈ I : ε ⊆ i . 0 1 0 2 0 2 1 2 (2) then Z2,0 ∩ pr−1 T0,2 is the only (3) If A = T0 ∩ P ol 0 0 1 0 2 1 2 2 maximal class of A, which is T0,2 -projectable. 0 1 0 2 0 2 1 2 (2) ∪ i with i ⊆ and (4) If A := T0 ∩ P ol 0 0 1 0 2 1 2 2 0 1 0 2 0 2 1 2 A = P ol , then A has exactly the following T0,2 0 0 1 0 2 1 2 2 projectable maximal classes: 0 1 0 A ∩ P ol ∪ δ , where δ fulfills the following conditions: 0 0 1

2 0 2 1 2 0 2 1 2 2 ∃i ∈ I : δ ⊆ i ∀ε ⊂ δ ∃i ∈ I : ε ⊆ i . δ⊆

(5) There is not any proper subset of Z2,i ∩ pr−1 T0,2 , i = 0, 1, whose projection agrees with T0,2 . With the aid of the above statements, our theorem can be proven as follows: 0 0 1 −1 One obtains the T0,2 -projectable maximal classes of pr T0,2 = P ol 0 1 0 by means of (2). Then, one obtains the T0,2 -projectable maximal classes of these classes by means of (1)–(4), etc.

13 Classes of Linear Functions

In this chapter, the elements of the clone Lk :=

n≥1

{f ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + n

n

ai · xi (mod k)}.

i=1

and the elements of a generalization of the set Lk are called linear functions. The lattice of the subclasses of Lk belongs to the earliest and best investigated sublattices of Lk . For the case that k is a prime number, all subclasses, which are no subsets of [L1k ], were determined by A. A. Salomaa in [Sal 64]. The results of [Sal 64] were also proven by J. Bagyinszki and J. Demetrovics in [Bag-D 82] and complemented with the remaining subclasses of [L1k ], p ∈ P. ´ Szendrei. For examMany results about linear functions were obtained by A. ple, she proven in [Sze 78] that Lk has only finitely many subclasses, if k is square-free.1 In addition, she showed in [Sze 78] that an arbitrary class has at most the order 2 (or 3), if k is square-free and k is an odd number (or an even number), respectively. One finds in [Sze 80] a short and fine determination of the subclasses not contained in [L1p ] of Lp (p prime). Similarly Theorem 13.2.1 is proven here. For the case that k is not square-free one easily find a class that does not have any finite basis (see Lemma 13.3.7). In Section 13.1, we start with properties of certain subclasses of the set Ud := n≥1 {f n ∈ Lk | ∃a0 , ..., an ∈ Ek ∃j ∈ {1, ..., n} : f (x) = a0 + aj · xj + d · i=1,i=j ai · xi }. This set is closed, if d is a divisor of k (notation: d | k). Then, with the aid of the results from the first section new proofs are given for the theorems of [Sal 64] and [Bag-D 82]. Section 13.3 gives a survey of further results, which were ´ Szendrei and the author, to classes from linear found by A. A. Bulatov, A. functions. 1

Through that, a presumption of A. A. Salomaa from [Sal 64] was confirmed.

384


13.1 Some Properties of the Subclasses of Ud That Contain rd In this section, let d ∈ Ek be a divisor of k. Further, the ternary function rd is defined by rd (x, y, z) := x + d · y − d · z (mod k). Lemma 13.1.1 For every subclass T of Ud , which contains the function rd , it holds: (a) T = [T 1 ∪ {rd }], (b) T = Lk ∩ P olk {(a, a + b) | a + bx ∈ T 1 }. Proof. (a): Let f n ∈ T be arbitrary and w.l.o.g. f (x1 , ..., xn ) = a0 + a1 x1 + d ·

n

ai xi .

i=2

Obviously, one can obtain every function of the type g(x1 , ..., xm ) := x1 + d ·

m

bi xi with b2 + ... + bm = 0 (mod k)

i=2

by identifying variables of a function of the form rd rd ... rd . In particular, the function f (x, y, x1 , ..., xn ) := x + d(−(a2 + ... + am )y +

n

ai xi )

i=1

is a superposition over rd . Consequently, by f (x1 , ..., xn ) = f (f (x1 , ..., x1 ), x1 , x2 , ..., xn ), we have f ∈ [T 1 ∪ {rd }]. (b): Because of (a), T = Lk ∩P olk G1 (T ) holds. The matrix G1 (T ) is, however, unambiguously determined by the first two rows (x = 0, x = 1). Thus we have T = Lk ∩ pr0,1 G1 (T ), where pr0,1 G1 (T ) = {(a, a + b) | a + bx ∈ T 1 }. n Lemma 13.1.2 Let f (x1 , ..., xn ) = a0 + a1 x1 + d · i=2 ai xi and a1 ' k = a2 ' k = 1. Then, the function rd is a superposition over f . Proof. Let f1 (x, y, z) := f (x1 , x2 , x3 , ..., x3 ). Then, the functions fi (x, y, z) := fi−1 (f1 (x, z, z), y, z), i = 2, 3, ... are superpositions over f . Since a2 ' k = 1, there exists a t with at1 = 1 (mod k). Consequently, ft (x, y, z) = a + x + a2 dy + bdz for certain a, b ∈ Ek .

13.1 Some Properties of the Subclasses of Ud That Contain rd

385

The functions ft,1 := ft and ft,i (x, y, z) := ft,i−1 (ft (x, y, z), y, z), i = 2, 3, ... belong to [{f }]. Since a2 ' k = 1, there exists an s with a2 s = 1 (mod k). Consequently, ft,s (x, y, z) = as + x + dy + sbdz is valid. Next we form the superpositions ft,s,1 := ft,s and ft,s,i (x, y, z) := ft,s,i−1 (ft,s (x, y, z), y, z), i = 2, 3, ... For i = k we obtain ft,s,k (x, y, z) = x+dy +(k−1)dz, i.e., rd is a superposition over f . Subsequently, the subclasses of Lk that contain the function r1 are determined. Obviously, a subset A ⊆ L1k determines a subclass T of the form T = [T 1 ∪{r1 }] if and only if [A ∪ {r1 }]1 = A holds. Further we have: Lemma 13.1.3 (a) If [A ∪ {r1 }]1 = A for a subset A of L1k , then the binary relation A := {(a, a + b − 1) | a + bx ∈ A} is a subgroup of the direct product (Ek ; +)2 . (b) Conversely, if γ is a subgroup of (Ek ; +)2 , then [Aγ ∪ {r1 }]1 = A, where Aγ := {a + bx ∈ L1k | (a, a + b − 1) ∈ γ}. Proof. (a): Let (a, a + b − 1), (a , a + b − 1) ∈ A be arbitrary, i.e., the functions a + bx and a + b x belong to A. Then, by assumption, r1 preserves the functions x, a + bx and a + b x of A. Consequently, r1 (a + bx, a + b x, x) = a + a + (b + b − 1)x ∈ A holds and therefore (a + a , a + a + b + b − 2) ∈ A . Thus, by definition of , we have (a, a + b − 1) + (a , a + b − 1) ∈ A for arbitrary (a, a + b − 1), (a , a + b − 1) ∈ A . Hence, A is a subgroup of (Ek ; +)2 . (b): We have to show that Aγ is closed in respect to the operation and that the function r1 preserves the functions of Aγ . Let f (x) := a + bx, g(x) := a + b x and h(x) := a + b x be arbitrary of Aγ . Then the function (f g)(x) = a + a b + bb x belongs to Aγ , since (a, a + b − 1) ∈ γ, (a , a + b − 1) ∈ γ and (a, a + b − 1) + b · (a , a + b − 1) = (a + a b, a + a b + bb − 1) ∈ γ.

386


Further r1 (a + bx, a + b x, a + b x) = (a + a − a ) + (b + b − b )x holds. Since (a + a − a , a + a − a + b + b − b − 1) = (a, a + b − 1) + (a , a + b − 1) − (a , a + b − 1) and γ is a group, the function (a + a − a ) + (b + b − b )x belongs to Aγ . Consequently, we have [Aγ ∪ {r1 }]1 = Aγ . When we summarize Lemmas 13.1.1–13.1.3, or as a consequence of [Sze 80], Lemma 4.3, we obtain: Theorem 13.1.4 The lattice of the closed subsets of Ud (d|k), which contain the function rd , is finite for arbitrary k ∈ N and is isomorphic to the subgroup lattice of the group (Ek ; +)2 for d = 1. One can describe the subgroups of the group (Ek ; +)2 easily. The subsequently given description is a special case of Theorem 4.3.1 from [Sco 64], p. 71: Theorem 13.1.5 (a) Let γ be a subgroup of (Ek ; +)2 . Then γ1 := {a | ∃b ∈ Ek : (a, b) ∈ γ}, γ2 := {b | ∃a ∈ Ek : (a, b) ∈ γ}, N2 := {b | (0, b) ∈ γ} N1 := {a | (a, 0) ∈ γ}, are subgroups of (Ek ; +), Ni ⊆ γi (i = 1, 2) and the mapping α from the factor set γ1 /N1 onto the factor set γ2 /N2 with α(a + N1 ) = b + N2 :⇐⇒ (a, b) ∈ γ is an isomorphism from γ1 /N1 onto γ2 /N2 . (b) Conversely: Let γ1 , γ2 , N1 , N2 be subgroups of (Ek ; +) with Ni ⊆ γi (i = 1, 2) and let α be an isomorphism from γ1 /N1 onto γ2 /N2 . Then γ := {(a, b) | a ∈ γ1 ∧ b ∈ γ2 ∧ α(a + N1 ) = b + N2 } is a universe of a subgroup of (Ek ; +)2 . With the help of Theorem 13.1.5 and with well-known theorems about cyclic groups, one can easily determine the cardinality of Lk ([{r1 }], Lk ). For this purpose, let ϕ be the Euler function (i.e., ϕ(n) is the number of all q ∈ {1, 2, ..., n − 1} with n ' q = 1) and let t(q, k) be the number of all n ∈ N with n|k and q|n.

13.2 The Subclasses of Linear Functions of Pk with k ∈ P

Theorem 13.1.6 It holds: |Lk ([{r1 }], Lk )| =

387

ϕ(q) · (t(q, k))2 .

q q ∈ N, q|k

Proof. Because of Theorem 13.1.4, we have to determine only the number of the subgroups of (Ek ; +)2 to the proof. Let γ ⊆ Ek2 be a subgroup of (Ek ; +)2 . This holds (by Theorem 13.1.5) if and only if there exist subgroups γi , Ni (Ni ⊆ γi , i = 1, 2) of (Ek ; +) and an isomorphism α from γ1 /N1 onto γ2 /N2 with γ = {(a, b) | α(a+N1 ) = b+N2 }. Moreover, it holds |γ1 /N1 | = |γ2 /N2 | =: q and q|k. Since (Ek ; +) is a cyclic group, the groups γi , Ni and γi /Ni (i = 1, 2) are cyclic. Therefore, for fixed q and k, there are exactly t(q, k) possibilities for the choice of γi /Ni (i ∈ {1, 2}). Further, if g is a generating element of γ1 /N1 , then the mapping α is completely determined by α(g), where α(g) is a generating element of γ2 /N2 . As is generally known, a cyclic group of the order q has exactly ϕ(q) generating elements. Thus, for determining the mapping α there are exactly ϕ(q) possibilities. Consequently, for γ with |γ1 /N1 | = |γ2 /N2 | = q there are exactly ϕ(q) · (t(q, k))2 possibilities, whereby our assertion is proven. We notice that one can prove further properties of the elements of Lk ([{r1 }], Lk ) with the aid of Lemma 13.1.1. For example, every subclass T of Lk with r1 ∈ T is finitely axiomatizable (see [McK 78]) and such a class has only finite many congruences, which are determined through the congruences on [{r1 }] and of (T 1 ; ) (see Chapter 9).

13.2 The Subclasses of Linear Functions of Pk with k ∈ P In this section, let p be an arbitrary prime number. Because of Lemma 13.1.2, every class A ⊆ Lp with A ⊆ [L1p ] contains the function r1 . Therefore, by Lemma 13.1.3 and Theorem 13.1.4, such a class A is determined by a certain subgroup of the group (Ek ; +)2 . Obviously, by Theorem 13.1.5, there are only the following p + 3 subgroups of (Ek ; +)2 : Ep2 , {(0, 0)}, {(0, x) | x ∈ Ep }, and {(x, t · x) | x ∈ Ep }, if t ∈ Ep . Consequently, by Section 13.1, we have

Theorem 13.2.1 ([Sal 64], [Bag-D 82], [Sze 80]) Lp has exactly p + 3 subclasses which are not subsets of [L1p ]:

388


Lp Lp ∩ P olp {(0, 1)} n (= n≥1 {f n ∈ Lp | f (x1 , ..., xn ) = i=1 ai xi ∧ a1 + ... + an = 1 } = [r1 ]), Lp ∩ n P olp {(x, x + 1) | x ∈ Ep } (= n≥1 {f n ∈ Lp | f (x1 , ..., xn ) = a0 + i=1 ai xi ∧ a1 + ... + an = 1 }), Lp ∩ P olp {0} and Lp ∩ P olp {(x, tx + 1) | x ∈ Ep } (= Lp ∩ P olp {(1 − t)−1 }), where t ∈ Ep \{1}. Since one obtains an arbitrary subclass A of [L1p ] from a subsemigroup A1 of (L1p ; ) by means of [A1 ]ζ,τ,∇ , it is sufficient to determine the subsemigroups of (L1p ; ) for the description of the remaining subclasses of Lp . Let A be an arbitrary subsemigroup of (L1p ; ). Then A = A ∪ CA , where A is either the empty set or a subgroup of the group S := ({ax+b | a ∈ Ep \{0}}; ) and CA ⊆ {ca | a ∈ Ep } =: C. If A = ∅ then the set Ua := {a | ∃b : ax + b ∈ A } is a subgroup of the group G := (Ep \{0}; ·). Now, one can easily see that for an arbitrary subgroup U of G, G/U =: {N1 , N2 , ..., Np−1 }, α ∈ Ep and 1 I ⊆ {1, 2, ..., p−1 |U | } the following subsets of Lp are closed in respect to : Tα,U := {ax + (1 − a)α | a ∈ U } (⊆ L1p ∩ P olp {α}), SU := {ax + b | a ∈ U ∧ b ∈ Ep }, Tα,U ∪ Cα,U,I with Cα,U,I := {cα+n (mod p) | n ∈ i∈N Ni } and Cα,U,∅ := {cα }, Tα,U ∪ Cα,U,I ∪ {cα } and SU ∪ C. Theorem 13.2.2 ([Bag-D 82]) An arbitrary subsemigroup of (L1p ; ) is either a subset of C or has the form Tα,U , SU , Tα,U ∪ Cα,U,I , Tα,U ∪ Cα,U,I ∪ {cα } or SU ∪ C. Proof. Let A be an arbitrary subsemigroup of (L1p ; ) with A = A ∪ CA (A ⊆ S, CA ⊆ C). The following cases are possible: Case 1: CA = ∅. 1.1.: A ⊆ L1p ∩ P ol{α} for a certain α ∈ Ep . In this case, A has the form Tα,U , since L1p ∩ P ol{α} is a cyclic group. 1.2.: For all α ∈ Ep it holds A ⊆ L1p ∩ {α}. Then A = {x} and A contains a function x + a (a = 0) or at least two functions of the form f (x) := bx + (1 − b)β, g(x) := cx + (1 − c)γ with c, b ∈ Ep \{0, 1} and β = γ, since a function ax + b with a ∈ Ep \{0, 1} has exactly a fixed point. Then, in the first case, all functions x, x + 1, ..., x + p − 1 belong to A , and in the second case, (f −1 g f g −1 )(x) = x + d ∈ A with

13.2 The Subclasses of Linear Functions of Pk with k ∈ P

389

d = 0 (because of β = γ). Thus, A = SU for certain U ≤ G. Case 2: CA = ∅. Because of considerations for Case 1, set A is either a subset of C or of the form SU ∪C or Tα,U ∪CA , where the functions of Tα,U preserve CA . The latter is valid if and only if CA has the form Cα,U ;I or Cα,U ;I ∪ {cα } for certain I. Theorem 13.2.3 ([Bag-D 82]) Lp has exactly 2 · t(p − 1) + 2p · (2 − p) + p + 3 + 2 · p ·

2h

h h | (p − 1)

closed subsets (including ∅), where t(p − 1) is the number of all divisors n ∈ N of p − 1. Proof. Because of Theorem 13.2.1, we only have to determine the number of all subsemigroups which were described in Theorem 13.2.2. Since Tα,U ⊆ L1p ∩P ol{α}, L1p ∩P ol{α}∩P ol{β} = {x} for α = β and U is a subgroup of the cyclic group G = (Ep \{0}; ·), there is exactly p · t(p − 1) − p + 1 subsemigroups of the form Tα,U . Obviously, there exist 2·t(p−1) sets of the form SU and SU ∪C (U ≤ G). Since there are exactly 2h possibilities for the choice of I (⊆ {1, 2, ..., h}, h := p−1 |U | ), one receives (considering the equivalence Tα,U ∪ Cα,U,I = Tβ,V,J ∪ Cβ,V,J ⇐⇒ (α, U, I) = (β, V, J) ∨ (U = V = {1} ∧ I = J) the following number of the subsemigroups of (L1p ; ) of the form Tα,U ∪Cα,U,I or Tα,U ∪ Cα,U,I ∪ {cα }: (2h+1 − 1) − (p + 1) · (2p − 1) p· h h | (p − 1)

= 2 · p · ((

h h | (p − 1)

2h ) − p · t(p − 1) − (p − 1) · (2p − 1).

By adding, one immediately obtains the number given in our theorem. Next, we provide some statements about the order of the subclasses of Lp . One finds generating systems for these classes in [Bag-D 82]. Theorem 13.2.4 ([Bag-D 82]) For an arbitrary subclass A ⊆ Lp it holds:

390


⎧ 1, if A ⊆ L1p , ⎪ ⎪ ⎨ 0 1 ord A = 3, if p = 2 and A ∈ {L2 , L2 ∩ P ol2 , [r1 ]}, 1 0 ⎪ ⎪ ⎩ 2 otherwise. Proof. For p = 2 the above statements were proven in Chapter 3. Because of Lemmas 13.1.1 and 13.1.2, we have [A1 ∪{r1 }] = A and therefore 1 < ord A ≤ 3 for arbitrary A ∈ L↓p (Lp ) with A ⊆ [L1p ]. If p = 2 then (by Lemma 13.1.2) r1 ∈ [∆r1 ]. This implies our theorem.

13.3 A Survey of Further Results on Linear Functions In this section, k ≥ 2 is arbitrary. Let n n { ai xi ∈ Lk | ai = 1} Lk,id := n≥1 i=1

i=1

the set of all idempotent functions of Lk . For d|k we set Ik,d := n≥1 {f n ∈ Lk,id | ∃i ∈ {1, ..., n} ∃a1 , ..., an ∈ Ek : f (x) = ai xi + d · j∈{1,...,i−1,i+1,...n} aj xj } Theorem 13.3.1 ([Sze 76], [Sze 82]; without proof ) Any non-trivial closed subset A of Lk,id has a unique representation of the form m Ik,di , (13.1) A= i=1

where m ≥ 1 and d1 , ..., dm (> 1) are pairwise relatively prime proper divisors of k. If k is odd, then any subclass of Lk,id has a order ≤ 2. If, in turn, k is even, then the order of any subclass is at most 3 and is equal to 3 if and only if in the representation (13.1) of A d1 · ... · dm is odd. To determine subclasses of Lk , the following lemmas are useful. We notice that the proof (in [Sze 78]) for Lemma 13.3.2 uses Theorem 13.3.1 and that the proof (in [Lau 88a]) for Lemma 13.3.3 requires Lemma 13.3.2, so that the proof of Theorem 13.3.1 forms the basis of the proofs for most statements of this section. Lemma 13.3.2 ([Sze 78]; without proof ) Let f (x) := a0 +

n

ai xi (mod k) ∈ Lk

i=1

with 0 ∈ {a1 , ..., an }, n ≥ 2 and a1 ' ... ' an ' k = 1. Then

13.3 A Survey of Further Results on Linear Functions

391

(a) rq1 ·...·qn ∈ [{f }], where qi := a1 ' a2 ' ... ' ai−1 ' ai+1 ' ... ' an ' k, i = 1, 2, ... ([Sze 78], Basic Lemma); (b) f ∈ [[f ]id ∪ [f ]1 ], where [A]id := [A] ∩ Lk,id for arbitrary A ⊆ Lk ([Sze 78], Corollary 3.1). Lemma 13.3.3 ([Lau 88a]; without proof ) Let A be a subclass of Lk which is no subset of Wq := Uq ∪ Qt t;t|q;t=1

for every divisor q = 1 of k. Then the function r1 belongs to A. For every k, d ∈ N d||k is a short notation for “d|k and d ' kd = 1”. For lack of a better name, d will be called a full divisor of k. The following lemma is easy to prove. Lemma 13.3.4 Let k, d ∈ N with d||k. Furthermore dZk denotes the set {d · z (mod k) | z ∈ Zk }. Then: (a) dZk := (dZk ; + (mod k), · (mod k)) is a ring which is isomorphic to Zk/d . (b) dZk has a unit which will be denoted by ed . (c) ed + en/d = 1 (mod k). For α, d ∈ Ek with d|k let Qd,α :=

{a0 · α +

n≥1

n

ai xi (mod k) | a0 +

i=1

n

ai xi ∈ Qd }.

i=1

Lemma 13.3.5 ([Sze 78]) Let k, d ∈ N with d||k. Then: (a) The mapping ψd : Lk/d −→ Qd,ed , a0 +

n

ai xi → ed · (a0 +

i=1

n

ai xi )

i=1

is an isomorphism. (b) The mapping ϕd : Qd −→ Qd,ed , a0 +

n i=1

is a homomorphism.

ai xi → ed · a0 +

n i=1

ai xi

392


(c) For any f ∈ Qd there exists a unique constant cd (f ) such that f = ϕd (f ) + cd (f ) (mod k). Moreover, if f, g ∈ Qd , the constant cd (f ) has the following properties: ∀α ∈ {ζ, τ, ∆, ∇} : cd (αf ) = cd (f ) cd (f g) = cd (f ). Proof. (a), (b), and (c) are easy to check with the aid of Lemma 13.3.4. Theorem 13.3.6 ([Sze 78]) Let k ∈ N\{1} be square-free. Then, for arbitrary subclass of Lk , 2 if k is odd, ord A ≤ 3 otherwise, i.e., there are only a finite number of subclasses of Lk . Proof. Let f n ∈ Lk \ [L1k ] be arbitrary. Set A := [f ]. To show our theorem, it suffices to prove the following statement: k is odd, [A2 ] if f∈ (13.2) 3 [A ] otherwise. W.l.o.g. we can assume that f is an n-ary function with n ≥ 4 and defined by f (x) := a0 +

n

ai · xi (mod k),

i=1

where 0 ∈ {a1 , a2 , ..., an }. For q := a1 ' a2 ' ... ' an ' k the following two cases are possible: Case 1: q = 1. In this case, we have [f ] = [[f ]id ∪ [{f }]1 ] by Lemma 13.3.2, (b), whereby our theorem follows from Theorem 13.3.1. Case 2: q = 1. With the aid of Lemma 13.3.5, one can reduce this case to the first case (see [Sze 78] for details). We notice that the paper [Sze 78] contains precise representations of the subclasses of Lk , where k is square-free. Lemma 13.3.7 If k ∈ N is not square-free, then there exists a subclass of Lk that is not finitely generated and does not have a basis. Proof. If k can not be represented as a product of different prime numbers, then there is a t ∈ Ek \{0} with t2 = 0 (mod k). Then, the set


Qt :=

{f n ∈ Lk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + t ·

n

393

ai · xi (mod k)}

i=1

n≥1

is a subclass of Lk , whose functions f n , g m for arbitrary n, m have the property that the number of the essential variables of f g is at most n−1. Therefore, Qt does not have a finite generating system. Since by Theorem 8.1.6 |L↓k (Lk )| ≤ ℵ0 holds, every class of linear functions that does not have a finite generating system, does not have a basis. The following theorem is a consequence of Theorems 13.3.6, 3.2.2.2, 13.2.3, and 8.1.6, Lemma 13.3.7, and [Lau-S 90] (for k = 6): Theorem 13.3.8 For arbitrary k ∈ N \ {1} it holds: < ℵ0 if k is square-free, |Lk (Lk )| = ℵ0 otherwise. In particular, we have: k 2 3 4 5 6 7 8 9 |L↓k (Lk )| 15 38 ℵ0 319 7524 470 ℵ0 ℵ0

A solution for the completeness problem for Lk , where k ∈ N is arbitrary, follows from αs 1 α2 Theorem 13.3.9 ([Lau 88a]; without proof ) Let k = pα (pi = 1 p2 ...ps j; pi ∈ P, αi ∈ N for i = 1, ..., s). Then, Lk has exactly pj for i = s s + 2s − 1 + i=1 pi pairwise different maximal classes: n (1) Sp := n≥1 {f n ∈ Lk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + i=1 ai xi (mod k) ∧ n i=1 ai = 1 (mod p)}, where p ∈ {p1 , ..., ps }; (2) Wq := Uq ∪ t|q,t=1 Qt , where q is a square-free divisor of k;

(3) Ta,p := Lk ∩ P olk {x ∈ Ek | p|(x − a)}, where a ∈ Ek and p ∈ {p1 , ..., ps }. As a consequence from the above theorem and from Lemma 13.3.3 we get: Lemma 13.3.10 Let A be an subclass of Lk . Then either r1 ∈ A or there exists a square-free divisor q of k with A ⊆ Wq . Since all subclasses of Lk that contain r1 were determined in Section 13.1, we need all subclasses of the maximal classes of the type Wq for a complete description of L↓k (Lk ). For k = 4 the following theorem gives a rough description of the missing classes.

394


Theorem 13.3.11 ([Lau 88a]; without proof ) It holds: (a) |L↓4 ([L14 ])| = 189. (b) |L4 ([r1 ], L4 )| = 15. (c) Exactly 20 subclasses of L4 contain the function r2 , but not the function r1 . (d) A subclass of L4 , which contains neither r1 nor r2 , is a subset of Q2 ∪[L14 ] and there exist a subclass B of [L14 ] and some t0 , t1 , t2 , t3 ∈ N ∪ {∞} with A = B ∪ K0,t0 ∪ K1,t1 ∪ K2,t2 ∪ K3,t3 ,

where Ka,r :=

∅

r

if r = 1,

[{a + 2 · i=1 xi (mod 4)}] if r > 1 (a ∈ E4 , r ∈ N) and Ka,∞ := r≥2 Ka,r . We notice that the lattice Lk (Lk ) for k = p2 (p prime) was studied in detail in [Bul-I 2002] and [Bul 2002]. To describe further results about classes of linear function, we generalize the set Lk as follows: Let R = (R; +, ·, −, 0, 1) be a finite unitary ring and let M := (M ; +, −, (r)r∈R , 0) be a finite faithful2 module over R. Further, we consider the mapping defined by : R × M −→ M, (r, x) → r x := r(x), which fulfills the axioms (M1 )–(M4 ) (see Part I, Section 1.2.8). Then one can define a closed subset LM of PM as follows: LM :=

n≥1

{f n ∈ PM | ∃ a0 ∈ M ∃ a1 , . . . , an ∈ R : f (x) = a0 +

n

ai xi }.

i=1

It is usual to call the elements of LM also linear functions over the (left) module M. We remark that one can define analogously a set LM for the case that M is a right module. (the ring of all (m, m)-matrices over the field Zp ; p If one sets R = Zm×m p (the module of all (m, 1)-matrices over R), then LM prime) and M = Zm×1 p is isomorphic to the set of all quasi-linear functions of Ppm (see Section 5.2.4). This matrix representation of the quasi-linear function was used in [Sze-S 81] to describe all subclasses that contain certain sets of unary quasi-linear functions: 2

A module M over a ring R is faithful iff {r ∈ R | ∀m ∈ M : r(m) = 0} = {0}.


395

Theorem 13.3.12 ([Sze-S 81]; without proof ) Let p prime, m ∈ N, pm ≥ 3, and M = Zm×1 . Moreover, set R = Zm×m p p I := {(i, j) ∈ N20 | 0 ≤ j ≤ i ≤ k − 1} ∪ {(−1, 0), (k − 1, k)}, H(r,s) := [{A · x ∈ L1M | rg(A) ≤ r ∨ rg(A) = k}] ∪ n≥2 {f n ∈ LM ∩ P olM {0} | dim(Im(f )) ≤ s}, where (r, s) ∈ I; 3 H(r,s) ; c := {f + c ∈ LM | f ∈ H(r,s) ∧ c ∈ M }, where (r, s) ∈ I. Then: (a) H(r,s) ⊆ H(r ,s ) iff r ≤ r and s≤ s . n } (b) H(k−1,k) = LM ∩ P olM {0} = n≥1 { i=1 Ai · xi | A1 , ..., An ∈ Zm×m p and H(k−1,k);c = LM . (c) The sets H(r,s) , (r, s) ∈ I, are the only subclasses of LM ∩ P olM {0} containing (LM ∩ P olM {0})1 . (d) The sets H(r,s);c , (r, s) ∈ I, are the only subclasses of LM containing L1M . For the case p = m = 2, one can find in [Kro-R 2003] all subclasses of quasi-linear selfdual4 functions, which contain all unary quasi-linear selfdual functions. In the case that R = M is a finite field F, one can find results on L↓F (LF ) in [Sze 80]. Among other things, the following theorem was proven in [Sze 80]: Theorem 13.3.13 (without proof ) For any finite field F, the class LF has only finitely many subclasses. Every subclass A of LF with A ⊆ [L1F ] contains the function x + y − z and is a class of the following form (a) or (b): n (a) I(E, S) := n≥1 {a0 + i=1 ai xi | {a1 , ..., an } ⊆ E ∧ (∃s, s ∈ S : n a0 = s − ( i=1 ai ) s }, where E is a universe of a subfield of F and S = V + a for an a ∈ F and a subspace V of F, considered as a vector space over E. n n (b) I(E, S0 ) := n≥1 {a0 + i=1 ai xi | {a1 , ..., an } ⊆ E ∧ i=1 ai = 1 ∧ a0 ∈ S0 }, where E is a universe of a subfield of F and S0 is a subspace of F, considered as a vector space over E. Theorem 13.3.14 ([Sze 80], without proof ) Let k = pm (p prime, m ≥ 1), F := (Ek ; +, ·) a field and let QLF be the set of all quasi-linear functions of Pk (see Section 5.2.4). Further, we set n LF := n≥1 {f n ∈ QLk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + i=1 ai · xi }, d

LF ;d := [LF ∪ {xp }] where d ∈ N is a divisor of k. 3 4

rg(A) denotes the range of the matrix A and dim V is the dimension of the vector space V . We remark that Im(f ) is a vector space, if f (x) = n i=1 Ai · xi . with respect to a fixed-point-free permutation π of order p

396


(In particular, LF ;1 = QLk and LF ;k = LF .) Then for any subclass A ⊂ Pk with LF ⊆ A, there exists a divisor d of k such that A = LF ;d . Theorem 13.3.15 ([Sze 80], without proof ) Let M be a faithful unitary module over a unitary ring R. Then for any subclass A of LM containing the operation x+y −z there exists a unique subring T of R and a unique T-submodule N of T × M such that A=

n≥1

{m +

n i=1

ri xi | {r1 , ..., rn } ⊆ T ∧ (1 −

n

ri , m) ∈ N }

i=1

(see also Section 9.6). Next we give some results by A. A. Bulatov about the elements of L↓M (LM ), where the unitary ring R is commutative (i.e., · is kommutative). The following theorem is the basis for a classification of the elements of L↓M (LM ). Theorem 13.3.16 ([Bul 98a]; without proof ) Let R be kommutative. Then: (1) The minimal classes 5 of L↓M (LM ) are exactly the following classes: (a) JM ; (b) [ca ], where a ∈ M ; (c) [ε x + (1 − ε) b], where b ∈ M , ε ∈ R \ {0, 1} and ε · ε = ε (e.g., ε is an idempotent of the ring R). (2) The minimal clones, which are elements of L↓M (LM ), are exactly the following clones: (a) JM ∪ A, where A is a minimal class of the form (b) or (c) (see (1)); (b) [x + b], where b ∈ M is of prime additive order; (c) [α x + b], where b ∈ M , α ∈ R \ {1} and there exists a prime number p with αp = 1 and (1 + α + α2 + ... + αp−1 ) b = 0; (d) [x − r y + r z], where r ∈ R \ {0} with r2 = 0 and r is of prime additive order; (e) [x + y − z], if char R is a prime number.6 In [Bul 98b], it was shown that any element of L↓M (LM ) can be assigned to one of four types, defined below. The definitions of these types use properties of coefficients of functions and, in addition, some special functions. Classes of the first type are those containing a function ε x + (1 − ε) y where ε is an idempotent of R (ε = 0, 1). Similarily, classes of the second type are those not of the first type and containing the function ε x + (1 − ε) b where ε is n an idempotent of R (ε = 0, 1) and b ∈ M . A function f = i=1 αi xi + a is called primitive, if there exists j ∈ {1, ..., n} such that αj is an invertible 5 6

See Chapter 19. If there is a least positive integer n with x + x + ... + x = 0 for all x ∈ R, then n times the ring R is said to have characteristic n (notation char R).


397

element and αi is a nilpotent element7 whenever i = j. Classes of the third type consist of primitive functions and functions whose coefficients are nilpotent elements. Finally, a class belongs to the fourth type if it contains the Mal’tsev function x + y − z and is not of the first type. ´ Szendrei in [Sze 80] (see The classes of the forth type were described by A. Theorem 13.3.15). Classes of the first and second type were studied in [Bul 98a]. The paper [Bul 98b] deals with the classes of primitive functions and with the lattice of classes of the third type. In particular, A. A. Bulatov completely describes the lattice of classes of primitive functions in [Bul 98b]. We need the following notations for the last theorem of this section: In [P¨ os-K 79], the direct product of functions and subclasses of Pk was defined in the following way. Let a set A := A1 ×A2 be Cartesian product of the sets A1 (n) (n) and A2 . The direct product of the functions f1 ∈ PA1 and f2 ∈ PA2 is (n)

the function f ∈ PA

defined by the equality

f ((b1 , c1 ), ..., (bn , cn )) = (f (b1 , ..., bn ), f (c1 , ..., cn )) for arbitrary b1 , ..., bn ∈ A1 , c1 , ..., cn ∈ A2 . In this case, we shall write f = f1 ⊗ f2 . The set (n)

(n)

C = {f1 ⊗ f2 | f1 ∈ C1 , f2 ∈ C2 , n ∈ N} is the direct product of the classes C1 ⊆ PA1 , C2 ⊆ PA2 and is denoted by C1 ⊗ C2 . It is well known that, for any idempotent ε ∈ R \ {0, 1}, we can represent the module M as the direct sum of two submodules (ε M) ⊕ ((1 − ε) M) (this is the so-called Peirce decomposition). Further, each function f (x) := n a + i=1 f = f1 ⊗ f2 where f1 (x) = nαi xi ∈ LM can be representedas n ε a + i=1 αi xi and f2 (x) = (1 − ε) a + i=1 αi xi are functions on the sets εM and (1−ε)M , respectively. To show this take (b1 , c1 ), . . . , (bn , cn ) ∈ M = (ε M ) × ((1 − ε) M ). Then f ((b1 , c1 ), . . . , (bn , cn )) = f (b1 + c1 , . . . , bn + cn ) n = a+ αi (bi + ci ) i=1

= εa+

n i=1

7

εαi (bi + ci ) + (1 − ε) a +

n

(1 − ε)αi (bi + ci )

i=1

An element a of a ring R = (R; +, ·, −, 0) is called nilpotent iff there exists an n ∈ N with an = 0.

398


= εa+

n

αi bi + (1 − ε) a +

i=1

n

αi ci

i=1

= (f1 (b1 , . . . , bn ), f2 (c1 , . . . , cn )). n Conversely, the function f = f1 ⊗ f2 = (b + c) + i=1 i + (1 − ε)ζi ) xi (εγ n corresponds to the given pair of functions f = b + γi xi ∈ LεM , 1 i=1 n f2 = c + i=1 ζi xi ∈ L(1−ε)M . The modules ε M , (1 − ε) M can be considered as εR- and (1 − ε)R-modules, respectively. So, LM = LM1 × LM2 where M1 = ε M is an εR-module and M2 = (1 − ε) M is an (1 − ε)Rmodule. Let C be a class of all functions preserving the kernels of projections os-K 79] (3.3.4, 3.3.5, p. 85) it was noted that from A onto A1 and A2 . In [P¨ the interval LA1 ×A2 (JA1 × JA2 , C) can be decomposed into the direct product [JA1 , PA1 ]×[JA2 , PA2 ]. Using this decomposition we obtain the next statement. Theorem 13.3.17 ([Bul 98a]; without proof ) For any idempotent ε ∈ R, (a) if the function ε x + (1 − ε) y belongs to a subclass C of LM , then C = C1 ⊗ C2 where C1 , C2 are subclasses of LεM and L(1−ε)M , respectively and L1 , L2 both contain all projections; (b) LM ([ε x + (1 − ε) y], LM ) ∼ = LεM (JεM , LεM ) × L(1−ε)M (J(1−ε)M , L(1−ε)M ). For a proper ideal I of R, let LIM be the set of all functions f ∈ LM whose coefficients belong to I. Further, let LP M be the set of all primitive functions of LM . For a ring R let idemp(R) be the set of all idempotents = 0. Moreover, put n n KM (R , M ) = { i xi + a | n ≥ 1, 1 , . . . , n ∈ R , (1 − i , a) ∈ M } i=1

i=1

where R is a unitary subring of R and M is a submodule of R -module R × M . Theorem 13.3.18 ([Bul 98a]; without proof ) The set of all maximal clones of LM comprise exactly the following clones: (a) (LIεM ∪ LP εM ) ⊗ L(1−ε)M where ε ∈ idemp(R); (b) KεM (R , R × ε M ) ⊗ L(1−ε)M where ε ∈ idemp(R), R is a maximal unitary subring of εR without proper idempotents; (c) KεM (εR, M ) ⊗ L(1−ε)M where ε ∈ idemp(R) is a minimal idempotent and M is a maximal submodule of εR × εM .

14 Submaximal Classes of P3

A subclass (or a subclone) of Pk is called submaximal if it is covered by a maximal class (clone).1 The concept submaximal class was introduced of I. G. Rosenberg in [Ros 74]. In [Ros 74] one also finds the first results about submaximal classes of Pk (see Chapter 17 for details). In general, the submaximal classes seem to be interesting for the following reasons.2 The largely unknown lattice Lk has intervals with antichains of cardinality c situated far down from the top. It is not unreasonable to assume that the lattice Lk is nicer near the top, and therefore the submaximal clones are good candidates. The problem of determining certain submaximal clones also came up in the study of shortest maximal chains in the lattice (see [Sze 83]). Given a maximal clone M , one can ask for a completeness criterion for M : under what conditions does the clone [F ] generated by some F ⊆ M coincide with M ? In the case that M is finitely generated, a full list of clones maximal in M would provide a general criterion, because then [F ] if and only if F is contained in no clone maximal in M . An application could be a characterization of Sheffer functions for M . V. B. Kudrjavcev and P. Schofield proven that they exist exactly for maximal clones of the form P olk with ∈ C1k ∪ Sk ∪ Uk (see Chapter 7), but the examples in the proofs have many variables. It would be interesting to have simple criteria of type Theorem 7.1.3, which, in its turn, could lead to the question: what is the minimum number of functional values 1

2

It is easy to see that every submaximal class is a clone: Assume there is a maximal class M ⊂ Pk and a submaximal class S ⊂ M with Jk ∩ S = ∅ and [S ∪ {f }] = M for all f ∈ M \ S. Then, M = S ∪ Jk , since M is a clone (see Theorem 5.1.1) and Jk = [e] for all e ∈ Jk . However, M = S ∪ Jk with S ∩ Jk = ∅ is not possible, since S ∪ Jk = Pk (k − 1) ∪ [Pk1 ] and every maximal class A (= Pk (k − 1) ∪ [Pk1 ]) of Pk contains an idempotent function g with g ∈ Jk . For example, if ∈ Mk and o is the smallest element of , then the function f 2 , defined by f (o, x) = f (x, o) = o for all x ∈ Ek and f (x, y) = x otherwise, is idempotent and belongs to (P olk ) \ Jk . With the help of the theorems from Chapter 5, one can easily find idempotent functions for the other possible cases. The following is an indirect quotation from [Ros-S 85].

400


whose knowledge can guarantee that a function is a Sheffer function? Finally, the submaximal clones may be of interest on their own, e.g., as a source of examples and counter-examples. For arbitrary k, the full list of the maximal classes of a maximal class A of Pk is only known, if A has the type S ([Ros-S 84], see Section 18.1), C1 (see 2 ∪ {(k − 1, k − 1)} (see Chapters 16 and 17) or A = P olk , where := Ek−1 Section 18.3). There are, however, some papers in which one finds submaximal classes for specific k or only such submaximal classes that contain certain functions. In Section 14.1, one finds a complete description of all submaximal classes of P3 and some remarks about generalization of the given theorems. The papers [Mac 79], [Mar-D-H 80], [Sal 64], [Bag-D 82], and [Lau 82a] form the basis of this description. The theorems from the first section are proven then in Sections 14.2–14.9. In Section 14.10, we will prove that there are 5 submaximal classes with finitely many subclasses, 7 with countably many subclasses, and the remaining 146 with uncountably many subclasses. All elements of the lattices L↓3 (A), where A is a submaximal class with|L↓3 (A)| ≤ ℵ0 are determined in Chapter 15.

14.1 A Survey of the Submaximal Classes of P3 The following theorem was proven 1958 by Jablonskij and is a special case of Rosenberg’s Theorem 6.1. Theorem 14.1.1 ([Jab 58]) P3 has exactly 18 maximal classes: (1) P ol3 {0}

(2) P ol3 {1}

(3) P ol3 {2}

(4) P ol3 {0, 1}

(5) P ol3 {0, 2} 0 1 (7) P ol3 1 2 0 1 (9) P ol3 0 1 0 1 (11) P ol3 0 1 0 1 (13) P ol3 0 1 0 1 (15) P ol3 0 1

2 0 2 2 2 2 2 2 2 2

0 2 0 1 1 0 1 0

2 0 0 2 1 2 0 1

1 2 0 2 1 2 2 1

(6) P ol3 {1, 2} 0 1 (8) P ol3 0 1 0 1 (10) P ol3 0 1 0 1 (12) P ol3 0 1 0 1 (14) P ol3 0 1 0 1 (16) P ol3 0 1

(17) P ol3 {(a, b, c, d) ∈ E34 | a + b = c + d (mod 3)} (18) P ol3 {(a, b, c) ∈ E33 | |{a, b, c}| ≤ 2}

2 2 2 2 2 2 2 2 2 2

0 1 1 2 0 2 0 1 2 0

1 0 2 1 0 1 1 0 0 2

2 1 0 2 2 1

2 0 1 2

14.1 A Survey of the Submaximal Classes of P3

401

In the following theorems, one finds all A-maximal classes for every maximal class A from Theorem 14.1.1. The next two theorems are special cases of general statements about the maximal classes of P olk E with 1 ≤ |E| ≤ k − 1 (see Chapters 16 and 17). Theorem 14.1.2 ([Lau 82a]) Let {a, b, c} := E3 . Then Ta := P ol3 {a} has exactly the following 12 maximal classes: (1) Ta ∩ P ol3 {b}

(2) Ta ∩ P ol3 {c}

(3) Ta ∩ P ol3 {a, b} (5) Ta ∩ P ol3 {b, c} a b (7) Ta ∩ P ol3 a b a b (9) Ta ∩ P ol3 a b a a b a (11) P ol3 a b a c

c c c c c a

a b a c

b a a b

a c c a c b

(4) Ta ∩ P ol3 {a, c} a b c b c (6) T ∩ P ol3 a b c c b a b c a a b (8) Ta ∩ P ol3 a b c b c c a b c (10) P ol3 a c b a a b a c b c (12) P ol3 a b a c a c b

Theorem 14.1.3 ([Lau 82a]) Let {a, b, c} := E3 . Then Ta,b := P ol3 {a, b} has exactly the following 15 maximal classes: (1) Ta,b ∩ P ol3 {a} (2) Ta,b ∩ P ol3 {b} a b a (3) Ta,b ∩ P ol3 a b b a b (4) Ta,b ∩ P ol3 ⎛b a a a a b ⎜a a b b ⎜ (5) Ta,b ∩ P ol3 ⎝ a b a a a b b a (6) Ta,b ∩ P ol3 {c} 0 1 2 a (7) Ta,b ∩ P ol3 0 1 2 b 0 1 2 a (8) Ta,b ∩ P ol3 0 1 2 c a b a (9) P ol3 a b c a b b (10) P ol3 a b c a a b b a (11) P ol3 a b a b c a a b b b (12) P ol3 a b a b c

b a b a

a b b a

b a a b

b a c b c a c b

⎞ b b⎟ ⎟ b⎠ b

402

14 Submaximal a a b (13) P ol3 ⎛a b a a b a (14) P ol3 ⎝ a b a ⎛a b b a b b (15) P ol3 ⎝ a b a a b a

Classes of P3 b a c b c b c a c b ⎞ b a b a b b b a a b⎠ a c c c c ⎞ a a b b a a b b a b a b a b⎠ a b a b b c c

The following theorem is a special case of a theorem from [Lar 93], in which B. Larose determined all maximal classes of P olk , where is a total order on Ek and 2 ≤ k ≤ 5. Theorem 14.1.4 ([Mac 79]) Let E3 := {a, b, c} and let min, max be defined by x a a a b b b c c c Then O := P ol

y max(x, y) min(x, y) a a a b b a c c a a b a . b b b c c b a c a b c b c c c

0 1 2 a a b 0 1 2 b c c

has exactly the following 13 maximal

classes: (1) O ∩ P ol{a}

(2) O ∩ P ol{c}

(3) O ∩ P ol{a, b} (5) O ∩ P ol{b, c} 0 (7) O ∩ P ol3 0 0 (9) O ∩ P ol3 0 (11) O ∩ P ol3 ι33

1 2 b c 1 2 c b

1 2 b b b c 1 2 a b c b

(4) O ∩ P ol{a, c} 0 (6) O ∩ P ol3 0 0 (8) O ∩ P ol3 0 0 (10) O ∩ P ol3 0

1 2 a b 1 2 b a

1 2 a b a c 1 2 b a c a 1 2 c a c b 1 2 a c b c

⎛

0 (12) P ol3 ⎝ 0 0 ⎛ 0 (13) P ol3 ⎝ 0 0

1 2 0 0 1 2 0 0 1 2 1 2 1 2 0 0 1 2 1 2 1 2 1 2

14.1 A Survey of the Submaximal Classes of P3 ⎞ 1 0 0 1 1 1 2 2 ⎠ = [{min} ∪ O1 ] 3 2 0 0 1 ⎞ 1 1 2 2 2 0 0 1 ⎠ = [{max} ∪ O1 ] 4 2 1 2 2

403

The following theorem is a special case of theorems from [Ros-S 84] and [Lau 84]: Theorem 14.1.5 ([Mar-D-H 80]) 0 1 2 S := P ol3 has exactly two maximal classes: 1 2 0 (1) S ∩ P ol3 {0} (2) S ∩ P ol3 λ3 . The following theorem follows from Section 13.2: Theorem 14.1.6 ([Bag-D 82]) L3 := P ol3 λ3 has exactly 5 maximal classes: (1) L3 ∩ P ol{0}. (2) L3 ∩ P ol{1}. (3) L3 ∩ P ol{2}. 0 1 2 (4) L3 ∩ P ol3 1 2 0 (5) [(L3 )1 ].

Theorem 14.1.7 ([Lau 82a]) Let E3 := {a, b, c}. Then C := P ol3

0 1 2 a b a c 0 1 2 b a c a

has exactly the fol-

lowing 7 maximal classes: (1) C ∩ P ol{a} (2) C ∩ P ol{a, b} 3

4

The equality of these two sets can be proven as follows: It is easily checked that [{min} ∪ O1 ] ⊆ P ol3 (...) is valid. Then one proves that [{min} ∪ O1 ] is maximal in O (see for this purpose also [Mac 79]). The equality of these two sets results from the considerations to the statement (12) and the fact that the classes (12) and (13) are isomorphic.

404


(3) C ∩ P ol{a, c} (4) C ∩ P ol{b, c} 0 (5) C ∩ P ol3 0 0 (6) C ∩ P ol3 0 ⎛ a (7) C ∩ P ol3 ⎝ b c

1 2 a a 1 2 b c

1 2 a b a c b 1 2 b a c a c

⎞ b a c a b a a b b a b c a a c c a c a c a a a b a b a b b a c a c a c c⎠ c b b a a a b a b b b a a c a c c c

Theorem 14.1.8 ([Lau 82a])

Let E3 := {a, b, c}. Then U := P ol3

0 1 2 a b 0 1 2 b a

13 maximal classes: (1) U ∩ P ol{c} (2) U ∩ P ol{a, b} (3) U ∩ P ol{a, c} (4) U ∩ P ol{b, c} 0 1 2 a b a c (5) U ∩ P ol3 0 1 2 b a c a 0 1 2 b a b c (6) U ∩ P ol3 0 1 2 a b c b 0 1 2 a (7) P ol3 0 1 2 b a c b c (8) P ol3 c a c b 0 1 2 a b a b (9) P ol3 0 1 2 b a c c ⎛ 0 1 2 a a b b c c a (10) P ol3 ⎝ 0 1 2 a a b b c c b 0 1 2 b c a c a b c ⎛ a a a a b b b b a b (11) P ol3 ⎝ a a b b a a b b a b a b a b a b a b c c ⎛ ⎞ a a a b b a b b c ⎜a a b b a b a b c⎟ ⎟ (12) P ol3 ⎜ ⎝a b a a b b a b c⎠ a b b a a a b b c

⎞ b a⎠ c ⎞ c c c c c c⎠ a b c

has exactly the following

14.1 A Survey of the Submaximal Classes of P3 ⎛ (13)

a ⎜a c c

P ol3 E24 ∪ ⎜ ⎝

a b c c

b a c c

b b c c

a c a c

a c b c

b c a c

b c b c

a c c a

a c c b

b c c a

b c c b

c a a c

c a b c

c b a c

c b b c

c a c a

c a c b

c b c a

c b c b

c c a a

c c a b

c c b a

c c b b

405 ⎞ c c⎟ ⎟. c⎠ c

Theorem 14.1.9 ([Lau 82a]) Let P3 (2) := {f ∈ P3 | |Im(f )| ≤ 2}. The set P ol3 ι33 (= P3 (2) ∪ [P31 ]) has exactly 5 maximal classes: (1) P3 (2) ∪ [{s1 , s2 }] (2) P3 (2) ∪ [{s1 , s3 }] (3) P3 (2) ∪ [{s1 , s6 }] (4) P3 (2) ∪ [{s1 , s4 , s5 }] (5) n≥1 {f n ∈ P3 |∃fi ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + ... + fn (xn )) (mod 2)} ∪ [P31 ]. (The definitions of the functions s1 , ..., s6 are given in Section 15.2, Table 15.1.)

Table 14.1 gives a summary of the above-described submaximal classes, where {a, b, c} := E3 and {α, β, γ} := E3 . The submaximal classes that can be described as intersections of maximal clones are labelled with numbers 1–17. The other classes are labelled with numbers 18–43 in the order that they occur in Theorems 14.1.2–14.1.9. With the aid of Table 14.1 and with a b a 0 1 2 a c = P ol{a, b} ∩ P ol3 , 5 {a, b, c} = E3 , P ol3 a b c 0 1 2 c a one can prove the following theorem: Theorem 14.1.10 P3 has exactly 158 submaximal classes. 5

This equationcan be proven as follows: a b a Because of ∆ = {a, b} and a b c a b c a b a 0 1 2 a c ◦ = it holds a b a a b c 0 1 2 c a a b a 0 1 2 a c ⊆ P ol{a, b} ∩ P ol3 . P ol3 a b c 0 1 2 c a a b a 0 1 2 a c is valid, we Since, in addition, ∆ {a, b} × = P ol3 a b c 0 1 2 c a have also 0 1 2 a c a b a P ol{a, b} ∩ P ol3 ⊆ P ol3 . 0 1 2 c a a b c

406

14 Submaximal Classes of P3 Table 14.1 Submaximal classes of P3

i

A

1 P ol{a} ∩ P ol{b} 2 P ol{a} ∩ P ol{a, b} 3 P ol{a} ∩ P ol{b, c} # $ 1 2 b c 4 P ol{a} ∩ P ol 0 #0 1 2 c b $ 1 2 a b a c 5 P ol{a} ∩ P ol 0 # 0 1 2 b a c $a 1 2 a a b 6 P ol{a} ∩ P ol 0 0 1 2 b c c 7 P ol{a} ∩ P olλ3 # $ 1 2 8 P ol{0} ∩ 0 1 2 #0 $ 1 2 a b 9 P ol{a, b} ∩ P ol 0 #0 1 2 b a$ 1 2 a c 10 P ol{a, b} ∩ P ol 0 #0 1 2 c a $ 1 2 α α β 11 P ol{a, b} ∩ P ol 0 #0 1 2 β γ γ $ 1 2 α β α γ 12 P ol{a, b} ∩ P ol 0 0$ 1 2 β# α γ α # 1 2 a b ∩ P ol 0 1 2 a a 13 P ol 0 #0 1 2 b a$ #0 1 2 b c 1 2 a b ∩ P ol 0 1 2 a b 14 P ol 0 0# 1 2 b a #0 1 2 b a $ 1 2 a a b ∩ P ol 0 1 2 α 15 P ol 0 0 1 2 β 0 1 2 b c c 0 1 2 a a b 3 16 P ol 0 1 2 b c c ∩ P olι3 # $ 1 2 ∩ P olλ 17 P ol 0 3 1 2 0 # $ a b c 18 P ol a c b # $ a b a c 19 P ol a #a b a c a $ a b a c b c 20 P ol a # a b a $c a c b b a 21 P ol a # a b $b 22 P ol ab ab ⎛ ⎞ a a a b b a b b a b b a b a b⎠ 23 P ol ⎝ a a b a a b b a b # a b b a a$ a b b a b b a 24 P ol a #a b a b c $ a b b a c b c 25 P ol a a b a b c a c b

6

6

ni (A) 3 6 3 3 3 6 3 1 3 6 9 b c a c β α

$ $ c a $ α γ γ α

ni (A) denotes the number of possibilities for A in case i.

9 6 6 9 3 1 3 3 3 3 3 3 6 3

14.1 A Survey of the Submaximal Classes of P3

i 26 27 28 29 30 31 32 33 34 35 36

%

a b a b a b a b P ol a b a b b a a b %a b b a c c c c a b b a a b b a P ol a b a b a b a b a b a a b a b b [{max} ∪ O1 ] [{min} ∪ O1 ] [(L3 )1 ] # $ 1 2 a P ol 0 #0 1 2 b$ P ol ac ac cb cb # $ 1 2 a b a b P ol 0 %0 1 2 b a c c 0 1 2 a a b b c P ol 0 1 2 a a b b c %0 1 2 b c a c a a a a a b b b b P ol a a b b a a b b ⎛a b a b a b a b a a a b b a b b a b b a b a b ⎝ P ol a a b a a b b a b a b b a a a b b

37 ⎛P olE24 ∪ a ⎜ a ⎜ ⎝ c c

a b c c

#

38 P ol # 39 P ol 40 P ol

b a c c

b b c c

a c a c

a c b c

b c a c

b c b c

0 1 2 a a 0 1 2 b c

a c c a

$

a c c b

b c c a

b c

b a c

a c b

c a b

a a a

b a a

ni (A) 3

a b a b c c

& 3 3 3 1 3 3

c a c b b c a b a b c⎞ c c c⎠ c c b c c b

c a a c

c a b c

3

&

b a c & c c c c c c a b c

3 3 3

c b a c

c b b c

c a c a

c a c b

c b c a

c b c b

c c a a

c c a b

c c b a

c c b b

⎞ c ⎟ c ⎟ c ⎠ c

3 3

0 1 2 a b a c b 0 1 2 b a c a c

#a

A

&

407

a b a

a a b

b b a

$ 3 b a b

a b b

b b b

c a a

a c a

a a c

c c a

c a c

a c c

$ c c c

3

41 P3 (2) ∪ [X], X ∈ {{s1 , s2 }, {s1 , s3 }, {s1 , s6 }}

3

42 P3 (2) ∪ [{s1 , s4 , s5 }]

1

43

n≥1 {f

n

∈ P3 |∃fi ∈ P31 : f (x1 , ..., xn ) = f0 (f1 (x1 ) + f2 (x2 ) + ... + fn (xn )) (mod 2)} ∪ [P31 ]

1

408


14.2 Some Declarations and Lemmas for Sections 14.3–14.9 In this chapter, we use the notations from Chapter 15, Table 15.1 for the unary functions of P3 . We prove Theorems 14.1.2–14.1.9 as follows in the next sections: Suppose in the theorem a list is indicated with subclasses (1), (2), ..., (rT ) for the class T . To prove that this list indicates all submaximal classes of T , we will show that every subset A ⊆ T , which is not contained in any subclass of the list, is a generating system for T . Then, it remains to show that the listed classes (1), (2), ..., (rT ) are proper subsets of the set T , and the set of all these classes is an antichain; i.e., in this set, no two elements are comparable in respect to ⊆. For this purpose, we give some functions (∈ T ) and a table that shows whether any function of these functions belongs to the indicated class. The sign + stands for “the function belongs to the class”. If the considered function does not belong to the class, we write -. We leave the readers to check the given table and, with the aid of the table, to prove that the classes (1), (2), ..., (rT ) are incomparable. If we assume in the proof that A ⊆ T and A is not a subset of the class (i), then there exists a function fi ∈ A which does not belong to the class (i). Since we can choose [A] instead of A, we can assume w.l.o.g. that the following holds: (∗ ):

If the class (i) is given in the form T ∩ P ol3 or P ol3 , where := (σ1 , σ2 , ..., σm ), then fi (σ1 , σ2 , ..., σm ) ∈ /.

Now some lemmas are given that we subsequently need more often and which are consequences from some theorems of Chapters 3 and 12. Already in Chapter 3 (see Theorem 3.3.1) the following was proven: Lemma 14.2.1 Let A be an arbitrary subset of P2 . Then, [A] = P2 if and only if A ⊆ B for every class B of the following list: (1) P ol2 {0} (2) P ol2 {1} 0 1 (3) P ol2 1 0 0 0 1 (4) P ol2 0 1 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 ⎜0 0 1 1 0 1 0 1⎟ ⎟ (5) P ol2 ⎜ ⎝ 0 1 0 0 1 1 0 1 ⎠. 0 1 1 0 0 0 1 1 Lemma 14.2.2 Let A be an arbitrary subset of T0 := P ol2 {0} ⊂ P2 . Then, [A] = T0 if and only if A ⊆ B for every class B of the following list: (1) T0 ∩ P ol2 {1}

14.2 Some Declarations and Lemmas for Sections 14.3–14.9

0 0 0 (3) T0 ∩ P ol2 0 ⎛ 0 ⎜0 (4) T0 ∩ P ol2 ⎜ ⎝0 0 (2) T0 ∩ P ol2

0 1 1 0 0 0 1 1

409

1 1 0 1 0 1 1 1 0 0 1 0

1 0 1 0

0 1 1 0

1 0 0 1

⎞ 1 1⎟ ⎟. 1⎠ 1

As a consequence of Theorem 12.4.3, we get: Lemma 14.2.3 Let A be an only if A ⊆ B for every class (1) P3,2 ∩ P ol3 {0} (2) P3,2 ∩ P ol3 {1} 0 1 (3) P3,2 ∩ P ol3 1 0 0 0 1 (4) P3,2 ∩ P ol ⎛0 1 1 0 0 0 1 1 ⎜0 0 1 1 0 (5) P3,2 ∩ P ol3 ⎜ ⎝0 1 0 0 1 0 1 1 0 0 0 1 0 (6) P3,2 ∩ P ol3 0 1 2 0 1 1 (7) P3,2 ∩ P ol3 . 0 1 2

arbitrary subset of P3,2 . Then [A] = P3,2 if and B of the following list:

0 1 1 0

1 0 0 1

⎞ 1 1⎟ ⎟ 1⎠ 1

The next lemma follows from Theorem 12.5.3: Lemma 14.2.4 Let A be an arbitrary subset of pr−1 T0 := P3,2 ∩ P ol3 {0} ⊂ P3 . Then [A] = pr−1 T0 if and only if A ⊆ B for every class B of the following list: (1) pr−1 T0 ∩ P ol3 {1} 0 0 1 −1 (2) pr T0 ∩ P ol3 0 1 1 0 1 0 (3) pr−1 T0 ∩ P ol3 0 0 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 ⎜0 0 1 1 0 1 0 1⎟ ⎟ (4) pr−1 T0 ∩ P ol3 ⎜ ⎝0 1 0 0 1 1 0 1⎠ 0 1 1 0 0 0 1 1 (5) pr−1 T0 ∩ P ol3 {0, 2} 0 1 1 (6) pr−1 T0 ∩ P ol3 . 0 1 2

410


14.3 Proof of Theorem 14.1.2 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of P ol{0}, which is not contained in any class ((1)–(12)) from the above list, which is given in Theorem 14.1.2. Then, there are some functions f1 , f2 , ..., f12 ∈ [A] with the above property (∗ ). First we prove that (P ol3 {0})1 ⊆ [A]. The function f1 belongs to {c0 , j2 , u2 , u1 , s2 , u5 }. If f1 ∈ {j2 , u1 } then c0 = f1 f1 ∈ [A]. If f1 = u2 then f2 f1 ∈ {c0 , j2 }. For f1 ∈ {s2 , u5 } we have f5 (x, f1 (x)) ∈ {c0 , j2 , u2 }. Therefore, c0 is a superposition over A. The function f3 (x) := f3 (co , x) ∈ {u1 , s2 , u5 } also belongs to [A]. Thus we have to distinguish three cases: Case 1: f3 = u1 . In this case j1 = f4 (c0 , u1 ) and a function f4 (x) = f4 (c0 , x) ∈ {j2 , j5 , s2 } belongs to [A]. Case 1.1: f4 = j2 . (x) := f11 (c0 , j2 , j1 , u2 , u1 ) ∈ [A] and f11 (x) := Then u1 j2 = u2 ∈ [A], f11 f11 (co , j1 , j2 , u1 , u2 ) ∈ [A], where {f11 , f11 } ∩ {s2 , j5 , u5 } = ∅. Because of f12 (c0 , j2 , j1 , u2 , u1 , x, s2 ) ∈ {j5 , u5 } and j2 u5 = j5 we can assume j5 ∈ [A]. Then j2 j5 = u5 and {f7 (c0 , j5 , u5 , j2 , j1 , u2 , u1 ), f7 (c0 , j5 , u5 , j1 , j2 , u1 , u2 )} = {s1 , s2 } ⊆ [A]. Therefore, (P ol3 {0})1 ⊆ [A] in Case 1.1. Case 1.2: f4 = s2 . Because of j1 s2 = j2 one can reduce this case to Case 1.1. Case 1.3: f4 = j5 . In this case, functions u5 = u1 j5 and f9 (c0 , j5 , u5 , u1 , j1 , x) ∈ {j2 , u2 , s2 } are superpositions over A. Therefore, because of j5 u2 = j2 , we have given either Case 1.1 or Case 1.2. Case 2: f3 = u5 . Here we can form the superpositions j5 = f4 (c0 , u5 ) and f6 (x1 , x2 ) := j5 (f6 (c0 , j5 (x1 ), u5 (x1 ), x1 , x2 )) 1 2 0 1 1 2 0 over A, where f6 ∈ . W.l.o.g. let f6 = . If 2 1 1 0 2 1 1 f6 (1, 1) = 1 we obtain Case 1 because of u5 (f6 (j5 (x), x)) = u1 . If f6 (1, 1) = 0 then it holds f6 (x, j5 (x)) = j2 , u5 j2 = u2 and f8 (c0 , j5 , u5 , j2 , u2 , x) ∈ {j1 , u1 , s2 }. Since u5 j1 = u1 and u2 s2 = u1 , we have also u1 ∈ [A] if f6 (1, 1) = 0, i.e., one can reduce Case 2 to Case 1. Case 3: f3 = s2 . Because of f10 (c0 , s2 , x) ∈ {j1 , j2 , j5 , u1 , u2 , u5 }, s2 ji = ui , i ∈ {1, 2, 5}, and u2 s2 = u1 , we obtain either Case 1 or Case 2. Consequently, (P ol{0})1 ⊆ [A] is proven. Next we prove that P3,2 ∩ P ol3 {0} is a subset of [A]. Since (P3,2 ∩ P ol3 {0})1 ⊆ [A] was already shown, we have to find a subset of [A] ∩ P3,2 that is not a

14.3 Proof of Theorem 14.1.2

subset of P ol

0 0 1 0 1 0

, P ol

0 0 1 0 1 1

411

and P olλ2 := {(a, b, c, d) ∈ E24 | a+b =

c + d (mod 2)}, to be able to use Lemma 14.2.4. Obviously, the function f11 (x1 , x2 ) := j5 (f11 (c0 , x1 , x2 , u1 (x1 ), u1 (x2 ))) ∈ [A] ∩ P3,2 0 0 1 does not preserve the relation . For the function 0 1 0 f7 (x1 , x2 ) := f7 (c0 , f11 (x1 , x2 ), u1 (f11 (x1 , x2 )), x1 , x2 , u1 (x1 ), u1 (x2 )) ∈ [A] 1 0 1 we can assume w.l.o.g. that f7 = . If f7 (1, 1) ∈ {0, 2} we have 1 0 0 0 0 1 j1 f7 ∈ P ol3 ∪ P ol3 λ2 . If f7 (1, 1) = 1 then this is valid for j2 f7 0 1 1 instead of j1 f7 . Thus by Lemma 14.2.4, we have P3,2 ∩ P ol3 {0} ⊆ [A]. After these preparations we can easily show that an arbitrary function f n ∈ P ol3 {0} is a superposition over A. For this purpose, we need the functions q1 , q2 ∈ P3,2 defined by 0 if f (x) ∈ {0, 1}, q1 (x) := 1 if f (x) = 2, 0 if f (x) ∈ {0, 2}, q2 (x) := 1 if f (x) = 1, Then f7 (q1 (x), q2 (x)) = f (x) ∈ [A] holds. Consequently, [A] = P ol3 {0}. As explained in Section 14.2, our theorem results from Table 14.2, where the binary functions h1 , h2 ,..., h8 are defined in Table 14.3.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

c0 − − + + − + + + + + + +

j1 + − + + − − + − + − + +

s2 − − − − + + + − − + + +

u5 − + − + + + + + + − + −

Table 14.2 h1 h2 h3 + + + + + − − + − + − − − + + − − + − − − − − − − − + − − − − − − − − −

h4 − − − − − − − − − + − −

h5 + + + + + + − + − − − −

h6 − − + − − − + + − − − −

h7 + − − + − − − − − − + +

h8 − − − + − − − − − − − +

412


x1 0 0 0 1 1 1 2 2 2

x2 0 1 2 0 1 2 0 1 2

h1 0 1 2 2 1 0 2 0 2

Table 14.3 h2 h3 h4 h5 0 0 0 0 1 1 1 1 1 1 2 2 1 2 1 1 1 1 2 1 1 1 0 2 0 1 2 2 1 1 0 2 2 1 1 2

h6 0 0 1 0 0 1 1 1 1

h7 0 0 0 2 1 0 0 0 0

h8 0 1 0 2 0 0 0 0 0

14.4 Proof of Theorem 14.1.3 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof let A be an arbitrary subset of P ol{0, 1}, which is not contained in any class ((1)–(15)) from the above list, which is given in Theorem 14.1.3. Then there are some functions f1 , f2 , ..., f15 ∈ [A] with the above property (∗ ). With the help of Lemma 14.2.1, one can easily prove that every function of P2 is a restriction of a function of [A]. Therefore, a function g ∈ {c0 , j2 , u2 } and a function h ∈ {j0 , j4 , s3 } are superpositions over A. If g ∈ {c0 , j2 } then the functions g g = c0 and h c0 = c1 belong to [A]. If g = u2 then f6 := f6 g ∈ {c0 , c1 , j2 , j3 } and {f6 f6 , h f6 f6 } = {c0 , c1 }. Thus the constant functions c0 and c1 are superpositions over A. Next we prove that the remaining unary functions of P ol3 {0, 1} and all functions of P3,2 are also superpositions over A. For this purpose, we distinguish two cases for the function h: Case 1: h ∈ {j0 , j4 }. Then h := h h ∈ {j1 , j5 } and the functions fi := h fi , i ∈ {1, 2, 3, 4, 5} belong to [A], where fi ∈ P3,2 and fi does not belong to the class (i). Further, := h(f10 (c0 , c1 , x)) belong to [A]. the functions f9 := h(f9 (c0 , c1 , x)) and f10 } ⊆ P3,2 and that It is easy to check that {h, f9 , f10 {h, f9 , f10 } is no subset 0 1 0 0 1 1 and P ol3 . Consequently, by Lemma of the classes P ol3 0 1 2 1 1 2 14.2.3, every function of P3,2 is a superposition over A. := f11 (c0 , j2 , j0 , c1 , x) ∈ {v2 , s3 }. A function of [A] is also f11 Case 1.1: f11 = v2 . := f12 (c0 , j2 , j3 , c1 , v2 ) = u2 ∈ [A] and In this case, we have f12 {f15 (c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 , u2 , v2 ), f15 (c0 , c1 , j1 , j0 , j2 , j3 , j5 , j4 , u2 , v2 )} = {s1 , s2 }. Consequently, (P ol3 {0, 1})1 ⊆ [A]. = s2 . Case 1.2: f11


First we form a ternary function f) 15 which ⎛ 0 0 ⎝1 0 f) 15 0 2

413

as a superposition over P3,2 ∪ {f15 }, for ⎞ ⎛ ⎞ 1 0 1⎠=⎝1⎠ 2 2

) ) holds. If f) 15 (0, 1, 0) = 0 then s3 (f15 (c0 , x, s3 )) = v2 ∈ [A]. If f15 (0, 1, 0) = 1 ) then f15 (j0 , x, s3 ) = v2 ∈ [A]. Consequently, we can reduce Case 1.2 to Case 1.1. Consequently, P3,2 ∪ (P ol3 {0, 1})1 ⊆ [A] was proven in Case 1. Case 2: h = s3 .

0 0 1 2 0 2 1 2 W.l.o.g. we can assume f8 = . Consequently, 0 1 2 2 0 2 1 1 f8 (x1 , x2, x3 ) :=f8 (c0 , c1, x1 , x2 , x3 , s3 (x2 ), s3 (x3 )) belongs to [A], and we 0 2 0 2 have f8 = . Then the functions f8 := f8 (s3 , c0 , x) and f8 := 2 2 0 1 f8 (x, s3 , c0 ) are superpositions over A, where {f8 , f8 } ∩ {j0 , j3 , j2 , j4 } = ∅. Thus, we have either Case 1 or j2 = s3 j3 and j3 = s3 j2 belong to [A]. A superposition over A is also a certain binary function q with the property ⎛ ⎞ ⎛ ⎞ 0 0 1 ⎜0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ q⎜ ⎝ 1 0 ⎠ = ⎝ 0 ⎠. 1 1 0 If {q(0, 2), q(2, 0), q(1, 2), q(2, 1), q(2, 2)} ∩ {0, 1} = ∅ then {q(c0 , x), q(x, c0 ), q(j2 , x), q(x, j2 ), q(x, x)} ∩ {j0 , j4 } = ∅, i.e., one can reduce Case 2 to Case 1 here. Therefore, we can assume q(a) = 2 for every a ∈ E32 \E22 in the following. := f14 (c0 , c1 , j2 , j3 , x, s3 , u2 , v2 ) ∈ Then q(j3 , x) = u2 , s3 u2 = v2 and f14 {j0 , j1 , j4 , j5 } belong to [A]. Since s3 j1 = j4 and s3 j5 = j0 , we have again Case 1. Hence P3,2 ∪ (P ol3 {0, 1})1 ⊆ [A]. 0 1 2 0 1 0 W.l.o.g. we can assume f7 = . Then f7 (x1 , x2 ) := 0 1 2 1 0 2 u2 (f7 (c0 , c1 , x1 , x2 , j0 (x2 ))) ∈ [A], where ⎞ ⎛ ⎞ ⎛ 0 0 0 ⎜0 1⎟ ⎜0⎟ ⎟ ⎜ ⎟ ⎜ ⎜1 0⎟ ⎜0⎟ ⎟ ⎜ ⎟ f7 ⎜ ⎜ 1 1 ⎟ = ⎜ 0 ⎟. ⎟ ⎜ ⎟ ⎜ ⎝2 0⎠ ⎝0⎠ 2 2 1

414


Further, f13 (x1 , x2 ) := u2 (f13 (c0 , j2 (x1 ), j2 (x2 ), c1 , x1 , x2 , s3 (x1 ), s3 (x2 ))) ∈ [A], where ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 0 2 ⎠ = ⎝ 2 ⎠. f13 2 0 2 It is easy to see that one obtains a binary function f15 with ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 1 0⎠=⎝1⎠ f15 2 0 2

as a superposition over P3,2 ∪ (P ol3 {0, 1})1 ∪ {f15 }. Let f n be an arbitrary function of P ol3 {0, 1}. We show f ∈ [A]. For this α1 , ..., α r ∈ E3n , β ∈ E3 ) be an n-ary function defined purpose, let fα 1 ,...,αr ;β ( by β if x ∈ { α1 , ..., α r }, fα1 ,..., (x) := αr ;β 0 otherwise. Obviously, the functions of the type fα ;1 are superpositions over A for every = (α1 , ..., αn ) ∈ E3n and αi = 2 then fα ;2 ∈ [A] follows from α ∈ E3n . If α fα ;2 = f7 (xi , fα ;1 (x)). Furthermore, we have fα1 ,..., αr ;2 (x) = f13 (fα 1 ;2 (x), fα 2 , α3 ,..., αr ;2 (x)).

Consequently, all functions of the type fα1 ,..., α1 , ..., α r } ⊆ E3n \E2n αr ;2 with { are superpositions over A. Then f ∈ [A] follows from f (x) = f15 (fβ1 ,...,βs ;1 (x), fγ1 ,...,γt ;2 (x)),

1 , ..., βs } := {x ∈ E n | f (x) = 1} and {γ1 , ..., γt } := {x ∈ where {β 3 n E3 | f (x) = 2}. Consequently, [A] = P ol3 {0, 1}. Then our theorem follows from Table 14.4. The functions g1 , ..., g10 of Table 14.4 are defined in Table 14.5. The function h3 is defined by ⎧ x ∈ E33 \E23 , ⎨ 2 if x ∈ {(0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 1, 1)}, h(x) := 1 if ⎩ 0 otherwise.


(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15)

j0 − − − + + − + − − + + + + − +

j4 − − − + + − + − + − + + + − +

x1 0 0 1 1 0 1 2 2 2

u2 + − + − + + + + + − + − + + +

x2 0 1 0 1 2 2 0 1 2

s3 − − − + + + + + − − − − + + +

g1 0 0 0 0 0 2 0 2 0

g1 + − + − + − − + + − + − − + +

g2 1 1 1 1 1 2 1 1 1

Table g2 g3 − + + − + − − − + − − + − − + − − − + − − − + − + − + − + −

14.4 g4 g5 − + − + − + + + + + − + − − − − + − − + − − − − + + − − − −

Table 14.5 g3 g4 g5 g6 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 1 0 1 0 1 2 2 2 0 1 1 1 2 2 1 1 2 2 1 2 2

g6 − + − − − + + − − − − + + − − g7 0 1 1 1 2 2 2 2 2

g7 + + + − − + + + − + − + − + − g8 0 0 0 0 2 2 2 2 2

g8 + − + − + + + + + − + − − + −

g9 + + + + + + − − − − − − − + −

g10 + + + − − − + − + − + + + − +

415

h + + + + − + + + − − − − − + −

g9 g10 0 0 0 0 1 0 1 1 1 0 2 0 2 0 2 0 2 0

14.5 Proof of Theorem 14.1.4 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of O, which is not contained in any class ((1)–(13)) from the above list, given in Theorem 14.1.4. Then, there are some functions f1 , ..., f13 ∈ [A] with the above property ( ). First we prove that O1 \{x} ⊆ [A] holds. For the unary function f1 , we have f1 (0) ∈ {1, 2}. Then, either f1 = c2 or we obtain c2 by forming f3 (x, f1 (x)). Consequently, {c0 , c1 , c2 } ⊆ [{f2 , f4 , f5 , c2 }] ⊆ [A]. W.l.o.g. we can assume 0 1 2 2 0 2 1 0 ∈ . f8 0 1 2 0 2 1 2 1 Since f8 is monotone, we have 0 1 2 0 0 1 0 0 ∈ . f8 0 1 2 0 2 1 2 1

416


Therefore, f8 (c0 , c1 , c2 , c0 , x, c1 , x) ∈ {j2 , j5 } is valid. Since we can assume w.l.o.g. 0 1 2 1 2 0 0 f10 ∈ , 0 1 2 2 1 1 2 and since f10 ∈ O, we have f10 (0, 1, 2, 0, 2) = 0 and f10 (0, 1, 2, 2, 2) ∈ {1, 2}. Thus f10 (c0 , c1 , c2 , x, c2 ) ∈ {j2 , u2 }. Since j5 u2 = j2 , j2 ∈ [A] holds. With the aid of functions f6 and f9 one can show analogously that v5 ∈ [A] also holds. Because of j5 = j2 v5 and v2 = v5 j2 we have j5 , v2 ∈ [A]. W.l.o.g. let 0 0 1 2 1 0 1 2 f7 ∈ . 0 1 2 0 1 2 1 2 Thus,

f7

0 1 2 1 0 1 1 0 1 2 1 1 2 1

∈

0 2

and therefore f7 (c0 , c1 , c2 , c1 , j5 , v5 , c1 ) = u5 ∈ [A]. Consequently, by u5 j2 = u2 , we have O1 \{x} ⊆ [A]. A conclusion of the Fundamental Lemma of Jablonskij (see Theorem 1.4.4, (a)) is the following fact: For the function f11 , there are some a, b, c ∈ E3 with the property (w.l.o.g.): ⎛ ⎞ ⎛ ⎞ 0 a c f11 ⎝ b c ⎠ ∈ ⎝ 1 ⎠ , 2 b d where (because of f11 ∈ O) (a, c) < (b, c) and (b, c) < (b, d). If one replaces the variables of function f11 by certain functions from O1 \{x}, one obtains a ∈ [A] with function f11 ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 2 0 ⎠ ∈ ⎝ 1 ⎠. f11 2 2 2 Now we can form gα (x, y) := f11 (u5 (x), u2 (y)), where ⎧ if (x, y) = (0, 2), ⎨α x ∈ {1, 2}, gα (x, y) = v2 (y) if ⎩ 0 otherwise.

Next, with the help of the function gα , we show that [A] ∩ {min, max} = ∅

(14.1)

is valid, where we use the notation x ∨ y := max(x, y) and x ∧ y := min(x, y). We distinguish three cases: Case 1: α = 0. In this case, we have g0 (x, y) = u5 (x) ∧ v2 (y) and


417

g0 (u5 (g0 (x, v5 (y))), v2 (g0 (u2 (x), y))) = x ∧ y ∈ [A] holds. Case 2: a = 1. Then g1 (x, y) = (u5 (x) ∧ v2 (y)) ∨ j2 (y), whereby g1 (u5 (g1 (u2 (x), u2 (y))), u5 (j5 (g1 (x, u5 (y))))) = x ∨ y ∈ [A] holds. Case 3: a = 2. Since g2 (x, y) = (u5 (x) ∧ v2 (y)) ∨ u2 (y) is valid in this case, we have g2 (u5 (g2 (x, u5 (y))), v2 (g2 (u2 (x), y))) = x ∨ y ∈ [A] Thus (14.1) is proven. Since the classes (12) and (13) are isomorphic, we can assume w.l.o.g. that min ∈ [A]. The function f12 has the property ⎛ ⎞ ⎛ ⎞ 0 1 2 0 0 1 0 0 1 0 0 0 0 1 f12 ⎝ 0 1 2 0 0 1 1 2 2 ⎠ ∈ ⎝ 1 2 1 2 2 ⎠ . 0 1 2 1 2 2 0 0 1 1 1 2 2 2 (x, y) := f12 (c0 , c1 , c2 , x, u2 (x), v2 (x), y, u5 (y), v5 (y)), either If one forms f12 (x, y)) or the function u2 (f12 (x, y)) is identical to the the function u5 (f12 function g(x, y) := u5 (x) ∨ u5 (y). Therefore, we get x ∨ y = g(x, y) ∧ (v5 (g(u2 (x), u2 (y)))) ∈ [A]. Since (O1 \{x}) ∪ {∧, ∨} is a generating system of O (see Section 11.4), we have shown [A] = O. Then our theorem follows from Table 14.6, where g1 := ∧, g2 := ∨, g3 (x, y) := (j5 (x) ∧ j5 (y)) ∨ j2 (y) and g4 := u5 g3 .

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

c0 + − + + − + + + + + + + +

c1 − − + − + + + + + + + + +

c2 − + − + + + + + + + + + +

j2 + − + − − + + − + − + + +

Table j 5 u2 + + − + + + − + + − + + + − − + + + + − + + + + + +

14.6 u5 v2 + − + + − + + − + + + − − + + + − + + + + + + + + +

v5 − + − − + − + + − + + + +

g1 + + + + + + + − + + − + −

g2 + + + + + − + + + + − − +

g3 + − + − + + + − + − + − −

g4 + + − + + + − + − + + − −

418


14.6 Proof of Theorem 14.1.5 Let A be an arbitrary subset of S, which is no subset of P ol3 {0} and no subset of L. Then there are two functions f1 , f2 ∈ [A] with the above property (∗ ). To prove [A] = S it is sufficient to show that α([A]) := {f c0 | f ∈ [A]} = P3 holds (see Theorem 8.3.2). α([A]) = P3 is proven, if one can show that α([A]) ⊆ B for every class B from the list of Theorem 14.1.1. Obviously, we have f11 ∈ {s4 , s5 }, S 1 = {s4 , s5 , e11 } ⊆ [f1 ] and f1 is no element of a maximal class of P3 with the marking (1), ..., (6), (8), ..., (16) of Theorem 14.1.1. Because of α(e21 ) = c10 , α([A]) is no subset of a maximal class of P3 with the marking (7). Further, α(f2 ) ∈ B for each maximal class B of P3 with the marking (17) or (18). Consequently, α([A]) = P3 and therefore [A] = S. Then, our Theorem 14.1.5 follows from S ∩L ⊆ S ∩P ol3 {0} and S ∩L S ∩P ol3 {0}.

14.7 Proof of Theorem 14.1.7 W.l.o.g. we assume a = 0, b = 1 and c = 2. Further, in this proof, let A be an arbitrary subset of C, which is not contained in any class with the marking (1)–(7) from the above list, which is given in Theorem 14.1.7. Then there are some functions f1 , ..., f7 ∈ [A] with the above property ( ). First we show that {c0 , c1 , c2 } ⊂ [A]. Obviously, f1 ∈ {j0 , j4 , j3 , c1 , u0 , u4 , u3 , c2 }. We distinguish the following three cases: Case 1: f1 ∈ {c1 , c2 , j3 , u4 }. In this case f1 := f1 f1 = cα , α ∈ {1, 2} holds. Since f2 , f3 ∈ C and f2 (0, 1) = 2 or f3 (0, 2) = 1, we have f2 (1, 1) ∈ {0, 2} or f3 (2, 2) ∈ {0, 1}. Therefore, {c1 , c2 } or {c0 , cα } is a subset of [A]. With the aid of functions f2 , f3 , f4 (∈ [A]) one can prove {c0 , c1 , c2 } ⊂ [A] easily. Case 2: f1 = j0 . In this case, the functions j0 j0 = j5 and f2 (x) := f2 (j5 , j0 ) ∈ {c2 , u2 } belong to [A]. If f2 = c2 , then we obtain by Case 1 that {c0 , c1 , c2 } ⊂ [A]. If f2 = u0 then u5 = u0 u0 ∈ [A]. For the function f7 is valid: ⎞ ⎛ ⎞ ⎛ 0 1 0 2 0 1 0 0 1 1 0 1 2 0 0 2 2 0 2 1 2 f7 ⎝ 1 0 2 0 0 0 1 0 1 0 0 1 0 2 0 2 0 2 2 ⎠ ∈ ⎝ 1 2 ⎠ . 2 2 1 1 0 0 0 1 0 1 1 1 0 0 2 0 2 2 2 2 1 0 1 2 0 1 0 2 Since f7 preserves the relation , we have 0 1 2 1 0 2 0 f7 Consequently,

0 2 0 1 0 0 . . . . . 0 0 0 . . . . 0 2 0 1 0 1 1 . . . . . 1 2 2 . . . . 2

=

0 0

.


419

j0 (f7 (u5 , u0 , j5 , j0 , j5 , j5 , ....., j5 , u5 , u5 , ...., u5 ) = c0 ∈ [A]. Therefore, one can reduce Case 2 to Case 1. Case 3: f1 ∈ {j4 , u0 , u3 }. It is easy to check that one can also reduce this case to the first case analogously to Case 2. Thus {c0 , c1 , c2 } ⊂ [A] is proven. Since f6 ∈ C and

f6

holds, we have

f6

0 1 2 0 1 0 2 1 0 1 2 1 0 2 2 2 0 1 2 0 1 0 2 0 0 1 2 1 0 2 2 0

Consequently, the functions f6 (x)

=

=

2 1 0 0

.

:=

f6 (c0 , c1 , c2 , c0 , c1 , c0 , c2 , x) and 0 0 := f6 (c0 , c1 , c2 , c0 , c1 , c2 , c0 , x) belong to [A], where f6 = 1 2 0 0 0 1 2 0 0 and f6 . Therefore, we can assume w.l.o.g. f5 = = 2 1 0 1 2 1 2 1 . Then 0 p(x) := f5 (c0 , c1 , c2 , x, f6 (x)) ∈ {j0 , j4 }. f6 (x)

It is easy to check that f6 (0, 1, 2, 1, 0, 2, 0, 1) ∈ {0, 2}. Consequently, f6 (x) := f6 (c0 , c1 , c2 , c1 , c0 , c2 , c0 , x) ∈ {j2 , s2 }. Therefore, we have to distinguish the following two cases for f6 : Case 1: f6 = j2 . Then, the functions p j2 = j3 , f6 j2 = u2 , f6 j3 = u3 , f6 (c0 , c1 , c2 , j2 , j3 , u2 , u3 , x) = s2 , j2 s2 = j1 , j3 s2 = j4 , s2 j1 = u1 and s2 j4 = u4 are superpositions over A. Consequently, the function f7 (x1 , x2 ) := f7 (x1 , x2 , s2 (x1 ), s2 (x2 ), c0 , j1 (x2 ), j1 (x1 ), j2 (x1 ), j3 (x1 ), j4 (x1 ), j4 (x2 ), c1 , u1 (x1 ), u2 (x1 ), u3 (x1 ), j4 (x1 ), j4 (x2 ), c2 ) belongs to [A], where

⎛

⎞ ⎛ ⎞ 0 1 α f7 ⎝ 1 0 ⎠ = ⎝ α ⎠ 2 2 β

and {α, β} = {1, 2}. Since s2 ∈ [A], we can assume α = 1 and β = 2. Because of f7 ∈ C, we have f7 (0, 0) = f7 (0, 2) = f7 (2, 0) = 0 and {f7 (1, 2), f7 (2, 1)} ⊆

420


{0, 1}. Consequently, the function f7 (x1 , x2 ) := j2 (f7 (u1 (x1 ), u1 (x2 ))) ∈ [A] has the properties ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎜ ⎟ ⎜ ⎟ 0 1 ⎟ ⎜0⎟ f7 ⎜ ⎝1 0⎠=⎝0⎠ 1 1 1 and f7 ∈ P3,2 . Obviously, the set {j2 , p, c0 , c1 , f7 } is a subset of P3,2 , however no subset of the maximal classes of P3,2 (see Lemma 14.2.3). Consequently, the sets P3,2 and {s2 r | r ∈ P3,2 } = {r ∈ P3 | Im(r ) ⊆ {0, 2} } are subsets of [A]. With the aid of the functions 2 if (2, i) = (x1 , x2 ), ti (x1 , x2 ) := 0 otherwise, i ∈ {1, 2}, of [A] we can form a function t ∈ [A] by t(x1 , x2 ) := f7 (s2 (f7 (s2 (x2 ), t1 (x1 , x2 ))), t2 (x1 , x2 )) ⎧ (x1 , x2 ) = (2, 1), ⎨ 1 if 2 if (x1 , x2 ) = (2, 2), = ⎩ 0 otherwise. Let S and T be nonempty disjoint subsets of E3n with the property that for different arbitrary tuples σ := (σ1 , ..., σn ) ∈ T and τ := (τ1 , ..., τn ) ∈ T there exists an i with {σi , τi } = {1, 2}. Then the function ⎧ x ∈ S, ⎨ 1 if x ∈ T, fS,T (x) := 2 if ⎩ 0 otherwise, belongs to C. [A] = C would be proven in Case 1, if we could show that every function of the form fS,T is an element of [A]. For σ := (σ1 , ..., σn ), τ := (τ1 , ..., τn ), {σi , τi } = {1, 2} and 2 if x ∈ {σ, τ }, gσ,τ (x) := 0 otherwise, we have

f{σ},{τ } (x) :=

t(g{σ},{τ } (x), xi ) if (σi , τi ) = (1, 2), t(g{σ},{τ } (x), s2 (xi )) if (σi , τi ) = (2, 1),

i.e., f{σ},{τ } ∈ [A]. Furthermore, fS,T (x) = f7 (f{σ},T (x), fS\{σ},T (x)), fS,T (x) = s2 (f7 (s2 (f{σ},T (x)), s2 (fS\{σ},T (x))))


421

is valid. Consequently, the functions of the type fS,T are superpositions over A (see Section 11.2). Thus, [A] = C holds in Case 1. Case 2: f6 = s2 . If the above function p is the function j4 , then we can reduce Case 2 to Case 1 because of j4 j4 s2 = j2 . Consequently, we can assume p = j0 ∈ [A]. Then we have {j0 j0 = j5 , s2 j0 = u0 , s2 j5 = u5 } ⊆ A. Further, a function of [A] is also the function f7 (x1 , x2 , x3 ) := f7 (x1 , x2 , s2 (x1 ), s2 (x2 ), c0 , j0 (x1 ), j0 (x2 ), x3 , j0 (x3 ), j5 (x2 ), j5 (x1 ), c1 , u0 (x1 ), u0 (x2 ), u5 (x3 ), u0 (x3 ), u5 (x2 ), u5 (x1 ), c2 ), ⎛

⎞ ⎛ ⎞ 0 1 0 1 2 f7 ⎝ 1 0 0 ⎠ ∈ ⎝ 1 2 ⎠ . 2 2 1 2 1

where

Since f7 ∈ C, we have f7 (2, 0, 0) = 0. Thus j0 (f7 (x, j0 , c0 )) = j2 ∈ [A], i.e., one can reduce Case 2 to Case 1. We have proven [A] = C with that. Then our theorem follows from Table 14.7 and Table 14.8. Table 14.8 Table 14.7 (1) (2) (3) (4) (5) (6) (7)

c1 − + − + + + +

p1 + − − − − − −

p2 + + − − − − −

p3 + − + + + − −

p4 − − + + − + +

p5 + + + − + + −

p6 + + − − + − +

x1 0 0 0 1 1 1 2 2 2

x2 0 1 2 0 1 2 0 1 2

p1 0 0 1 0 2 0 1 0 0

p2 0 1 0 0 0 2 1 0 0

p3 0 2 0 0 2 1 2 2 2

p4 2 0 0 2 2 2 0 2 2

p5 0 1 0 1 1 0 0 0 2

p6 0 0 0 0 0 0 0 2 1

14.8 Proof of Theorem 14.1.8 7

W.l.o.g. we can assume a = 0, b = 1, and c = 2. Further, in this proof let A be an arbitrary subset of U , which is not contained in any class from the above list, which is given in Theorem 14.1.8. Then there are some functions f1 , ..., f13 ∈ [A] with the above property ( ). First we show {c0 , c1 , c2 } ⊂ [A]. 7

Essential parts of this proof come from the proof of the following theorem, found by W. Harnau: Let A ⊆ U be arbitrary. Then [A] = U if and only if U 1 \{u3 , v3 } ⊆ [A] and A ⊆ B for every class B with the marking (9), (11), (12), (13) in Theorem 14.1.8.

422


Because of f2 (0, 1) = 2 and f2 ∈ U we have f2 (x, x) ∈ {u3 , v3 , c2 }. Obviously, c0 , c1 and c2 are superpositions over {c2 , f1 , f3 , f4 }. If f2 (x, x) = c2 then we can assume either {c1 , u2 } ⊆ [A] or {co , v2 } ⊆ [A] because of u3 u3 = u2 , v3 v3 = v2 , f3 (u2 , u3 ) ∈ {j3 , c1 , v2 }, f4 (v2 , v3 ) ∈ {c0 , j2 , u2 }, j3 j3 = c1 , j2 j2 = c0 , v2 u3 = v3 , u2 v3 = u3 , f8 (u2 , u3 , v2 , v3 ) ∈ {c0 , c1 , c2 , j2 , j3 } and u3 c2 = c0 . Since u2 c1 = c0 , v2 c0 = c1 and f2 (c0 , c1 ) = c2 , we have {c0 , c1 , c2 } ⊂ [A]. Next we prove U 1 ∪ P3,2 ⊆ [A]. By definition of f9 we have 0 1 2 0 1 0 1 2 2 ∈ , f9 0 1 2 1 0 2 2 0 1 i.e.,

f9

0 1 2 1 0 1 1 0 1 2 1 0 0 0

=

2 2

and therefore f9 (c0 , c1 , c2 , c1 , c0 , x, x) ∈ {u3 , v3 }. Since v3 v3 = v2 , we have that either {u2 , u3 } or {v2 , v3 } is a subset of [A]. W.l.o.g. let 0 1 2 0 1 0 2 1 f5 = 0 1 2 1 0 2 0 2

and f6

0 1 2 1 0 1 2 0 1 2 1 0 2 1

=

0 2

,

i.e., we have f5 (c0 , c1 , c2 , c0 , c1 , u2 , u3 ) = v2 and f6 (c0 , c1 , c2 , c1 , c0 , v2 , v3 ) = u2 . Thus, because of v2 u3 = v3 and u2 v3 = u3 , the set {u2 , u3 , v2 , v3 } is a subset of [A]. Further, the function f7 with f7 (x) := f7 (c0 , c1 , c2 , x) ∈ {j0 , j4 , s3 } belongs to [A]. We distinguish two cases for f7 : Case 1: f7 ∈ {j0 , j4 }. In this case, the set {f7 , c0 , c1 , f7 (v2 ), f7 (f12 (x1 , ..., x8 , c2 ))} is a subset of [A] ∩ P3,2 , however, no subset of a maximal class of P3,2 (see Lemma 14.2.3). Consequently, P3,2 is a subset of [A]. Obviously, the functions s1 and s3 are superpositions over {f11 , c2 , u2 , u3 , v2 , v3 } ∪ P3,2 . Therefore, P3,2 ∪ U 1 ⊆ [A] in Case 1. Case 2: f7 = s3 . W.l.o.g. we can assume ⎛ ⎞ ⎛ ⎞ 0 1 2 0 0 1 1 2 2 0 1 1 1 f10 ⎝ 0 1 2 0 0 1 1 2 2 1 0 ⎠ ∈ ⎝ 0 0 ⎠ . 0 1 2 1 2 0 2 0 1 2 2 0 1 Since f10 ∈ U , we have f10 (0, 1, 2, 0, 2, 1, 2, 0, 1, 2, 2) ∈ {0, 1}, i.e.,


423

f10 (c0 , c1 , c2 , c0 , u2 , c1 , u2 , c1 , v2 , u3 , v3 , x, s3 ) ∈ {j0 , j4 }. Therefore, one can reduce Case 2 to Case 1. Thus P3,2 ∪ U 1 ⊆ [A]. with Then, a binary function f11 ⎛ ⎞ ⎛ ⎞ 0 0 0 ⎝ 1 0⎠=⎝1⎠ f11 2 0 2 is a superposition over A. It is ⎛ 0 ⎜ ⎜ 1 f13 ⎝ 2 2

easy to prove that a 6-ary function f13 with ⎞ ⎛ ⎞ 0 0 2 2 2 0 ⎜2⎟ 2 2 0 0 2⎟ ⎟=⎜ ⎟ 1 2 1 2 0⎠ ⎝2⎠ 2 1 2 1 1 2

∈ U , we have is a superposition over P3,2 ∪ U 1 ∪ {f13 }. Since f13 ⎛ ⎞ ⎛ ⎞ 0 0 0 2 2 2 0 ⎜ ⎟ ⎜2⎟ ⎜ 0 2 2 0 0 2 ⎟ f13 ⎝ = ⎜ ⎟. 2 0 2 0 2 0⎠ ⎝2⎠ 2 2 0 2 0 0 2

We form (x1 , x2 , x3 ) := u2 (f13 (x1 , x2 , x3 , u3 (x3 ), u3 (x2 ), u3 (x1 ))) ∈ [A]. f13 Because of {f13 (0, 0, 2), f13 (0, 2, 0), f13 (2, 0, 0)} ⊆ {0, 2} we can assume w.l.o.g. f13 (0, 2, 0) = f13 (2, 0, 0). Then: f13 (x1 , x2 , c0 ) ∈ {d1 , d2 }, where d1 and d2 are defined in Table 14.9.

x1 0 0 1 1 0 1 2 2 2

x2 0 1 0 1 2 2 0 1 1

d1 0 0 0 0 0 0 0 0 2

d2 0 0 0 0 2 2 2 2 2

d3 0 0 0 1 0 1 0 1 0

Table 14.9 d4 min max 0 0 0 1 0 1 1 0 1 1 1 1 0 0 2 0 1 2 0 0 2 0 1 2 0 2 2

d5 0 1 1 0 2 2 2 2 0

d6 0 1 0 0 0 2 1 1 2

d7 2 2 2 2 0 1 2 2 2

Thus because of u5 (d2 (u3 (x1 ), u3 (x2 ))) = d1 (x1 , x2 ) and

d8 1 0 0 1 1 1 1 1 1

d9 0 0 0 1 2 2 2 2 2

424


u3 (d1 (u3 (x1 ), u3 (x2 ))) = d2 (x1 , x2 ), the functions d1 and d2 are superpositions over A. Further, we have min(x1 , x2 ) = f11 (d3 (x1 , x2 ), d1 (x1 , x2 )) ∈ [A]

and

max(x1 , x2 ) = f11 (d4 (x1 , x2 ), d2 (x1 , x2 )) ∈ [A]

(see Table 14.9). With that, it was shown that the basis functions, determined in [Gni 65] 8 for the class U , belong to [A]. Therefore, [A] = U . Then, our theorem follows from Tables 14.9 and 14.10, where the ternary function d ∈ U is defined by 2 if x1 = x2 = 2 ∨ x1 = x3 = 2 ∨ x2 = x3 = 2, d(x1 , x2 , x3 ) := 0 otherwise.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

c1 − + − + + + + − + + + + +

c2 + − + + + + + − + + + + +

j2 − + − − + + + − + + + + +

u3 − − + − + − + + − + + + +

v3 − − − + − + + + − + + + +

Table s3 d5 + − + + − + − − − − − − − − + − + − + − + − + + + +

14.10 d6 d7 + + + − − + − + − − − − − + + − + − − + − − − + + −

d8 − + − + + + − − + − + + +

d9 + + + + + − + − + + − − −

d max min + + + + + + + + + − + + + − + − + + + + + + − − + + + + + − + − + + − − − − −

14.9 Proof of Theorem 14.1.9 Let A be an arbitrary subset of B := P olι33 , which is not contained in any class from the list, which is given in Theorem 14.1.9. Then, for each i ∈ {1, 2, ..., 5} there is a function fi ∈ A which does not belong to the class with marking (i). Obviously, S := {s1 , s2 , ..., s6 } is a subset of [{f1 , f2 , f3 , f4 }]. For the function f5 (x) := f (x, x, ..., x) ∈ [A], the following cases are possible: Case 1: f5 ∈ B 1 \{c0 , c1 , c2 }. In this case, one can easily prove (P3 (2))1 ⊆ [A] when one forms the superpositions of the form sf5 s and sf5 f5 s , where {s, s } ⊂ S. Consequently, P31 ⊆ [A]. Then, by Theorem 4.3, we have [P31 ∪ {f5 }] = P olι33 = [A]. 8

See Chapter 11.

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A

425

Case 2: f5 ∈ {c0 , c1 , c2 }. Obviously, we have {co , c1 , c2 } ⊆ [S ∪{f5 }] in this case. Since f5 ∈ P3 (2)\[P31 ], there exist some tuples a := (a1 , ..., an ) and a := (a1 , ..., ai−1 , ai , ai+1 , ..., an ) with ai = ai and f5 (a) = f5 (a ). Therefore, a unary function of B 1 \{c0 , c1 , c2 } belongs to [A]. Hence, one can reduce Case 2 to Case 1. Consequently, [A] = P olι33 is proven. Since one can easily prove that the classes with the marking (1)–(5) are Bmaximal, our theorem holds.

14.10 On the Cardinality of L↓3 (A) for Submaximal Clones A The aim of this section is to prove Theorem 14.10.1 ([Bul-L-S 96]) There are 5 submaximal clones (⊂ P3 ) with finitely many subclasses and 7 submaximal clones (⊂ P3 ) whose subclass lattice is infinite but countable. The subclass lattices of the remaining 146 submaximal clones of P3 have the cardinality of continuum. More precise: Let F1 , ...., F43 be the 43 families of submaximal clones of P3 given in Table 14.1 and let A ∈ Fi with i ∈ {1, 2, ..., 43}. Then ⎧ 5 if ⎪ ⎪ ⎪ ⎪ ⎪ 8 if ⎪ ⎪ ⎨ ↓ 32 if |L3 (A)| = ⎪ ⎪ ⎪ ⎪ ℵ0 if ⎪ ⎪ ⎪ ⎩ c otherwise.

i = 17, i = 7, i = 30, i ∈ {28, 29, 43},

We need the following lemmas for proof of the above theorem. Lemma 14.10.2 Let n ∈ N \ {1, 2} and pn ∈ P3n be defined by 1 if ∃i : (xi = 1 ∧ ∀i = j : xi = 2), pn (x1 , ..., xn ) := 0 otherwise. Let πn be the n-ary relation αn := {(1, 2, 2, ..., 2), (2, 1, 2, ..., 2), ..., (2, 2, , ..., 2, 1)} ∪ (E3n \{1, 2}n ). Then (a) ∀n ≥ 3 : pn ∈ P ol3 αn .

426


(b) ∀m = n : pm ∈ P ol3 αn . (c) (pi )i≥3 is an infinite basis for the clone [{pi | i ≥ 3}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 1 2 2 ... 2 1 ⎜ 2 1 2 ... 2 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ pn ⎜ ⎝ . . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ αn . 2 2 2 ... 1 1

(b) is easy to check by considering the two cases m < n and m > n. (c) Let j ≥ 3. By (b), we have that pi ∈ P ol3 αj for all i ≥ 3, i = j, and thus [{p3 , p4 , ..., }\{pj }] ⊆ P ol3 αj . Since pj ∈ P ol3 αj , we deduce that pj ∈ [{p3 , p4 , ..., }\{pj }]. Consequently, {p3 , p4 , ...} is an independent set of operations, proving (c). Lemma 14.10.3 Let n ∈ N \ {1} and qn ∈ P3n be defined by 1 if ∀i : xi = 2, qn (x1 , ..., xn ) := pn (x) otherwise. Then {qi | i ≥ 2} is an infinite basis for the clone [{qi | i ≥ 2}]. Proof. The proof can be found in [Jan-M 59] (see also Lemma 8.1.1). Lemma 14.10.4 Let n ∈ N \ {1} and rn ∈ P3n be defined by ⎧ ∃i : (xi = 2 ∧ ∀j = i : xj = 1), ⎨ 1 if 2 if x1 = ... = xn = 2, rn (x1 , ..., xn ) := ⎩ 0 otherwise. Let n be the n-ary relation n := {(a1 , ..., an ) ∈ E3n | (∃i : ai = 2 ∧ ∀j = i : aj = 1) ∨ (∃i : ai = 0)}. Then (a) ∀n ≥ 2 : rn ∈ P ol3 n . (b) ∀n = m : rm ∈ P ol3 n . (c) {ri | i ≥ 2} is an infinite basis for [{ri | i ≥ 2}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 2 1 1 ... 1 1 ⎜ 1 2 1 ... 1 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ rn ⎜ ⎝ ...................... ⎠ = ⎝ ... ⎠ . 1 1 1 ... 2 1 (b): Let m = n, m, n ≥ 2, A = (aij )n,m be an n×m matrix on E3 with columns a1 , ..., am ∈ n , where ai = (a1i , ..., ani ) (i = 1, ..., n). If aij = 0 for some i, j


427

then rm (ai1 , ..., aim ) = 0 and by definition of n , rn (A) = rn (a1 , ..., an ) ∈ . So assume that a1 , ..., am ∈ {1, 2}n . We distinguish two cases: Case 1: m < n. At least one row of the matrix A consists of 1s, hence rm (A) ∈ n . Case 2: m > n. In this case, there are i = j with ai = aj . Hence, one row of the matrix A consists of 1s or contains at least two 2s, and so rm (A) ∈ n . Therefore (b) holds. (c) follows straightforward from (a) and (b). Lemma 14.10.5 Let n ∈ N \ {1, 2} and sn ∈ P3n be defined by sn (x1 , ..., xn ) := x1 if ∃i : (xi = 1 ∧ ∀j = i : xj = 0) ∨ (xi = 0 ∧ ∀j = i : xj = 1), 2 otherwise. Let σn be the n-ary relation σn := {(a1 , ..., an , an+1 ) ∈ E3n | (∃i ∈ {1, 2, ..., n} : (xi = 1 ∧ ∀j = i : xi = 0) ∨ ai = 2}. Then (a) ∀n ≥ 3 : sn ∈ P ol3 σn . (b) ∀n = m : sm ∈ P ol3 σn . (c) {si | i ≥ 3} is an infinite basis for [{si | i ≥ 3}]. Proof. (a) follows from ⎞ ⎛ ⎞ 1 1 0 0 ... 0 ⎜ 0 1 0 ... 0 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ sn ⎜ ⎜ ...................... ⎟ = ⎜ ... ⎟ . ⎝ 0 0 0 ... 1 ⎠ ⎝ 0 ⎠ 2 0 0 0 ... 0 ⎛

(b) and (c) are easy to verify. The following lemma was found by B. Strauch. Lemma 14.10.6 Let n ∈ N \ {1, 2} and let the n-ary function tn ∈ P3n be defined by ⎧ ∃i : (xi ∈ E2 ∧ ∀j = i : xj = 0), ⎨ 0 if x = 1 ∨ (∃i : xi = 2), tn (x1 , ..., xn ) := 2 if ⎩ 1 otherwise. Moreover, let

428


αm := {(a1 , ..., am , a1 , ..., am ) ∈ E22·m | (∃i : ai = 1) ∧ ∀j = i : aj = 0)}, x := x + 1 (mod 2), βm := (E2m \{0}m ) × {1}m , γm := E32·m \E22·m , τm := αm ∪ βm ∪ γm . Then (a) ∀n ≥ 3 : tn ∈ P ol3 τn . (b) ∀m = n : tn ∈ P ol3 τm . (c) {tn | n ≥ 3} is an infinite basis for [{tn | s ≥ 3}]. Proof. (a) follows from ⎛

⎞ ⎛ ⎞ 0 1 0 0 ... 0 0 ⎜ 0 1 0 ... 0 0 ⎟ ⎜ 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 ... 0 1 ⎟ ⎜ 0 ⎟ ⎟ = ⎜ ⎟ ∈ τn . ⎜ sn ⎜ ⎟ ⎜ ⎟ ⎜ 0 1 1 ... 1 1 ⎟ ⎜ 1 ⎟ ⎜ 1 0 1 ... 1 1 ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ . . . . . . . . . . . . . . ⎠ ⎝ ... ⎠ 1 1 1 1 ... 1 0

(b): Let m, n ≥ 3, m = n and r1 , ..., rn ∈ τm . If {r1 , ..., rn } ⊆ αm ∪ βm or {r1 , ..., rn } ⊆ βm , we have tn (r1 , ..., rn ) ∈ τm . So assume {r1 , ..., rn } ⊆ αm ∪βm and {r1 , ..., rn } ⊆ βm . It is easy to see that ri = rj for some i = j implies tn (r1 , ..., rn ) ∈ τm . Further, by the definition of tn tn (r1 , ..., rn ) = {o}m × {1}m

(14.2)

is possible only if tn (r1 , ..., rn ) ∈ τm . For pairwise distinct r1 , ..., rn ∈ αm ∪βm , the condition (14.2) can be valid only if m = n. Since we assume m = n, tn (r1 , ..., rn ) ∈ τm holds. (c) follows from (a) and (b). The following two lemmas were found by A. Bulatov. For these lemmas, we need some notations. For every n ∈ N let n := {1, 2, ..., n}. Furthermore, let

:=

and for n ≥ 3

0 1 0 2 1 0 2 0

(14.3)


429

n := {(a1 , a2 , ..., an ) | ∃i : (ai ∈ {1, 2} ∧ (∀j = i : aj = 0)}∪({1, 2}n \{2}n ). (14.4) We want to prove that the co-clone [{} ∪ {m | m ≥ 3}]

(14.5)

has an infinite basis. Lemma 14.10.7 For a finite index-set I and every i ∈ I let πi be an mi -ary relation of {} ∪ {m | m ≥ 3}. Moreover, let ϕi be a mapping of mi in n, i ∈ I. If the relation l , l ≥ 3 has the form l = {(a1 , ...., al ) ∈ E3l | ∃al+1 , ..., an ∈ E3 : ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi }

(14.6)

for a some n and if there exists an i ∈ I with πi = , then: (a) If a tuple (a1 , ..., an ) ∈ E3n fulfills the condition ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi

(14.7)

(we say : “(a1 , ..., an ) is permissible”) and if aj = 2 for a some j ∈ n, then the n-tuple (a1 , ..., aj−1 , 1, aj+1 , .., an ) is also permissible. (b) ∃α ∈ I : ϕα (mα ) ⊆ l ∧ πα = . (c) Let ϕα (mα ) ⊆ l and πα = . Then ϕα is bijective and πα = l holds. Proof. (a) follows straightforwad from the observation that, if we replace a 2 with a 1 in a tuple of l or , we obtain again a tuple from l or , respectively. (b): Suppose that ∀i ∈ I : (πi = =⇒ ϕi (mi ) ⊆ l).

(14.8)

Because of (14.4) and (14.6), there exists a permissible n-tuple of the form a := (a1 , ..., al , al+1 , ..., an ) with a1 = 1 and a2 = ... = al = 2; i.e., ∀i ∈ I : (aϕi (1) , ..., aϕi (mi ) ) ∈ πi .

(14.9)

By (a) we can suppose w.l.o.g. that ∀i ∈ {l + 1, l + 2, ..., n} : ai ∈ {0, 1}. We now show that (14.8) and (14.9) imply the permissibility of the tuple b := (b1 , ..., bn ), where b1 = ... = bl = 2 and bi = ai for all i ∈ {l+1, l+2, ..., n}, which leads to the contradiction (2, 2, ..., 2) ∈ l . The following cases are possible for α ∈ I:

430


Case 1: 1 ∈ ϕα (mα ). In this case, the permissibility of b follows from (14.9). Case 2: 1 ∈ ϕα (mα ); i.e., there exists an i ≤ mα with ϕα (i) = 1. Case 2.1: πα = . W.l.o.g. let ϕα (1) = 1. By (14.8) we have ϕα (2) ∈ {3, ..., n}, hence (bϕα (1) , bϕα (2) ) ∈ {(2, 1), (2, 0)} ⊆ . Case 2.2: πα = s . Case 2.2.1: 0 ∈ {aϕα (1) , ..., aϕα (1) }. By (14.8) and the definition of s we obtain (aϕα (1) , ..., aϕα (i−1) , a1 , aϕα (i+1) , ..., aϕα (s) ) = (0, ..., 0, 1, 0, ..., 0).

(14.10)

Since ak = bk for all k ≥ 2 and b1 = 2, the inclusion (bϕα (1) , ..., bϕα (s) ) ∈ s follows from (14.10). Case 2.2.2: {aϕα (1) , ..., aϕα (s) } ⊆ {1, 2}. Since ϕα (mα ) ⊆ m there exists a j such that ϕα (j) ∈ m, bϕα (j) = aϕα (j) = 1 and {bϕα (1) , ..., bϕα (s) } ⊆ {1, 2}, hence (bϕα (1) , ..., bϕα (s) ) belongs to πα and therefore b is permissible as required. (c): By assumption, there is an s ≥ 3 with πα = s . Then mα = s. First we prove that ϕα is injective. Suppose ϕα is not injective. Then the following two cases are possible: Case 1: |ϕα (mα )| = 1, i.e., ϕα (s) = {j} for a some j ≤ m. Then for every permissible tuple a := (a1 , ..., as ) we have (aϕα (1) , aϕα (2) , ..., aϕα (s) ) = (aj , aj , ..., aj ) = (1, 1, ..., 1) which contradicts (14.6) because all elements of {0, 1, 2} occur in every row of the relation l . Case 2: ∃i1 , i2 , i3 ∈ s : i1 = i2 ∧ ϕα (i1 ) = ϕα (i2 ) ∧ ϕα (i3 ) = ϕα (i1 ). W.l.o.g. let ϕ(i1 ) = 1. Then for the tuple (a1 , ..., al ) := (1, 0, 0, ..., 0) ∈ l we obtain (aϕα (1) , aϕα (2) , ..., aϕα (s) ) ∈ s , which is not possible, because of aϕ(i1 ) = aϕα (i2 ) = 1 and aϕα (i3 ) = 0. Consequently, ϕα is injective. We now show that ϕα is surjective. Suppose that ϕα (mα ) ⊂ m. We may assume w.l.o.g. that 1 ∈ ϕα (mα ). Consider the tuple a := (a1 , a2 , ..., al ) = (1, 2, 2, ..., 2) ∈ l . Then (aϕα (1) , ..., aϕα (s) ) = (2, 2, 2, ..., 2) ∈ s , in contradiction with (14.6). We have shown that ϕα is a bijection. Combining this with the properties of l , we deduce that πα = l . Lemma 14.10.8 Let n ∈ N \ {1, 2} and let and n be as in (14.3) and (14.4). Then: (a) ∈ [{n | n ≥ 3}]. (b) ∀l ≥ 3 : l ∈ [{} ∪ {n | n ∈ N\{1, 2, l}}]. (c) The co-clone [{} ∪ {n | n ≥ 3}] has the infinite basis {, 3 , 4 , ...}. Proof. (a) follows from the fact that the constant function c1 preserves the relation n for all n ≥ 3 but does not preserve the relation .


431

(b): Suppose for an l ≥ 3 we have l ∈ [{} ∪ {n | n ∈ N\{1, 2, l}}]. Then, by Theorem 2.11.2,(a),(b) l = {(aϕ(1) , , ...., aϕ(l) ) ∈ E3l | ∃al+1 , ..., an ∈ E3 : ∀i ∈ I : (aϕi (1) , aϕi (2) , ..., aϕi (mi ) ) ∈ πi },

(14.11)

holds, where I is an index-set, πi are some mi -ary relations of {}∪{m | m ≥ 3, m = l} for all i ∈ I and ϕ : l −→ n and ϕi : mi −→ n are mappings. There must be at least a relation under the relations πi , which is different from the relation , since in the opposite case, l ∈ [{}] holds and this implies P ol3 ∪ {c1 } ⊆ P ol 3 l , which by means of Theorem 14.1.8 0 1 2 1 2 [P ol3 ∪ {c1 }] = P ol3 ⊆ P ol3 l results from, which contra0 1 2 2 1 0 1 2 1 2 dicts c2 ∈ P ol3 l and c2 ∈ P ol3 . 0 1 2 2 1 It results now from the definition of the relation l that ϕ is an injective mapping. W.l.o.g. we can assume that ϕ(x) = x for all x ∈ l. Therefore (14.11) agrees with (14.6) and, with the help of Lemma 14.10.7, we have a contradiction to our assumption. (c) follows directly from (a) and (b). Proof of Theorem 14.10.1: The statements for the classes with the numbers 7, 17, or 30 of Table 14.1 are consequences from Chapter 13 (see also Theorem 15.1.1). For the classes with the number 28, 29, or 43, our assertion follows from Theorem 8.1.6 and Sections 15.3 and 15.4. 0 1 2 Since the class of Theorem 8.2.2 is a subclass of A8 = P ol3 {0}∩P ol3 1 2 0 we have |L↓ (A8 )| = c. The clones A ∈ Fj with j ∈ J := {9, 14, 24, 25, 27, 33, 35, 37, 39, 40, 41, 42} obviously satisfy the condition ∃a, b ∈ E3 : a = b ∧ P3,{a,b} ⊆ Ai . Hence for every j ∈ J there exists a subclass of A with infinite basis such that the range of its functions is a 2-element set (see for instance Lemma 14.10.2). An example of a subclass of A ∈ F22 with infinite basis is given in the proof of Theorem 12.3.8. Let A ∈ F32 with a = 1, b = 2 and c = 0. We have shown in Lemma 14.10.8 that there are uncountable many co-clones containing Inv3 A. This means that |L3 (J3 , A32 )| = c (see Chapter 2).

432


Finally, we use constructions given in Lemmas 14.10.2–14.10.6 to show that |L↓3 (A)| = c with A ∈ Fi for the remaining i. All results are presented in Table 14.11. Table 14.11 Let A ∈ Fi , where Considered cases Use the construci= for a, b, c, α, β, γ tion given in Lemma 14.10.j, where j = (w.l.o.g.)

1 2,4 3 5,19,20,21,23,26,31 6,13,16,36,38 10 11 11 11 12 12 15 18 34

a 0 0 2 0 0 0 0 0 0 0 0 0 2 0

b 2 1 0 1 1 2 1 2 1 1 2 1 0 1

c α 1 2 1 2 2 1 0 0 2 2 0 1 1 2 0 1 2

β γ

1 1 1 1 0 1

2 2 0 2 2 2

4 4 4 2 3 2 3 8 3 2 2 3 5 5

15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3

We say that the class A ∈ Lk (or the lattice L↓k (A) of subclasses of A = [A] ⊆ Pk ) has the depth t, if t is the least integer for which there are some classes A1 , ..., At−1 ∈ Lk with A ⊂ A1 ⊂ A2 ⊂ ... ⊂ At−1 ⊂ Pk . In particular, it holds that the maximal classes of Pk have the depth 1 and the submaximal classes of Pk have the depth 2. For k = 3 by Theorems 13.2.3 and 8.1.6, there are finite and countably infinite sublattices of depth 1 or 2. In this chapter, these sublattices will be determined. The finite lattice L↓3 (L3 ) of depth 1 can be found in Section 15.1. This lattice is a conclusion from Chapter 13. In addition, by Section 14.10, this lattice is the only finite lattice of depth 1, and furthermore, this lattice contains all finite sublattices of L3 of depth 2. In Section 15.2 is a description of all subsemigroups (or subclasses) of (Ps1 ; ) (or [P31 ]), respectively. These subclasses are also subclasses of the submaximal class L of all quasilinear functions of P3 . The list of all subsemigroups of (Ps1 ; ) is then an important aid in Section 15.3 during the determination of the remaining elements of lattice L↓3 (L). Because of Theorem 8.1.6, a further countable sublattice of depth 2 is L↓3 ([O1 ∪ {max}]). This sublattice is given in Section 15.4. Except for isomorphic lattices, all countable sublattices of L3 of depth 2 are traced with that (see Section 14.10).

15.1 The Lattice of Subclasses of P3 of Linear Functions Let L3 be the set of all linear functions of P3 . As a consequence of Theorem 13.2.1, one can see that L↓3 (L3 ) has exactly 6 elements, which are not subsets of [L13 ]. The subclasses of [L13 ] is obtained from Theorem 13.2.2 or from Section 15.2. Thus it holds:

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Theorem 15.1.1 The class L3 of all linear functions of P3 has exactly 38 subclasses: 012 0 , L3 ∩ , [Hi ], L3 , L3 ∩ P ol3 {a}(a ∈ E3 ), L3 ∩ 120 1 where i ∈ {1, 2, 3, 4, 5, 6, 7, 8, 601, 602, 603, 604, 605, 606, 607, 608, 1201, 1202, 1203, 1204, 1232, 1233, 1234, 1235, 1263, 1264, 1265, 1266, 1294, 1295, 1297, 1298} (see Table 15.10 of Section 15.2).

15.2 The Subsemigroups of (P31 ; ) In this section, we determine the subclasses of [P31 ]. It suffices to describe all subsemigroups of (P31 ; ) for this. Since the subsemigroups of (P31 ; ) play a role in many investigations of P3 , it is particularly a question of clarifying the construction of these subsemigroups. To facilitate checking the following considerations without a lot of expenditure also by hand, some tables are given at the end of this section. The functions of P31 and their notations are given in Table 15.1. As usual the operation is defined by (f g)(x) := f (g(x)) P31

(see Table 15.2). for all f, g ∈ The number statement of the following theorem was published by G. Wilde and Sh. Raney in 1972 without proof (as a result of a computer calculation). Theorem 15.2.1 ([Wil-R 72], [Lau 84a]) (P31 ; ∗) has exactly 1299 subsemigroups (including ∅), which are listed in Table 15.10. Proof.1 In preparation for the proof of the above theorem, some notations, which we use during the description of the subsemigroups of (P31 , ∗), are given. Let C := {c0 , c1 , c2 }, J := {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 }, U := {c0 , c2 , u0 , u1 , u2 , u3 , u4 , u5 }, V := {c1 , c2 , v0 , v1 , v2 , v3 , v4 , v5 }, S := {s1 , s2 , s3 , s4 , s5 , s6 }. Starting from the possible subsemigroups Ji (i ∈ {−3, −2, −1, 0, 1, 2, ..., 41}) of J (= J41 ), given in Table 15.4, one can construct, with the aid of the following mappings, 1

Basically, the following proof comes from [Lau 84a], which required correction in some places. I owe K. Todorov (Sofia) and Anne Fearnley (Montreal) indications of the mistakes.

15.2 The Subsemigroups of (P31 ; )

435

ϕi : f → s−1 i ∗ f ∗ si (i ∈ {2, 3, ..., 6}) and the notation 1 ϕi (A) := { s−1 i ∗ f ∗ si | f ∈ A } (A ⊆ P3 )

the classes isomorphic to Ji . These isomorphic classes are given in Tables 15.5–15.9, where we use the notations Ui := ϕ2 (Ji ) and Vi := ϕ6 (Ji ) (i ∈ {−3, −2, −1, 0, 1, 2, ..., 41}). Furthermore we use the notations: S0 := ∅, S1 := {s1 }, S2 := {s1 , s3 }, S3 := {s1 , s2 }, S4 := {s1 , s6 }, S5 := {s1 , s4 , s5 }, S6 := S. To make the structure of the subsemigroups recognizable short, we assign a tuple τ (H) as follows to every subsemigroup H: τ (H) := (a, b, c, d) :⇐⇒ H ∩ J = Ja ∧ H ∩ U = Ub ∧ H ∩ V = Vc ∧ H ∩ S = Sd . Obviously, it holds H = H ⇐⇒ τ (H) = τ (H ). H always denotes an arbitrary subsemigroup of (P31 ; ∗). The number of possibilities for H in one of the below cases i for H is denoted with ni . In Cases 4.1 and 4.2 some possibilities are already included in other cases for H, so that we must change the number ni into the number ni , i.e., ni denotes the number of possibilities for H in Case i, which were not included in the preceded cases. In most of the following cases i for H, the numbers ni or/and ni and the possibilities for H are given. To check the following proof is an arduous matter that should be carried out only with the aid of a computer. A summary of all possibilities for H together with the corresponding characteristic tuples τ (H) can be found in Table 15.10. Case 1: H ⊆ C. Obviously, n1 = 8 and H = Hi (i = 1, ..., 8; see Table 15.10) is an arbitrary subset of C. Case 2: H ⊆ C and H ⊆ A ∈ {J, U, V }. Case 2.1: H ⊆ J. It is not hard to check that n2.1 = 41 and H ∈ {J1 , J2 , ..., J41 } (or H = Ht , t ∈ {9, 10, ..., 49}, see Table 15.10).

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Case 2.2: H ⊆ U . By ϕ2 (J) = U and by Case 2.1, we have n2.2 = 41, and H is one of the following sets: Ui := ϕ2 (Ji ) with i ∈ {1, 2, ..., 41} (or H = Ht for t ∈ {50, 51, ..., 90}). Case 2.3: H ⊆ V . By ϕ6 (J) = V and by Case 2.1, we have n2.3 = 41, and for H only, the following sets are possible: Vi := ϕ6 (Ji ) with i ∈ {1, 2, ..., 41} (or H = Hi , i ∈ {91, 92, ..., 131}). Case 3: H ⊆ J ∪ U , H ⊆ J and H ⊆ U . Case 3.1: H ⊆ C ∪ {j1 , j4 , u2 , u3 } = J25 ∪ U25 . The possibilities for H are: J3 ∪ {c0 , c2 }, {c0 , c1 } ∪ U3 , J12 ∪ {c0 , c2 }, {c0 , c1 } ∪ U12 , J25 ∪ {c0 , c2 }, {c0 , c1 } ∪ U25 and Jp ∪ Uq , where (p, q) ∈ {(3, 3), (3, 12), (12, 3), (12, 12), (3, 25), (25, 3), (12, 25), (25, 12), (25, 25)} (or H = Hi with i ∈ {132, 133, ..., 146}); i.e., we have n3.1 = 15. Case 3.2: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c2 , u2 , u3 }. For H ∩ U only the following sets are possible: {c2 }, {c0 , c2 }, {c0 , u2 } = U3 , {c0 , c2 , u2 } = U12 and {c0 , c2 , u2 , u3 } = U25 . Case 3.2.1: H ∩ U = {c2 }. Obviously, H = J8 ∪ {c2 } ( = H147 ). Case 3.2.2: H ∩ U = {c0 , c2 }. In this case, H ∩ J is an arbitrary subsemigroup of J, which contains {c0 , c1 }, but no subset is of J25 , i.e., it holds n3.2.2 = 19 and H = {c0 , c2 } ∪ Ji with i ∈ {13, 14, 15, 22, 23, 24, 26, 27, 28, 29, 33, 34, 35, 36, 37, 38, 39, 40, 41} (or H = Ht with t ∈ {148, 149, ..., 166}). Case 3.2.3: H ∩ U = U3 . In this case, we have n3.2.3 = 17 and H = U3 ∪ Ji with i ∈ {4, 13, 14, 16, 18, 23, 24, 27, 28, 30, 33, 34, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {167, 168, ..., 183}). Case 3.2.4: H ∩ U = U12 . It is easy to check that n3.2.4 = 13 and H = U12 ∪ Ji with i ∈ {13, 14, 23, 24, 27, 28, 33, 34, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {184, 185, ..., 196}). Case 3.2.5: H ∩ U = U25 . In this case, H ∩ J is an arbitrary subsemigroup of J, which contains {j2 , j3 }. Thus, we have n3.2.5 = 7 and H = U25 ∪ Ji with i ∈ {27, 33, 36, 38, 39, 40, 41} (or H = Ht with t ∈ {197, 198, ..., 203}). In summary, we get: n3.2 = 57. Case 3.3: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c1 , u2 , u3 }. In this case, H is isomorphic to a subsemigroup of J ∪ U which fulfills the conditions of Case 3.2. Consequently, we have n3.3 = 57 and H = ϕ2 (Hi ) with i ∈ {147, 148, ..., 203} (or H = Ht with t ∈ {204, 205, ..., 260}). We remark that ϕ2 (Ja ∪ Ub ) = Jb ∪ Ua . Case 3.4: H ∩ J ⊆ {c0 , c1 , j1 , j4 } and H ∩ U ⊆ {c0 , c1 , u2 , u3 }. Then there are two functions f and g of H with 0 02 0 01 f ∈ and g ∈ . 1 20 2 10


437

This implies |H ∩ J| = |H ∩ U | and, if (H\C) ∩ J = {ji | i ∈ I} with I ⊆ {0, 1, ..., 5}, (H\C)∩U ∈ {{ui | i ∈ I}, {u5−i | i ∈ I}}. Examining the possible cases yields n3.4 = 19 and H = Jp ∪ Uq with (p, q) ∈ {(2, 2), (5, 5), (8, 8), (9, 9), (15, 15), (16, 16), (17, 18, (18, 17), (22, 22), (23, 23), (24, 24), (26, 28), (28, 26), (30, 30), (34, 34), (37, 38), (38, 37), (39, 39), (41, 41)} . Summing up, we get n3 = 148. Case 4: H ⊆ A ∪ B ∈ {J ∪ V, U ∪ V }, H ⊆ A and H ⊆ B. Case 4.1: A = J and B = V . By ϕ3 (J ∪ U ) = J ∪ V we have n4.1 = 148, and H is isomorphic to a subsemigroup which we have already determined in Case 3. Hence, one receives a list of the subsemigroups with the aid of the results from the third case, where one has to consider ϕ3 (Ja ∪ Ub ) = ϕ3 (Ja ) ∪ ϕ5 (Jb ) (see Table 15.7 and 15.9). Twenty-three of the 148 subsemigroups were already determined, so that we have n4.1 = 125. 2 In Table 15.10 (for the Case 4.1), 148 sets are given, where every already listed set, is characterized by their first number; this first number is given in bold point. Case 4.2: A = U and B = V . By ϕ4 (J ∪ U ) = U ∪ V , we have n4.2 = 148 and H is isomorphic to a subsemigroup that we have already determined in Case 3. While listing the subsemigroups, one notices that ϕ4 (Ja ∪ Ub ) = ϕ4 (Ja ) ∪ Vb holds. With the aid of Table 15.10 one sees that 48 of the subsemigroups were determined in previous cases. Thus n4.2 = 102. Case 5: H ⊆ J ∪ U ∪ V , H ⊆ J ∪ U , H ⊆ J ∪ V and H ⊆ U ∪ V . Case 5.1: H ∩ (U ∪ V ) ⊆ C ∪ {u2 , u3 , v2 , v3 }. For H ∩ (U ∪ V ), only the sets U3 ∪ V8 = {c0 , c1 , u2 , v2 }, U12 ∪ V15 = {c0 , c1 , c2 , u2 , v2 } and U25 ∪ V22 = {c0 , c1 , c2 , u2 , u3 , v2 , v3 } come into consideration. Case 5.1.1: H ∩ (U ∪ V ) = U3 ∪ V8 . Then, all the subsets Ji of J which contain the two functions c0 , c1 and which have the properties j0 ∈ Ji ∨ j4 ∈ Ji =⇒ {j2 , j3 } ⊆ Ji , j1 ∈ Ji =⇒ j3 ∈ Ji , j5 ∈ Ji =⇒ j2 ∈ Ji . are possible for H ∩ J. Thus n5.1.1 = 11 and H = Ji ∪ U3 ∪ V8 with i ∈ {13, 14, 24, 27, 28, 33, 36, 38, 39, 40, 41}. Case 5.1.2: H ∩ (U ∪ V ) = U12 ∪ V15 . With the aid of Case 5.1.1, we obtain n5.1.2 = 11 and H = Ji ∪ U12 ∪ V15 , i ∈ {13, 14, 24, 27, 28, 33, 36, 38, 39, 40, 41}. Case 5.1.3: H ∩ (U ∪ V ) = U25 ∪ V22 . Then H ∩ J is a subsemigroup of J, which contains j2 and j3 . Consequently, n5.1.3 = 7 and H = Ji ∪ U25 ∪ V22 , i ∈ {27, 33, 36, 38, 39, 40, 41}. In summary, we obtain n5.1 = 29. 2

This was not taken into account in [Lau 84a]!

438


Case 5.2: H ∩ (J ∪ V ) ⊆ C ∪ {j1 , j4 , v1 , v4 }. Because of ϕ2 (C ∪ {j1 , j4 , v1 , v4 }) = U25 ∪ V22 , we have n5.2 = 29 and H = ϕ2 (A), where A is a set which we have already determined in Case 5.1. More exactly: If A = Ja ∪ Ub ∪ Vc then ϕ2 (A) = Jb ∪ Ua ∪ ϕ5 (Jc ). Case 5.3: H ∩ (J ∪ U ) ⊆ C ∪ {j0 , j5 , u0 , u5 }. This case is also isomorphic to Case 5.1. One can obtain all 29 possibilities for H with the aid of the mapping ϕ6 from the sets determined in Case 5.1, where ϕ6 (Ja ∪ Ub ∪ Vc ) = Jc ∪ ϕ4 (Jb ) ∪ Va must be pointed out. Case 5.4: H fulfills none of the conditions from Cases 5.1, 5.2, or 5.3. In this case, there are functions f, g, h ∈ H with 0 0212 0 0112 1 0102 f ∈ ,g ∈ and h ∈ . 1 2021 2 1021 2 1020 Since certain functions ja , ub , vc for certain a, b, c belong to H, one easily checks that C ⊂ H and, if (H\C) ∩ J = {ji | i ∈ I}, it holds (H\C) ∩ U ∈ {{ui | i ∈ I}, {u5−i | i ∈ I}} and (H\C) ∩ V ∈ {{vi | i ∈ I}, {v5−i | i ∈ I}}. Examining the possible cases using the results of Case 2 yields n5.4 = 7 and H = Jp ∪ Uq ∪ Vr with (p, q, r) ∈ {(24, 24, 26), (26, 28, 24), (28, 26, 28), (37, 38, 39), (38, 37, 38), (39, 39, 37), (41, 41, 41)}. Consequently, n5 = 94. Case 6: S ∩ H = ∅. Since S as is well-known has 6 subgroups, the following cases are possible: Case 6.1: S ∩ H = {s1 }. Then we have H = A ∪ {s1 }, where A is one of the 600 subsemigroups of J ∪ U ∪ V determined above. Case 6.2: S ∩ H = {s1 , s3 }. In this case, H\S is one of the following 31 sets: ∅, {c2 }, {c0 , c1 }, C, J27 , J32 , J37 , J, J27 ∪ {c2 }, J37 ∪ {c2 }, C ∪ J, U1 ∪ V2 , U3 ∪ V8 , U6 ∪ V5 , U12 ∪ V15 , U10 ∪ V9 , U25 ∪ V22 , U21 ∪ V16 , U29 ∪ V23 , U31 ∪ V30 , U35 ∪ V34 , U38 ∪ V39 , U ∪ V , J27 ∪ U3 ∪ V8 , J27 ∪ U12 ∪ V15 , J27 ∪ U25 ∪ V22 , J37 ∪ U38 ∪ V39 , J ∪ U12 ∪ V15 , J ∪ U3 ∪ V8 , J ∪ U25 ∪ V22 , J ∪ U ∪ V . Case 6.3: S ∩ H = {s1 , s2 }. In this case, n6.3 = 31 and the possibilities for H be determined with the results from Case 6.2 and with the mapping ϕ6 . Case 6.4: S ∩ H = {s1 , s6 }. In this case, n6.4 = 31 and one can determine the possibilities for H, using Case 6.2 and the mapping ϕ2 . Case 6.5: S ∩ H ∈ {{s1 , s4 , s5 }, S}. For H\S only the sets ∅, C and J ∪ U ∪ V are possible, whereby n6.5 = 6. Consequently, we have n6 = 699. In summary, we get that (P31 ; ∗) has exactly 1299 different subsemigroups. We remark that one finds further information about the given subsemigroups in [Bij-T 91].


439

Table 15.1

x j0 (x) j1 (x) j2 (x) j3 (x) j4 (x) j5 (x) u0 (x) u1 (x) u2 (x) u3 (x) u4 (x) u5 (x) 0 1 0 0 1 1 0 2 0 0 2 2 0 1 0 1 0 1 0 1 0 2 0 2 0 2 2 0 0 1 0 1 1 0 0 2 0 2 2 x v0 (x) v1 (x) v2 (x) v3 (x) v4 (x) v5 (x) s1 (x) s2 (x) s3 (x) s4 (x) s5 (x) s6 (x) 0 2 1 1 2 2 1 0 0 1 1 2 2 1 1 2 1 2 1 2 1 2 0 2 0 1 2 1 1 2 1 2 2 2 1 2 0 1 0

Table 15.2

f ∗g j0 j1 j2 j3 j4 j5 u0 u1 u2 u3 u4 u5 v0 v1 v2 v3 v4 v5 s2 s3 s4 s5 s6

ji j5−i ji c0 c1 j5−i ji u5−i ui c0 c2 u5−i ui v5−i vi c1 c2 v5−i vi ui j5−i vi u5−i v5−i

ui j5−i c0 ji j5−i c1 ji u5−i c0 ui u5−i c2 ui v5−i c1 vi v5−i c2 vi ji vi j5−i v5−i u5−i

vi s2 s3 c0 j0 j1 j5−i j2 j0 ji j1 j2 j5−i j4 j3 ji j3 j5 c1 j5 j4 c0 u0 u1 u5−i u2 u0 ui u1 u2 u5−i u4 u3 ui u3 u5 c2 u5 u4 c1 v0 v1 v5−i v2 v0 vi v1 v2 v5−i v4 v3 vi v3 v5 c2 v5 v4 v5−i s1 s5 ui s4 s1 u5−i s3 s6 ji s6 s2 j5−i s5 s4

s4 j2 j0 j1 j4 j5 j3 u2 u0 u1 u4 u5 u3 v2 v0 v1 v4 v5 v3 s6 s2 s5 s1 s3



440

15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3 Table 15.3

ϕ2 (f ) = ϕ3 (f ) = ϕ5 (f ) = ϕ4 (f ) = ϕ6 (f ) = f s2 ∗ f ∗ s2 s3 ∗ f ∗ s3 s4 ∗ f ∗ s5 s5 ∗ f ∗ s4 s6 ∗ f ∗ s6 c0 c0 c1 c1 c2 c2 c1 c2 c0 c2 c0 c1 c2 c1 c2 c0 c1 c0 j0 u0 j4 v1 u3 v3 j1 u2 j5 v2 u5 v4 j2 u1 j3 v0 u4 v5 j3 u4 j2 v5 u1 v0 j4 u3 j0 v3 u0 v1 j5 u5 j1 v4 u2 v2 u0 j0 v1 j4 v3 u3 u1 j2 v0 j3 v5 u4 u2 j1 v2 j5 v4 u5 u3 j4 v3 j0 v1 u0 u4 j3 v5 j2 v0 u1 u5 j5 v4 j1 v2 u2 v0 v5 u1 u4 j2 j3 v1 v3 u0 u3 j0 j4 v2 v4 u2 u5 j1 j5 v3 v1 u3 u0 j4 j0 v4 v2 u5 u2 j5 j1 v5 v0 u4 u1 j3 j2 s1 s1 s1 s1 s1 s1 s2 s2 s6 s6 s3 s3 s3 s6 s3 s2 s6 s2 s4 s5 s5 s4 s4 s5 s5 s4 s4 s5 s5 s4 s6 s3 s2 s3 s2 s6

15.2 The Subsemigroups of (P31 ; ) Table 15.4 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Ji {c0 , c1 } {c1 } {c0 } ∅ {j1 } {j5 } {c0 , j1 } {c0 , j2 } {c0 , j5 } {c1 , j1 } {c1 , j3 } {c1 , j5 } {j0 , j5 } {j1 , j4 } {j1 , j5 } {c0 , c1 , j1 } {c0 , c1 , j2 } {c0 , c1 , j3 } {c0 , c1 , j5 } {c0 , j1 , j2 } {c0 , j1 , j5 } {c0 , j2 , j5 } {c1 , j1 , j3 } {c1 , j1 , j5 } {c1 , j3 , j5 } {c0 , c1 , j0 , j5 } {c0 , c1 , j1 , j2 } {c0 , c1 , j1 , j3 } {c0 , c1 , j1 , j4 } {c0 , c1 , j1 , j5 } {c0 , c1 , j2 , j3 } {c0 , c1 , j2 , j5 } {c0 , c1 , j3 , j5 } {c0 , j1 , j2 , j5 } {c1 , j1 , j3 , j5 } {j0 , j1 , j4 , j5 } {c0 , c1 , j1 , j2 , j3 } {c0 , c1 , j1 , j2 , j5 } {c0 , c1 , j1 , j3 , j5 } {c0 , c1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j4 , j5 } {c0 , c1 , j0 , j2 , j3 , j5 } {c0 , c1 , j1 , j2 , j3 , j4 } {c0 , c1 , j1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 }

Table 15.5 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Ui := ϕ2 (Ji ) {c0 , c2 } {c2 } {c0 } ∅ {u2 } {u5 } {c0 , u2 } {c0 , u1 } {c0 , u5 } {c2 , u2 } {c2 , u4 } {c2 , u5 } {u0 , u5 } {u2 , u3 } {u2 , u5 } {c0 , c2 , u2 } {c0 , c2 , u1 } {c0 , c2 , u4 } {c0 , c2 , u5 } {c0 , u2 , u1 } {c0 , u2 , u5 } {c0 , u1 , u5 } {c2 , u2 , u4 } {c2 , u2 , u5 } {c2 , u4 , u5 } {c0 , c2 , u0 , u5 } {c0 , c2 , u2 , u1 } {c0 , c2 , u2 , u4 } {c0 , c2 , u2 , u3 } {c0 , c2 , u2 , u5 } {c0 , c2 , u1 , u4 } {c0 , c2 , u1 , u5 } {c0 , c2 , u4 , u5 } {c0 , u2 , u1 , u5 } {c2 , u2 , u4 , u5 } {u0 , u2 , u3 , u5 } {c0 , c2 , u2 , u1 , u4 } {c0 , c2 , u2 , u1 , u5 } {c0 , c2 , u2 , u4 , u5 } {c0 , c2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u1 , u4 , u3 , u5 }

441

442

15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3 Table 15.6 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

Vi := ϕ6 (Ji ) {c2 , c1 } {c1 } {c2 } ∅ {v4 } {v2 } {c2 , v4 } {c2 , v5 } {c2 , v2 } {c1 , v4 } {c1 , v0 } {c1 , v2 } {v3 , v2 } {v4 , v1 } {v4 , v2 } {c2 , c1 , v4 } {c2 , c1 , v5 } {c2 , c1 , v0 } {c2 , c1 , v2 } {c2 , v4 , v5 } {c2 , v4 , v2 } {c2 , v5 , v2 } {c1 , v4 , v0 } {c1 , v4 , v2 } {c1 , v0 , v2 } {c2 , c1 , v3 , v2 } {c2 , c1 , v4 , v5 } {c2 , c1 , v4 , v0 } {c2 , c1 , v4 , v1 } {c2 , c1 , v4 , v2 } {c2 , c1 , v5 , v0 } {c2 , c1 , v5 , v2 } {c2 , c1 , v0 , v2 } {c2 , v4 , v5 , v2 } {c1 , v4 , v0 , v2 } {v3 , v4 , v1 , v2 } {c2 , c1 , v4 , v5 , v0 } {c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v4 , v0 , v2 } {c2 , c1 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 } {c2 , c1 , v3 , v5 , v0 , v2 } {c2 , c1 , v4 , v5 , v0 , v1 } {c2 , c1 , v4 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v5 , v0 , v1 , v2 }

Table 15.7 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ3 (Ji ) = Jt {c1 , c0 } {c0 } {c1 } ∅ {j5 } {j1 } {c1 , j5 } {c1 , j3 } {c1 , j1 } {c0 , j5 } {c0 , j2 } {c0 , j1 } {j4 , j1 } {j5 , j0 } {j5 , j1 } {c1 , c0 , j5 } {c1 , c0 , j3 } {c1 , c0 , j2 } {c1 , c0 , j1 } {c1 , j5 , j3 } {c1 , j5 , j1 } {c1 , j3 , j1 } {c0 , j5 , j2 } {c0 , j5 , j1 } {c0 , j2 , j1 } {c1 , c0 , j4 , j1 } {c1 , c0 , j5 , j3 } {c1 , c0 , j5 , j2 } {c1 , c0 , j5 , j0 } {c1 , c0 , j5 , j1 } {c1 , c0 , j3 , j2 } {c1 , c0 , j3 , j1 } {c1 , c0 , j2 , j1 } {c1 , j5 , j3 , j1 } {c0 , j5 , j2 , j1 } {j4 , j5 , j0 , j1 } {c1 , c0 , j5 , j3 , j2 } {c1 , c0 , j5 , j3 , j1 } {c1 , c0 , j5 , j2 , j1 } {c1 , c0 , j3 , j2 , j1 } {c1 , c0 , j4 , j5 , j0 , j1 } {c1 , c0 , j4 , j3 , j2 , j1 } {c1 , c0 , j5 , j3 , j2 , j0 } {c1 , c0 , j5 , j3 , j2 , j1 } {c1 , c0 , j4 , j5 , j3 , j2 , j0 , j1 }

t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

15.2 The Subsemigroups of (P31 ; ) Table 15.8 i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ4 (Ji ) = Ut {c2 , c0 } {c0 } {c2 } ∅ {u5 } {u2 } {c2 , u5 } {c2 , u4 } {c2 , u2 } {c0 , u5 } {c0 , u1 } {c0 , u2 } {u3 , u2 } {u5 , u0 } {u5 , u2 } {c2 , c0 , u5 } {c2 , c0 , u4 } {c2 , c0 , u1 } {c2 , c0 , u2 } {c2 , u5 , u4 } {c2 , u5 , u2 } {c2 , u4 , u2 } {c0 , u5 , u1 } {c0 , u5 , u2 } {c0 , u1 , u2 } {c2 , c0 , u3 , u2 } {c2 , c0 , u5 , u4 } {c2 , c0 , u5 , u1 } {c2 , c0 , u5 , u0 } {c2 , c0 , u5 , u2 } {c2 , c0 , u4 , u1 } {c2 , c0 , u4 , u2 } {c2 , c0 , u1 , u2 } {c2 , u5 , u4 , u2 } {c0 , u5 , u1 , u2 } {u3 , u5 , u0 , u2 } {c2 , c0 , u5 , u4 , u1 } {c2 , c0 , u5 , u4 , u2 } {c2 , c0 , u5 , u1 , u2 } {c2 , c0 , u4 , u1 , u2 } {c2 , c0 , u3 , u5 , u0 , u2 } {c2 , c0 , u3 , u4 , u1 , u2 } {c2 , c0 , u5 , u4 , u1 , u0 } {c2 , c0 , u5 , u4 , u1 , u2 } {c2 , c0 , u3 , u5 , u4 , u1 , u0 , u2 }

Table 15.9 t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

i −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

ϕ5 (Ji ) = Vt {c1 , c2 } {c2 } {c1 } ∅ {v2 } {v4 } {c1 , v2 } {c1 , v0 } {c1 , v4 } {c2 , v2 } {c2 , v5 } {c2 , v4 } {v1 , v4 } {v2 , v3 } {v2 , v4 } {c1 , c2 , v2 } {c1 , c2 , v0 } {c1 , c2 , v5 } {c1 , c2 , v4 } {c1 , v2 , v0 } {c1 , v2 , v4 } {c1 , v0 , v4 } {c2 , v2 , v5 } {c2 , v2 , v4 } {c2 , v5 , v4 } {c1 , c2 , v1 , v4 } {c1 , c2 , v2 , v0 } {c1 , c2 , v2 , v5 } {c1 , c2 , v2 , v3 } {c1 , c2 , v2 , v4 } {c1 , c2 , v0 , v5 } {c1 , c2 , v0 , v4 } {c1 , c2 , v5 , v4 } {c1 , v2 , v0 , v4 } {c2 , v2 , v5 , v4 } {v1 , v2 , v3 , v4 } {c1 , c2 , v2 , v0 , v5 } {c1 , c2 , v2 , v0 , v4 } {c1 , c2 , v2 , v5 , v4 } {c1 , c2 , v0 , v5 , v4 } {c1 , c2 , v1 , v2 , v3 , v4 } {c1 , c2 , v1 , v0 , v5 , v4 } {c1 , c2 , v2 , v0 , v5 , v3 } {c1 , c2 , v2 , v0 , v5 , v4 } {c1 , c2 , v1 , v2 , v0 , v5 , v3 , v4 }

t −3 −1 −2 0 2 1 8 7 6 5 4 3 10 9 11 15 14 13 12 21 20 19 18 17 16 25 29 28 22 26 27 24 23 31 30 32 36 35 34 33 37 39 38 40 41

443


444

Table 15.103 i 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 3

τ (Hi ) (0, 0, 0, 0) (−1, −1, 0, 0) (−1, 0, −1, 0) (0, −2, −2, 0) (−3, −1, −1, 0) (−1, −3, −2, 0) (−2, −2, −3, 0) (−3, −3, −3, 0) (1, 0, 0, 0) (2, 0, 0, 0) (3, −1, 0, 0) (4, −1, 0, 0) (5, −1, 0, 0) (6, 0, −2, 0) (7, 0, −2, 0) (8, 0, −2, 0) (9, 0, 0, 0) (10, 0, 0, 0) (11, 0, 0, 0) (12, −1, −2, 0) (13, −1, −2, 0) (14, −1, −2, 0) (15, −1, −2, 0) (16, −1, −2, 0) (17, −1, −2, 0) (18, −1, 0, 0) (19, 0, −2, 0) (20, 0, −2, 0) (21, 0, −2, 0) (22, −1, −2, 0) (23, −1, −2, 0) (24, −1, −2, 0) (25, −1, −2, 0) (26, −1, −2, 0) (27, −1, −2, 0) (28, −1, −2, 0) (29, −1, −2, 0) (30, −1, −2, 0) (31, 0, −2, 0) (32, 0, 0, 0) (33, −1, −2, 0) (34, −1, −2, 0) (35, −1, −2, 0) (36, −1, −2, 0) (37, −1, −2, 0) (38, −1, −2, 0) (39, −1, −2, 0) (40, −1, −2, 0) (41, −1, −2, 0) (0, 1, 0, 0) (0, 2, 0, 0) (−1, 3, 0, 0) (−1, 4, 0, 0) (−1, 5, 0, 0) (0, 6, −1, 0) (0, 7, −1, 0) (0, 8, −1, 0) (0, 9, 0, 0) (0, 10, 0, 0) (0, 11, 0, 0)

Hi ∅ {c0 } {c1 } {c2 } {c0 , c1 } {c0 , c2 } {c1 , c2 } {c0 , c1 , c2 } {j1 } {j5 } {c0 , j1 } {c0 , j2 } {c0 , j5 } {c1 , j1 } {c1 , j3 } {c1 , j5 } {j0 , j5 } {j1 , j4 } {j1 , j5 } {c0 , c1 , j1 } {c0 , c1 , j2 } {c0 , c1 , j3 } {c0 , c1 , j5 } {c0 , j1 , j2 } {c0 , j1 , j5 } {c0 , j2 , j5 } {c1 , j1 , j3 } {c1 , j1 , j5 } {c1 , j3 , j5 } {c0 , c1 , j0 , j5 } {c0 , c1 , j1 , j2 } {c0 , c1 , j1 , j3 } {c0 , c1 , j1 , j4 } {c0 , c1 , j1 , j5 } {c0 , c1 , j2 , j3 } {c0 , c1 , j2 , j5 } {c0 , c1 , j3 , j5 } {c0 , j1 , j2 , j5 } {c1 , j1 , j3 , j5 } {j0 , j1 , j4 , j5 } {c0 , c1 , j1 , j2 , j3 } {c0 , c1 , j1 , j2 , j5 } {c0 , c1 , j1 , j3 , j5 } {c0 , c1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j4 , j5 } {c0 , c1 , j0 , j2 , j3 , j5 } {c0 , c1 , j1 , j2 , j3 , j4 } {c0 , c1 , j1 , j2 , j3 , j5 } {c0 , c1 , j0 , j1 , j2 , j3 , j4 , j5 } {u2 } {u5 } {c0 , u2 } {c0 , u1 } {c0 , u5 } {c2 , u2 } {c2 , u4 } {c2 , u5 } {u0 , u5 } {u2 , u3 } {u2 , u5 }

Case 1

2.1

2.2

If the sets Hi are unions of certain sets, repeated constants are not removed in the descriptions of the sets to make the construction of the sets Hi recognizable.

15.2 The Subsemigroups of (P31 ; ) i 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

τ (Hi ) (−1, 12, −1, 0) (−1, 13, −1, 0) (−1, 14, −1, 0) (−1, 15, −1, 0) (−1, 16, −1, 0) (−1, 17, −1, 0) (−1, 18, 0, 0) (0, 19, −1, 0) (0, 20, −1, 0) (0, 21, −1, 0) (−1, 22, −1, 0) (−1, 23, −1, 0) (−1, 24, −1, 0) (−1, 25, −1, 0) (−1, 26, −1, 0) (−1, 27, −1, 0) (−1, 28, −1, 0) (−1, 29, −1, 0) (−1, 30, −1, 0) (0, 31, −1, 0) (0, 32, 0, 0) (−1, 33, −1, 0) (−1, 34, −1, 0) (−1, 35, −1, 0) (−1, 36, −1, 0) (−1, 37, −1, 0) (−1, 38, −1, 0) (−1, 39, −1, 0) (−1, 40, −1, 0) (−1, 41, −1, 0) (0, 0, 1, 0) (0, 0, 2, 0) (0, −2, 3, 0) (0, −2, 4, 0) (0, −2, 5, 0) (−2, 0, 6, 0) (−2, 0, 7, 0) (−2, 0, 8, 0) (0, 0, 9, 0) (0, 0, 10, 0) (0, 0, 11, 0) (−2, −2, 12, 0) (−2, −2, 13, 0) (−2, −2, 14, 0) (−2, −2, 15, 0) (0, −2, 16, 0) (0, −2, 17, 0) (0, −2, 18, 0) (−2, 0, 19, 0) (−2, 0, 20, 0) (−2, 0, 21, 0) (−2, −2, 22, 0) (−2, −2, 23, 0) (−2, −2, 24, 0) (−2, −2, 25, 0) (−2, −2, 26, 0) (−2, −2, 27, 0) (−2, −2, 28, 0) (−2, −2, 29, 0) (0, −2, 30, 0) (−2, 0, 31, 0) (0, 0, 32, 0) (−2, −2, 33, 0) (−2, −2, 34, 0) (−2, −2, 35, 0)

Hi {c0 , c2 , u2 } {c0 , c2 , u1 } {c0 , c2 , u4 } {c0 , c2 , u5 } {c0 , u2 , u1 } {c0 , u2 , u5 } {c0 , u1 , u5 } {c2 , u2 , u4 } {c2 , u2 , u5 } {c2 , u4 , u5 } {c0 , c2 , u0 , u5 } {c0 , c2 , u2 , u1 } {c0 , c2 , u2 , u4 } {c0 , c2 , u2 , u3 } {c0 , c2 , u2 , u5 } {c0 , c2 , u1 , u4 } {c0 , c2 , u1 , u5 } {c0 , c2 , u4 , u5 } {c0 , u2 , u1 , u5 } {c2 , u2 , u4 , u5 } {u0 , u2 , u3 , u5 } {c0 , c2 , u2 , u1 , u4 } {c0 , c2 , u2 , u1 , u5 } {c0 , c2 , u2 , u4 , u5 } {c0 , c2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c2 , u0 , u2 , u1 , u4 , u3 , u5 } {v4 } {v2 } {c2 , v4 } {c2 , v5 } {c2 , v2 } {c1 , v4 } {c1 , v0 } {c1 , v2 } {v3 , v2 } {v4 , v1 } {v4 , v2 } {c2 , c1 , v4 } {c2 , c1 , v5 } {c2 , c1 , v0 } {c2 , c1 , v2 } {c2 , v4 , v5 } {c2 , v4 , v2 } {c2 , v5 , v2 } {c1 , v4 , v0 } {c1 , v4 , v2 } {c1 , v0 , v2 } {c2 , c1 , v3 , v2 } {c2 , c1 , v4 , v5 } {c2 , c1 , v4 , v0 } {c2 , c1 , v4 , v1 } {c2 , c1 , v4 , v2 } {c2 , c1 , v5 , v0 } {c2 , c1 , v5 , v2 } {c2 , c1 , v0 , v2 } {c2 , v4 , v5 , v2 } {c1 , v4 , v0 , v2 } {v3 , v4 , v1 , v2 } {c2 , c1 , v4 , v5 , v0 } {c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v4 , v0 , v2 }

445 Case

2.3


446 i 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190

τ (Hi ) (−2, −2, 36, 0) (−2, −2, 37, 0) (−2, −2, 38, 0) (−2, −2, 39, 0) (−2, −2, 40, 0) (−2, −2, 41, 0) (3, −3, −1, 0) (−3, 3, −2, 0) (12, −3, −3, 0) (−3, 12, −3, 0) (25, −3, −3, 0) (−3, 25, −3, 0) (3, 3, 0, 0) (3, 12, −1, 0) (12, 3, −2, 0) (12, 12, −3, 0) (3, 25, −1, 0) (25, 3, −2, 0) (12, 25, −3, 0) (25, 12, −3, 0) (25, 25, −3, 0) (8, −2, −3, 0) (13, −3, −3, 0) (14, −3, −3, 0) (15, −3, −3, 0) (22, −3, −3, 0) (23, −3, −3, 0) (24, −3, −3, 0) (26, −3, −3, 0) (27, −3, −3, 0) (28, −3, −3, 0) (29, −3, −3, 0) (33, −3, −3, 0) (34, −3, −3, 0) (35, −3, −3, 0) (36, −3, −3, 0) (37, −3, −3, 0) (38, −3, −3, 0) (39, −3, −3, 0) (40, −3, −3, 0) (41, −3, −3, 0) (4, 3, 0, 0) (13, 3, −2, 0) (14, 3, −2, 0) (16, 3, 0, 0) (18, 3, 0, 0) (23, 3, −2, 0) (24, 3, −2, 0) (27, 3, −2, 0) (28, 3, −2, 0) (30, 3, 0, 0) (33, 3, −2, 0) (34, 3, −2, 0) (36, 3, −2, 0) (38, 3, −2, 0) (39, 3, −2, 0) (40, 3, −2, 0) (41, 3, −2, 0) (13, 12, −3, 0) (14, 12, −3, 0) (23, 12, −3, 0) (24, 12, −3, 0) (27, 12, −3, 0) (28, 12, −3, 0) (33, 12, −3, 0)

Hi {c2 , c1 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 } {c2 , c1 , v3 , v5 , v0 , v2 } {c2 , c1 , v4 , v5 , v0 , v1 } {c2 , c1 , v4 , v5 , v0 , v2 } {c2 , c1 , v3 , v4 , v5 , v0 , v1 , v2 } {c0 , j1 , c2 } {c0 , c1 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 } {c0 , c1 , c0 , c2 , u2 } {c0 , c1 , j1 , j4 , c0 , c2 } {c0 , c1 , c0 , c2 , u2 , u3 } {c0 , j1 , c0 , u2 } {c0 , j1 , c0 , c2 , u2 } {c0 , c1 , j1 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 , u2 } {c0 , j1 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j4 , c0 , u2 } {c0 , c1 , j1 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u3 } {c1 , j5 , c2 } {c0 , c1 , j2 , c0 , c2 } {c0 , c1 , j3 , c0 , c2 } {c0 , c1 , j5 , c0 , c2 } {c0 , c1 , j0 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , c0 , c2 } {c0 , c1 , j1 , j3 , c0 , c2 } {c0 , c1 , j1 , j5 , c0 , c2 } {c0 , c1 , j2 , j3 , c0 , c2 } {c0 , c1 , j2 , j5 , c0 , c2 } {c0 , c1 , j3 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 } {c0 , c1 , j1 , j3 , j5 , c0 , c2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 } J ∪ {c0 , c2 } {c0 , j2 , c0 , u2 } {c0 , c1 , j2 , c0 , u2 } {c0 , c1 , j3 , c0 , u2 } {c0 , j1 , j2 , c0 , u2 } {c0 , j2 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , c0 , u2 } {c0 , c1 , j1 , j3 , c0 , u2 } {c0 , c1 , j2 , j3 , c0 , u2 } {c0 , c1 , j2 , j5 , c0 , u2 } {c0 , j1 , j2 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , c0 , u2 } {c0 , c1 , j1 , j2 , j5 , c0 , u2 } {c0 , c1 , j2 , j3 , j5 , c0 , u2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , u2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , u2 } J ∪ {c0 , u2 } {c0 , c1 , j2 , c0 , c2 , u2 } {c0 , c1 , j3 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 }

Case

3.1

3.2.1 3.2.2

3.2.3

3.2.4


τ (Hi ) (34, 12, −3, 0) (36, 12, −3, 0) (38, 12, −3, 0) (39, 12, −3, 0) (40, 12, −3, 0) (41, 12, −3, 0) (27, 25, −3, 0) (33, 25, −3, 0) (36, 25, −3, 0) (38, 25, −3, 0) (39, 25, −3, 0) (40, 25, −3, 0) (41, 25, −3, 0) (−2, 8, −3, 0) (−3, 13, −3, 0) (−3, 14, −3, 0) (−3, 15, −3, 0) (−3, 22, −3, 0) (−3, 23, −3, 0) (−3, 24, −3, 0) (−3, 26, −3, 0) (−3, 27, −3, 0) (−3, 28, −3, 0) (−3, 29, −3, 0) (−3, 33, −3, 0) (−3, 34, −3, 0) (−3, 35, −3, 0) (−3, 36, −3, 0) (−3, 37, −3, 0) (−3, 38, −3, 0) (−3, 39, −3, 0) (−3, 40, −3, 0) (−3, 41, −3, 0) (3, 4, 0, 0) (3, 13, −1, 0) (3, 14, −1, 0) (3, 16, 0, 0) (3, 18, 0, 0) (3, 23, −1, 0) (3, 24, −1, 0) (3, 27, −1, 0) (3, 28, −1, 0) (3, 30, 0, 0) (3, 33, −1, 0) (3, 34, −1, 0) (3, 36, −1, 0) (3, 38, −1, 0) (3, 39, −1, 0) (3, 40, −1, 0) (3, 41, −1, 0) (12, 13, −3, 0) (12, 14, −3, 0) (12, 23, −3, 0) (12, 24, −3, 0) (12, 27, −3, 0) (12, 28, −3, 0) (12, 33, −3, 0) (12, 34, −3, 0) (12, 36, −3, 0) (12, 38, −3, 0) (12, 39, −3, 0) (12, 40, −3, 0) (12, 41, −3, 0) (25, 27, −3, 0) (25, 33, −3, 0)

Hi {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 } J ∪ {c0 , c2 , u2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , u3 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u3 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 } J ∪ {c0 , c2 , u2 , u3 } {c1 , c2 , u5 } {c0 , c1 , c0 , c2 , u1 } {c0 , c1 , c0 , c2 , u4 } {c0 , c1 , c0 , c2 , u5 } {c0 , c1 , c0 , c2 , u0 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 } {c0 , c1 , c0 , c2 , u2 , u4 } {c0 , c1 , c0 , c2 , u2 , u5 } {c0 , c1 , c0 , c2 , u1 , u4 } {c0 , c1 , c0 , c2 , u1 , u5 } {c0 , c1 , c0 , c2 , u4 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 } {c0 , c1 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , c0 , c2 , u2 , u4 , u5 } {c0 , c1 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 } ∪ U {c0 , j1 , c0 , u1 } {c0 , j1 , c0 , c2 , u1 } {c0 , j1 , c0 , c2 , u4 } {c0 , j1 , c0 , u2 , u1 } {c0 , j1 , c0 , u1 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 } {c0 , j1 , c0 , c2 , u2 , u4 } {c0 , j1 , c0 , c2 , u1 , u4 } {c0 , j1 , c0 , c2 , u1 , u5 } {c0 , j1 , c0 , u2 , u1 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 } {c0 , j1 , c0 , c2 , u2 , u1 , u5 } {c0 , j1 , c0 , c2 , u1 , u4 , u5 } {c0 , j1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , j1 } ∪ U {c0 , c1 , j1 , c0 , c2 , u1 } {c0 , c1 , j1 , c0 , c2 , u4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 } {c0 , c1 , j1 , c0 , c2 , u2 , u4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 } {c0 , c1 , j1 , c0 , c2 , u1 , u5 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , j1 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 , j1 } ∪ U {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 }

447 Case

3.2.5

3.3


448 i 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 147 280 150 281 151 282 283 284 285 286 287 288 289 290 291 132 149 148 134 136 157 156 154 155 153 152 161 160 159 158 162 164 163 165 166 292 293 294 295 296 297

τ (Hi ) (25, 36, −3, 0) (25, 38, −3, 0) (25, 39, −3, 0) (25, 40, −3, 0) (25, 41, −3, 0) (2, 2, 0, 0) (5, 5, 0, 0) (8, 8, −3, 0) (9, 9, 0, 0) (15, 15, −2, 0) (16, 16, 0, 0) (17, 18, 0, 0) (18, 17, 0, 0) (22, 22, −3, 0) (23, 23, −3, 0) (24, 24, −3, 0) (26, 28, −3, 0) (28, 26, −3, 0) (30, 30, 0, 0) (34, 34, −3, 0) (37, 38, −3, 0) (38, 37, −3, 0) (39, 39, −3, 0) (41, 41, −3, 0) (8, −2, −3, 0) (−3, −1, 8, 0) (15, −3, −3, 0) (−3, −3, 15, 0) (22, −3, −3, 0) (−3, −3, 22, 0) (8, 0, 8, 0) (8, −1, 15, 0) (15, −2, 8, 0) (15, −3, 15, 0) (8, −1, 22, 0) (22, −2, 8, 0) (15, −3, 22, 0) (22, −3, 15, 0) (22, −3, 22, 0) (3, −3, −1, 0) (14, −3, −3, 0) (13, −3, −3, 0) (12, −3, −3, 0) (25, −3, −3, 0) (29, −3, −3, 0) (28, −3, −3, 0) (26, −3, −3, 0) (27, −3, −3, 0) (24, −3, −3, 0) (23, −3, −3, 0) (36, −3, −3, 0) (35, −3, −3, 0) (34, −3, −3, 0) (33, −3, −3, 0) (37, −3, −3, 0) (39, −3, −3, 0) (38, −3, −3, 0) (40, −3, −3, 0) (41, −3, −3, 0) (7, 0, 8, 0) (14, −2, 8, 0) (13, −2, 8, 0) (21, 0, 8, 0) (19, 0, 8, 0) (29, −2, 8, 0)

Hi {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u3 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u5 } {c0 , c1 , j1 , j4 } ∪ U {j5 , u5 } {c0 , j5 , c0 , u5 } {c1 , j5 , c2 , u5 } {j0 , j5 , u0 , u5 } {c0 , c1 , j5 , c0 , c2 , u5 } {c0 , j1 , j2 , c0 , u2 , u1 } {c0 , j1 , j5 , c0 , u1 , u5 } {c0 , j2 , j5 , c0 , u2 , u5 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 , u1 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 , u4 } {c0 , c1 , j1 , j5 , c0 , c2 , u1 , u5 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , u5 } {c0 , j1 , j2 , j5 , c0 , u2 , u1 , u5 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 , u1 , u5 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u0 , u2 , u3 , u5 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 } J ∪U {c1 , j5 , c2 , c1 } {c0 , c1 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 } {c0 , c1 , c2 , c1 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 } {c0 , c1 , c2 , c1 , v3 , v2 } {c1 , j5 , c1 , v2 } {c1 , j5 , c2 , c1 , v2 } {c0 , c1 , j5 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 , v2 } {c1 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j5 , c1 , v2 } {c0 , c1 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v3 , v2 } {c0 , j1 , c2 } {c0 , c1 , j3 , c2 , c1 } {c0 , c1 , j2 , c2 , c1 } {c0 , c1 , j1 , c2 , c1 } {c0 , c1 , j1 , j4 , c2 , c1 } {c0 , c1 , j3 , j5 , c2 , c1 } {c0 , c1 , j2 , j5 , c2 , c1 } {c0 , c1 , j1 , j5 , c2 , c1 } {c0 , c1 , j2 , j3 , c2 , c1 } {c0 , c1 , j1 , j3 , c2 , c1 } {c0 , c1 , j1 , j2 , c2 , c1 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 } J ∪ {c2 , c1 } {c1 , j3 , c1 , v2 } {c0 , c1 , j3 , c1 , v2 } {c0 , c1 , j2 , c1 , v2 } {c1 , j3 , j5 , c1 , v2 } {c1 , j1 , j3 , c1 , v2 } {c0 , c1 , j3 , j5 , c1 , v2 }

Case

3.4

4.1


τ (Hi ) (28, −2, 8, 0) (27, −2, 8, 0) (24, −2, 8, 0) (31, 0, 8, 0) (36, −2, 8, 0) (35, −2, 8, 0) (33, −2, 8, 0) (39, −2, 8, 0) (38, −2, 8, 0) (40, −2, 8, 0) (41, −2, 8, 0) (14, −3, 15, 0) (13, −3, 15, 0) (29, −3, 15, 0) (28, −3, 15, 0) (27, −3, 15, 0) (24, −3, 15, 0) (36, −3, 15, 0) (35, −3, 15, 0) (33, −3, 15, 0) (39, −3, 15, 0) (38, −3, 15, 0) (40, −3, 15, 0) (41, −3, 15, 0) (27, −3, 22, 0) (36, −3, 22, 0) (33, −3, 22, 0) (39, −3, 22, 0) (38, −3, 22, 0) (40, −3, 22, 0) (41, −3, 22, 0) (−1, −3, 3, 0) (−3, −3, 14, 0) (−3, −3, 13, 0) (−3, −3, 12, 0) (−3, −3, 25, 0) (−3, −3, 29, 0) (−3, −3, 28, 0) (−3, −3, 26, 0) (−3, −3, 27, 0) (−3, −3, 24, 0) (−3, −3, 23, 0) (−3, −3, 36, 0) (−3, −3, 35, 0) (−3, −3, 34, 0) (−3, −3, 33, 0) (−3, −3, 37, 0) (−3, −3, 39, 0) (−3, −3, 38, 0) (−3, −3, 40, 0) (−3, −3, 41, 0) (8, 0, 7, 0) (8, −1, 14, 0) (8, −1, 13, 0) (8, 0, 21, 0) (8, 0, 19, 0) (8, −1, 29, 0) (8, −1, 28, 0) (8, −1, 27, 0) (8, −1, 24, 0) (8, 0, 31, 0) (8, −1, 36, 0) (8, −1, 35, 0) (8, −1, 33, 0) (8, −1, 39, 0)

Hi {c0 , c1 , j2 , j5 , c1 , v2 } {c0 , c1 , j2 , j3 , c1 , v2 } {c0 , c1 , j1 , j3 , c1 , v2 } {c1 , j1 , j3 , j5 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c1 , v2 } J ∪ {c1 , v2 } {c0 , c1 , j3 , c2 , c1 , v2 } {c0 , c1 , j2 , c2 , c1 , v2 } {c0 , c1 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j2 , j5 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 , v2 } J ∪ {c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c2 , c1 , v3 , v2 } {c0 , c1 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c2 , c1 , v3 , v2 } J ∪ {c2 , c1 , v3 , v2 } {c0 , c2 , v4 } {c0 , c1 , c2 , c1 , v0 } {c0 , c1 , c2 , c1 , v5 } {c0 , c1 , c2 , c1 , v4 } {c0 , c1 , c2 , c1 , v4 , v1 } {c0 , c1 , c2 , c1 , v0 , v2 } {c0 , c1 , c2 , c1 , v5 , v2 } {c0 , c1 , c2 , c1 , v4 , v2 } {c0 , c1 , c2 , c1 , v5 , v0 } {c0 , c1 , c2 , c1 , v4 , v0 } {c0 , c1 , c2 , c1 , v4 , v5 } {c0 , c1 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 } ∪ V {c1 , j5 , c1 , v0 } {c1 , j5 , c2 , c1 , v0 } {c1 , j5 , c2 , c1 , v5 } {c1 , j5 , c1 , v0 , v2 } {c1 , j5 , c1 , v4 , v0 } {c1 , j5 , c2 , c1 , v0 , v2 } {c1 , j5 , c2 , c1 , v5 , v2 } {c1 , j5 , c2 , c1 , v5 , v0 } {c1 , j5 , c2 , c1 , v4 , v0 } {c1 , j5 , c1 , v4 , v0 , v2 } {c1 , j5 , c2 , c1 , v5 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 , v1 }

449 Case


450 i 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 204 352 207 355 208 356 405 406 407 408 409 410 411 412 413 414 206 205 135 137 214 213 211

τ (Hi ) (8, −1, 38, 0) (8, −1, 40, 0) (8, −1, 41, 0) (15, −3, 14, 0) (15, −3, 13, 0) (15, −3, 29, 0) (15, −3, 28, 0) (15, −3, 27, 0) (15, −3, 24, 0) (15, −3, 36, 0) (15, −3, 35, 0) (15, −3, 33, 0) (15, −3, 39, 0) (15, −3, 38, 0) (15, −3, 40, 0) (15, −3, 41, 0) (22, −3, 27, 0) (22, −3, 36, 0) (22, −3, 33, 0) (22, −3, 39, 0) (22, −3, 38, 0) (22, −3, 40, 0) (22, −3, 41, 0) (1, 0, 1, 0) (6, 0, 6, 0) (3, −3, 3, 0) (10, 0, 10, 0) (12, −3, 12, 0) (21, 0, 21, 0) (20, 0, 19, 0) (19, 0, 20, 0) (25, −3, 25, 0) (29, −3, 29, 0) (28, −3, 28, 0) (26, −3, 24, 0) (24, −3, 26, 0) (31, 0, 31, 0) (35, −3, 35, 0) (37, −3, 39, 0) (39, −3, 37, 0) (38, −3, 38, 0) (41, −3, 41, 0) (−2, 8, −3, 0) (−1, −3, 3, 0) (−3, 15, −3, 0) (−3, −3, 12, 0) (−3, 22, −3, 0) (−3, −3, 25, 0) (0, 8, 3, 0) (−2, 8, 12, 0) (−1, 15, 3, 0) (−3, 15, 12, 0) (−2, 8, 25, 0) (−1, 22, 3, 0) (−3, 15, 25, 0) (−3, 22, 12, 0) (−3, 22, 25, 0) (−2, 3, −2, 0) (−3, 14, −3, 0) (−3, 13, −3, 0) (−3, 12, −3, 0) (−3, 25, −3, 0) (−3, 29, −3, 0) (−3, 28, −3, 0) (−3, 26, −3, 0)

Hi {c1 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c1 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c1 , j5 } ∪ V {c0 , c1 , j5 , c2 , c1 , v0 } {c0 , c1 , j5 , c2 , c1 , v5 } {c0 , c1 , j5 , c2 , c1 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v5 , v2 } {c0 , c1 , j5 , c2 , c1 , v5 , v0 } {c0 , c1 , j5 , c2 , c1 , v4 , v0 } {c0 , c1 , j5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j5 } ∪ V {c0 , c1 , j0 , j5 , c2 , c1 , v5 , v0 } {c0 , c1 , j0 , j5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 } ∪ V {j1 , v4 } {c1 , j1 , c1 , v4 } {c0 , j1 , c2 , v4 } {j1 , j4 , v4 , v1 } {c0 , c1 , j1 , c2 , c1 , v4 } {c1 , j3 , j5 , c1 , v0 , v2 } {c1 , j1 , j5 , c1 , v4 , v0 } {c1 , j1 , j3 , c1 , v4 , v2 } {c0 , c1 , j1 , j4 , c2 , c1 , v4 , v1 } {c0 , c1 , j3 , j5 , c2 , c1 , v0 , v2 } {c0 , c1 , j2 , j5 , c2 , c1 , v5 , v2 } {c0 , c1 , j1 , j5 , c2 , c1 , v4 , v0 } {c0 , c1 , j1 , j3 , c2 , c1 , v4 , v2 } {c1 , j1 , j3 , j5 , c1 , v4 , v0 , v2 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v4 , v0 , v2 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j1 , j2 , j3 , j4 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v5 , v0 , v2 } J ∪V {c2 , u5 , c2 , c1 } {c0 , c2 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 } {c0 , c2 , c2 , c1 , v4 } {c0 , c2 , u0 , u5 , c2 , c1 } {c0 , c2 , c2 , c1 , v4 , v1 } {c2 , u5 , c2 , v4 } {c2 , u5 , c2 , c1 , v4 } {c0 , c2 , u5 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 , v4 } {c2 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u5 , c2 , v4 } {c0 , c2 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v1 } {c2 , u2 , c1 } {c0 , c2 , u4 , c2 , c1 } {c0 , c2 , u1 , c2 , c1 } {c0 , c2 , u2 , c2 , c1 } {c0 , c2 , u2 , u3 , c2 , c1 } {c0 , c2 , u4 , u5 , c2 , c1 } {c0 , c2 , u1 , u5 , c2 , c1 } {c0 , c2 , u2 , u5 , c2 , c1 }

Case

4.2


τ (Hi ) (−3, 27, −3, 0) (−3, 24, −3, 0) (−3, 23, −3, 0) (−3, 36, −3, 0) (−3, 35, −3, 0) (−3, 34, −3, 0) (−3, 33, −3, 0) (−3, 37, −3, 0) (−3, 39, −3, 0) (−3, 38, −3, 0) (−3, 40, −3, 0) (−3, 41, −3, 0) (0, 7, 3, 0) (−1, 14, 3, 0) (−1, 13, 3, 0) (0, 21, 3, 0) (0, 19, 3, 0) (−1, 29, 3, 0) (−1, 28, 3, 0) (−1, 27, 3, 0) (−1, 24, 3, 0) (0, 31, 3, 0) (−1, 36, 3, 0) (−1, 35, 3, 0) (−1, 33, 3, 0) (−1, 39, 3, 0) (−1, 38, 3, 0) (−1, 40, 3, 0) (−1, 41, 3, 0) (−3, 14, 12, 0) (−3, 13, 12, 0) (−3, 29, 12, 0) (−3, 28, 12, 0) (−3, 27, 12, 0) (−3, 24, 12, 0) (−3, 36, 12, 0) (−3, 35, 12, 0) (−3, 33, 12, 0) (−3, 39, 12, 0) (−3, 38, 12, 0) (−3, 40, 12, 0) (−3, 41, 12, 0) (−3, 27, 25, 0) (−3, 36, 25, 0) (−3, 33, 25, 0) (−3, 39, 25, 0) (−3, 38, 25, 0) (−3, 40, 25, 0) (−3, 41, 25, 0) (−3, −1, 8, 0) (−3, −3, 13, 0) (−3, −3, 14, 0) (−3, −3, 15, 0) (−3, −3, 22, 0) (−3, −3, 23, 0) (−3, −3, 24, 0) (−3, −3, 26, 0) (−3, −3, 27, 0) (−3, −3, 28, 0) (−3, −3, 29, 0) (−3, −3, 33, 0) (−3, −3, 34, 0) (−3, −3, 35, 0) (−3, −3, 36, 0) (−3, −3, 37, 0)

Hi {c0 , c2 , u1 , u4 , c2 , c1 } {c0 , c2 , u2 , u4 , c2 , c1 } {c0 , c2 , u2 , u1 , c2 , c1 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 } {c2 , c1 , v3 , v4 , v1 , v2 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 } U ∪ {c2 , c1 } {c2 , u4 , c2 , v4 } {c0 , c2 , u4 , c2 , v4 } {c0 , c2 , u1 , c2 , v4 } {c2 , u4 , u5 , c2 , v4 } {c2 , u2 , u4 , c2 , v4 } {c0 , c2 , u4 , u5 , c2 , v4 } {c0 , c2 , u1 , u5 , c2 , v4 } {c0 , c2 , u1 , u4 , c2 , v4 } {c0 , c2 , u2 , u4 , c2 , v4 } {c2 , u2 , u4 , u5 , c2 , v4 } {c0 , c2 , u1 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , v4 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , v4 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , v4 } U ∪ {c2 , v4 } {c0 , c2 , u4 , c2 , c1 , v4 } {c0 , c2 , u1 , c2 , c1 , v4 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u1 , u5 , c2 , c1 , v4 } {c0 , c2 , u1 , u4 , c2 , c1 , v4 } {c0 , c2 , u2 , u4 , c2 , c1 , v4 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 } U ∪ {c2 , c1 , v4 } {c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 , v1 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } U ∪ {c2 , c1 , v4 , v1 } {c0 , c1 , v2 } {c0 , c2 , c2 , c1 , v5 } {c0 , c2 , c2 , c1 , v0 } {c0 , c2 , c2 , c1 , v2 } {c0 , c2 , c2 , c1 , v3 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 } {c0 , c2 , c2 , c1 , v4 , v0 } {c0 , c2 , c2 , c1 , v4 , v2 } {c0 , c2 , c2 , c1 , v5 , v0 } {c0 , c2 , c2 , c1 , v5 , v2 } {c0 , c2 , c2 , c1 , v0 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , c2 , c1 , v4 , v5 , v2 } {c0 , c2 , c2 , c1 , v4 , v0 , v2 } {c0 , c2 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , c2 , c1 , v3 , v4 , v1 , v2 }

451 Case


452 i 369 368 370 371 452 453 454 455 456 457 458 459 450 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512

τ (Hi ) (−3, −3, 38, 0) (−3, −3, 39, 0) (−3, −3, 40, 0) (−3, −3, 41, 0) (0, 8, 4, 0) (−2, 8, 13, 0) (−2, 8, 14, 0) (0, 8, 16, 0) (0, 8, 18, 0) (−2, 8, 23, 0) (−2, 8, 24, 0) (−2, 8, 27, 0) (−2, 8, 28, 0) (0, 8, 30, 0) (−2, 8, 33, 0) (−2, 8, 34, 0) (−2, 8, 36, 0) (−2, 8, 38, 0) (−2, 8, 39, 0) (−2, 8, 40, 0) (−2, 8, 41, 0) (−3, 15, 13, 0) (−3, 15, 14, 0) (−3, 15, 23, 0) (−3, 15, 24, 0) (−3, 15, 27, 0) (−3, 15, 28, 0) (−3, 15, 33, 0) (−3, 15, 34, 0) (−3, 15, 36, 0) (−3, 15, 38, 0) (−3, 15, 39, 0) (−3, 15, 40, 0) (−3, 15, 41, 0) (−3, 22, 27, 0) (−3, 22, 33, 0) (−3, 22, 36, 0) (−3, 22, 38, 0) (−3, 22, 39, 0) (−3, 22, 40, 0) (−3, 22, 41, 0) (0, 1, 2, 0) (0, 6, 5, 0) (−3, 3, 8, 0) (0, 10, 9, 0) (−3, 12, 15, 0) (0, 21, 16, 0) (0, 20, 18, 0) (0, 19, 17, 0) (−3, 25, 22, 0) (−3, 29, 23, 0) (−3, 28, 24, 0) (−3, 26, 28, 0) (−3, 24, 26, 0) (0, 31, 30, 0) (−3, 35, 34, 0) (−3, 37, 38, 0) (−3, 39, 37, 0) (−3, 38, 39, 0) (−3, 41, 41, 0) (13, 3, 8, 0) (14, 3, 8, 0) (24, 3, 8, 0) (27, 3, 8, 0) (28, 3, 8, 0)

Hi {c0 , c2 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 } ∪ V {c2 , u5 , c2 , v5 } {c2 , u5 , c2 , c1 , v5 } {c2 , u5 , c2 , c1 , v0 } {c2 , u5 , c2 , v4 , v5 } {c2 , u5 , c2 , v5 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 } {c2 , u5 , c2 , c1 , v4 , v0 } {c2 , u5 , c2 , c1 , v5 , v0 } {c2 , u5 , c2 , c1 , v5 , v2 } {c2 , u5 , c2 , v4 , v5 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c2 , u5 , c2 , c1 , v4 , v5 , v2 } {c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c2 , u5 } ∪ V {c0 , c2 , u5 , c2 , c1 , v5 } {c0 , c2 , u5 , c2 , c1 , v0 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 } {c0 , c2 , u5 , c2 , c1 , v4 , v0 } {c0 , c2 , u5 , c2 , c1 , v5 , v0 } {c0 , c2 , u5 , c2 , c1 , v5 , v2 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v2 } {c0 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 , u5 } ∪ V {c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c2 , u0 , u5 } ∪ V {u2 , v2 } {c2 , u2 , c2 , v2 } {c0 , u2 , c1 , v2 } {u2 , u3 , v3 , v2 } {c0 , c2 , u2 , c2 , c1 , v2 } {c2 , u4 , u5 , c2 , v4 , v5 } {c2 , u2 , u5 , c2 , v5 , v2 } {c2 , u2 , u4 , c2 , v4 , v2 } {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 , v5 } {c0 , c2 , u1 , u5 , c2 , c1 , v4 , v0 } {c0 , c2 , u2 , u5 , c2 , c1 , v5 , v2 } {c0 , c2 , u2 , u4 , c2 , c1 , v4 , v2 } {c2 , u2 , u4 , u5 , c2 , v4 , v5 , v2 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 , v5 , v2 } {c2 , c1 , v3 , v4 , v1 , v2 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v1 , v2 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } U ∪V {c0 , c1 , j2 , c0 , u2 , c1 , v2 } {c0 , c1 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j5 , c0 , u2 , c1 , v2 }

Case

5.1.1


τ (Hi ) (33, 3, 8, 0) (36, 3, 8, 0) (38, 3, 8, 0) (39, 3, 8, 0) (40, 3, 8, 0) (41, 3, 8, 0) (13, 12, 15, 0) (14, 12, 15, 0) (24, 12, 15, 0) (27, 12, 15, 0) (28, 12, 15, 0) (33, 12, 15, 0) (36, 12, 15, 0) (38, 12, 15, 0) (39, 12, 15, 0) (40, 12, 15, 0) (41, 12, 15, 0) (27, 25, 22, 0) (33, 25, 22, 0) (36, 25, 22, 0) (38, 25, 22, 0) (39, 25, 22, 0) (40, 25, 22, 0) (41, 25, 22, 0) (3, 13, 3, 0) (3, 14, 3, 0) (3, 24, 3, 0) (3, 27, 3, 0) (3, 28, 3, 0) (3, 33, 3, 0) (3, 36, 3, 0) (3, 38, 3, 0) (3, 39, 3, 0) (3, 40, 3, 0) (3, 41, 3, 0) (12, 13, 12, 0) (12, 14, 12, 0) (12, 24, 12, 0) (12, 27, 12, 0) (12, 28, 12, 0) (12, 33, 12, 0) (12, 36, 12, 0) (12, 38, 12, 0) (12, 39, 12, 0) (12, 40, 12, 0) (12, 41, 12, 0) (25, 27, 25, 0) (25, 33, 25, 0) (25, 36, 25, 0) (25, 38, 25, 0) (25, 39, 25, 0) (25, 40, 25, 0) (25, 41, 25, 0) (8, 8, 13, 0) (8, 8, 14, 0) (8, 8, 24, 0) (8, 8, 27, 0) (8, 8, 28, 0) (8, 8, 33, 0) (8, 8, 36, 0) (8, 8, 38, 0) (8, 8, 39, 0) (8, 8, 40, 0) (8, 8, 41, 0) (15, 15, 13, 0)

Hi {c0 , c1 , j1 , j2 , j3 , c0 , u2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , u2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , u2 , c1 , v2 } J ∪ {c0 , u2 , c1 , v2 } {c0 , c1 , j2 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , c2 , c1 , v2 } J ∪ {c0 , c2 , u2 , c2 , c1 , v2 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , c1 , j1 , j2 , j3 , j5 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } J ∪ {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 } {c0 , j1 , c0 , c2 , u1 , c2 , v4 } {c0 , j1 , c0 , c2 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , c2 , v4 } {c0 , j1 , c0 , c2 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , v4 } {c0 , j1 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , v4 } {c0 , j1 } ∪ U ∪ {c2 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 } {c0 , c1 , j1 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 } {c0 , c1 , j1 } ∪ U ∪ {c2 , c1 , v4 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 , c0 , c2 , u2 , u1 , u4 , u5 , c2 , c1 , v4 , v1 } {c0 , c1 , j1 , j4 } ∪ U ∪ {c2 , c1 , v4 , v1 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 } {c1 , j5 , c2 , u5 , c2 , c1 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c1 , j5 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c1 , j5 , c2 , u5 } ∪ V {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 }

453 Case

5.1.2

5.1.3

5.2

5.3

454 i 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 600 + t

15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3 τ (Hi ) (15, 15, 14, 0) (15, 15, 24, 0) (15, 15, 27, 0) (15, 15, 28, 0) (15, 15, 33, 0) (15, 15, 36, 0) (15, 15, 38, 0) (15, 15, 39, 0) (15, 15, 40, 0) (15, 15, 41, 0) (22, 22, 27, 0) (22, 22, 33, 0) (22, 22, 36, 0) (22, 22, 38, 0) (22, 22, 39, 0) (22, 22, 40, 0) (22, 22, 41, 0) (24, 24, 26, 0) (26, 28, 24, 0) (28, 26, 28, 0) (37, 38, 39, 0) (38, 37, 38, 0) (39, 39, 37, 0) (41, 41, 41, 0) (..., 1)

1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227

(0, 0, 0, 2) (0, −2, −1, 2) (−3 − 1, −2, 2) (−3, −3, −3, 2) (27, −1, −2, 2) (32, 0, 0, 2) (37, −1, −2, 2) (41, −1, −2, 2) (27, −3, −3, 2) (37, −3, −3, 2) (41, −3, −3, 2) (0, 1, 2, 2) (−3, 3, 8, 2) (0, 6, 5, 2) (−3, 12, 15, 2) (0, 10, 9, 2) (−3, 25, 22, 2) (0, 21, 16, 2) (−3, 29, 23, 2) (0, 31, 30, 2) (−3, 35, 34, 2) (−3, 38, 39, 2) (−3, 41, 41, 2) (27, 3, 8, 2) (27, 12, 15, 2) (27, 25, 22, 2) (37, 38, 39, 2)

1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238

(41, 12, 15, 2) (41, 3, 8, 2) (41, 25, 22, 2) (41, 41, 41, 2) (0, 0, 0, 3) (−1, −1, 0, 3) (−2, −2, −3, 3) (−3, −3, −3, 3) (−2, −2, 27, 3) (0, 0, 32, 3) (−2, −2, 37, 3)

Hi Case {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j5 , c0 , c2 , u5 } ∪ V {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v4 , v5 , v0 , v2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } ∪ V {c0 , c1 , j1 , j3 , c0 , c2 , u2 , u4 , c2 , c1 , v4 , v2 } 5.4 {c0 , c1 , j1 , j5 , c0 , c2 , u1 , u5 , c2 , c1 , v4 , v0 } {c0 , c1 , j2 , j5 , c0 , c2 , u2 , u5 , c2 , c1 , v5 , v2 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 } {c0 , c1 , j0 , j2 , j3 , j5 , c0 , c2 , u0 , u2 , u3 , u5 , c2 , c1 , v3 , v5 , v0 , v2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v1 , v2 } J ∪U ∪V Ht ∪ {s1 } 6.1 (t ∈ {1, 2, ..., 600}) {s1 , s3 } 6.2 {c2 , s1 , s3 } {c0 , c1 , s1 , s3 } {c0 , c1 , c2 , s1 , s3 } {c0 , c1 , j2 , j3 , s1 , s3 } {j0 , j1 , j4 , j5 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , s1 , s3 } J ∪ {s1 , s3 } {c0 , c1 , j2 , j3 , c2 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , c2 , s1 , s3 } J ∪ {c0 , c2 , s1 , s3 } {u2 , v2 , s1 , s3 } {c0 , c1 , u2 , v2 , s1 , s3 } {c2 , u2 , c2 , v2 , s1 , s3 } {c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } {u2 , u3 , v3 , v2 , s1 , s3 } {c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 , s1 , s3 } {c2 , u4 , u5 , c2 , v4 , v5 , s1 , s3 } {c0 , c2 , u4 , u5 , c2 , c1 , v4 , v5 , s1 , s3 } {c2 , u2 , u4 , u5 , c2 , v4 , v5 , v2 , s1 , s3 } {c0 , c2 , u2 , u4 , u5 , c2 , c1 , v4 , v5 , v2 , s1 , s3 } {c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v0 , v1 , s1 , s3 } U ∪ V ∪ {s1 , s3 } {c0 , c1 , j2 , j3 , c0 , u2 , c1 , v2 , s1 , s3 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } {c0 , c1 , j2 , j3 , c0 , c2 , u2 , u3 , c2 , c1 , v3 , v2 , s1 , s3 } {c0 , c1 , j0 , j1 , j4 , j5 , c0 , c2 , u0 , u1 , u4 , u5 , c2 , c1 , v4 , v5 , v 0 , v 1 , s 1 , s 3 } J ∪ {c0 , c2 , u2 , c2 , c1 , v2 , s1 , s3 } J ∪ {c0 , u2 , c1 , v2 , s1 , s3 } J ∪ {c0 , c2 , u2 , u3 , c2 , c1 , v2 , v3 , s1 , s3 } J ∪ U ∪ V ∪ {s1 , s3 } {s1 , s2 } 6.3 {c0 , s1 , s2 } {c2 , c1 , s1 , s2 } {c0 , c1 , c2 , s1 , s2 } {c2 , c1 , v5 , v0 , s1 , s2 } {v3 , v4 , v1 , v2 , s1 , s2 } {c2 , c1 , v3 , v4 , v1 , v2 , s1 , s2 }

15.2 The Subsemigroups of (P31 ; ) i 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258

τ (Hi ) (−2, −2, 41, 3) (−3, −1, 27, 3) (−3, −1, 37, 3) (−3, −3, 41, 3) (2, 2, 0, 3) (8, 8, −3, 3) (5, 5, 0, 3) (15, 15, −3, 3) (9, 9, 0, 3) (22, 22, −3, 3) (16, 16, 0, 3) (23, 23, −3, 3) (30, 30, 0, 3) (34, 34, −3, 3) (39, 39, −3, 3) (41, 41, −3, 3) (8, 8, 27, 3) (15, 15, 27, 3) (22, 22, 27, 3) (39, 39, 37, 3)

1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289

(15, 15, 41, 3) (8, 8, 41, 3) (22, 22, 41, 3) (41, 41, 41, 3) (0, 0, 0, 4) (−2, 0, −2, 4) (−1, −3, −1, 4) (−3, −3, −3, 4) (−1, 27, −1, 4) (0, 32, 0, 4) (−1, 37, −1, 4) (−1, 41, −1, 4) (−2, 27, −3, 4) (−2, 37, −3, 4) (−3, 41, −3, 4) (1, 0, 1, 4) (6, 0, 6, 4) (3, −3, 3, 4) (12, −3, 12, 4) (10, 0, 10, 4) (25, −3, 25, 4) (21, 0, 21, 4) (29, −3, 29, 4) (31, 0, 31, 4) (35, −3, 35, 4) (38, −3, 38, 4) (41, −3, 41, 4) (3, 27, 3, 4) (12, 27, 12, 4) (25, 27, 25, 4) (38, 37, 38, 4)

1290 1291 1292 1293 1294 1295 1296 1297 1298 1299

(12, 41, 12, 4) (3, 41, 3, 4) (25, 41, 25, 4) (41, 41, 41, 4) (0, 0, 0, 5) (−3, −3, −3, 5) (41, 41, 41, 5) (0, 0, 0, 6) (−3, −3, −3, 6) (41, 41, 41, 6)

Hi V ∪ {s1 , s2 } {c0 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c2 , c1 , v3 , v4 , v1 , v2 , s1 , s2 } {c0 , c2 } ∪ V ∪ {s1 , s2 } {j5 , u5 , s1 , s2 } {c1 , j5 , c2 , u5 , s1 , s2 } {c0 , j5 , c0 , u5 , s1 , s2 } {c0 , c1 , j5 , c0 , c2 , u5 , s1 , s2 } {j0 , j5 , u0 , u5 , s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , s1 , s2 } {c0 , j1 , j2 , c0 , u2 , u1 , s1 , s2 } {c0 , c1 , j1 , j2 , c0 , c2 , u2 , u1 , s1 , s2 } {c0 , j1 , j2 , j5 , c0 , u2 , u1 , u5 , s1 , s2 } {c0 , c1 , j1 , j2 , j5 , c0 , c2 , u2 , u1 , u5 , s1 , s2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , s1 , s2 } J ∪ U ∪ {s1 , s2 } {c1 , j5 , c2 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j5 , c0 , c2 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 , c2 , c1 , v5 , v0 , s1 , s2 } {c0 , c1 , j1 , j2 , j3 , j4 , c0 , c2 , u2 , u1 , u4 , u3 , c2 , c1 , v3 , v4 , v 1 , v 2 , s 1 , s 2 } {c0 , c1 , j5 , c0 , c2 , u5 } ∪ V ∪ {s1 , s2 } {c1 , j5 , c2 , u5 } ∪ V ∪ {s1 , s2 } {c0 , c1 , j0 , j5 , c0 , c2 , u0 , u5 } ∪ V ∪ {s1 , s2 } J ∪ U ∪ V ∪ {s1 , s2 } {s1 , s6 } {c1 , s1 , s6 } {c0 , c2 , s1 , s6 } {c0 , c1 , c2 , s1 , s6 } {c0 , c2 , u1 , u4 , s1 , s6 } {u0 , u2 , u3 , u5 , s1 , s6 } {c0 , c2 , u0 , u2 , u3 , u5 , s1 , s6 } U ∪ {s1 , s6 } {c1 , c0 , c2 , u1 , u4 , s1 , s6 } {c1 , c0 , c2 , u0 , u2 , u3 , u5 , s1 , s6 } {c0 , c1 } ∪ U ∪ {s1 , s6 } {j1 , v4 , s1 , s6 } {c1 , j1 , c1 , v4 , s1 , s6 } {c0 , j1 , c2 , v4 , s1 , s6 } {c0 , c1 , j1 , c2 , c1 , v4 , s1 , s6 } {j1 , j4 , v4 , v1 , s1 , s6 } {c0 , c1 , j1 , j4 , c2 , c1 , v4 , v1 , s1 , s6 } {c1 , j3 , j5 , c1 , v0 , v2 , s1 , s6 } {c0 , c1 , j3 , j5 , c2 , c1 , v0 , v2 , s1 , s6 } {c1 , j1 , j3 , j5 , c1 , v4 , v0 , v2 , s1 , s6 } {c0 , c1 , j1 , j3 , j5 , c2 , c1 , v4 , v0 , v2 , s1 , s6 } {c0 , c1 , j0 , j2 , j3 , j5 , c2 , c1 , v3 , v5 , v0 , v2 , s1 , s6 } J ∪ V ∪ {s1 , s6 } {c0 , j1 , c0 , c2 , u1 , u4 , c2 , v4 , s1 , s6 } {c0 , c1 , j1 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , s1 , s6 } {c0 , c1 , j1 , j4 , c0 , c2 , u1 , u4 , c2 , c1 , v4 , v1 , s1 , s6 } {c0 , c1 , j0 , j2 , j3 , j5 , co , c2 , u0 , u2 , u3 , u5 , c2 , c1 , v3 , v5 , v 0 , v 2 , s 1 , s 6 } {c0 , c1 , j1 } ∪ U ∪ {c2 , c1 , v4 , s1 , s6 } {c0 , j1 } ∪ U ∪ {c2 , v4 , s1 , s6 } {c0 , c1 , j1 , j4 } ∪ U ∪ {c2 , c1 , v4 , v1 , s1 , s6 } J ∪ U ∪ V ∪ {s1 , s6 } {s1 , s4 , s5 } {c0 , c1 , c2 , s1 , s4 , s5 } J ∪ U ∪ V ∪ {s1 , s4 , s5 } {s1 , s2 , s3 , s4 , s5 , s6 } {c0 , c1 , c2 , s1 , s2 , s3 , s4 , s5 , s6 } J ∪ U ∪ V ∪ {s1 , s2 , s3 , s4 , s5 , s6 }

455 Case

6.4

6.5 6.6

456


15.3 Classes of Quasilinear Functions of P3 that are not Subsets of [P31 ] The elements of the set L := [P31 ] ∪

n≥1 {f

n

1 ∈ P3 | ∃a ∈ E2 ∃f0 ∈ Pk1 ∃f1 , ..., fn ∈ P3,2 : f (x) = f0 (a+f1 (x1 )+f2 (x2 )+...+fn (xn ) (mod 2))},

are called quasilinear functions in this section. The set L agrees with the set L3 from Chapter 4. In Lemma 4.1, it was proven that L = P ol3 λ3 with λ3 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E3 } holds and that L is a submaximal class of P3 . L was examined already by I. A. Mal’tsev in [Mal 72] and [Mal 73b]. In [Mal 72], one also finds proof that L has exactly countable-infinite-many subclasses (see Theorem 8.1.5). As in [Lau 85] we will determine all subclasses of L with the aid of the 1299 subsemigroups of (P31 ; ). Further, we determine the order of theses subclasses. After introducing some notations in Section 15.3.1, we determine in Section 15.3.2 the subclasses of L whose functions take only values from the set {0, 1}. With the aid of an isomorphic mapping, we obtain all subclasses of L whose functions take only values from the set {0, 2}. In Section 15.3.3, all subclasses of L, whose functions take only values from {0, 1} or {0, 2}, are determined then as follows: We begin with the proof of certain (necessary and sufficient) criteria with which one can easily determine whether the union of certain classes, which are given in Section 15.3.3, is again a class. Then we determine the remaining classes, which do not fulfill the conditions of the criteria. With the aid of the subclasses of {f ∈ L | Im(f ) ⊆ {0, 1} ∨ Im(f ) ⊆ {0, 2}}, the remaining classes are easily described, as shown in Section 15.3.4. Notably, another kind of derivation of the subclasses of {f ∈ L | Im(f ) ⊆ {0, 1} ∨ Im(f ) ⊆ {0, 2}} in [Dem-M 89] can be found. In [Dem-M 89], one can also find rough drafts of lattices from such classes. 15.3.1 Some Notations We arrange to write only + instead of + (mod 2). Denote L the set of all linear functions of P2 , and let La,b := {f ∈ L | Im(f ) ⊆ {a, b}}, {a, b} ⊆ E3 , a = b. Let pra,b be a mapping from La,b onto L (⊆ P2 ), which is defined by

15.3 Classes of Quasilinear Functions of P3

457

pra,b f n = F n :⇐⇒ ∀x ∈ {a, b}n : g(f (x1 , ..., xn )) = F (g(x1 ), ..., g(xn )), a 0 g = , f n ∈ La,b , and F ∈ L. b 1

where

With the aid of an arbitrary subclass A of L (see Theorem 3.2.2) one can describe a subclass of La,b by −1 pra,b A := {f ∈ La,b | pra,b f ∈ A}.

Further, let Za,b be the notation for the set bcb P ol3 , {a, b, c} = E3 . bca For the description of certain isomorphic classes, we use the automorphisms ϕi : L −→ L, f n → s−1 i (f (si (x1 ), ..., si (xn ))), (i = 1, 2, ..., 6) from Section 15.2. Obviously, one can describes every subclass A of L in the form A = (A ∩ L0,1 ) ∪ (A ∩ L0,2 ) ∪ (A ∩ L1,2 ) ∪ (A ∩ [P31 [3]]). We determine, therefore, the subclasses of L0,1 in Section 15.3.2 and then in Section 15.3.3 the subclasses of L0,1 ∪ L0,2 that are not contained in L0,1 or L0,2 . With the aid of these results, it is easy to determine the remaining subclasses of L in Section 15.3.4. 15.3.2 Subclasses of L0,1 The following lemma results from the definition of the class L0,1 and the facts j0 = 1 + j1 + j2 , j3 = 1 + j2 , j4 = 1 + j1 , j5 = j1 + j2 immediately. Lemma 15.3.2.1 It holds: L0,1 = n≥1 {f n ∈ P3 | ∃a0 , ..., an , b1 , ..., bn ∈ E2 : n f (x) = a0 + i=1 (ai j1 (xi ) + bi j2 (xi ))}.

With the aid of this lemma, one can see that the identities −1 A= {f n ∈ L0,1 | ∃a0 , ..., an , b1 , ..., bn ∈ E2 : pr0,1 n n≥1 f (x) = a0 + i=1 (a i j1 (xi ) + bi j2 (xi )) ∧ n (pr0,1 f )(y) = a0 + i=1 ai yi ∈ A}

458


and −1 Z2,c ∩pr0,1 A=

{f n ∈ L0,1 | ∃a0 , ..., an , c ∈ 2 : E n n≥1 f (x) = a0 + i=1 (a i (j1 (xi ) + cj2 (xi ))) ∧ n (pr0,1 f )(y) = a0 + i=1 ai yi ∈ A}

hold, where c ∈ E2 and A denotes a closed subset of L. The sets {f n ∈ L0,1 | ∃a1 , ..., an ∈E2 : Bc,r := n n≥1 f (x) = c + i=1 ai j2 (xi ) ∧ (at most r of the ai are 1) } and Br := B0,r ∪ B1,r , where r ∈ {∞, 1, 2, ...}, are also subclasses of L0,1 . Obviously, Bc,∞ and B∞ do not have any basis. Therefore, they are also not finitely generated. Lemma 15.3.2.2 Let A be a subclass of L0,1 , which contains a function −1 (pr0,1 A) and i(x1 ) + j2 (x2 ) with i ∈ {j1 , j5 }. Then, it holds that A = pr0,1 A = [A ∪ {j1 (x1 ) + j2 (x2 )}] for every A with A ⊆ A and [pr0,1 A ] = pr0,1 A. −1 −1 Proof. Obviously, A ⊆ pr0,1 (pr0,1 A). Let f n ∈ pr0,1 (pr0,1 A) and f (x) = n a0 + i=1 (ai j1 (xi ) + bi j2 (xi )). We want to show that f belongs to A. Be cause n of pr0,1 f ∈ pr0,1 A there is a function f ∈ A with f (x) = a0 + (a j (x ) + b j (x )). A superposition over i(x ) + j (x ) 1 2 2 is obviously i 2 i i=1 i 1 i n the function q(x, x1 , ..., xn ) = i(x) + i=1 (bi + bi )j2 (xi ). Consequently, we −1 (pr0,1 A). have f (x) = q(f (x), x1 , ..., xn ) ∈ A and therefore A = pr0,1 Since f ∈ [A ] holds for every A with [pr0,1 A ] = pr0,1 A, the equation A = [A ∪ {j1 (x1 ) + j2 (x2 )}] also results from the one shown above.

Lemma 15.3.2.3 Let A be a subclass of L0,1 and pr0,1 A ⊆ [L1 ]. Then, −1 −1 (pr0,1 A) or Z2,0 ∩ pr0,1 (pr0,1 A) or the set Z2,1 ∩ A is either the set pr0,1 −1 pr0,1 (pr0,1 A). n Proof. For f (x) = a0 + i=1 (ai j1 (xi ) + bi j2 (xi )) denote Ch(f ) the set {(ai , bi ) | i = 1, 2, ..., n}. We distinguish the following three cases for A: Case 1: There exists an a ∈ E2 so that Ch(f ) ⊆ {(1, a), (0, 0)} for every f ∈ A holds. 01a In this case, f ∈ A preserves the relation , and we have A = 012 −1 pr0,1 (pr0,1 A) ∩ Z2,a . Case 2: A contains a function f with (0, 1) ∈ Ch(f ). W.l.o.g. let (a1 , b1 ) = (0, 1). Then, f (x1 , x2 ) := f (x1 , x2 , ..., x2 ) = a0 + j2 (x1 ) + r(x2 ), where r ∈ {c0 , j1 , j2 , j5 }. If r ∈ {j1 , j5 } then we have f (x1 , f (x2 , x2 )) = j2 (x1 ) + i(x2 ) ∈ A, i ∈ {j1 , j5 }; thus Lemma 15.3.2.3 follows from Lemma 15.3.2.2. If r ∈ {c0 , j2 } then we obtain f (x) :=


459

f (x, f (x, x)) ∈ {j2 , j3 }. Since pr0,1 A ⊆ [L1 ], an inverse image h of the function H with H(y1 , y2 , y3 ) = y1 + y2 + y3 belongs to A because of Lemma 3.2.2.1. Consequently, h(x1 , f (x2 ), f (f (x2 ))) = i(x1 ) + j2 (x2 ), i ∈ {j1 , j5 }, is a function of A. Therefore, Lemma 15.3.2.3 also follows from Lemma 15.3.2.2 for r ∈ {c0 , j2 }. Case 3: A contains a function f with {(1, 0), (1, 1)} ⊆ Ch(f ). W.l.o.g. let (a1 , b1 ) = (1, 0) and (a2 , b2 ) = (1, 1). Then the function f (x1 , x1 , x2 , ..., x2 ) fulfills the condition of Case 2. Theorem 15.3.2.4 Let A be a subclass of L0,1 with A ⊆ [L10,1 ] ∪ B∞ . Then −1 −1 −1 A ∈ {pr0,1 (pr0,1 A), Z2,0 ∩ pr0,1 (pr0,1 A), Z2,1 ∩ pr0,1 (pr0,1 A)}.

Proof. If A ⊆ [L10,1 ] ∪ B∞ , there is a function f ∈ A with pr0,1 f ∈ [P21 ]. Let w.l.o.g. n (ai j1 (xi ) + bi j2 (xi )) ∈ A, f (x) = a0 + i=1

where a1 = 1 and (a2 , b2 ) = (0, 0). The following two cases are possible: Case 1: (a2 , b2 ) = (0, 1). One can obtain the function f (x1 , x2 , x3 ) = a0 + j1 (x1 ) + b1 j2 (x1 ) + j2 (x2 ) + p(x3 ) with p ∈ {c0 , j1 , j2 , j5 } by identifying the variables x3 , ..., xn from f . Then f (f (x1 , x2 , x2 ), x1 , x2 ) = i(x1 ) + j2 (x2 ), i ∈ {j1 , j5 }. −1 Thus by Lemma 15.3.2.2, A = pr0,1 (pr0,1 A). Case 1: a2 = 1. In this case, we have pr0,1 A ⊆ [L1 ] and our theorem follows from Lemma 15.3.2.3.

Theorem 15.3.2.5 The subclasses (= ∅) of [L10,1 ] ∪ B∞ are exactly [c0 ], [c1 ], [c0 , c1 ], [Ji ], [Ja ] ∪ Br , [Jb ] ∪ B0,r ∪ B1,s , [Jc ] ∪ B0,r , [Jd ] ∪ B1,r , where {r, s} ⊂ {∞, 2, 3, ...}, 1 ≤ i ≤ 41, a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {27, 33, 36, 40}, c ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40} and d ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}. Proof. One obtains the statements of the theorem easily by means of the properties of the functions of [L10,1 ] ∪ B∞ and the results from Section 15.2 about the subsemigroups of (P31 ; ). Lemma 15.3.2.6 Let A be a subclass of L0,1 and pr0,1 A ⊆ [L1 ]. Then there is a function h3 ∈ A with h(x1 , x2 , x3 ) := i(x1 ) + i(x2 ) + i(x3 ), i ∈ {j1 , j5 }, and it holds A = [A1 ∪ {h}].

460


Proof. By Lemma 3.2.2.1 we have pr0,1 A = [(pr0,1 A)1 ∪ {pr0,1 h}]. If A = −1 −1 (pr0,1 A), a ∈ E2 , then Lemma 15.3.2.6 holds. If A = pr0,1 (pr0,1 A) Z2,a ∩ pr0,1 then j1 (x1 ) + j2 (x2 ) ∈ A. By Lemma 15.3.2.2 and the above remark, we have A = [A1 ∪ {h, j1 (x1 ) + j2 (x2 )}], where h ∈ A is an inverse image of the function H 3 with H(y1 , y2 , y3 ) = y1 + y2 + y3 . Because of {j1 , j5 } ⊆ A, we can assume h(x1 , x2 , x3 ) = j1 (x1 ) + j1 (x2 ) + j1 (x3 ), and it holds h(x1 , x2 , j5 (x2 )) = j1 (x1 ) + j1 (x2 ) ∈ [A1 ∪ {h}]; i.e., we have A = [A1 ∪ {h}] −1 (pr0,1 A). Since there are no other possibilities because in the case A = pr0,1 of Theorem 15.3.2.4, our Lemma 15.3.2.6 is valid. Theorem 15.3.2.7 Let A be a subclass of L0,1 . Then ⎧ 2, if A ⊆ [L10,1 ] ∪ B∞ ∧ (C ∩ A1 = ∅ ∨ pr0,1 A ⊆ [L1 ]), ⎪ ⎪ ⎨ 3, if pr0,1 A ⊆ [L1 ] ∧ C ∩ A1 = ∅, ord A = 1 1 ⎪ ⎪ ⎩ t, if A ⊆ [L0,1 ] ∪ B∞ ∧ A ∩ (Bt \[A ]) = ∅ ∧ 1 (∀ r ≥ t : A ∩ (Br \[A ]) = ∅). Proof. By Theorem 15.3.2.4, the following three cases are possible for A: −1 Case 1: A = Z2,a ∩ pr0,1 (pr0,1 A), a ∈ E2 . In this case, ord A = ord pr0,1 A. Then, the statements of the theorem follow from Theorem 3.1.1. −1 (pr0,1 A) and A ⊆ [L10,1 ] ∪ B∞ . Case 2: A = pr0,1 By Lemma 15.3.2.6 and 15.3.2.2, it holds in this case: ≤ 3 if pr0,1 A ⊆ [L1 ], ord pr0,1 A ≤ ord A = 2 if pr0,1 A ⊆ [L1 ]. As is generally known, only the classes of L, which do not contain any constant functions and which are not subsets of [L1 ], have the order 3. If a constant function belongs to pr0,1 A then the function h from Lemma 15.3.2.6 is obviously a superposition over binary functions of A. Thus the statements of the theorem hold in Case 2. Case 3: A ⊆ [L10,1 ∪ B∞ . In this case, the order of A follows from Theorem 15.3.2.5 and from [L10,1 ] ∪ Br ⊂ [L10,1 ] ∪ Br+1 , 2 ≤ r ≤ ∞. One can obtain the following lemma as a direct consequence from Theorem 15.3.2.7: Theorem 15.3.2.8 The only finitely generated subclasses of L0,1 are the classes [Ja ]∪B∞ , [Jb ]∪B0,∞ , [Jc ]∪B1,∞ , [Jd ]∪B0,∞ ∪B1,s and [Jd ]∪B0,s ∪B1,∞ , where a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40}, c ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}, d ∈ {27, 33, 36, 40} and 2 ≤ r, s ≤ ∞.


461

15.3.3 The Subclasses of L0,1 ∪ L0,2 That Are Not Subclasses of L0,1 or L0,2 The aim of this section is first of all the derivation of a necessary and sufficient criterion with which one can find out whether a set A1 ∪ A2 (A1 ⊆ L0,1 , A2 ⊆ L0,2 ) is closed. Put A A := {f g | f ∈ A ∧ g ∈ A }. Obviously, it holds: Lemma 15.3.3.1 Let A1 and A2 be subclasses of L with A1 ⊆ L0,1 and A2 ⊆ L0,2 . Then, A1 ∪ A2 is closed if and only if A1 A2 ⊆ A1 and A2 A1 ⊆ A2 . Lemma 15.3.3.2 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , (A1 ∪ A2 )1 a subsemigroup of (P31 ; ) and {i, j} = {1, 2}. Then (a) A11 ⊆ {c0 , c1 , j1 , j4 } =⇒ A1 A2 ⊆ A1 , (b) A12 ⊆ {c0 , c2 , u2 , u3 } =⇒ A2 A1 ⊆ A2 , (c) pr0,i Ai ⊆ [L1 ] =⇒ Ai Aj ⊆ Ai , (d) (pr0,1 A1 ⊆ [L1 ] ∧ j1 (x1 ) + j2 (x2 ) ∈ A1 ∧ A12 ⊆ {c0 , c2 , u2 , u3 }) =⇒ A1 A2 ⊆ A1 . Proof. (a): By Section 15.3.2 and by the assumptions over A1, A 1 is asub 0 0 class of Z2,0 ∩ L0,1 = [c1 , j1 (x1 ) + j1 (x2 )]. Thus, because of j1 , = 2 0 we have A1 A2 = A1 . (b): By assumption 15.3.2, A2 is a subset of ϕ2 (Z2,0 ∩L0,1 ). Thus, and Section 0 0 , we have A2 A1 ⊆ A2 . = because of u2 1 0 (c): Let w.l.o.g. pr0,1 A1 ⊆ [L1 ]. Then, because of Lemma 15.3.2.6, A1 = [A11 ∪ {h}], where h ∈ A1 is an arbitrary inverse image of the function H(y1 , y2 , y3 ) = y1 + y2 + y3 . Suppose, A1 A2 ⊆ A1 holds. Then, A1 := [A1 ∪ (A1 A2 )] is a closed set with A1 ⊃ A1 and A1 = [(A1 )1 ∪ {h}]. Consequently, we have (A1 )1 ⊃ A11 , i.e., there is a function g ∈ A1 and there are certain functions p1 , ..., pn ∈ (A1 ∪ A2 )1 with g (x) := g(p1 (x), p2 (x), ..., pn (x)) ∈ A11 . However, we have also g(x1 , ..., xn ) = g0 (g1 (x1 ), g2 (x2 ), ..., gn (xn )) for certain functions g0 ∈ [{h}] and g1 , ..., gn ∈ A11 . Therefore, we obtain g = g0 (g1 p1 , g2 p2 , ..., gn pn ). Moreover, by assumption, gi pi ∈ (A1 ∪A2 )1 , i = 1, 2, ..., n, and A1 = [A1 ]. Therefore, we have g ∈ A1 , contrary to the assumption. (d): Let pr0,1 A1 ⊆ [L1 ], j1 (x1 ) + j2 (x2 ) ∈ A1 and A12 ⊆ {c0 , c2 , u2 , u3 }. Then, by Section 15.3.2, A1 = [A11 ∪ {j1 (x1 ) + j2 (x2 )}] and A2 ⊆ ϕ2 (Z2,0 ∩ L0,1 ) = [c2 , u2 (x1 ) ⊕ u2 (x2 )] with xy x⊕y 00 0 02 2 . 20 2 22 0

462


0 0 = , j2 (a ⊕ u2 (x1 ) ⊕ u2 (x2 ) ⊕ ... ⊕ um (xm )) = 2 0 j2 (a) + j2 (x1 ) + j2 (x2 ) + ... + j2 (xm ) and (A1 ∪ A2 )1 is closed in respect to , it follows A1 A2 ⊆ A1 .

Since j1

Theorem 15.3.3.3 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , A1 ⊆ [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ), A2 ⊆ ϕ2 ([L10,1 ] ∪ B∞ ) ∪ (Z2,0 ∩ L0,1 ) and A11 ⊆ {c0 , c1 , j1 , j4 } or A12 ⊆ {c0 , c1 , u2 , u3 }. Then, A1 ∪ A2 is closed if and only if (A1 ∪ A2 )1 is a subsemigroup of (P31 ; ). Proof. If A1 ∪ A2 is closed, then (A1 ∪ A2 )1 is obviously a subsemigroup of (P31 ; ). Let (A1 ∪ A2 )1 be a closed set in respect to . Then the following three cases are possible: Case 1: (A1 ∪ A2 )1 ⊆ {c0 , c1 , c2 , j1 , j4 , u2 , u3 }. In this case, by Lemma 15.3.3.2, (a), (b), and Lemma 15.3.3.1, A1 ∪ A2 is closed. Case 2: A11 ⊆ {c0 , c1 , j1 , j4 } and A12 ⊆ {c0 , c2 , u2 , u3 }. Because of Lemma 15.3.3.2, (b) it holds that A2 A1 ⊆ A2 . Since A1 is not a subset of [L10,1 ] ∪ B∞ , we have pr0,1 A1 ⊆ [L1 ] or pr0,1 A1 ⊆ [L1 ] and j1 (x1 ) + j2 (x2 ) ∈ A1 . With the aid of Lemma 15.3.3.2, (c), (d): A1 A2 ⊆ A1 . Thus, because of Lemma 15.3.3.1, A1 A2 is closed. Case 3: (A11 ⊆ {c0 , c1 , j1 , j4 } and (A12 ⊆ {c0 , c2 , u2 , u3 }. Because of ϕ2 ({c0 , c1 , j1 , j4 }) = {c0 , c2 , u2 , u3 } the classes that fulfill the conditions of the third case are isomorphic to such classes that fulfill the conditions of the second case. Therefore, Theorem 15.3.3.3 is also valid in the third case. Theorem 15.3.3.4 Let A1 and A2 be subclasses of L, A1 ⊆ L0,1 , A2 ⊆ L0,2 , A1 ∪ A2 ⊆ [L1 ], A11 ⊆ {c0 , c1 , j1 , j4 } and A12 ⊆ {c0 , c2 , u2 , u3 }. Then, A1 ∪ A2 is closed if and only if pr0,1 A1 ⊆ [L1 ] and pr0,2 A2 ⊆ [L1 ] hold and (A1 ∪ A2 )1 is closed in respect to . Proof. If pr0,i Ai ⊆ [L1 ] for i = 1, 2 and (A1 ∪ A2 )1 is a subsemigroup of (P31 ; ), then Theorem 15.3.3.2, (c) and Lemma 15.3.3.1 imply that A1 ∪ A2 is closed. } Let A1 ∪ A2 be a subclass of L. Since, by assumption, A11 ⊆ {c0 , c1 ,j1 , j4 0 0 1 and A12 ⊆ {c0 , c2 , u2 , u3 }, there is a function p1 ∈ A11 with p ∈ 2 10 0 0 2 and a function q ∈ A12 with p ∈ . Because A1 ∪ A2 is closed, we 1 20 have p A2 ⊆ A1 and q A1 ⊆ A2 . Obviously, from this and the assumption A1 ∪ A2 ⊆ [L1 ], it follows that A1 ⊆ [L1 ] and A2 ⊆ [L1 ]. Suppose pr0,1 A1 ⊆


463

[L1 ]. Then, by Section 15.3.2, a function t(x1 , x2 ) = g(x1 )+j2 (x2 ) with g ∈ A11 belongs to A1 . Therefore the function if g ∈ {j0 , j1 , j4 , j5 }, t(x1 , q(x2 )) t (x1 , x2 ) := t(q(x1 ), q(x2 )) if g ∈ {j2 , j3 }, belongs to A1 and it holds pr0,1 t ∈ [L1 ] obviously, contrary to the assumption. Consequently, pr0,1 A1 ⊆ [L1 ]. Similarly, one can show that pr0,2 A2 ⊆ [L1 ]. To complete the description of all subclasses of L0,1 ∪ L0,2 , which are not subclasses of L0,1 , L0,2 or [L1 ], we are only still missing the subclasses of [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ) and of ϕ2 (L10,1 ∪ B∞ ) ∪ (Z2,0 ∩ L0,1 ) because of Theorems 15.3.3.3 and 15.3.3.4. Since the two sets to be examined are isomorphic, determining only the subclasses of one suffices, as in the following theorem. Theorem 15.3.3.5 The subclasses of [L10,1 ] ∪ B∞ ∪ ϕ2 (Z2,0 ∩ L0,1 ), which are not subclasses of [L1 ], L0,1 or L0,2 , are the following classes: A ∪ A , where A ∈ {[c0 , c1 ], [Ja ] ∪ B∞ , [Jb ] | a ∈ {27, 33, 36, 38, 39, 40, 41}, b ∈ {3, 12, 25} } and A ∈ { ϕ2 (Z2,0 ∩ L0,1 ), ϕ2 ([c0 , j1 (x1 ) + j1 (x2 )]) }, [Jc ] ∪ B0,∞ ∪ ϕ([c0 , j1 (x1 ) + j1 (x2 )]), c ∈ {4, 13, 16, 18, 23, 28, 30, 34}, [Jd ] ∪ B1,∞ ∪ ϕ2 ([c0 , j1 (x1 ) + j1 (x2 )]), d ∈ {14, 24}, and [A1 ] ∪ Br , [Je ] ∪ B0,r ∪ [c0 , u2 ], [Jf ] ∪ B1,r ∪ [c0 , u2 ], [Jg ] ∪ Bα,r ∪ Bβ,r−1 ∪ [A2 ], [Jg ] ∪ B0,r ∪ B1,s ∪ [c0 , u2 ], where e ∈ {4, 13, 16, 18, 23, 27, 28, 30, 33, 34, 36, 40}, f ∈ {7, 14, 19, 21, 24, 27, 29, 31, 33, 35, 36, 40}, g ∈ {27, 33, 36, 40}, 2 ≤ r, s ≤ ∞, {α, β} = E2 , A1 is a subsemigroup of J ∪ U25 with (A1 \{c0 }) ∩ U25 = ∅, which contains {j2 , j3 }, and A2 ∈ { [c2 ], [c0 , c2 , u2 ] }. Proof. The theorem follows from Sections 15.2 and 15.3.2. 15.3.4 The Remaining Subclasses of L Obviously, Lemma 15.3.3.1 implies the following Lemma 15.3.4.1 Let A1 , A2 and A3 be subclasses of L with A1 ⊆ L0,1 , A2 ⊆ L0,2 and A3 ⊆ L1,2 . Then A1 ∪ A2 ∪ A3 is closed if and only if A1 ∪ A2 , A1 ∪ A3 and A2 ∪ A3 are closed sets and (A1 ∪ A2 ∪ A3 )1 is a semigroup. Since the sets L0,1 ∪L0,2 , L0,1 ∪L1,2 , L0,2 ∪L1,2 are isomorphic, one has a complete description of the subclasses of L0,1 ∪L0,2 ∪L1,2 through Lemma 15.3.4.1


464

and Sections 15.3.2 and 15.3.3. The following theorem now gives information about the missing subclasses of L. Theorem 15.3.4.2 The subclasses of L, which are not subsets of [L]1 and which contain at least a permutation, are A1 ∪ [s1 ] (A1 is a subclass of L0,1 ∪ L0,2 ∪ L1,2 ), A2 ∪[s1 , si ] (i ∈ {2, 3, 6}, A2 is a subclass of L0,1 ∪L0,2 ∪L1,2 with si A2 ⊆ A2 , A12 ∪ {s1 , si } is a subsemigroup of (P31 ; ), L0,1 ∪ L0,2 ∪ L1,2 ∪ [s1 , s4 , s5 ] and L. Proof. The theorem follows from Section 15.2 and from the properties of the functions of L.

15.4 The Subclasses of [O 1 ∪ {max}] 15.4.1 Some Descriptions of the Class M Some functions of P3 , which are used in Sections 15.4.2–15.4.6, are defined in the following two tables (see also Table 15.1). Table 15.12

Table 15.11 x 0 1 2

j2 0 0 1

j5 0 1 1

u2 0 0 2

u5 0 2 2

v2 1 1 2

v5 1 2 2

s1 0 1 2

x 0 0 0 1 1 1 2 2 2

y x ∨ y := max(x, y) 0 0 1 1 2 2 0 1 1 1 2 2 0 2 1 2 2 2

Our object of investigation is the set M := [{c0 , c1 , c2 , j2 , j5 , u2 , u5 , v2 , v5 , max}], which, by [Mac 79] (see Theorem 14.1.4), is a maximal class of the class 012001 O := P ol3 . 012122 M can also be described in the form [O1 ∪ {max}] or

n≥1

{f n ∈ P3 | ∃f1 , ..., fn ∈ O1 : f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (xn )}.

15.4 The Subclasses of [O1 ∪ {max}]

465

15.4.2 Some Lemmas and a Rough Partition of the Subclasses of M One checks the following lemma easily: Lemma 15.4.2.1 (a) For every function f n ∈ M there is exactly a representation of the following form: f (x1 , ..., xn ) = f1 (x1 ) ∨ ... ∨ fn (xn )

(15.1)

fi (x) := f (0, ..., 0, x, 0, ..., 0)

(15.2)

with i−1

(i = 1, ..., n). (b) If the function f is given in the form (15.1) with (15.2), for every g ∈ M 1 it holds: (g f )(x1 , ..., xn ) = (g f1 )(x1 ) ∨ ... ∨ (g fn )(xn ). We agree to represent functions f n of M in the form (15.1) in which the functions fi are given by (15.2). Further, denote F (f ) the set of all functions fi of (15.2) that describe the function f . Let numf (fi ) be the number of occurrence of function fi in (15.1) for f , where fi is defined by (15.2). If f arises from the context, instead of F (f ), we will write only F , and instead of numf (fi ), we will write only num(fi ). Lemma 15.4.2.2 is the basis for the following theorems about bases and generating systems for subclasses of M . Lemma 15.4.2.2 Let f n ∈ M , n ≥ 2, f (x1 , ..., xn ) := f1 (x1 ) ∨ ... ∨ fn (xn ),

(15.3)

F := {f1 , ..., fn }, {c0 , c1 , c2 } ∩ F = ∅ (i.e., all variables of f are essential) and denote num(g) the number of functions g ∈ F that occur in (15.3). Then (a) f ∈ [[{f }]2 ] if and only if the function f fulfills at least one of the following conditions: 1) s1 ∈ F and u5 ∈ F ; 2) num(s1 ) = 1 and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}; 3) F ⊆ {u2 , u5 }; 4) F ∈ {{j5 }, {j2 , u5 }, {j5 , u2 }, {j5 , u5 }, {j2 , j5 , u2 }, {j2 , j5 , u5 }, {j2 , u2 , u5 }, {j5 , u2 , u5 }, {j2 , j5 , u2 , u5 }}; 5) num(j5 ) = 1 and F = {j2 , j5 }; 6) num(v2 ) = 1 and F = {v2 , v5 }; 7) num(u2 ) ≥ 2 and F = {j2 , u2 }; 8) n = 2 and F ∈ {{j2 }, {v5 }, {j2 , u2 }}.

466


(b) f ∈ [[{f }]3 ] and f ∈ [[{f }]2 ] if and only if the function f fulfills at least one of the following conditions: 1) num(s1 ) ≥ 2 and u5 ∈ F ; 2) num(s1 ) = 1, u5 ∈ F and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}; 3) num(j5 ) ≥ 2 and F = {j2 , j5 }; 4) num(u2 ) = 1 and F = {j2 , u2 }; 5) num(v2 ) ≥ 2 and F = {v2 , v5 }; 6) n = 3 and F ∈ {{j2 }, {v5 }}. (c) f ∈ [[{f }]n−1 ] and n ≥ 4 if and only if F ∈ {{j2 }, {v5 }, {j2 , u2 }} and num(u2 ) ≤ 1. Proof. Since, by assumption, all variables of f are essential, we have F ⊆ {j2 , j5 , u2 , u5 , v2 , v5 }. If {v2 , v5 } ∩ F = ∅ then F ⊆ {v2 , v5 }. Thus for f the following cases are possible: Case 1: s1 ∈ F . Case 1.1: u5 ∈ F . Let w.l.o.g. f1 = s1 . The functions ht (x, y) := f (x, ..., x, y, x, ..., x) = x ∨ ft (y) t−1

are superpositions over f for every t ∈ {2, 3, ..., n}. Then one can obtain function f as a superposition over these functions as follows: (...((hn hn−1 ) hn−2 )... h3 ) h2 . Therefore, f ∈ [[{f }]2 ] in Case 1. Case 1.2: u5 ∈ F . Let w.l.o.g. f1 = s1 and f2 = u5 . Then the ternary functions x ∨ u5 (y) ∨ fi (z) (i = 3, ..., n) are superpositions over f , and these functions form a generating system for f . Therefore, f ∈ [[{f }]3 ]. This is reducible to f ∈ [[{f }]2 ] iff num(s1 ) = 1 and F ∈ {{s1 , u5 }, {s1 , u2 , u5 }}. Case 2: F ⊆ {j2 , j5 , u2 , u5 }. Case 2.1: F = {j2 }. Because of j2 j2 = c0 , we have j2 (x1 ) ∨ ... ∨ j2 (xr ) ∈ [{j2 (x1 ) ∨ ... ∨ j2 (xr−1 )}] for all r ≥ 2. Case 2.2: F = {j5 }. In this case, we have f ∈ [{j5 (x) ∨ j5 (x2 )}], i.e., f ∈ [[{f }]2 ]. Case 2.3: F = {j2 , j5 }. By (...((g g) g)...) g, where g(x, y) := j5 (x) ∨ j2 (y), one can obtain all functions of the form j5 (x1 ) ∨ j2 (x2 ) ∨ ... ∨ j2 (xt ) for t ≥ 2. Thus f ∈ [[{f }]2 ], if num(j5 ) = 1. If num(j5 ) ≥ 2 then j5 (x) ∨ j5 (y) ∨ j2 (z) belongs to [{f }]3 and we have f ∈ [[{f }]3 ] and f ∈ [[{f }]2 ]. Case 2.4: F ⊆ {u2 , u5 }.


467

Since up uq = uq for all p, q ∈ {2, 5}, it follows from Lemma 15.4.2.1, (b) that f ∈ [[{f }]2 ]. Case 2.5: F = {j2 , u2 }. Because of j2 u2 = j2 , u2 j2 = c0 , u2 u2 = u2 and u2 (x) ∨ j2 (x) = u2 we have in this case: num(u2 ) ≥ 2 =⇒ {j2 (x) ∨ u2 (y), u2 (x) ∨ u2 (y)} ⊆ [{f }]2 =⇒ f ∈ [[{f }]2 ] and num(u2 ) = 1 =⇒ f ∈ [[{f }]n−1 ]. Case 2.6: F = {j2 , u5 }. Since ∆n−1 f = u5 , the functions u5 and j2 (u5 (x)) ∨ u5 (y) = j5 (x) ∨ u5 (y) belong to [{f }]2 . Thus by j5 u5 = j5 and u5 j5 = u5 , functions of the form j5 (x1 ) ∨ ... ∨ j5 (xr ) ∨ u5 (xr+1 ) ∨ ... ∨ u5 (xm ) are superpositions over [{f }]2 for arbitrary m > r ≥ 1. From this and by j5 (j2 (x1 ) ∨ u5 (x2 )) ∨ u5 (x2 ) = j2 (x1 ) ∨ u5 (x2 ) ) follows then that f ∈ [[{f }]2 ]. Case 2.7: F = {j5 , u2 }. If num(u2 ) = 1 or num(j5 ) ≥ 2, we have n = 2 or h(x, y) := j5 (x) ∨ j5 (y) ∨ u2 (z) ∈ [{f }]3 . Thus, h(x, y, y) = j5 (x) ∨ y ∈ [{f }]2 . Then functions of the form x1 ∨j5 (x2 )∨...∨j5 (xt ) (t ≥ 2) are superpositions over h. Replacing x1 by j5 (x1 ) ∨ u2 (x2 ) ∈ [{f }]2 and then identifying variables shows that f ∈ [[{f }]2 ] in the case num(u2 ) = 1. If we have num(u2 ) ≥ 2 and num(j5 ) = 1, then the function j5 (x) ∨ u2 (y) ∨ u2 (z) belongs to [{f }]3 and, therefore, the function x ∨ u2 (y) also belongs to [{f }]3 . From this, f ∈ [[{f }]2 ]. Case 2.8: F = {j5 , u5 }. Because of j5 (j5 (x1 )∨u5 (x2 ))∨u5 (j5 (x3 )∨u5 (x4 )) = j5 (x1 )∨j5 (x2 )∨u5 (x3 )∨ u5 (x4 ) we have f ∈ [[{f }]2 ]. Case 2.9: F = {j2 , j5 , u2 }. In this case, j2 (x) ∨ y and j5 (x) ∨ u2 (y) are some superpositions over f . Thus f ∈ [[{f }]2 ] is an easy conclusion from our considerations of Cases 1 and 2.7. Case 2.10: F = {j2 , j5 , u5 }. Some superpositions over f are u5 = ∆n−1 f , j2 (x) ∨ u5 (y) and j5 (x) ∨ u5 (y). Hence, and by Case 2.8, we have f ∈ [[{f }]2 ]. Case 2.11: F = {j2 , u2 , u5 }. Then, the functions u5 = ∆n−1 f , u2 (x) ∨ u5 (y), j2 (x) ∨ u5 (y) and j2 (u5 (x)) ∨ u5 (y) = j5 (x) ∨ u5 (y) are superpositions over f . Using considerations from Cases 2.4, 2.6, and 2.8, we obtain f ∈ [[{f }]2 ]. Case 2.12: F = {j5 , u2 , u5 }. Then the functions x∨u5 (y), u2 (x)∨u5 (y) and j5 (x)∨u5 (y) are superpositions over f . By Cases 2.4 and 2.8, we get f ∈ [[{f }]2 ]. Case 2.13: F = {j2 , j5 , u2 , u5 }. Since j2 (x) ∨ u5 (y), j5 (x) ∨ u5 (y), u2 (x) ∨ u5 (y) ∈ [{f }]2 , one can obtain f ∈ [[{f }]2 ] by proceeding to the above cases analogously. Case 3: F ⊆ {v2 , v5 }.

468


Case 3.1: F = {v2 }. Obviously, f ∈ [{v2 (x) ∨ v2 (y)}], i.e., f ∈ [[{f }]2 ]. Case 3.2: F = {v5 }. Because of v5 v5 = c2 we have f ∈ [[{f }]n−1 ]. Case 3.3: F = {v2 , v5 }. This case resembles Case 2.3. Thus we have: num(v2 ) = 1 =⇒ f ∈ [[{f }]2 ] and

num(v2 ) ≥ 2 =⇒ f ∈ [[{f }]3 ] ∧ f ∈ [[{f }]2 ].

Next, we want to consider a rough partition of the lattice L3 (M ). Let R := {f ∈ M | |Im(f )| ≤ 2}. Then every subclass T of M has the form (T ∩ R) ∪ (T ∩ P ol3

0 ), 2

(15.4)

since all functions of M with |Im(f )| ≥ 3 have the property f (0, ..., 0) = 0 and f (2, ..., 2) = 2. Thus one can obtain all subclasses of M if every subclass T that fulfills exactly one of the following conditions (I)–(IV ) is determined: (I) T ⊆ [M 1 ]; (II) T ⊆ R and T ⊆ [M1 ]; 0 , T ⊆ R and T ⊆ [M 1 ]; (III) T ⊆ M ∩ P ol3 2 0 (IV ) ∃T1 ∈ L3 (R)\{∅} ∃T2 ∈ L3 (M ∩ P ol3 )\L3 (R) : 2 1 T = T1 ∪ T2 ∧ T ⊆ [M ]. We determine the subclasses of M in Sections 15.4.3–15.4.6, in compliance with the above partition (I)–(IV) of the subclasses. First, we determine the maximal classes of M . Theorem 15.4.2.3 M has exactly 8 maximal classes. These classes are: (1) M ∩ P ol{0} = {f ∈ M | F (f ) ⊆ {c0 , j2 , j5 , u2 , u5 , s1 }}; (2) M ∩ P ol{2} = {f ∈ M | F (f ) ∩ {c2 , u2 , u5 , v2 , v5 , s1 } = ∅}; (3) M ∩ P ol{1, 2} = {f ∈ M | F (f ) ∩ {c1 , c2 , j5 , u5 , v2 , v5 , s1 } = ∅};

(4) M ∩ P ol

0120 0121


469

= {f ∈ M | F (f ) ⊆ {c0 , c1 , j2 , j5 , u2 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 }}; 0121 (5) M ∩ P ol 0122 = {f ∈ M |F (f ) ⊆ {c0 , c1 , j5 , u5 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 , v5 }}; 01201 (6) M ∩ P ol 01222 = {f ∈ M |F (f ) ⊆ {c0 , c2 , u2 , u5 , s1 } ∨ F (f ) ⊆ {c1 , v2 , v5 }}; 01201 (7) M ∩ P ol 01212 = {f ∈ M | F (f ) ⊆ {c0 , c1 , j2 , j5 , s1 } ∨ F (f ) ⊆ {c1 , c2 , v2 , v5 }}; (8) [{s1 }] ∪ R := [{s1 }] ∪ {f ∈ M | |Im(f )| ≤ 2}. Proof. Since M = [M 1 ∪ {max}] and max ∈ T for all classes T , which are defined by (1)–(7), it is easy to prove that the classes (1)–(7) are M -maximal. One can show the M -maximality of [{s1 }] ∪ R as follows: Let f ∈ M \([{s1 }] ∪ R). Since c0 ∈ R, one of the following 9 functions is a superposition over R ∪ {f }: x ∨ y, x ∨ g(y) (g ∈ {j2 , j5 , u2 , u5 }), h1 (x) ∨ h2 (y) (h1 ∈ {j2 , j5 }, h2 ∈ {u2 , u3 }). Through substitution of x, y by certain functions of {j2 , j5 , u2 , u5 }, one can reduce these 9 cases to the case: t(x, y) := j5 (x)∨u2 (y) ∈ [R∪{f }]. Because of j5 (t(x, u5 (y)) = j5 (x)∨j5 (y) and u5 (j2 (x)∨j2 (y)) = u2 (x)∨u2 (y) it holds x∨y = t(j5 (x)∨j5 (y), u2 (x)∨u2 (y)). Hence [R ∪ {f }] = M and [{s1 }] ∪ R is M -maximal. We still have to show that the class M does not have any further maximal classes. Suppose T ⊂ M is an M -maximal class that is different from the 8 M -maximal classes listed above. Then, there is for every i ∈ {1, 2, ..., 8}, a certain function fi ∈ T that does not belong to the class (i). Consequently, it holds f2 ∈ {c0 , c1 , j2 , j5 } for f2 (x) := f2 (x, x, ..., x), and f3 (a1 , a2 , ..., an ) = 0 for certain a1 , a2 , ..., an ∈ {1, 2}. Hence, c0 ∈ T , since j2 j2 = c0 and f3 (g1 (x), ..., gn (x)) = c0 , if f2 ∈ {c1 , j5 } and x if ai = 2, gi (x) := f2 (x) if ai = 1 (i = 1, 2, ..., n). Some unary functions ft with f4 ∈ {u5 , v5 }, f5 ∈ {j2 , u2 }, f6 ∈ {j2 , j5 } and f7 ∈ {u2 , u5 } are superpositions over {ft , c0 } (t = 4, 5, 6, 7). It is easy to check that {j2 , j5 , u2 , u5 } ⊂ [{f4 , f5 , f6 , f7 }] holds. In the above proof of the M -maximality of [{s1 }] ∪ R, we have shown already that max ∈ [{j2 , j5 , u2 , u5 , f8 }] holds. Since in addition ui ∨ c1 = vi (i = 2, 5), v2 (c0 ) = c1 and v5 (c1 ) = c2 , we have M 1 ∪{max} ⊆ T . Consequently, T = M , in contradiction to T ⊂ M .

470


15.4.3 The Subclasses of [M 1 ] Since all subclasses of [P31 ] were already determined in Section 15.2, one obtains the following theorem as a consequence of Theorem 15.2.1: Theorem 15.4.3.1 [M 1 ] has exactly 190 pairwise distinct subclasses Hi . These classes Hi are defined in Table 15.13 through Hi1 for i ∈ {1, 2, ..., 95}, and through H95+i := Hi ∪ [{s1 }] for i = 1, 2, ..., 95.

i 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 7O 73 76 79 82 85 88 91 94

Hi1 ∅ {c2 } {c1 , c2 } {j2 , c0 } {j2 , c0 , c1 } {j2 , j5 , c0 , c1 } {u2 , c0 } {u5 , c2 } {u5 , c0 , c2 } {u2 , u5 , c0 , c2 } {v2 , c2 } {v2 , c1 , c2 } {j5 , c1 , c2 } {j2 , j5 , c0 , c1 , c2 } {u2 , c0 , c1 , c2 } {v2 , c0 , c1 } {v2 , v5 , c0 , c1 , c2 } {j5 , u5 , c0 } {j5 , u5 , c0 , c1 , c2 } {j2 , j5 , u2 , c0 , c1 } {j2 , j5 , u2 , u5 , c0 , c1 , c2 } {j5 , v2 , c1 } {j5 , v2 , c0 , c1 , c2 } {j2 , j5 , v2 , c0 , c1 } {j5 , v2 , v5 , c0 , c1 , c2 } {u2 , v2 , c0 , c1 } {u5 , v5 , c2 } {u5 , v2 , v5 , c2 } {u2 , u5 , v2 , v5 , c2 } {j2 , u2 , v2 , c0 , c1 , c2 } {j2 , j5 , u2 , v2 , c0 , c1 } {j5 , u5 , v2 , v5 , c0 , c1 , c2 }

i 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95

Table 15.13 Hi1 {c0 } {c0 , c1 } {c0 , c1 , c2 } {j5 , c0 } {j5 , c0 , c1 } {u2 } {u5 , c0 } {u2 , u5 } {u2 , u5 , c0 } {v2 } {v2 , c1 } {v2 , v5 , c2 } {j2 , c0 , c1 , c2 } {u2 , c0 , c1 } {u5 , c0 , c1 , c2 } {v5 , c0 , c1 , c2 } {j5 , u5 } {j2 , u2 , c0 , c1 } {j2 , u2 , c0 , c1 , c2 } {j2 , j5 , u2 , c0 , c1 , c2 } {j2 , v2 , c0 , c1 } {j5 , v2 , c0 , c1 } {j5 , v5 , c1 , c2 } {j2 , j5 , v2 , c0 , c1 , c2 } {j2 , j5 , v2 , v5 , c0 , c1 , c2 } {u2 , v2 , c2 } {u5 , v5 , c1 , c2 } {u5 , v2 , v5 , c1 , c2 } {u2 , u5 , v2 , v5 , c0 , c1 , c2 } {j5 , u5 , v5 , c1 , c2 } {j2 , j5 , u2 , v2 , c0 , c1 , c2 } {j2 , j5 , u2 , u5 , v2 , v5 } ∪ H81

i 3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93

Hi1 {c1 } {c0 , c2 } {j5 } {j5 , c1 } {j2 , j5 , c0 } {u5 } {u2 , c2 } {u2 , c0 , c2 } {u2 , u5 , c2 } {v5 , c2 } {v5 , c1 , c2 } {v2 , v5 , c1 , c2 } {j5 , c0 , c1 , c2 } {u5 , c1 , c2 } {u2 , u5 , c0 , c1 , c2 } {v2 , c0 , c1 , c2 } {j2 , u2 , c0 } {j5 , u5 , c1 , c2 } {j2 , j5 , u2 , c0 } {j2 , j5 , u2 , u5 , c0 } {j2 , v2 , c0 , c1 , c2 } {j5 , v2 , c1 , c2 } {j5 , v5 , c0 , c1 , c2 } {j5 , v2 , v5 , c1 , c2 } {u2 , v2 } {u2 , v2 , c0 , c1 , c2 } {u5 , v5 , c0 , c1 , c2 } {u5 , v2 , v5 , c0 , c1 , c2 } {j2 , u2 , v2 , c0 , c1 } {j5 , u5 , v5 , c0 , c1 , c2 } {j5 , u5 , v2 , v5 , c1 , c2 }


471

15.4.4 The Subclasses of R

Theorem 15.4.4.1 The class J1 := {f ∈ M | Im(f ) ⊆ {0, 1}} has exactly the following (countable infinite-many) subclasses, which are not subclasses of [M 1 ]: J1 , J2 := J1 ∩ P ol{0} = {f ∈ J1 | f ∈ [{c1 }]}, J3 := J1 ∩ P ol{1} = {f ∈ J1 | f ∈ [{c0 }]}, J4 := J1 ∩ P ol{0} ∩ P ol{1}, J5 := {f ∈ J1 | numf (j5 ) ≤ 1}, J6 := J2 ∩ J5 , J7 := J3 ∩ J5 , J8 := J4 ∩ J5 , J9 := {f ∈ J1 | f ∈ [{j5 }] ∨ numf (j5 ) = 0}, J10 := {f ∈ J9 | f ∈ [{c1 }]}, J11 := {f ∈ J9 | f ∈ [{j5 }]}, J12 := {f ∈ J9 | f ∈ [{c1 , j5 }], J9,r := {f ∈ J9 | numf (j2 ) ≤ r}, J10,r := J10 ∩ J9,r , J11,r := J11 ∩ J9,r , J12,r := J12 ∩ J9,r , J13 := {f ∈ J1 | numf (j2 ) = 0}, J14 := J13 ∩ J2 , J15 := J13 ∩ J3 , J16 := J13 ∩ J4 , where r = 1, 2, 3, 4, .... . Proof. With the aid of the Hasse diagram of the above classes (see Figure 15.1) and the generating systems of these classes (see Table 15.14), one can prove the theorem.

472


Jr1 J r3

J2 r

Jr5

J4 r

J6

r

rJ7 r J8 rJ9

J10 r

rJ11 J12 r J9,r r

J10,r r

rJ11,r r J12,r

J13 r J14 r

rH16 rJ15

r J16

H15 r H11

rH14 rH13

r

rH10 rH5 r H12

H9

r

r H2 r ∅ = H1

Fig. 15.1. The subclasses of J1

rH3


473

Table 15.14 A J1 J2 J3 J4 J5 J6 J7 J8 J9 J10 J11 J12 J9,r J10,r J11,r J12,r J13 J14 J15 J16

generating system (or basis, if it exists) for A {j5 (x) ∨ j5 (y), j2 , c1 } {j5 (x) ∨ j5 (y), j2 } {j5 (x) ∨ j5 (y), j2 (x) ∨ j5 (y), c1 } {j5 (x) ∨ j5 (y), j2 (x) ∨ j5 (y)} {j5 (x) ∨ j2 (y), c0 , c1 } {j5 (x) ∨ j2 (y), c0 } {j5 (x) ∨ j2 (y), c1 } {j5 (x) ∨ j2 (y)} {j5 , c1 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j5 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {c1 , j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j2 (x1 ) ∨ ... ∨ j2 (xr ) | r ∈ N} {j5 , c1 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {j5 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {c1 , j2 (x1 ) ∨ ... ∨ j2 (xr )} {j2 (x1 ) ∨ ... ∨ j2 (xr )} {j5 (x) ∨ j5 (y), c0 , c1 } {j5 (x) ∨ j5 (y), c0 } {j5 (x) ∨ j5 (y), c1 } {j5 (x) ∨ j5 (y)}

A1 {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j5 , c1 } {j5 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j5 , c1 } {j5 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j2 , c0 , c1 } {j2 , c0 } {j2 , j5 , c0 , c1 } {j2 , j5 , c0 } {j2 , c0 , c1 } {j2 , c0 } {j5 , c0 , c1 } {j5 , c0 } {j5 , c1 } {j5 }

In analog mode, the following two theorems can be proven when one uses Figures 15.2 and 15.3 and Tables 15.15 and 15.16. Theorem 15.4.4.2 The class U1 := {f ∈ M | Im(f ) ⊆ {0, 2}} has exactly the following 14 pairwise different subclasses, which are not subclasses of [M 1 ]: U1 , U2 := U1 ∩ P ol{0} = {f ∈ U1 | f ∈ [{c2 }]}, U3 := {f ∈ U1 | F (f ) ⊆ {c0 , c2 , u2 }}, U4 := {f ∈ U1 | F (f ) ⊆ {c0 , c2 , u5 }}, U5 := U1 ∩ P ol{2} = {f ∈ U1 | f ∈ [{c0 }]}, U6 := {f ∈ U3 | f ∈ [{c2 }]}, U7 := {f ∈ U4 | f ∈ [{c2 }]}, U8 := {f ∈ U3 | f ∈ [{c0 }]}, U9 := {f ∈ U1 | f ∈ [{c2 }] ∨ numf (u5 ) ≥ 1}, U10 := {f ∈ U4 | f ∈ [{c0 }]}, U11 := {f ∈ U1 | f ∈ [{c0 , c2 }]}, U12 := {f ∈ U3 | f ∈ [{c0 , c2 }]}, U13 := {f ∈ U9 | f ∈ [{c2 }]}, U14 := {f ∈ U4 | f ∈ [{c0 , c2 }]}.

H4

H22


U10

474

H6

H26

U12

H17

∅

U13

U11

H24 H25

H28

H20

U7

U3

Fig. 15.2. The subclasses of U1

H2

H19

U6

U2

U1

U14

H18

H27

H23

U4

U8

H21

U5

U9

q q q BB

B q

B

B q

B

B q q

B

B

B q

B

B

B q q

B

B q B q

B q q

B

B q q q

Bq @ J J @ q J @ J @ J q q @ J @ J @ q q J A@ A @ J A @ A Jq A @ A J A @ A J A A @ J A A @q q J A A @ @ J A A @ J A A q @ J A A H HH @ A J A HH J @A A HH Aq @Aq Jq


475

Table 15.15 A U1 U2 U3 U4 U5 U6 U7 U8 U9 U10 U11 U12 U13 U14

Basis for A {u2 (x) ∨ u2 (y), u5 , c0 , c1 } {u2 (x) ∨ u2 (y), u5 , c0 } {u2 (x) ∨ u2 (y), c0 , c2 } {u5 (x) ∨ u5 (y), c0 , c2 } {u2 (x) ∨ u2 (y), u5 , c2 } {u2 (x) ∨ u2 (y), c0 } {u5 (x) ∨ u5 (y), c0 } {u2 (x) ∨ u2 (y), c2 } {u2 (x) ∨ u5 (y), c2 } {u5 (x) ∨ u5 (y), c2 } {u2 (x) ∨ u2 (y), u5 } {u2 (x) ∨ u2 (y)} {u2 (x) ∨ u5 (y)} {u5 (x) ∨ u5 (y)}

A1 {u2 , u5 , c0 , c2 } {u2 , u5 , c0 } {u2 , c0 , c2 } {u5 , c0 , c2 } {u2 , u5 , c2 } {u2 , c0 } {u5 , c0 } {u2 , c2 } {u5 , c2 } {u5 , c2 } {u2 , u5 } {u2 } {u5 } {u5 }

Theorem 15.4.4.3 The class V1 := {f ∈ M | Im(f ) ⊆ {1, 2}} has exactly the following (countable infinite-many) subclasses, which are not subclasses of [M 1 ] : V1 , V2 := V1 ∩ P ol(2) = {f ∈ V1 | f ∈ [{c1 }]}, V3 := {f ∈ V1 | f ∈ [{c1 , c2 , v2 }] ∨ numf (v5 ) ≥ 1}, V4 := {f ∈ V3 | f ∈ [{c1 }]}, V5 := {f ∈ V1 | f ∈ [{c1 , c2 , v2 }] ∨ numf (v2 ) ≤ 1}, V6 := {f ∈ V5 | f ∈ [{c1 }]}, V7 := {f ∈ V1 | f ∈ [{c2 , v2 }] ∨ F (f ) ⊆ {c1 , v5 }}, V8 := {f ∈ V7 | f ∈ [{c1 }]}, V9 := {f ∈ V7 | f ∈ [{v2 }]}, V10 := {f ∈ V7 | f ∈ [{c1 , v2 }]}, V7,r := {f ∈ V7 | numf (v5 ) ≤ r}, V8,r := {f ∈ V7,r | f ∈ [{c1 }]}, V9,r := {f ∈ V7,r | f ∈ [{v2 }]}, V10,r := {f ∈ V7,r | f ∈ [{c1 , v2 }]}, V11 := {f ∈ V3 | f ∈ [{v2 }]}, V12 := V5 ∩ V11 , V13 := {f ∈ V11 | f ∈ [{c1 }]}, V14 := {f ∈ V12 | f ∈ [{c1 }]}, V15 := {f ∈ V1 | F (f ) ⊆ {c1 , c2 , v2 }}, V16 := {f ∈ V15 | f ∈ [{c1 }]}, V17 := {f ∈ V15 | f ∈ [{c2 }]}, V18 := {f ∈ V15 | f ∈ [{c1 , c2 }]}, where r = 1, 2, 3, 4, ... .

476


Vr1 rV3

V2 r

V11 r

rV5

r V4 V6

rV7

r r

V8

V12 r V9 r

V10 r

V7,r+1 r V8,r+1 r

rV9,r+1 rV10,r+1 V7,r r

V8,r r

rV9,r r V10,r

r V15 V16 r

H36 r

V17 r H35 r V18

r

r

H31 H29

r r H30 r

r

r H4

rH33 rH7 r H3

r ∅ Fig. 15.3. The subclasses of V1

rV13 r V14


477

Table 15.16 A V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V7,r V8,r V9,r V10,r V11 V12 V13 V14 V15 V16 V17 V18

Generating system (or basis, if it exists) for A {v2 (x) ∨ v2 (y), v5 , c1 } {v2 (x) ∨ v2 (y), v5 } {v2 (x) ∨ v2 (y) ∨ v5 (z), v2 , c1 } {v2 (x) ∨ v2 (y) ∨ v5 (z), v5 } {v2 (x) ∨ v5 (y), v2 , c1 } {v2 (x) ∨ v5 (y), v2 } {v2 , c1 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v2 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {c1 , v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v5 (x1 ) ∨ ... ∨ v5 (xr ) | r ∈ N} {v2 , c1 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {v2 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {c1 , v5 (x1 ) ∨ ... ∨ v5 (xr )} {v5 (x1 ) ∨ ... ∨ v5 (xr )} {v2 (x) ∨ v2 (y) ∨ v5 (z), c1 } {v2 (x) ∨ v5 (y), c1 } {v2 (x) ∨ v2 (y) ∨ v5 (z)} {v2 (x) ∨ v5 (y)} {v2 (x) ∨ v2 (y), c1 , c2 } {v2 (x) ∨ v2 (y), c2 } {v2 (x) ∨ v2 (y), c1 } {v2 (x) ∨ v2 (y)}

A1 {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v2 , v5 , c1 , c2 } {v2 , v5 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v5 , c1 , c2 } {v5 , c1 , c2 } {v5 , c2 } {v5 , c2 } {v2 , c1 , c2 } {v2 , c2 } {v2 , c1 } {v2 }

Theorem 15.4.4.4 Class J1 ∪U1 has exactly the following subclasses (ordered with respect to equality of the unary functions), which are not subclasses of J1 or U1 or [(J1 ∪ U1 )1 ]: J3 ∪ H4 , J7 ∪ H4 , J15 ∪ H4 ; J11 ∪ H6 , J11,r ∪ H6 ; J13 ∪ H6 ; J1 ∪ H6 , J5 ∪ H6 , J9 ∪ H6 , J9,r ∪ H6 ; H5 ∪ U6 ; H3 ∪ U9 , H3 ∪ U10 ; H5 ∪ U3 ; H5 ∪ U4 ; H5 ∪ U1 ; J4 ∪ U13 , J16 ∪ U14 ; J12 ∪ H19 , J12 ∪ U6 , J12,r ∪ H19 ; J14 ∪ U7 ; J11 ∪ H19 , J11 ∪ U6 , J11,r ∪ H19 ; J3 ∪ U9 ; J13 ∪ U4 ; J11 ∪ H24 , J11,r ∪ H24 , J11 ∪ U3 ; J10 ∪ H19 , J10,r ∪ H19 , J10 ∪ U6 , J6 ∪ H19 , J6 ∪ U6 , J2 ∪ H19 , J2 ∪ U6 ;

478


J9 ∪ H19 , J9,r ∪ H19 , J5 ∪ H19 , J1 ∪ H19 , J9 ∪ U6 , J5 ∪ U6 , J1 ∪ U6 ; J9 ∪ H24 , J9,r ∪ H24 , J5 ∪ H24 , J1 ∪ H24 , J5 ∪ U3 , J9 ∪ U3 , J1 ∪ U3 ; J2 ∪ U2 ; J1 ∪ U1 ; where r = 1, 2, 3, .... Proof. Let T be an arbitrary subclass of J1 ∪ U1 , which was not described in Theorems 15.4.3.1, 15.4.4.1, or 15.4.4.2. Then, obviously T1 := T ∩ J1 and T2 := T ∩ U1 are subclasses of J1 or U1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.1, and 15.4.4.2. These possibilities imply the possibilities for T1 and T2 , which are given in Table 15.17. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.

Table 15.17 [T 1 ] H37 H38 H39 H40 H41 H42 H43 H44 H45 H50 H51 H52 H53 H54 H55 H56 H57 H58 H59 H60 H61

Possibilities for T ∩ J1 T ∩ U1 J3 , J7 , J15 H4 J11 , J11,r H6 J13 H6 J1 , J5 , J9 , J9,r H6 H5 U6 H3 U9 , U10 H5 U3 H5 U4 H5 U1 H9 , J4 , J8 , J16 H18 , U13 , U14 H10 , J12 , J12,r H19 , U6 H11 , J14 H20 , U7 H13 , J11 , J11,r H19 , U6 H12 , J3 , J7 , J15 H22 , U9 H14 , J13 H25 , U4 H13 , J11 , J11,r H24 , U3 H15 , J2 , J6 , J10 , J10,r H19 , U6 H16 , J1 , J5 , J9 , J9,r H19 , U6 H16 , J1 , J5 , J9 , J9,r H24 , U3 H15 , J2 , J6 , J10 , J10,r H26 , U2 H16 , J1 , J5 , J9 , J9,r H28 , U1

Theorem 15.4.4.5 Class J1 ∪ V1 has exactly the following subclasses, which are not subclasses of J1 or V1 or [(J1 ∪ V1 )1 ]: J3 ∪ H7 , J7 ∪ H7 , J15 ∪ H7 ; J11 ∪ H7 , J11,r ∪ H7 ; J13 ∪ H7 ; J1 ∪ H7 , J5 ∪ H7 , J9 ∪ H7 , J9,r ∪ H7 ;


479

H5 ∪ V17 ; H5 ∪ V9 , H5 ∪ V9,r , H5 ∪ V11 , H5 ∪ V12 ; H5 ∪ V15 ; H5 ∪ V1 , H5 ∪ V3 , H5 ∪ V5 , H5 ∪ V7 , H5 ∪ V7,r ; J11 ∪ H32 , J11 ∪ V17 , J11,r ∪ H32 ; J11 ∪ H34 , J11 ∪ V15 , J11,r ∪ H34 ; H12 ∪ V17 , J3 ∪ H32 , J3 ∪ V17 , J7 ∪ H32 , J7 ∪ V17 , J15 ∪ H32 , J15 ∪ V17 ; H14 ∪ V17 , J13 ∪ H32 , J13 ∪ V17 ; H12 ∪ V15 , J3 ∪ H34 , J3 ∪ V15 , J7 ∪ H34 , J7 ∪ V15 , J15 ∪ H34 , J15 ∪ V15 ; H14 ∪ V15 , J13 ∪ H34 , J13 ∪ V15 ; H12 ∪V9 , H12 ∪V11 , H12 ∪V12 , H12 ∪V9,r , J15 ∪V9 , J15 ∪V12 , J15 ∪V11 , J7 ∪V12 , J7 ∪ V11 , J3 ∪ V11 ; H14 ∪ V9 , H14 ∪ V11 , H14 ∪ V12 , H14 ∪ V9,r , J13 ∪ V9 , J13 ∪ V11 , J13 ∪ V12 ; J1 ∪ H32 , J1 ∪ V17 , J5 ∪ H32 , J5 ∪ V17 , J9 ∪ H32 , J9 ∪ V17 , J9,r ∪ H32 ; J1 ∪ H34 , J1 ∪ V15 , J5 ∪ H34 , J5 ∪ V15 , J9 ∪ H34 , J9 ∪ V15 , J9,r ∪ H34 ; H12 ∪ V1 , H12 ∪ V3 , H12 ∪ V5 , H12 ∪ V7 , H12 ∪ V7,r , J15 ∪ V7 , J15 ∪ V5 , J15 ∪ V3 , J15 ∪ V1 , J7 ∪ V5 , J7 ∪ V3 , J7 ∪ V1 , J3 ∪ V3 , J3 ∪ V1 ; H14 ∪ V1 , H14 ∪ V3 , H14 ∪ V5 , H14 ∪ V7 , H14 ∪ V7,r , J13 ∪ V1 , J13 ∪ V3 , J13 ∪ V5 , J13 ∪ V7 ; J1 ∪ V1 ; where r = 1, 2, 3, ... . Table 15.18 [T 1 ] H37 H38 H39 H40 H46 H47 H48 H49 H62 H63 H64 H65 H66 H67 H68 H69 H70 H71 H72 H73 H74

Possibilities for T ∩ J1 T ∩ V1 H12 , J3 , J7 , J15 H7 H13 , J11 , J11,r H7 H14 , J13 H7 H16 , J1 , J5 , J9 , J9,r H7 H5 H32 , V17 H5 H33 , V9 , V9,r , V11 , V12 H5 H34 , V15 H5 H36 , V1 , V3 , V5 , V7 , V7,r H13 , J11 , J11,r H32 , V17 H13 , J11 , J11,r H34 , V15 H12 , J3 , J7 , J15 H32 , V17 H14 , J13 H32 , V17 H12 , J3 , J7 , J15 H34 , V15 H14 , J13 H34 , V15 H12 , J3 , J7 , J15 H33 , V9 , V9,r , V11 , V12 H14 , J13 H33 , V9 , V9,r , V11 , V12 H16 , J1 , J5 , J9 , J9,r H32 , V17 H16 , J1 , J5 , J9 , J9,r H34 , V15 H12 , J3 , J7 , J15 H36 , V1 , V3 , V5 , V7 , V7,r H14 , J13 H36 , V1 , V3 , V5 , V7 , V7,r H16 , J1 , J5 , J9 , J9,r H36 , V1 , V3 , V5 , V7 , V7,r

480


Proof. Let T be an arbitrary subclass of J1 ∪ V1 , which was not described in Theorems 15.4.3.1, 15.4.4.1, or 15.4.4.3. Then obviously T1 := T ∩ J1 and T2 := T ∩ V1 are subclasses of J1 or V1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.1, and 15.4.4.3. These possibilities imply the possibilities for T1 and T2 , which are given in Table 15.18. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.

Theorem 15.4.4.6 Class U1 ∪ V1 has exactly the following subclasses, which are not subclasses of U1 or V1 or [(U1 ∪ V1 )1 ]: U6 ∪ H3 ; U9 ∪ H7 ; U3 ∪ H7 ; U4 ∪ H7 ; U1 ∪ H7 ; H2 ∪ V17 ; H6 ∪ V9 , H6 ∪ V9,r , H6 ∪ V11 , H6 ∪ V12 ; H6 ∪ V15 ; H6 ∪ V1 , H6 ∪ V3 , H6 ∪ V5 , H6 ∪ V7 , H6 ∪ V7,r ; U12 ∪ V18 ; U6 ∪ V17 ; U8 ∪ V16 ; U3 ∪ V15 ; H22 ∪ V10 , H22 ∪ V10,r , H22 ∪ V13 , H22 ∪ V14 , U9 ∪ V13 ; H22 ∪ V9 , H22 ∪ V9,r , H22 ∪ V11 , H22 ∪ V12 , U9 ∪ V11 ; H25 ∪ V9 , H25 ∪ V9,r , H25 ∪ V11 , H25 ∪ V12 , U4 ∪ V9 ; H22 ∪ V2 , H22 ∪ V4 , H22 ∪ V6 , H22 ∪ V8 , H22 ∪ V8,r , U9 ∪ V2 , U9 ∪ V4 ; H22 ∪ V1 , H22 ∪ V3 , H22 ∪ V5 , H22 ∪ V7 , H22 ∪ V7,r , U9 ∪ V1 , U9 ∪ V3 ; H25 ∪V1 , H25 ∪V3 , H25 ∪V5 , H25 ∪V7 , H25 ∪V7,r , U4 ∪V1 , U4 ∪V3 , U4 ∪V5 , U4 ∪V7 ; U5 ∪ V2 ; U1 ∪ V1 ; where r = 1, 2, 3, ... . Proof. Let T be an arbitrary subclass of U1 ∪ V1 , which was not described in Theorems 15.4.3.1, 15.4.4.2, or 15.4.4.3. Then obviously T1 := T ∩ U1 and T2 := T ∩ V1 are subclasses of U1 or V1 with T = T1 ∪ T2 . Further, [T 1 ] is a subclass H of [M 1 ] with H 1 = T 1 . The possibilities for H follow from Theorems 15.4.3.1, 15.4.4.2, and 15.4.4.5, which are given in Table 15.19. When one selects the sets T1 ∪ T2 ⊆ [M 1 ], which are closed, one receives the statement of the theorem.


481

Table 15.19 [T 1 ] H41 H42 H43 H44 H45 H46 H47 H48 H49 H75 H76 H77 H78 H79 H80 H81 H82 H83 H84 H85 H86

Possibilities for T ∩ U1 T ∩ V1 H19 , U6 H3 H22 , U9 H7 H24 , U3 H7 H25 , U4 H7 H28 , U1 H7 H2 H32 , V17 H6 H33 , V9 , V9,r , V11 , V12 H6 H34 , V15 H6 H36 , V1 , V3 , V5 , V7 , V7,r H17 , U12 H29 , V18 H19 , U6 H32 , V17 H21 , U8 H31 , V16 H24 , U3 H34 , V15 H22 , U9 H30 , V10 , V10,r , V13 , V14 H22 , U9 H33 , V9 , V9,r , V11 , V12 H25 , U4 H33 , V9 , V9,r , V11 , V12 H22 , U9 H35 , V2 , V4 , V6 , V8 , V8,r H22 , U9 H36 , V1 , V3 , V5 , V7 , V7,r H25 , U4 H36 , V1 , V3 , V5 , V7 , V7,r H27 , U5 H35 , V2 , V4 , V6 , V8 , V8,r H28 , U1 H36 , V1 , V3 , V5 , V7 , V7,r

Theorem 15.4.4.7 Class J1 ∪ U1 ∪ V1 has the following subclasses, which are not subclasses of J1 ∪ U1 , J1 ∪ V1 , U1 ∪ V1 or [M 1 ]: J11 ∪ H19 ∪ H32 , J11,r ∪ H19 ∪ H32 , J11,r ∪ U6 ∪ V17 ; J11 ∪ H24 ∪ H34 , J11,r ∪ H24 ∪ H34 , J11 ∪ U3 ∪ V15 ; H12 ∪ H22 ∪ V9 , H12 ∪ H22 ∪ V9,r , H12 ∪ H22 ∪ V11 , H12 ∪ H22 ∪ V12 , J3 ∪ U9 ∪ V11 ; H14 ∪ H25 ∪ V9 , H14 ∪ H25 ∪ V9,r , H14 ∪ H25 ∪ V11 , H14 ∪ H25 ∪ V12 , J13 ∪ U4 ∪ V9 ; J9 ∪ H19 ∪ H32 , J9,r ∪ H19 ∪ H32 , J5 ∪ H19 ∪ H32 , J1 ∪ H19 ∪ H32 , J5 ∪ U6 ∪ V17 , J1 ∪ U6 ∪ V17 , J9 ∪ U6 ∪ V17 ; J9 ∪ H24 ∪ H34 , J9,r ∪ H24 ∪ H34 , J5 ∪ H24 ∪ H34 , J1 ∪ H24 ∪ H34 , J1 ∪ U3 ∪ V15 , J5 ∪ U3 ∪ V15 , J9 ∪ U3 ∪ V15 ; H12 ∪ H22 ∪ V1 , H12 ∪ H22 ∪ V3 , H12 ∪ H22 ∪ V5 , H12 ∪ H22 ∪ V7 , H12 ∪ H22 ∪ V7,r , J3 ∪ U9 ∪ V3 , J3 ∪ U9 ∪ V1 ; H14 ∪ H25 ∪ V1 , H14 ∪ H25 ∪ V3 , H14 ∪ H25 ∪ V5 , H14 ∪ H25 ∪ V7 , H14 ∪ H25 ∪ V7,r , J13 ∪ U4 ∪ V1 , J13 ∪ U4 ∪ V3 , J13 ∪ U4 ∪ V5 , J13 ∪ U4 ∪ V7 ; J1 ∪ U1 ∪ V1 ; where r = 1, 2, 3, .... Proof. It is easy to check that a class T of the form T1 ∪ T2 ∪ T3 with T1 ∈ L3 (J1 )\{∅}, T2 ∈ L3 (U1 )\{∅} and T3 ∈ L3 (V1 )\{∅} is closed if and only if the sets Ti ∪ Tj are closed for all i, j ∈ {1, 2, 3} and i = j. Thus, our theorem

482


follows from Theorems 15.4.4.4–15.4.4.6 with the aid of Theorem 15.4.3.1, in which one can find the possibilities for T 1 . 15.4.5 The Subclasses of M ∩ P ol3 {(0, 2)} To receive a coarse partition of the lattice of the subclasses of 0 A1 := M ∩ P ol3 2 = {f ∈ M | f ∈ [{c0 , c1 , c2 }] ∧ F (f ) ∩ {v2 , v5 } = ∅ ∧ ({j2 , j5 } ∩ F (f ) = ∅ =⇒ {s1 , u2 , u5 } ∩ F (f ) = ∅)}, we determine the maximal classes of A1 first. Lemma 15.4.5.1 A1 has exactly three maximal classes: (1) A2 := A1 ∩ P ol3 {0, 1} = {f ∈ A1 | u5 ∈ F (f )}; (2) A3 := A1 ∩ P ol3 {1, 2} = {f ∈ A1 | {j2 , u2 } ∩ F (f ) = ∅ =⇒ {s1 , j5 , u5 } ∩ F (f ) = ∅}; (3) B1 := {f ∈ A1 | F (f ) ⊆ {c0 , u2 , u5 , s1 }}. Proof. One can conclude from Theorem 15.4.2.2 that A1 A2 A3 B1

= [{max, x ∨ j2 (y), u2 , u5 }], = [{max, x ∨ j5 (y), u2 }], = [{max, x ∨ j2 (y), x ∨ u2 (y), u5 }], = [{max, u2 , u5 }].

With the aid of the above statements, it is easy to prove the A1 -maximality of A2 , A3 , and B. Denote now T an arbitrary subset of A1 , which is not a subset of X for all X ∈ {A2 , A3 , B}. Then there are some functions qi (i = 1, 2, 3) with q1 ∈ T \ B1 , q2 ∈ T \ A2 and q3 ∈ T \ A3 . By identifying the variables in the functions q2 and q3 , one obtains the functions u5 and u2 . The function q1 has at least two essential variables and it holds that F (q1 )∩{j2 , j5 } = ∅. By identifying certain variables of q1 and substituting certain variables of q1 through the functions u2 , u5 we obtain the functions j5 (x) ∨ u2 (y) and j5 (x) ∨ u5 (y). In the proof of Lemma 15.4.2.2, we showed that an arbitrary function t ∈ M with F (t) ∈ {{j5 , u2 }, {j5 , u5 }} is a superposition over the above-constructed functions. Then, by identifying variables in a function t4 ∈ M with F (t) = {j5 , u2 } and numt (j5 ) = numt (u2 ) = 2, we obtain max ∈ [{q1 , q2 , q3 }]. Since in addition x ∨ (j5 (u2 (y)) ∨ u2 (x)) = x ∨ j2 (y) holds, we have [{q1 , q2 , q3 }] = A3 , whereby our lemma is proven. Obviously, it holds I1 := A2 ∩ A3 = A1 ∩ P ol3 {1},


483

and the functions f of I1 are idempotent; i.e., f (x, x, ..., x) = x. Subsequently, we determine the elements of L3 (A1 )\(L3 (R) ∪ L3 ([M 1 ]), which belong to B1 and I1 , and then the remaining elements of L3 (A1 )\(L3 (R) ∪ L3 ([M 1 ]) (see Figure 15.4). With the aid of Theorems 15.4.3.1 and 15.4.4.2, one obtains a complete description of L3 (A1 ).

Theorem 15.4.5.2 B1 has exactly the following 20 subclasses, which are not subsets of [M 1 ] or U1 : B1 = [{max, u2 , u5 }], B2 := {f ∈ B1 | F (f ) ∩ {s1 , u5 } = ∅} = [{max, x ∨ u2 (y), u5 }], B3 := {f ∈ B1 | numf (s1 ) ≤ 1} = [{x ∨ u2 (y), x ∨ u5 (y), u2 }], B4 := {f ∈ B1 | u5 ∈ F (f )} = [{max, u2 }], B5 := {f ∈ B2 | numf (s1 ) ≥ 2 =⇒ u5 ∈ F (f )} = [{x ∨ y ∨ u5 (z), x ∨ u2 (y)}], B6 := {f ∈ B2 | u2 ∈ F (f ) =⇒ u5 ∈ F (f )} = [{max, u2 (x) ∨ u5 (y)}], B7 := B5 ∩ B6 = [{x ∨ y ∨ u5 (z), u2 (x) ∨ u5 (y), s1 }], B8 := {f ∈ B1 | u2 ∈ F (f )} = [{max, u5 }], B9 := B2 ∩ B3 = [{x ∨ u2 (y), u5 }], B10 := B2 ∩ B4 = [{max, x ∨ u2 (y)}], B11 := B7 \[{s1 }], B12 := B7 ∩ B8 = [{x ∨ y ∨ u5 (z), s1 }], B13 := B7 ∩ B3 = [{x ∨ u5 (y), u2 (x) ∨ u5 (y), s1 }], B14 := B3 ∩ B4 = [{x ∨ u2 (y), u2 }], B15 := B11 ∩ B12 = [{x ∨ y ∨ u5 (z)}], B16 := B12 ∩ B13 = [{x ∨ u5 (y), s1 }], B17 := B9 ∩ B13 = [{x ∨ u5 (y), u2 (x) ∨ u5 (y)}], B18 := B9 ∩ B14 = [{x ∨ u2 (y)}], B19 := [{max}], B20 := [{x ∨ u5 (y)}] (See Figure 15.5). Proof. Except for the functions f with u5 ∈ F (f ) and numf (s1 ) ≥ 2, for which f ∈ [[{f }]3 ] is valid, we have by Lemma 15.4.2.2 f ∈ [{f }]2 for all other functions f ∈ B1 \[M 1 ]. Consequently, one can describe an arbitrary subclass B of B1 in the form of a closure of a certain subset of {x ∨ y ∨ u5 (z), max, x ∨ u2 (y), x ∨ u5 , u2 (y) ∨ u5 (y), u2 , u5 , s1 }. When one examines the possible cases for B (⊆ U1 or ⊆ [M 1 ]) with the aid of Figure 15.5, one obtains our theorem.

r

r

r

r

r

r

A15,r−1

r

A15,r

Fig. 15.4

r r

r

r

r

r

r

A25

A24

r

A20

r

A14,1

A19

r

r

A15,1

A18

r A23

A17

r

A16 A22

r A37

A21

r

A14,r−1

A35

A34

r

A14

r A36

r

see Figure 15.5

A32 r

A31

r

r

A14,r

r A33

r

A15

B1

A30

A29

A28

A27

r

r

A13

r

A12

see Figure 15.6

A26 r

A11

A10

r

r

A9

r r

A8

I1 r

A7

A6

r

A5

A3 r

A4

A2

A1

484 15 Finite and Countably Infinite Sublattices of Depth 1 or 2 of L3

r

r

r

r


485

B1

q Q

Q

Q Q Q

Q Q Q B4 B2 B3 Q Qq q q P

J @PPP PP J

@ P B5 B6 P

PP J @ @ q X q XX PP JP

X X X P XXX

J PP B7 B8 PP B10 J

XXXXqB9 qP qP Pq PP J

A @PPP P P J

P @ B PPP PP A B11 B13

12 14 P JB A P @ P Jq q ` q q P

` PP ` ``` @ APP @ @ `` A PPP @``` @ @B B16 ``` B17 B PP B19 AA18 @ @ ` @ 15 q @ q q @q ` Pq @ @ @ @ B20

Fig. 15.5

Theorem 15.4.5.3 I1 has exactly 17 subclasses, which are not subsets of [M 1 ]: I1 = [{max, u2 (x) ∨ j5 (y)}], I2 := [{max, x ∨ j5 (y), x ∨ u2 (y)}], I3 := [{x ∨ j5 (y), j5 (x) ∨ u2 (y)}], I4 := [{x ∨ u2 (y), j5 (x) ∨ u2 (y)}], I5 := [{max, x ∨ j2 (y), x ∨ j5 (y)}], I6 := [{max, x ∨ j2 (y), x ∨ u2 (y)}], I7 := [{x ∨ j2 (y), j5 (y) ∨ u2 (y)}], I8 := [{x ∨ j5 (y), x ∨ j2 (y)}], I9 := [{x ∨ j2 (y), x ∨ u2 (y)}], I10 := [{max, x ∨ j5 (y)}], I11 := [{max, x ∨ j2 (y)}], B10 = [{max, x ∨ u2 (y)}], I12 := [{j5 (x) ∨ u2 (y)}], I13 := [{x ∨ j5 (y)}], I14 := [{x ∨ j2 (y)}], B18 = [{x ∨ u2 (y)}], B19 = [{max}]. Proof. Because of Theorem 15.4.2.2, every subclass of I1 has a generating system from binary functions of I1 . Consequently, one obtains the subclasses


486

of I1 , which are not subclasses of [M 1 ], through closure of the subsets of {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)}. (During the forming of these classes one notices that the equations x ∨ j5 (j5 (x) ∨ u2 (y)) = x ∨ j2 (y) and u2 (x) ∨ j5 (x ∨ u2 (y)) = x ∨ j2 (y) are valid.) The Figure 15.6 gives the Hasse diagram of the classes constructed in this manner. I1

q P P PP PP P

PP

PPI2 @ @ @ I6 I3 I4 I5 q q q @q P P PP @PPP @ P PP PP @ @ P PP P @ I7 PP I8 PP I9 I10 @ I11 B10 P P q @q q q @P q q PP @ @ PP @ PP @ PP @ @ @ @q q q PP q q

I12

I13

I14

B18

B19

Fig. 15.6

Theorem 15.4.5.4 The following classes are the only subclasses of A2 that are not contained in I1 , B1 or [M 1 ]: A2 = [{max, x ∨ j5 (y), u2 }], A4 := [{max, j2 (x) ∨ u2 (y)}], A5 := [{x ∨ u2 (y), j5 (x) ∨ u2 (y), u2 }], A6 := [{x ∨ j5 (y), u2 }], A7 := [{x ∨ j2 (y), x ∨ u2 (y), u2 }], A8 := [{j5 (x) ∨ u2 (y), u2 }], A9 := [{x ∨ u2 (y), j2 (x) ∨ u2 (y)}], A10 := [{x ∨ j2 (y), u2 (x) ∨ u2 (y)}], A11 := [{s1 , j2 (x) ∨ u2 (y), u2 (x) ∨ u2 (y)}], A12 := [{x ∨ j2 (y), u2 }], A13 := A11 \[{s1 }], A14 := [{u2 (x1 ) ∨ j2 (x2 ) ∨ ... ∨ j2 (xn ) | n ∈ N\{1}}],


487

A14,r := [{f ∈ A10 | numf (j2 ) ≤ r}], A15 := A14 ∪ [{s1 }], A15,r := A14,r ∪ [{s1 }], where r = 1, 2, ... . Proof. Let A be a subclass of A2 , which is not a subset of I1 , B1 or [M 1 ]. Because of A ⊆ I1 = A2 ∩ A3 , u2 belongs to A. Then, by A ⊆ [M 1 ] and A ⊆ B1 , we have j2 (x) ∨ u2 (y) ∈ A. Thus A contains the class A14,1 . One can verify the remaining statements of our theorem easily with the aid of the above-noted generating systems of the classes, Figure 15.4, and Theorem 15.4.2.2. Theorem 15.4.5.5 The following classes are the only subclasses of A3 that are not contained in I1 , B1 or [M 1 ]: A3 = [{max, j5 (x) ∨ u2 (y), u5 }], A16 := {f ∈ A1 | u5 ∈ F (f )} = [{j5 (x) ∨ u5 (y), j2 (x) ∨ u5 (y)}], A17 := {f ∈ A16 | {j2 , u2 } ∩ F (f ) = ∅} = [{u5 (x) ∨ j5 (y), x ∨ y ∨ u5 (z)}], A18 := {f ∈ A17 | numf (s1 ) ≤ 1} = [{x ∨ j5 (y) ∨ u5 (z)}], A19 := {f ∈ A18 | numf (s1 ) = 1 =⇒ j5 ∈ F (f )} = [{x∨u5 (y), j5 (x)∨u5 (y)}], A20 := {f ∈ A19 | numf (s1 ) = 0} = [{j5 (x) ∨ u5 (y)}], A21 := A16 ∪ [{s1 }], A22 := A17 ∪ [{s1 }], A23 := A18 ∪ [{s1 }], A24 := A19 ∪ [{s1 }], A25 := A20 ∪ [{s1 }], A26 := I3 ∪ A16 , A27 := I2 ∪ A16 , A28 := I5 ∪ A16 , A29 := B10 ∪ A16 , A30 := I8 ∪ A16 , A31 := I10 ∪ A17 , A32 := B19 ∪ A16 , A33 := I13 ∪ A17 , A34 := B18 ∪ A16 , A35 := B19 ∪ A17 , A36 := I13 ∪ A18 , A37 := I13 ∪ A20 . Proof. Let T be a subclass of A3 with T ⊆ I1 , T ⊆ B1 and T ⊆ [M 1 ]. It follows from T ⊆ I1 , T ⊆ B1 , T ⊆ [M 1 ] and j2 u5 = j5 that the functions u5 and j5 (x) ∨ u5 (y) belong to T . Hence we have A20 ⊆ T . Since for an arbitrary function f ∈ A3 either f (1, ..., 1) = 1 or f (1, ..., 1) = 2 holds, and the case f (1, ..., 1) = 2 and f ∈ A3 is only possible, if u5 ∈ F (f ),

488


the class T has the form (T ∩ I1 ) ∪ (T ∩ A16 ).

(15.5)

First we show the possibilities for T ∩ A16 . Because of j2 (u5 (y)) ∨ u5 (j2 (x) ∨ u5 (y)) = u2 (x) ∨ u5 (y) and j5 (u2 (x) ∨ u5 (y)) ∨ u5 (y) = j2 (x) ∨ u5 (y) we have j2 (x) ∨ u5 (y) ∈ T ⇐⇒ u2 (x) ∨ u5 (y) ∈ T.

(15.6)

Further, it holds that: j5 (x1 ) ∨ j5 (x2 ) ∨ u5 (x3 ) ∨ u5 (x4 ) ∈ [{j5 (x) ∨ u5 (y)}] This implies x ∨ y ∨ u5 (z) = j5 (g(z, x)) ∨ j5 (g(z, y)) ∨ u5 (g(x, z)) ∨ u5 (g(y, z)) ∈ [{j5 (x) ∨ u5 (y), g}]

(15.7)

for every g(x, y) ∈ {j2 (x) ∨ u5 (y), u2 (x) ∨ u5 (y)}. With the aid of (15.6) and (15.7), it is easy to check that the classes A16 , ..., A20 are the only possibilities for T if T ⊆ A16 holds. The remaining possibilities for T can be obtained with the help of (15.5) and Theorem 15.4.5.3, when one determines the classes I ∈ L3 (I1 )\{∅} and A ∈ {A16 , ..., A20 } with I ∪ A = [I ∪ A].

15.4.6 The Remaining Subclasses of M The following lemma is the basis for determining the subclasses of M that are still missing. Lemma 15.4.6.1 Let T be a subclass of M , which is not a subset of [M 1 ], R or A1 . Furthermore, let T1 := T ∩ R and T2 := T ∩ P ol

0 . 2

Then T fulfills one of the two following conditions: (a) T ∈ {T1 ∪ [{s1 }] | T1 ∈ L3 (R)\L3 ([M 1 ])}. (b) T2 ∩ {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅.


489

Proof. For T the following two cases are possible: Case 1: T2 ⊆ [M 1 ]. 0 1 Because of T ⊆ R and (T ∩P ol ) ⊆ {u2 , u5 , s1 }, this case is only possible 2 for s1 ∈ T2 . Since T2 \[{s1 }] ⊆ R and classes of the form T1 ∪ [{s1 }] are closed for all T1 ∈ L3 (R), T fulfills the condition (a). Case 2: T2 ⊆ [M 1 ]. By Section 15.4.5, T fulfills the condition (b). In the following, denote T a subclass of M with the properties: T ∩ R = ∅ and T ∩ {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), = ∅. j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} Furthermore, we use the notations: T1 := T ∩ R and T2 := T ∩ P ol

0 (= T ∩ A1 ). 2

Theorem 15.4.6.2 If T ⊆ J1 ∪ A1 then T is one of the following classes: 1) H2 ∪ B19 , J12 ∪ I11 , J14 ∪ I10 , J2 ∪ I5 (classes that contain the set {c0 , max}); 2) J2 ∪ I8 , J14 ∪ I13 , Ji ∪ I14 (i ∈ {1, 2, 5, 6, 11, 12}) (classes T with c0 ∈ T , max ∈ T and {x∨j5 (y), x∨j2 (y)}∩T = ∅); 3) H3 ∪ I14 , J7 ∪ I14 , J3 ∪ I14 (classes T with c1 ∈ T and c0 ∈ T ); 4) J16 ∪ I13 , J16 ∪ I10 , J8 ∪ I4 , J8 ∪ I7 , J8 ∪ I14 , J4 ∪ I5 , J4 ∪ I1 , J4 ∪ I8 , J4 ∪ I3 , J4 ∪ I14 (classes that do not contain constant functions). Proof. Because of x∨c1 (x) = v2 (x) the set J1 ∪A1 is not closed. To determine all closed subsets T of J1 ∪ A1 , we distinguish the following cases: Case 1: c0 ∈ T1 . Then for every function f ∈ T , we have F (f ) ⊆ T 1 . Thus T2 is a subset of I1 , and we have {max, x ∨ j5 (y), x ∨ j2 (y)} ∩ T = ∅ and {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅. If {max, x ∨ j5 (y)} ∩ T = ∅, T does not contain c1 , since x ∨ c1 (x) = v2 (x) ∈ T . Case 1.1: max ∈ T2 . In this case, the class T is also describable in the form T = [T 1 ∪[max}]. With

490


the aid of Theorems 15.4.3.1, 15.4.4.1, 15.4.5.3 and Table 15.14, one obtains the classes given in 1). Case 1.2: max ∈ T2 . Case 1.2.1: x ∨ j5 (y) ∈ T2 . If one examines the possibilities that arise from Theorems 15.4.4.1 and 15.4.5.3 for T , then one sees that only the sets J14 ∪ I13 and J2 ∪ I8 are closed. Case 1.2.2: x ∨ j5 (y) ∈ T2 and x ∨ j2 (y) ∈ T2 . In this case, we have T2 = I14 and T1 contains J12 . Then, the possibilities for T are: Ji ∪ I14 , where i ∈ {1, 2, 5, 6, 11, 12}). Case 2: c0 ∈ T1 and c1 ∈ T1 . Because of x ∨ c1 (x) = v2 (x), c1 ∨ ui = vi (i = 2, 5) and j2 (c1 ) ∨ u2 = u2 , only classes T are possible with {max, x ∨ j5 (y), x ∨ u2 (y), x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} ∩ T = ∅. Hence, by Theorem 15.4.4.1 and Section 15.4.5, T1 ∈ {H3 , J7 , J3 } and T2 = I14 , where every possibility supplies a closed class, which is given in 3). Case 3: {c0 , c1 } ∩ T = ∅. In this case, by Theorem 15.4.4.1, T1 ∈ {H9 , J16 , J8 , J4 }. Further, we have {max, x ∨ j2 (y), x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)} ∩ T = ∅ and {x ∨ u5 (y), j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} ∩ T = ∅. Thus T1 = H9 is not possible. We must only continue to examine, therefore, the following three cases: Case 3.1: T1 = J16 . Then, T can not contain the functions x ∨ j2 (y), x ∨ u2 (y) and j5 (x) ∨ u2 (y); thus T2 ∈ {I13 , B19 , I10 } and T ∈ {J16 ∪ I13 , J16 ∪ I10 }. Case 3.2: T1 = J8 . In this case, we have T ∩ {max, x ∨ j5 (y)} = ∅ and T2 ⊆ I4 . By scrutinizing the possibilities that result, one receives T ∈ {J8 ∪ I4 , J8 ∪ I7 , J8 ∪ I14 }. Case 3.3: T1 = J4 . The following cases are still possible, then: Case 3.3.1: max ∈ T2 . Then T contains the class J4 ∪ I5 . Because of j5 (x) ∨ u2 (y) ∈ [{j5 , x ∨ u2 (y)}] the set J4 ∪ I2 is not closed, and only the classes T with T = J4 ∪ I5 or T = J4 ∪ I1 fulfill the conditions of this case. Case 3.3.2: max ∈ T2 . Case 3.3.2.1: x ∨ j5 (y) ∈ T . Since x ∨ j5 (j5 (x) ∨ j2 (y)) = x ∨ j2 (y), T contains the class I8 and it holds that T ∈ {J4 ∪ I8 , J4 ∪ I3 }. Case 3.3.2.2: {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅ and x ∨ j5 (y) ∈ T . Because of (j5 (x) ∨ j5 (y)) ∨ u2 (x) = x ∨ j5 (y), this case is not possible. Case 3.3.2.3: x ∨ j2 (y) ∈ T and {x ∨ j5 (y), x ∨ u2 (y), j5 (x) ∨ u2 (y)} ∩ T = ∅. Only the class J4 ∪ I14 can be T in this case.


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Theorem 15.4.6.3 If T ⊆ U1 ∪ A1 , then T is exactly one of the following classes: 1) H2 ∪ B19 , H6 ∪ B19 , U6 ∪ B4 , U7 ∪ B8 , U3 ∪ B4 , U4 ∪ B8 , U2 ∪ B1 , U1 ∪ B1 (classes that contain {c0 , max}; 2) U1 ∪ B3 , U2 ∪ B3 , U3 ∪ B14 , U6 ∪ B14 , U4 ∪ B16 , U7 ∪ B16 (classes T with c0 ∈ T and max ∈ T ); 3) H4 ∪ B19 , H4 ∪ B10 , U5 ∪ B1 , U8 ∪ B4 , U9 ∪ B2 , U9 ∪ B6 , U10 ∪ B8 (classes T with {c2 , max} ⊆ T and c0 ∈ T ); 4) H4 ∪ B18 , U5 ∪ B3 , U8 ∪ B14 , U9 ∪ B5 , U9 ∪ B9 (classes T with {c2 , x ∨ u2 (y)}, c0 ∈ T and max ∈ T ); 5) U9 ∪ Bi (i ∈ {7, 11, 13, 17}), U10 ∪ Bt (t ∈ {12, 15, 16, 20}) (classes T with {c2 , x ∨ u5 (y)} ⊆ T and {c0 , max, x ∨ u2 (y)} ∩ T = ∅). Proof. Since U1 \[{c0 , c1 }] ⊆ A1 , we have T ∩ {c0 , c2 } = ∅. Case 1: c0 ∈ T1 . In this case, it holds that F (f ) ⊆ T 1 for all f ∈ T . Because of Section 15.4.5, then, we have either B18 or B19 or B20 as a subset of T . Case 1.1: max ∈ T2 (B19 ⊆ T ). Then [T 1 ∪ {max}] = T , and because of Theorems 15.4.3.1, 15.4.4.2, and 15.4.5.2, for T there are only the possibilities given in 1). Case 1.2: max ∈ T2 . If x ∨ u2 (y) ∈ T2 (i.e., B18 ⊆ T2 ) then u2 belongs to T1 , and by Theorem 15.4.5.2, we have T2 ∈ {B3 , B14 }. Thus T ∈ {U6 ∪ B14 , U3 ∪ B14 , U2 ∪ B3 , U1 ∪ B3 }. In the case x∨u5 (y) ∈ T2 (i.e., B20 ⊆ T2 ) and x∨u2 (y) ∈ T , only T1 ∈ {U4 , U7 } and T2 = B16 are possible (because of c0 ∈ T ). Therefore, in Case 1.2, only the classes given in 2) are possible. Case 2: c0 ∈ T1 and c2 ∈ T1 . Since jr (c2 ) ∨ us = vs and x ∨ jr (c2 ) = v2 (r, s ∈ {2, 5}), we have T ∩ {jr (x) ∨ us (y), x ∨ jr (y) | r, s ∈ {2, 5}} = ∅ and, by Section 15.4.5, T contains either B18 , B19 or B20 . Case 2.1: max ∈ T2 . Then T1 belongs to {H4 , U5 , U8 , U9 , U10 } and T2 belongs to {B1 , B2 , B4 , B6 , B8 , B10 , B19 }. In 3) those classes of the form T1 ∪ T2 are given, which are closed. Case 2.2: max ∈ T and x ∨ u2 (y) ∈ T . In this case, by Theorem 15.4.5.2, we have T2 ∈ {B3 , B5 , B9 , B14 , B18 }. Then T1 belongs to the set {H4 , U5 , U8 , U9 }. When one considers the possible cases, one receives the classes given in 4). Case 2.3: {max, x ∨ u2 (y)} ∩ T = ∅ and x ∨ u5 (y) ∈ T .

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Then it holds that T2 ∈ {U9 , U10 } and T2 ∈ {B7 , B11 , B12 , B13 , B15 , B16 , B17 , B20 }. The closed sets of the form T1 ∪ T2 are given in 5). Theorem 15.4.6.4 If T ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) V1 ∪ I5 , V1 ∪ I12 , V1 ∪ B19 , V15 ∪ Is (s ∈ {2, 5, 6, 10, 11}), V15 ∪ B10 , V15 ∪ B19 , V17 ∪ It (t ∈ {2, 5, 6, 10, 11}), V17 ∪ B10 , V17 ∪ B19 (classes that belong {c1 , max}); 2) V15 ∪ B18 , V15 ∪ I9 , V17 ∪ B18 , V17 ∪ I9 (classes T with {c1 , x ∨ u2 (y)} ⊂ T and {max, x ∨ j5 (y)} ∩ T = ∅); 3) V ∪ I8 (V ∈ {H32 , H34 , V1 , V3 , V15 , V17 }), V ∪ I13 (V ∈ {H32 , H34 , V1 , V3 , V5 , V7 , V15 , V17 }) (classes T with {c1 , x ∨ j5 (y)} ⊂ T and {max, x ∨ u2 (y)} ∩ T = ∅); 4) V ∪ I14 (V ∈ {H3 , H32 , H34 , V15 , V17 }) (classes T with x ∨ j2 (y) ∈ T and {max, x ∨ u2 (y), x ∨ j5 (y)} ∩ T = ∅); 5) V2 ∪ B19 , V10 ∪ B19 , V13 ∪ B19 , V2 ∪ I13 , V4 ∪ I13 , V8 ∪ I13 , V2 ∪ I8 , V4 ∪ I8 , V2 ∪ I10 , V2 ∪ I5 (classes T with {c2 , v5 } ⊂ T and c1 ∈ T ); 6) H4 ∪ B10 , H4 ∪ B19 , H31 ∪ Ir (r ∈ {3, 7, 8, 12, 13, 14}), V16 ∪ Bs (s ∈ {10, 18, 19}), V16 ∪ It (t ∈ {1, 2, 4, 5, 6, 9, 10, 11}) (classes T with c2 ∈ T and T ∩ {c1 , v5 } = ∅); 7) H29 ∪ Ir (r ∈ {3, 7, 8, 12, 13, 14}), V18 ∪ Bs (s ∈ {10, 18, 19}), V18 ∪ It (t ∈ {1, 2, 4, 5, 6, 9, 10, 11}) (classes T with T ∩ {c1 , c2 } = ∅). Proof. Because of U1 ∩ A1 = ∅, the set A1 ∪ V1 is not closed and, therefore, T ⊂ V1 ∪ A1 . Since u5 ∈ A for every A ∈ {B20 , A20 }, u2 ∈ A14,1 and u2 ∈ B14 , T2 is a certain subclass of I1 . Case 1: c1 ∈ T1 . Then, because of j5 ∨ u2 (c1 ) = j5 , we have j5 (x) ∨ u2 (y) ∈ T2 . Case 1.1: max ∈ T2 . Since x∨c1 = v2 (x), it holds that V17 ⊆ T1 . With the aid of Theorems 15.4.4.3 and 15.4.5.3, one obtains only the possibilities given in 1) for T .


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Case 1.2: max ∈ T2 . By Theorem 15.4.5.3 and because of j5 (x) ∨ u2 (y) ∈ T , only the following three cases are possible: Case 1.2.1: x ∨ u2 (y) ∈ T2 . Because of c1 ∨u2 = v2 it holds V17 ⊆ T1 . Further, because x∨u2 (v5 ) = u5 (x), v5 ∈ T . Hence, T ∈ {V15 ∪ B18 , V15 ∪ I9 , V17 ∪ B18 , V17 ∪ I9 }. Case 1.2.2: x ∨ u2 (y) ∈ T2 and x ∨ j5 (y) ∈ T2 . Because of Theorem 15.4.5.3, this case is only possible for T2 ∈ {I8 , I13 }. With the aid of Theorem 15.4.4.3, this implies that T is a class given in 3). Case 1.2.3: T2 = I14 (= [{x ∨ j2 (y)}]). Because of x ∨ j2 (v5 (y)) = x ∨ j5 (y), the function v5 does not belong to T1 . With the aid of Theorem 15.4.4.3 this implies that T is a class which is given in 4). Case 2: c1 ∈ T1 and c2 ∈ T1 . We distinguish two cases: Case 2.1: v5 ∈ T1 . Because of j5 (x) ∨ u2 (v5 (x)) = u5 (x) ∈ T and x ∨ u2 (v5 (x)) = u5 (x) ∈ T , T2 cannot contain j5 (x) ∨ u2 (y) or x ∨ u2 (y). Further, we have: x ∨ j2 (y) ∈ T =⇒ x ∨ j2 (v5 (y)) = x ∨ j5 (y) ∈ T. Thus T2 ∈ {B19 , I5 , I8 , I10 , I13 } and V10 ⊆ T1 . Because of c1 ∈ T , this implies T1 ∈ {V2 , V4 , V6 , V8 , V10 , V13 , V14 }. The possibilities resulting for T are given in 5). Case 2.2: v5 ∈ T1 . Then, because of Theorem 15.4.4.3, T1 ∈ {H4 , H31 , V16 }. The possibilities resulting for T are given in 6). Case 3: {c1 , c2 } ∩ T = ∅. Because of v5 v5 = c2 , v5 ∈ T . Thus, either T1 = H29 = [{v2 }] or T1 = V18 = [{v2 (x) ∨ v2 (y)}]. With the aid of Theorem 15.4.5.3, this implies that T is a class which is given in 7).

Theorem 15.4.6.5 If T ⊆ (J1 ∪ U1 ) ∪ A1 , T1 ⊆ J1 ∪ A1 and T1 ⊆ U1 ∪ A1 , then T is exactly one of the following classes: 1) (J4 ∪ U13 ) ∪ Ai (i ∈ {3, 16, 21, 26, 30}), (J16 ∪ U14 ) ∪ Am (m ∈ {17, 18, 19, 20, 22, 23, 24, 25, 33, 35, 36, 37}), (J12 ∪ U6 ) ∪ Ap (p ∈ {4, 7, 9, 10, 11, 13}), (J12 ∪ H19 ) ∪ An (n ∈ {12, 14, 15}), (J12 ∪ H19 ) ∪ As,r (s ∈ {14, 15}), r ∈ N), (J12,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J14 ∪ U7 ) ∪ Aq (q ∈ {17, 18, 19, 20, 22, 23, 24, 25, 33, 35, 36, 37}),

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(J10,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J10 ∪ H19 ) ∪ At (t ∈ {12, 14, 15}), (J10 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J16 ∪ H19 ) ∪ Ai (i ∈ {8, 12, 14, 15}), (J16 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J10 ∪ U6 ) ∪ Am (m ∈ {11, 13}), (J6 ∪ U6 ) ∪ An (n ∈ {5, 10, 11, 13}), (J2 ∪ H19 ) ∪ Ap (p ∈ {6, 12, 14, 15}), (J2 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J2 ∪ U6 ) ∪ Aq (q ∈ {2, 10, 11, 13}), (J2 ∪ U2 ) ∪ A1 (classes T with {c0 , c1 } ∩ T = ∅); 2) (J11 ∪ H19 ) ∪ Ai (i ∈ {12, 14, 15}), (J11 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J11 ∪ U6 ) ∪ Am (m ∈ {10, 11, 13}), (J11,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J9 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J9,r ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (J5 ∪ H19 ) ∪ An (n ∈ {12, 14, 15}), (J5 ∪ H19 ) ∪ As,r (s ∈ {14, 15}; r ∈ N), (J1 ∪ H19 ) ∪ Ap (p ∈ {12, 14, 15}), (J1 ∪ H19 ) ∪ As,r (s ∈ {14, 15}, r ∈ N), (J9 ∪ U6 ) ∪ Aq (q ∈ {11, 13}), (Ji ∪ U6 ) ∪ At (i ∈ {1, 5}, t ∈ {10, 11, 13}) (classes T with c1 ∈ T and c2 ∈ T ). Proof. Case 1: c2 ∈ T1 . Case 1.1: c1 ∈ T1 . In this case, T is a subset of (J2 ∪ U2 ) ∪ A1 . Table 15.20 indicates the possibilities for T1 that result from Theorem 15.4.4.4: Table 15.20 T11 1 H50 1 H51 1 H52 1 H57 1 H60

T1 J4 ∪ U13 , J16 ∪ U14 J12 ∪ H19 , J12 ∪ U6 , J12,r ∪ H19 J14 ∪ U7 J10 ∪ H19 , J10,r ∪ H19 , J10 ∪ U6 , J6 ∪ H19 , J6 ∪ U6 , J2 ∪ H19 , J2 ∪ U6 J2 ∪ U2

Then, by Section 14.4.5, we have T ∩ {j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅.


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Consequently, with the aid of Theorems 15.4.5.4 and 15.4.5.5, one obtains the classes of 1) as possibilities for T , where these classes are sorted after the cases that result from Table 15.20. Case 1.2: c1 ∈ T1 . Because of T ∩ R ⊆ V1 , this case is only possible for T ∩ {max, x ∨ u2 (y), x ∨ u5 (y), x ∨ j5 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y)} = ∅. Consequently, we have A14,1 ⊆ T2 ⊆ A10 or T2 = A16 . Because of Theorem 15.4.4.4, T1 can be only a class from Table 15.21. Table 15.21 T21 1 H14 1 H53 1 H58

T2 H5 ∪ U6 J11 ∪ H19 , J11 ∪ U6 , J11,r ∪ H19 J9 ∪ H19 , J9,r ∪ H19 , J5 ∪ H19 , J1 ∪ H19 , J9 ∪ U6 , J5 ∪ U6 , J1 ∪ U6

When one scrutinizes the cases resulting from that, one receives the classes of 2). Case 2: c2 ∈ T1 . Then, by T ∩ R ⊆ V1 , we have {j2 (x) ∨ u2 (y), j5 (x) ∨ u2 (y), j5 (x) ∨ u5 (y), x ∨ j5 (y)} ∩ T = ∅ and T2 ⊆ B1 . Thus T contains x ∨ u2 (y) or x ∨ u5 (y). If T ∩ J1 = [{c1 }], then we have T ∩ {j2 , j5 } = ∅. However, by T2 ⊆ B1 , this cannot be possible. Equally, the case T ∩ J1 = [{c1 }] is not possible, since v2 ∈ [{c1 , x ∨ u2 (y)}] and v5 ∈ [{c1 , x ∨ u5 (y)}]. Thus the second case cannot occur.

Theorem 15.4.6.6 If T ⊆ (J1 ∪ V1 ) ∪ A1 , T1 ⊆ J1 ∪ A1 and T1 ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) (H14 ∪ Vi ) ∪ B19 (i ∈ {1, 15, 17}), (J11 ∪ Vm ) ∪ I11 (m ∈ {15, 17}), (J13 ∪ Vn ) ∪ I10 (n ∈ {1, 15, 17}), (J1 ∪ Vp ) ∪ I5 (p ∈ {1, 15, 17}) (classes that contain {c0 , max}); 2) (J13 ∪ V ) ∪ I13 (V ∈ {H32 , H34 , V1 , V3 , V5 , V7 , V17 }), (J11 ∪ V ) ∪ I14 (V ∈ {H32 , V17 , H34 , V15 }), (Ji ∪ V ) ∪ I14 (i ∈ {1, 5}, V ∈ {H32 , V17 , H34 , V15 }), (J1 ∪ V ) ∪ I8 (V ∈ {H32 , V17 , H34 , V15 , V1 }) (classes that contain c0 but not max); 3) (J15 ∪ V15 ) ∪ I10 , (J3 ∪ V15 ) ∪ I5 , J3 ∪ V15 ) ∪ I1 , (J7 ∪ V15 ) ∪ I4 ,

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(J15 ∪ H34 ) ∪ I15 , (J15 ∪ V15 ) ∪ I15 , (J3 ∪ H34 ) ∪ I10 , (J3 ∪ V15 ) ∪ I10 , (J7 ∪ H34 ) ∪ I7 , (J7 ∪ V15 ) ∪ I7 , (J3 ∪ H34 ) ∪ I14 , (J3 ∪ V15 ) ∪ I14 , (J7 ∪ H34 ) ∪ I14 , (J7 ∪ V15 ) ∪ I14 , (J15 ∪ V1 ) ∪ I10 , (J3 ∪ V1 ) ∪ I5 , (J3 ∪ V1 ) ∪ I1 , (J3 ∪ V1 ) ∪ I8 , (J15 ∪ V7 ) ∪ I13 , (J15 ∪ V5 ) ∪ I13 , (J15 ∪ V3 ) ∪ I13 , (J15 ∪ V1 ) ∪ I13 (classes that contain c2 but not c0 ); 4) (J15 ∪ V17 ) ∪ I10 , (J15 ∪ V17 ) ∪ I5 , (J15 ∪ V17 ) ∪ I1 , (J7 ∪ V17 ) ∪ I4 , (J7 ∪ V17 ) ∪ I1 , (J15 ∪ H32 ) ∪ I13 , (J15 ∪ V17 ) ∪ I13 , (J3 ∪ H32 ) ∪ I8 , (J3 ∪ V17 ) ∪ I3 , (J7 ∪ H32 ) ∪ I12 , (J7 ∪ H32 ) ∪ I7 , (J7 ∪ H32 ) ∪ I14 , (J7 ∪ V17 ) ∪ I14 , (J3 ∪ H32 ) ∪ I14 , (J3 ∪ V17 ) ∪ I14 (classes T with {c0 , c1 , c2 } ∩ T = {c1 }). Proof. Because of T ∩ U1 = ∅, we have {j2 (x) ∨ u2 (y), j5 (x) ∨ u5 (y), x ∨ u5 (y)} ∩ T = ∅, Thus T2 ⊆ I1 . Then the following cases are possible: Case 1: c0 ∈ T1 . Then {j5 (x) ∨ u2 (y), x ∨ u2 (y)} ∩ T = ∅. Case 1.1: max ∈ T2 . Because of T = [T 1 ∪ {max}], T is one of the classes given in 1) (see also Theorems 15.4.3.1 and 15.4.5.3). Case 1.2: max ∈ T2 . In this case, by Theorem 15.4.5.3, T2 belongs to {I13 , I8 , I14 } and it holds that 1 1 1 , H67 , H73 }, T2 = I13 =⇒ T11 ∈ {H65 1 1 1 1 1 T2 = I14 =⇒ T1 ∈ {H62 , H63 , H70 , H71 }, 1 1 1 1 T2 = I8 =⇒ T1 ∈ {H70 , H71 , H74 }.

From this and from Theorem 15.4.4.5, we get the possibilities given in 2) for T. Case 2: c0 ∈ T1 . Then, because of T ∩ J1 = ∅ and T ∩ V1 = ∅, the function c1 belongs to T . Case 2.1: c2 ∈ T1 . 1 1 , H72 }. Because of Theorems 15.4.4.5 and 15.4.5.3, the possiThen T11 ∈ {H66 bilities 3) for T follow. Case 2.2: c2 ∈ T1 . 1 . Then, with the help of Theorems 15.4.4.5 and In this case, we have T11 = H64 15.4.5.3, one obtains the possibilities given in 4) for T .


497

Theorem 15.4.6.7 If T ⊆ (U1 ∪ V1 ) ∪ A1 , T1 ⊆ U1 ∪ A1 and T1 ⊆ V1 ∪ A1 , then T is exactly one of the following classes: 1) (H2 ∪ V17 ) ∪ B19 , (H6 ∪ V15 ) ∪ B19 , (H6 ∪ V1 ) ∪ B19 , (U6 ∪ V17 ) ∪ B4 , (U3 ∪ V15 ) ∪ B4 , (U4 ∪ V1 ) ∪ B8 , (U1 ∪ V1 ) ∪ B1 (classes that contain max); 2) (U6 ∪ V17 ) ∪ B14 , (U3 ∪ V15 ) ∪ B14 , (U1 ∪ V1 ) ∪ B3 (classes that contain x ∨ u2 (y) but not max); 3) A ∪ Bi (A ∈ {U4 ∪ V9 , U4 ∪ V1 , U4 ∪ V3 , U4 ∪ V5 , U4 ∪ V7 }, i ∈ {16, 20}) (classes T with T ∩ {max, x ∨ u2 (y)} = ∅ and x ∨ j5 (y) ∈ T ). Proof. Because of T ⊆ V1 ∪ A1 , we have c0 ∈ T1 . T ∩ V1 = [{c2 }] implies c1 ∈ T1 . Hence, T2 ⊆ B1 . By Theorem 15.4.5.2, only the following cases are possible: Case 1: max ∈ T2 . Then T = [T 1 ∪ {max}] and one obtains the possibilities given in 1) of T with the aid of Theorems 15.4.3.1, 15.4.4.6, and 15.4.5.2. Case 2: max ∈ T2 . Then {x ∨ u2 (y), x ∨ u5 (y)} ∩ T = ∅. Case 2.1: x ∨ u2 (y) ∈ T2 . Every class T that satisfies this condition is given in 2). Case 2.2: x ∨ u5 (y) ∈ T and x ∨ u2 (y) ∈ T . 1 1 1 , H84 , H86 } and the possibilities for T are given Then T11 is an element of {H81 in 3). Theorem 15.4.6.8 If T ⊆ R ∪ A1 , T1 ⊆ J1 ∪ U1 ∪ A1 , T1 ⊆ J1 ∪ V1 ∪ A1 and T1 ⊆ U1 ∪ V1 ∪ A1 , then T is exactly one of the following classes: 1) (J11 ∪ U6 ∪ V17 ) ∪ A4 , (J11 ∪ U3 ∪ V5 ) ∪ A4 , (J1 ∪ U6 ∪ V17 ) ∪ A2 , (J1 ∪ U3 ∪ V15 ) ∪ A2 , (J13 ∪ U4 ∪ V1 ) ∪ A31 , (J1 ∪ U1 ∪ V1 ) ∪ A1 (classes that contain {c0 , max}); 2) (J11 ∪ U6 ∪ V17 ) ∪ A7 , (J11 ∪ H19 ∪ H32 ) ∪ A12 , (J11 ∪ U6 ∪ V17 ) ∪ A10 , (J11 ∪ H19 ∪ H32 ) ∪ A (A ∈ {A14 , A15 , A14,r , A15,r | r ∈ N}), (J11 ∪ U6 ∪ V17 ) ∪ Ai (i ∈ {11, 13}), (J11,r ∪ H19 ∪ H32 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r) 1 ); (classes T with max ∈ T and T11 = H87 3) (J11 ∪ U3 ∪ V15 ) ∪ A7 , (J11 ∪ H24 ∪ H34 ) ∪ A12 , (J11 ∪ U3 ∪ V17 ) ∪ A10 , (J11 ∪ H24 ∪ H34 ) ∪ A (A ∈ {A14 , A15 , A14,r , A15,r | r ∈ N}), (J11 ∪ U3 ∪ V15 ) ∪ Ai (i ∈ {11, 13}),

498

4) 5)

6)

7)

8)


(J11,r ∪ H24 ∪ H34 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r) 1 ); (classes T with max ∈ T and T11 = H88 (J13 ∪ U4 ∪ V9 ) ∪ Ai (i ∈ {19, 20, 24, 25, 36, 37}) 1 ); (classes T with max ∈ T and T11 = H90 (J5 ∪ U6 ∪ V17 ) ∪ A5 , (J1 ∪ H19 ∪ H32 ) ∪ A6 , (J5 ∪ H19 ∪ H32 ) ∪ A12 , (J1 ∪ H19 ∪ H32 ) ∪ A12 , (J5 ∪ U6 ∪ V17 ) ∪ A10 , (J1 ∪ U6 ∪ V17 ) ∪ A10 , (J5 ∪ H19 ∪ H32 ) ∪ A8 , (Ji ∪ U6 ∪ V17 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {11, 13}), (Ji ∪ H19 ∪ H32 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {14, 15}), (J9,r ∪ H19 ∪ H32 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (Ji ∪ H19 ∪ H32 ) ∪ As,r (i ∈ {1, 5, 9}, s ∈ {14, 15}, r ∈ N) 1 ); (classes T with max ∈ T and T11 = H91 (J5 ∪ U3 ∪ V15 ) ∪ A5 , (J1 ∪ H24 ∪ H34 ) ∪ A6 , (Ji ∪ H24 ∪ H34 ) ∪ A12 (i ∈ {1, 5}), (Ji ∪ U3 ∪ V15 ) ∪ A10 (i ∈ {1, 5}), (J5 ∪ H24 ∪ H34 ) ∪ A8 , (Ji ∪ U3 ∪ V15 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {11, 13}), (Ji ∪ H24 ∪ H34 ) ∪ At (i ∈ {1, 5, 9}, t ∈ {14, 15}), (J9,r ∪ H24 ∪ H34 ) ∪ As,r (s ∈ {14, 15}; r, r ∈ N; r ≤ r), (Ji ∪ H24 ∪ H34 ) ∪ As,r (i ∈ {1, 5, 9}, s ∈ {14, 15}, r ∈ N) 1 ); (classes T with max ∈ T and T11 = H94 (J13 ∪ U4 ∪ Vi ) ∪ At (i ∈ {1, 3, 5}, t ∈ {18, 19, 23, 24, 36}), (J13 ∪ U4 ∪ Vi ) ∪ At (i ∈ {1, 3, 5, 7}, t ∈ {20, 25, 37}) 1 ); (classes T with max ∈ T and T11 = H94 M 1 ). (class T with max ∈ T and T11 = H91

Proof. Because of T ⊆ (J1 ∪ V1 ) ∪ A1 , we have c0 ∈ T1 . Case 1: max ∈ T2 . Then T = [T 1 ∪ {max}] and one obtains the possibilities for T given in 1) with the aid of Theorem 15.4.3.1. Case 2: max ∈ T2 . Since c0 ∈ T , we have T11 = Hi1 with i ∈ {87, 88, 90, 91, 92, 94, 95} by Theorems 15.4.3.1 and 15.4.4.7. With the aid of Theorems 15.4..4.7, 15.4.5.2– 15.4.5.5, one obtains the possibilities given in 2)–8) for T .

16 The Maximal Classes of Q ⊆ Ek

a∈Q

P olk {a} for

In this section we describe all maximal classes of the subclass TQ := P olk {a} a∈Q

of Pk for arbitrary Q with ∅ = Q ⊆ Ek , k ≥ 2. With the aid of these classes, a completeness criterion for TQ can easily be formulated. This criterion implies necessary and sufficient conditions regarding whether a finite algebra is semiprimal and has only trivial subalgebras. Moreover, if |Q| ≥ 2, we prove that every maximal class of TQ is an intersection of TQ with a certain maximal classes of Pk or P olk {a} (a ∈ Q).Presumably, something similar is valid for the maximal classes of TQ := ∈Q P olk , where Q ⊆ P(Ek ) and |Q | ≥ 2. For k = 3 this presumption was proven in [Lau 95b].

16.1 Notations We say that a relation is derivable from the relation with the aid of Invk TQ (or briefly is -derivable), if ∈ [{} ∪ Invk TQ ] (see Section 2.4). In this case we also write {} ∪ TQ or briefly

.

The following lemma provides the basis of later proofs and summarizes some well-known statements that can easily be checked. Lemma 16.1.1 (a) ∀, ∈ Rk : (P ol ⊆ TQ ∧ ({} ∪ Invk TQ ) =⇒ P ol ⊆ TQ ∩ P ol ); (b) Invk Pk = Dk (see Chapter 2);

500

16 The Maximal Classes of

a∈Q

P olk {a} for Q ⊆ Ek

(c) For every relation ∈ Invk TQ there are some relations 1 , ..., r ∈ {{a} | a ∈ Q} ∪ Dk and a relation ∈ Dk with = (1 × 2 × ... × r ) ∩ . Next we define some relation sets that we need to describe of the maximal classes of TQ (Q ⊆ Ek ). In this case, we also use the notations from Chapter 5. ⎧ { ∈ Mk | a is greatest or smallest element of Ek in respect to ⎪ ⎪ ⎪ ⎪ }, ⎪ ⎪ ⎪ ⎪ ⎨ if Q = {a}, Mk;Q := { ∈ Mk | a is greatest (smallest) and b is smallest (greatest) ⎪ ⎪ ⎪ element of Ek in respect to }, if Q = {a} and ⎪ ⎪ ⎪ ⎪ a = b, ⎪ ⎩ ∅ otherwise; ⎧ ⎪ ⎨ { ∈ Uk | ∀x ∈ Ek ∀q ∈ Q : (x, q) ∈ =⇒ x = q}, Uk;Q := if |Q| ≤ k − 2, ⎪ ⎩ ∅ otherwise; ⎧ ⎪ ⎨ { ∈ Sk | ∀x, y ∈ Ek : (x, y) ∈ ∧ x ∈ Q =⇒ y ∈ Q}, if |Q| = s · p, k = t · p, p prime, Sk;Q := ⎪ ⎩ ∅ otherwise; Pk,Q := { {(x, s(x)) | x ∈ Ek } | s is a permutation on Ek with exactly one fixed point (∈ Q); all proper cycles of s have the same prime number length, and s preserves Q }; ⎧ 3 { {(a, b, c) ∈ Ek | a +G b = c} | (Ek ; +G ) is an elementar Abelean ⎪ ⎪ ⎪ 2-group with the neutral element ⎨ Lk;Q := q }, ⎪ ⎪ if k = 2m , m ≥ 1 and Q = {q}, ⎪ ⎩ ∅ otherwise; Ck;Q := (C1k \{{q} | q ∈ Q}) ∪

k−1

h=2 {

∈ Chk | ∀q ∈ Q : q is a central element of }

(in particular, we have Ck;Ek = C1k \{{q} | q ∈ Ek }) and ⎧ 2 2 2 ⎪ ⎨ { (\ιk ) ∪ {(q, q)} | ∈ Ck;Q ∪ {Ek }}, if Q = {q}, Nk;Q := { ⊆ E 2 |∃q ∈ Q : {(x, q), (q, x) | x ∈ Ek } ⊆ ∧ τ ρ = ρ ∧ k ⎪ ⎩ ∩ ι2k = {(q, q)} ∧ ∩ ((Q\{q}) × (Ek \{q})) = ∅} , if 2 ≤ |Q|. Obviously, Nk;Q ⊆ a∈Q Nk;{a} . An element c ∈ Q with {(x, c), (c, x) | x ∈ Ek \{c}} ⊆ (∈ Nk;Q ) is called central element of . If one considers an arbitrary other finite set A instead of Ek , then the relation

16.2 Results of Chapter 16

501

sets MA , MA;Q , UA , UA;Q , SA , SA;Q , PA;Q , LA , LA;Q , CA , CA;Q , NA;Q and BA for Q ⊆ A can be defined when one replaces the set Ek by A in the above definitions or in the definitions of Chapter 5.

16.2 Results of Chapter 16 Our aim is to prove the following Theorem 16.2.1 ( [Sze 91], [Lau 82b], [[Lau 95a]) Let Rmax (TQ ) := Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Lk;Q ∪ Ck;Q ∪ Nk;Q ∪ Pk;Q . Then {TQ ∩ P olk | ∈ Rmax (TQ )} is the set of all maximal classes of TQ for ∅ = Q ⊆ Ek . The following theorem is a direct conclusion from the above theorem and from the fact that TQ is finitely generating (see Lemma 16.3.1): Theorem 16.2.2 (Completeness Criterion for TQ ) For an arbitrary subset M of TQ is valid: [M ] = TQ ⇐⇒ ∀ ∈ Rmax (TQ ) : M ⊆ TQ ∩ P olk .

The next theorem is a special case of Theorem 16.2.2: Theorem 16.2.3 (Completeness Criterion for the Class of all Idempotent Functions of Pk ) For an arbitrary subset M of TEk ( = n≥1 {f n ∈ Pk | f (x, ..., x) = x}) with k ≥ 3 is valid: [M ] = TEk ⇐⇒ ∀ ∈ C1k;Ek ∪ Sk;Ek ∪ Pk;Ek ∪ Nk;Ek : M ⊆ TEk ∩ P olk . Theorem 16.2.1 can also be formulated in the language of the Universal Algebra: 1 F A finite algebra (A; F ) (F ⊆ PA ) is called semi-primal, if [F ] = P olA InvA holds (see [Fos-P 64], [Den 82], [P¨ os-K 79], p. 143) or [Den-W 2002]. Let 1 F is the set of all uniSub(A) be the set of all subalgebras of A. Then InvA verses of algebras of Sub(A). If Sub(A)\{A} contains only 1-element algebras and Q := {a | ({a}; F ) ∈ Sub(A)} holds, then every ∈ Inv F ∩(PA;Q ∪Sk;Q )

502


a∈Q


defines a non-trivial automorphism s (s(a) = b :⇐⇒ (a, b) ∈ ) of the algebra A and the relations of Uk;Q are some non-trivial congruences of A. Then the following theorem follows from Theorem 16.2.1: Theorem 16.2.4 ([Sze 91]) Let A = (A; F ) be a finite algebra with the property that (SubA)\{A} contains only 1-element algebras. Then the following conditions are equivalent: (a) (A; F ) is semi-primal with {a ∈ A | ({a}; F ) ∈ SubA} = Q. (b) A has no proper automorphisms, is simple (i.e., A has only trivial congruences) and the direct product Ah of A for h = 2, 3, ..., |A|−1 has no subalgebra whose universe is an element of the set MA;Q ∪LA;Q ∪NA;Q ∪CA;Q .

16.3 Some Lemmas Lemma 16.3.1 TQ = [TQ3 ] for all Q with ∅ = Q ⊆ Ek . Proof. For k = 2, our assertion is valid by Chapter 3. Let k ≥ 3. Then the below-defined functions ∨, ·, w, ra;b , if {a, b} ⊂ Ek , qa,b;c , if a = b or {a, b} ⊆ Q and {a, b, c} ⊆ Ek , belong to TQ : x ∨ y := max(x, y), x · y := min(x, y) in respect to the total order 0 < 1 < 2 < . . . < k − 1; x if x = y ∈ Q, w(x, y) := 0 otherwise; b if x = a, ja;b (x) := 0 otherwise; ra;b (x, y, z) := x ∨ ja;b (y) · z; qa,b;c (x, y, z) := x ∨ ja;c (y) · jb;c (z). We show that the above-defined functions form a generating system for TQ . ... w with The function wn := w w n − 1 times x if x1 = x2 = ... = xn = x ∈ Q , wn (x1 , ..., xn ) = 0 otherwise

(n ≥ 1) is a superposition over w. Let f n be an arbitrary function of TQ , which is different from wn . Then one can represent f as follows:

16.3 Some Lemmas

f (x1 , ..., xn ) = wn (x1 , ..., xn )∨ ∈

a = (a1 , ..., an ) Ekn \{(q, q, ..., q) | q f (a) = 0

503

ja1 ;f (a) (x1 ) · ... · jan ;f (a) (xn ). ∈ Q}

Therefore, f is a superposition over the set B := {wn } ∪ {x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ) | (a1 , ..., an ) ∈ Ekn \{(q, q, ..., q) | q ∈ Q} ∧ b ∈ Ek }. An arbitrary function of B\{wn } is generated from functions of the type ra;b and qa,b;c , since ra1 ,...,ai ,ai+1 ;b (x, x1 , ..., xi , xi+1 , y) := ra1 ,...,ai ;b (x, x1 , ..., xi , rai+1 ;b (x, xi+1 , y)) = x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ) · y (i = 1, ..., n) and ra1 ,...,an ;b (x, x1 , ..., xn , qai ,aj ;b (x, xi , xj )) = x ∨ ja1 ;b (x1 ) · ... · jan ;b (xn ). Consequently, [T 3 ] = T. The next lemma is a conclusion of the above lemma: Lemma 16.3.2 For every Q with ∅ = Q ⊆ Ek , the lattice of the subclasses of TQ is dual atomar and TQ has only finite-many maximal classes. Since the maximal classes of TQ are clones1 , one can easily show the following Lemma 16.3.3 For every maximal class M of TQ there exists an h-ary relation M with M = P olk M and 1 ≤ h ≤ k 3 . Lemma 16.3.4 Let ∅ = Q ⊆ Ek . Then (a) ∀ ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Bk ∪ (Ck \{{q} | q ∈ Q}) : TQ ⊆ P olk ; (b) Vk↑ (TQ ) = {TQ | Q ⊆ Q} (T∅ := Pk ); (c) ∀ ∈ a∈Q (Pk;{a} ∪ Nk;{a} ) : TQ ∩ P olk ⊂ TQ . Proof. (a) and (c) are easy to check. (b): It is sufficient to prove the following statement for ∅ = Q ⊆ Ek and f ∈ Pk : (a ∈ Q ∧ f (a, a, ..., a) = a) =⇒ TQ\{a} ⊆ [TQ ∪ {f }]. Let f (a, a, ..., a) = a for a certain a ∈ Q, Q ⊆ Ek and f (x) := f (x, x, ..., x). To prove TQ\{a} ⊆ [TQ ∪{f }], let g m ∈ TQ\{a} be arbitrary. Then the function hm+1 defined by g 1

One can prove this analogous to the proof of footnote 1 of Chapter 14.

504


a∈Q


⎧ x, if x1 = ... = xm+1 = x ∈ Q, ⎪ ⎪ ⎨ u, if x1 = ... = xm = u ∈ Q\{a} ∧ hg (x1 , .., xm+1 ) := xm+1 = f (u), ⎪ ⎪ ⎩ g(x1 , ..., xm ) otherwise, belongs to TQ and g(x1 , ..., xm ) = hg (x1 , ..., xm , f (x1 )) is valid. Therefore, g ∈ [TQ ∪ {f }] holds. Lemma 16.3.5 For every relation γ ∈ R := Mk ∪ Uk ∪ Sk ∪ Lk ∪ Bk ∪ (Ck \{{q} | q ∈ Q}) there exists a γ ∈ [{γ} ∪ Invk TQ ], which belongs to Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Lk;Q ∪ Ck;Q ∪ (Pk;{q} ∪ Nk;{q} ). q∈Q

Proof. Examining the case γ ∈ R\(Mk;Q ∪ Uk;Q ∪ Sk;Q ∪ Ck;Q )

(16.1)

suffices. Further, we can assume that γ is h-ary, where h ≥ 2 because of (16.1). First we form the (h − 1)-ary relation γa := pr1,2,...,h−1 (∆({a} × γ))

(16.2)

for a ∈ Q. If h = 2 (i.e., γ ∈ (C2k \C2k;Q )∪(Mk \Mk;Q )∪(Uk \Uk;Q )∪(Sk \Sk;Q )), then we have (16.3) γa = {x ∈ Ek | (a, x) ∈ γ}. Because of (16.1), it is easy to check that there is an a ∈ Q such that the relation γa belongs to C1k;Q = {E ⊂ Ek | |E| ≥ 2 ∨ ∃b ∈ Ek \Q : E = {b}}: If γ ∈ Mk \Mk;Q , then for this purpose, one can choose a ∈ Q \ {o, e}, where o is the smallest element and e is the greatest element of Ek in respect to γ. If γ ∈ Uk \Uk;Q , then an a ∈ Q of an at least 2-element equivalence class of γ fulfills the above condition. If γ ∈ Sk \Sk;Q , then there exists an a ∈ Q and a b ∈ Ek \Q with (a, b) ∈ γ; thus γa = {b} ∈ C1k;Q . If γ ∈ C2k \C2k;Q , there is an a ∈ Q, which is no central element of γ. Consequently, we also have γa ∈ C1k;Q in this case. Now, let h ≥ 3 and γ ∈ Lk , i.e., γ ∈ Chk ∪ Bhk . Choose a ∈ Q \ C, if γ ∈ Ck and C is the set of all central elements of γ. For γ ∈ Bhk let a ∈ Q be arbitrary. Then the relation γa defined by (16.2) is an (h − 1)-ary reflexive, totally symmetric relation with the same central elements as γ and the new central element a. Since a is no central element of γ, there exist d1 , ..., dh−1 ∈ Ek with (a, d1 , ..., dh−1 ) ∈ γ. Consequently, we have γa = Ekh−1 and γa is a central relation. Through repetitions of the above construction, one obtains a γ-derivable relation of Ck;Q . Finally, let γ ∈ Lk , where k = pm , p prime and m ≥ 1. Then there is an elementar Abelean p-group (Ek ; +) with γ = {(x, y, u, v) ∈ Ek4 | x+y = u+v}. By Lemma 5.2.4.2, we can assume w.l.o.g. that the neutral element o of the p-group (Ek ; +) belongs to Q. For p = 2 we now form the γ-derivable relation

16.3 Some Lemmas

505

5 γ := pr3,4 (δ{0,1,2} ∩ ({o} × γ)) = {(x, y) ∈ Ek2 | x + y = o}.

Obviously, γ ∈ Pk;{o} , since for p = 2 the equation x + x = o is valid only if x = o. If p = 2, then one can form the relation γ := pr1,2,3 (∆({o} × γ)) = {(x, y, z) ∈ Ek3 | x + y = z} ∈ [{γ} ∪ Invk TQ ]. Thus γ ∈ Lk;{o} in the case {o} = Q. If |Q| ≥ 2 there is an a ∈ Q\{o} and the γ-derivable relation 4 γ := pr0,1 ((γ × {a}) ∩ δ{2,3} ) = {(x, a − x) | x ∈ Ek }

belongs to Sk . Further, we have that either γ ∈ Sk;Q is valid or (see above) it is possible to derive from γ a relation of C1k;Q . Lemma 16.3.6 Let |Q| ≥ 2 and γ ∈ a∈Q Pk;{a} ∪ Nk;{a} . Then there exists a ∈ [{γ} ∪ Invk TQ ] with ∈ Pk;Q ∪ Nk;Q ∪ C1k;Q . Proof. First let γ ∈ ( a∈Q Pk;{a} )\Pk;Q . Then there is (b, c) ∈ γ with b ∈ Q and c ∈ Ek \Q. The γ-derivable relation γb := pr1 (∆({b} × γ)) = {x ∈ Ek | (b, x) ∈ γ}

(16.4)

belong to Ck;Q (because of γb = {c}). Finally, let γ ∈ ( a∈Q Nk;{a} )\Nk;Q ; i.e., γ has the following properties: • ∃q ∈ Q : ι2k ∩ γ = {(q, q)} ∧ {(x, q), (q, x) | x ∈ Ek } ⊆ γ; • γ is symmetric and • ∃b ∈ Q \ {q} ∃c ∈ Ek \{b, q} : (b, c) ∈ γ. We form the γ-derivable relation γb (see (16.4)). Then we have {q, c} ⊆ γb and b ∈ γb . Thus γb ∈ C1k;Q . Lemma 16.3.7 Let A := P olk be TQ -maximal, where ∈ Rkt . Moreover, Vk↑ (A)\{A} = Vk↑ (TQ ). Then |Q| = 1 is valid and there exists a relation γ ∈ Pk;Q ∪ Nk;Q with A ⊆ P olk γ. Proof. Because of Lemma 16.3.4, the relation has the following properties: (I) If |Q| ≥ 2, then there is a relation γ ∈ [{} ∪ Invk TQ ] with γ ∈ (Rmax (Pk )\{{q} | q ∈ Q}) ∪ Rmax (T{a} ). a∈Q

In particular, it follows from (I) that t ≥ 2. W.l.o.g. we can assume the following three properties of :

506


a∈Q


(II) does not have any double rows. (III) Every -derivable (t−1)-ary relation belongs to Invk TQ (see 16.1.1,(c)). (IV) No -derivable t-ary relation with the properties P olk ⊂ TQ and | | < || exists. From assumptions (II) - (IV), some further properties of the relation follow: (V) does not have a constant row. (Suppose, has a constant row (a, a, ..., a). Because of (I) we have a ∈ Q and it is valid = pr0,...,i−1 × {a} × pri+1,...,t−1 for certain i ∈ Et . Then, with the help of (III) and 16.1.1,(c) one can prove that is an invariant of TQ , contrary to our assumptions about .) A direct consequence from (V) and (III) is: (VI) ∀i ∈ Et : pr0,...,i−1,i+1,...,t−1 = Ekt−1 . Next we show that t (VII) (t ≥ 3 ∨ |Q| = 1) =⇒ ∩ δ{0,1,...,t−1} ∈ {{(q, q, ..., q)} | q ∈ Q}; t (t = 2) =⇒ ∩ δ{0,1} ∈ {∅, {(q, q)} | q ∈ Q}

holds. By (III) we have t t ∩ δ{0,1,...,t−1} ∈ {∅, {(q, q, ..., q)}, δ{0,1,...,t−1} | q ∈ Q}. t t ∩ δ{0,1,...,t−1} = δ{0,1,...,t−1} is not possible, since A = P olk ⊂ TQ and TQ contains at most a constant. t = ∅, the equation pr0,...,t−2 = Ekt−1 (see (VI)) If Q = {a} and ∩ δ{0,1,...,t−1} t+1 implies prt (({a} × ) ∩ δ{0,...,t−1} ∈ C1k;{a} , a contradiction to (I). t = ∅ that prt−2,t−1 ( ∩ In the case that t ≥ 3, it follows from ∩ δ{0,1,...,t−1} t δ{0,...,t−2} ) ∈ Invk TQ , which contradicts (III). Therefore (VII) holds.

As generally known (see Chapter 6), the relation σi () := {(a1 , ..., ai ) ∈ Eki | ∃u ∈ Ek : {(a1 , u), ..., (ai , u)} ⊆ } is derivable from the relation for t = 2, and it is valid that (VIII) (t = 2 ∧ ◦ (τ ) = Ek2 ) =⇒ ∀i ≥ 2 : σi () = Eki . For the relation , the following three cases are possible: Case 1: t = 2. We consider the -derivable relation ∩ (τ ). Case 1.1: ∩ (τ ) ∈ {∅, {(q, q)}} for a certain q ∈ Q. Then is antisymmetric. Because of (VI) the relation ◦(τ ) has the property ι2k ⊆ ◦ (τ ). Case 1.1.1: ◦ (τ ) = ι2k . Because of pr0 = pr1 = Ek (see (VI)) we have || ≥ k. If || > k, there exists a, b, c ∈ Ek with (a, c), (b, c) ∈ and a = b. Thus (a, b) ∈ ◦ (τ ),

16.3 Some Lemmas

507

which is not possible because of ◦ (τ ) = ι2k . Therefore || = k, and has the form {(x, s(x)) | x ∈ Ek }, where s = e11 is a permutation on Ek , which has at most a fixed point (namely q). Assume the permutation has proper cycles of different length. Let r (≥ 2) be the length of a smallest proper cycle. Then we have pr0 (( ◦ ◦ ... ◦ ) ∩ ι2k ) ∈ C1k;Q , r times a contradiction to (I). Therefore, all proper cycles of s have the very same length l. Suppose l = p · m, p prime and m ≥ 2. Then ◦ ◦ ... ◦ has proper m times cycles of the length p; i.e., a relation of the set Sk ∪ Pk;{q} is derivable from the relation . But, this contradicts the condition (I) for |Q| ≥ 2 or ∈ Sk . Therefore, Q = {q} and A ⊆ P olk γ for a certain γ ∈ Pk;{q} in Case 1.1.1. Case 1.1.2: ι2k ⊂ ◦ (τ ) ⊂ Ek2 . In this case, we see that the relation = ◦ (τ ) is not an invariant of TQ and that P olk ⊆ TQ holds, which is not possible because of Vk↑ (A)\{A} = Vk↑ (TQ ). Case 1.1.3: ◦ (τ ) = Ek2 . By (VIII) we have σk () = Ekk . Consequently, there exists a u ∈ Ek with (x, u) ∈ for all x ∈ Ek . Because of ∩ ι2k ∈ {∅, {(q, q)}} (q ∈ Q), this is only possible for ∩ ι2k = {(q, q)} and u = q. If (τ ) ◦ = Ek2 is valid, we can prove {(x, q) | x ∈ Ek } ⊆ τ in analog mode too, through which we receive a contradiction to the antisymmetry of the relation . Therefore ι2k ⊆ (τ ) ◦ ⊂ Ek2 . Consequently, Case 1.1.3 is reducible to Cases 1.1.1 and 1.1.2. Case 1.2: {(q, q)} ⊂ ∩ (τ ) ⊂ . Because of condition (III) this case is not possible. Case 1.3: ∩ (τ ) = . In this case, is symmetric. Further, we have ι2k ⊆ ◦ . We distinguish three cases: Case 1.3.1: ◦ = ι2k . Then is a permutation with at most a fixed point q (if ∩ ι2k = {(q, q)}), and every proper cycle of has the length 2 because of symmetry of , i.e., ∈ Sk ∪ Pk;{q} . Thus, as in Case 1.1.1, we obtain a contradiction to the condition (I) either or |Q| = 1 and A ⊆ P olk γ for certain γ ∈ Pk;Q are valid. Case 1.3.2: ι2k ⊂ ◦ ⊂ Ek2 . This case can be excluded as Case 1.1.2. Case 1.3.3: ◦ = Ek2 . Because of ◦ = ◦ (τ ) = Ek2 and by (VIII), we have σk () = Ekk ,; i.e., there is u ∈ Ek with {(x, u) | x ∈ Ek } ⊆ . Consequently, ∩ ι2k = {(q, q)} and u = q. This and the symmetry of implies that q is a central element of . Therefore, belongs to Nk;{q} . Because of (I), this is only possible for |Q| = 1.

508


a∈Q


Case 2: t = 3. 3 = {(q, q, q)} for certain q ∈ Q (see Since pr0,1 = Ek2 (by (VI)), ∩ δ{0,1,2} (VII)) and (III) is valid, we have ∆ = Ek ×{q}, i.e., (a, a, q) ∈ for all a ∈ Ek . Analogously, one can prove that the tuples (a, q, a) and (q, a, a) belong to for every a ∈ Ek . Next we prove that is totally symmetric. Assume the relation is not totally symmetric. Then the -derivable relation {(as(0) , as(1) , as(2) ) | (a0 , a1 , a2 ) ∈ } := s s is permutation on E3 is totally symmetric. Because of ⎛ ⎞ a a q ⎝ a q a ⎠ ⊆ ⊂ (a = q) q a a the relation is not, however, an invariant of TQ . This is a contradiction to the condition (IV). Next we prove (IX) ∀a, b ∈ Ek : {(a, b, c), (a, b, c )} ⊆ =⇒ c = c . Suppose there are a, b, c, c with {(a, b, c), (a, b, c )} ⊆ , and c = c . Then the -derivable relation 6 ) 1 := pr0,2,5 (( × ) ∩ δ{0,3},{1,4} = {(x, y, z) | ∃u ∈ Ek : {(x, u, y), (x, u, z)} ⊆ } 3 has the property P olk 1 ⊆ TQ because of δ{0,1,2} ⊆ 1 . Consequently, 1 is a diagonal relation. Further, it is easy to check that (a, c, c ) ∈ 1 and (d, q, q) ∈ 1 for all d ∈ Ek hold. Therefore, 1 is the diagonal relation Ek3 . For our relation , this means that for arbitrary (x, y, z) ∈ Ek3 there is a u with (x, u, y) ∈ and (x, u, z) ∈ . Then, when one chooses x = y = q and z = q and considers the total symmetry of the relation , there exists an r with r = q and (q, q, r) ∈ . Above ∆ = Ek × {q} was, however, proven. Therefore, our assumption c = c was false. With that, (IX) is valid. Now we consider the -derivable relation

2 := ◦ = {(a, b, c, d) | ∃u ∈ Ek : {(a, b, u), (u, c, d)} ⊆ }. 4 Because of {(a, a, q), (q, a, a) | a ∈ Ek } ⊆ we have δ{0,1,2,3} ⊆ 2 and thus P olk 2 ⊆ TQ . Obviously, the relation 2 does not have any double rows. Because of our assumptions about the relation , however, this is possible only for 2 = Ek4 . Then, by definition of 2 , we have {(a, b, u), (u, c, a)} ⊆ for arbitrary a, b, c ∈ Ek , and certain u. Since is totally symmetric, however,

16.3 Some Lemmas

509

{(a, u, b), (a, u, c)} ⊆ , where b = c is possible, contrary to (IX). Therefore, Case 2 is not possible. Case 3: t ≥ 4. Because of pr0,1,...,t−2 = Ekt−1 (see (VI)), (II), (III) and (VII) we have ∩ t−1 t = δ{0,...,h−3} × {q} for certain q ∈ Q. Since pr0,...,h−3,h−1 = Ekt−1 δ{0,...,h−3} t−2 t is also valid, one can analogously prove ∩ δ{0,...,h−3} = δ{0,..,h−3} × {q} × Ek , contrary to that just shown. Therefore, there is no t-ary relation with the above-demanded properties and t ≥ 4. An equivalence relation ∼ is defined by ∼ :⇐⇒ TQ ∩ P olk = TQ ∩ P olk on a set A ⊆ Rk . We select a representative from every equivalence class of ∼ now and obtain a certain subset of A, which we denote with A∼ , where A ∈ {Pk;Q , Mk;Q , Sk;Q } in the following. Lemma 16.3.8 Let ∅ = Q ⊆ Ek . Then (a) ∀, ∈ Mk;Q : ∼ ⇐⇒ ( = τ ∨ = ); (b) ∀, ∈ Pk;Q ∪ Sk;Q : ∼ ⇐⇒ (∃t : = ◦ ◦ ... ◦ ); t times ∼ ∼ (c) ∀, ∈ M∼ k;Q ∪Sk;Q ∪Pk;Q ∪Uk;Q ∪Nk;Q ∪Lk;Q ∪Ck;Q : ∼ ⇐⇒ = . Proof. The statements (a) and (b) are direct conclusions from the proof of corresponding statements about relations of Mk ∪ Sk (see Chapter 5). To prove (c), we agree that o denotes the smallest element of Ek and that e denotes the greatest element of Ek (in respect to ∈ Mk;Q ). Because of (a), we can assume w.l.o.g. that o ∈ Q. Now, let and be two different relations ∼ ∼ of M∼ k;Q ∪ Sk;Q ∪ Pk;Q ∪ Uk;Q ∪ Nk;Q ∪ Lk;Q ∪ Ck;Q . To prove TQ ∩ P olk ⊆ TQ ∩ P olk

(16.5)

we distinguish the following 9 cases: ∼ Case 1: {, } ⊆ C1k;Q ∪ Uk;Q ∪ P∼ k;Q ∪ Sk;Q . If |Q| ≥ k − 1 then Uk;Q = Pk;Q = Sk;Q = ∅ and (16.5) is obviously valid for the relations , . Let now |Q| ≤ k − 2. Denote ω the relation {(q, q) | q ∈ Q} ∪ (Ek \Q)2 of Uk;Q . It is easy to check that the set A(γ) := {f n ∈ PEk \Q | ∃f ∈ TQ ∩P olk γ : (∀a ∈ (Ek \Q)n : f (a) = f (a))} n≥1

is a maximal class of PEk \Q for γ ∈ (C1k;Q \{γ | γ ⊆ Q ∨ γ = Ek \Q}) ∪ ∼ 1 (Uk;Q \{ω}) ∪ P∼ k;Q ∪ Sk;Q and that A(γ) = PEk \Q holds for γ ∈ {γ ∈ Ck | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}. With the aid of Chapters 5 and 6, our assertion (16.5) results from that for ∈ {γ ∈ C1k | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}. If ∈ {γ ∈ C1k | γ ⊆ Q ∨ γ = Ek \Q} ∪ {ω}, then it is also easy to prove that

510


a∈Q


there is a | |-ary function f1 ∈ TQ ∩ P olk with f1 ( ) ∈ . Consequently, (16.5) is also valid for the remaining relations of the Case 1. 1 Case 2: ∈ C1k;Q and ∈ M∼ k;Q ∪ (Ck;Q \Ck;Q ) ∪ Nk;Q ∪ Lk;Q . The following | |-ary function f2 with the properties f2 ( ) ∈ , f2 (q, ..., q) = q for all q ∈ Q and f2 (a) = c for the remaining tuples a, where c ∈ , belongs to TQ ∩ P olk . Thus (16.5) holds. ∼ ∼ 1 Case 3: ∈ M∼ k;Q ∪ Uk;Q ∪ Pk;Q ∪ Sk;Q and ∈ (Ck;Q \Ck;Q ) ∪ Nk;Q ∪ Lk;Q . Let be an h -ary relation. Further, let q ∈ Q and a ∈ Ek \Q. In addition, if ∈ Nk;Q , then we choose q ∈ Q so that (q, q) ∈ holds. Then the h -tuples (q, a, a, . . . , a), (a, q, a, . . . , a), . . . , (a, a, . . . , a, q) belong to , though not all belong to . Consequently, there exists an h -ary function f3 ∈ TQ ∩ P olk with ⎞ ⎛ q a a . . . a ⎟ ⎜ ⎜a q a . . . a⎟ ⎟∈ ⎜ f3 ⎜ ⎟ / . . . . . . . . ⎠ ⎝ a a a . . . q Hence (16.5) holds in Case 3. ∼ ∼ Case 4: ∈ P∼ k;Q ∪ Sk;Q and ∈ Mk;Q . This case can occur only for |Q| ≤ 2. Further, we have S∼ k;Q = ∅, if |Q| = 1 or k = 3. It is easy to check that (TQ ∩ P olk ) ⊆ (TQ ∩ P olk )1 is valid for k ≥ 3. Case 5: ∈ Uk;Q and ∈ M∼ k;Q . For o ∈ Q the binary function f4 defined by y if x = o , f4 (x, y) := o otherwise, belongs to TQ ∩ P olk . However, f4 does not preserve the relation with the smallest element o , since (f4 (o , e ), f (α, e )) = (e , o ) if α ∈ Ek \{o , e } . 1 Case 6: ∈ M∼ k;Q and ∈ Ck;Q . Let a ∈ Ek \ and

ta (x, y) :=

x if x = y ∈ Q, a otherwise.

The function ta ∈ TQ preserves and does not preserve . Therefore, (16.5) holds.

16.3 Some Lemmas

511

∼ ∼ Case 7: ∈ M∼ and ∈ M∼ k;Q ∪ Uk;Q ∪ Pk;Q ∪ Sk;Q . k;Q r | | , {r, s} ⊂ Ek and f5 a | |-ary function defined by Let = s if ⊆ , ∈ α f5 ( ) := β ∈ \ if ⊂

and, if r h. Set A = P olk El × . Then A contains every function f ∈ P olk El with

538

17 Maximal Classes of P olk El for 2 ≤ l < k

Im(f ) ⊆ T ∈ {{a1 , .., ar , b1 , ..., bh −r } | (a1 , ..., ar , b1 , ..., bh −r ) ∈ }. Then, because of h < h , it is not possible to describe A by a relation ⊆ Elr × Ekh−1 , contrary to our assumptions. Case r = 1 cannot occur, as shown by the following lemma: h Lemma 17.5.12 If r = 1 then ∈ Ck;E [1]. l

Proof. Because of Lemma 17.5.10,(a),(c) we must only show that has an (l; 1)-central element of El . The following relation is a -derivable relation: γt := {(a1 , ..., at ) ∈ El × Ekt−1 | ∃c ∈ El : ∀i1 , ..., ih−1 ∈ {1, ..., t} : (c, ai1 , ..., aih−1 ) ∈ }. First we show that γh = El × Ekh−1 holds. For this purpose, let (a1 , ..., ah ) be an arbitrary element of El × Ekh−1 . By (V II) for (a2 , ..., ah ) there is a certain α ∈ El with (α, a2 , ..., ah ) ∈ . Since is weakly (l, 1)-central because of Lemma 17.5.10,(c), we have (α, ai−1 , ..., aih−1 ) ∈ for all {i1 , ..., ih−1 } ⊆ {1, 2, ..., h} and 1 ∈ {i1 , ..., ih−1 }. Consequently, (a1 , ..., ah ) belongs to γh , whereby γh = El × Ekh−1 holds. Because of γh = El × Ekh−1 it is easy to prove that the relation γh+1 is (l; 1)-reflexive, which is valid only for γh+1 = El × Ekh because of Lemma 17.5.11. Therefore, by induction, we have γk = El × Ekk−1 . The existence of an (l; 1)-central element follows immediately from that. We can, therefore, always presuppose that r ≥ 2. By Lemma 17.5.10, the relation γr,s := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | (∃i ∈ {1, ..., s} : bi ∈ El ) ∨ (a1 , ..., ar , b1 , ..., bs ) ∈ ιr+s k [r]} is a subset of . Next we clarify for which r, s the equation = γr,s is valid. Lemma 17.5.13 The relation γr,s has the following properties: (a) (4 ≤ r ≤ l ∧ 1 ≤ s ≤ k − l) =⇒ γr,s ∈ [{γr−1,s } ∪ Invk P olk El ]; (b) (2 ≤ r ≤ l ∧ 3 ≤ s ≤ k − l) =⇒ γr,s ∈ [{γr,s−1 } ∪ Invk P olk El ]; (c) (3 ≤ r ≤ l ∧ s = 1) =⇒ γl,1 ∈ [{γr,1 } ∪ Invk P olk El ]; (d) (r = 2 ∧ 2 ≤ s ≤ k − l) =⇒ γ2,k−l ∈ [{γ2,s } ∪ Invk P olk El ]; (e) (3 ≤ r ≤ l ∧ 2 ≤ s ≤ k − l) =⇒ γl,k−l ∈ [{γr,s } ∪ Invk P olk El ]; Proof. (a): Obviously, the (r + s)-ary relation α1 := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | ∃u ∈ El : (a1 , a2 , ..., ar−2 , u, b1 , ..., bs ) ∈ γr−1,s ∧ (∀j1 , ..., jr−4 ∈ {1, 2, .., r − 2} : (u, aj1 , aj2 , ..., ajr−4 , ar−1 , ar , b1 , ..., bs ) ∈ γr−1,s )}

17.5 Not Through Relations of Rmax (Pl ) ∪ Rmax (Pk ) Describable Classes

539

is γr−1,s -derivable. Consequently, to prove (a), it suffices to show the equation α1 = γr,s : Let x := (a1 , ..., ar , b1 , ..., bs ) ∈ γr,s be arbitrary. The following table shows that γr,s ⊆ α1 . i 1 2 2.1 2.2 2.3 3

Case i for x ∃i ∈ {1, ..., s} : bi ∈ El ∃{i, j} ⊆ {1, ..., r} : i < j ∧ ai = aj {i, j} ⊆ {1, 2, ..., r − 2} i ∈ {1, 2, ..., r − 2} ∧ j ∈ {r − 1, r} i=r−1 ∧ j =r ∃{i, j} ⊆ {1, ..., s} : i < j ∧ bi = bj

u ∈ El such that x ∈ α1 u ∈ El u = ar−1 u = aj u = a1 u ∈ El

Suppose there is a tuple (x1 , ..., xr , y1 , ..., ys ) ∈ α1 \γr,s . Since the elements of this tuple are pairwise different, an element u ∈ El , for which (x1 , x2 ..., xr−2 , u, y1 ...ys ) ∈ γr−1,s holds, must belong to the set {x1 ..., xr−2 }. Analogously, one can show that the remaining conditions of the definition of α1 are only fulfilled if u ∈ {xr−1 , xr }. Therefore, no element u that fulfills all conditions from the definition of α1 exists, whereby α1 = γr,s is proven. (b): The relation α2 := {(a1 , ..., ar , b1 , ..., bs ) ∈ Elr × Eks | ∃u1 , u2 ∈ El : ∀i1 , ..., ir−2 ∈ {1, ..., r} : (u1 , a1 , ai1 , ..., air−2 , b1 , ..., bs−2 , bs−1 ) ∈ γr,s−1 ∧ (u2 , a2 , a11 , ..., air−2 , b1 , ..., bs−2 , bs ) ∈ γr,s−1 ∧ (u1 , u2 , ai1 , ..., air−2 , b2 , b3 , ..., bs−1 , bs ) ∈ γr,s−1 } is an γr,s−1 -derivable relation. Analogous to the proof of (a), one can show that α2 = γr,s . We only prove here that γr,s ⊆ α2 is valid. However, this follows from the following table, where x := (a1 , ..., ar , b1 , ..., bs ) ∈ γr,s is arbitrary. i 1 2 3 4 4.1 4.2 4.3 5

Case i for x ∃i ∈ {1, ..., s − 2} : bi ∈ El bs−1 ∈ El bs ∈ El ∃{i, j} ⊆ {1, ..., s} : i < j ∧ bi = bj {i, j} ⊆ {1, 2, ..., s − 2, s − 1} {i, j} ⊆ {1, 2, ..., s − 2, s} {i, j} ⊆ {2, 3, ..., s − 1, s} ∃{i, j} ⊆ {1, ..., r} : i < j ∧ ai = aj

u1 , u2 ∈ El such that x ∈ α1 u1 = u2 u2 = a2 u1 = a1 u1 u1 u1 u1

= u2 = a2 = u2 = a1 = a1 , u2 = a2 = a1 , u2 = a2

The statements (c)–(e) follow from (a) and (b). Then Lemma 17.5.13, (c)–(e) implies Lemma 17.5.14 If = γr,s then (r, s) ∈ {(2, 1), (l, 1), (2, k − l), (l, k − l)}.

540


Note: As is generally known, every relation that is preserved from all unary functions of Pk , is representable through the union of certain diagonal relations on Ek . Then, as one can easily prove, every relation that is preserved from all unary functions of P olk El is representable through the union of certain El -diagonal relations on Ek . Consequently, because of Lemma 17.5.14 (and Lemmas 17.3.9 and 17.3.10) all maximal classes of P olk El , which all functions of (P olk El )1 contain and fulfill the conditions of this section and of Case 2, were determined. We say that an h-ary, totally reflexive and totally symmetric relation γ (⊆ Elr × Eks ) is strongly (l; r)-homogeneous, if the following is valid: ∀a0 , ..., ar−1 ∈ El ∀b0 , ..., bs−1 ∈ Ek : ( (∃v0 , ..., vr−1 ∈ El : (v0 , ..., vr−1 , b0 , ..., bs−1 ) ∈ ∧ (∀i ∈ Er ∀j ∈ Er \{i} : (a0 , ..., aj−1 , vi , aj+1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ )) =⇒ (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ) Lemma 17.5.15 is either an (l; r)-central or a strong (l; r)-homogeneous relation. Proof. First, for t ∈ {r, r + 1, ..., l}, we consider the -derivable relation t := {(a0 , ..., at−1 , b0 , ..., bs−1 ) ∈ Elt × Eks | ∃c ∈ El : (∀i0 , ..., ir−2 ∈ Et : (ai0 , ..., air−2 , c, b0 , ..., bs−1 ) ∈ ))}. Since is totally (l; r)-symmetric, this relation is also totally (l; r)-symmetric. If one put c = a0 for t = r, then one can see that ⊆ r holds. Therefore, because of our assumption (V I), the following two cases are possible: Case 1: r = Elr × Eks . This case is only possible for r < l, since, in the opposite case, the definition of t implies = Ell × Eks . For r < l the -derivable relation r+1 is obvious totally (l; r + 1)-reflexive, whereby r+1 = Elr+1 × Eks follows with the aid of Lemma 17.5.11. One easily shows by induction that i = Eli × Eks is valid for h [r]. every i ∈ {r + 1, ..., l}. Then, l = Ell × Eks implies ∈ Ck;E l Case 2: = r . In this case, the relation is (l; r)-homogeneous; that is, it fulfills the condition (∃v ∈ El : ∀i ∈ Er : (a0 , ..., ai−1 , v, ai+1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ) =⇒ (a0 , a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ for arbitrary a0 , a1 , ..., ar−1 ∈ El and arbitrary b0 , ..., bs−1 ∈ Ek . The total (l; r)-symmetric relation


541

γt := {(a0 , ..., at−1 , b0 , ..., bs−1 ) | ∃v0 , v1 , ..., vt−1 ∈ El : (∀i1 , ..., ir ∈ Et : (vi1 , ..., vir , b0 , ..., bs−1 ) ∈ ) ∧ (∀n ∈ Et ∀j1 , ..., jr−2 ∈ Et \{n} : (aj1 , ..., ajr−2 , an , vn , b0 , ..., bs−1 ) ∈ )} (t ∈ {r, r + 1, ..., l}) is -derivable and, if γt−1 = Elt−1 × Eks , totally (l; t)reflexive. In particular, for r = t we have ⊆ γr . (To see this, one chooses vα = aα in the definition of γr for every α ∈ Er .) And = γr means that is strongly (l; r)-homogeneous. Now, by assumption (V I), we have that γr ∈ {, Elr × Eks } holds. Thus our lemma would be proven, if we could show that the case γr = Elr × Eks does not occur. Suppose γr = Elr × Eks and r < l. Then γr+1 is totally (l; r + 1)-reflexive and therefore γr+1 = Elr+1 × Eks because of Lemma 17.5.11. Thus, one can prove by induction that γl = Ell × Eks . Therefore, for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek , there are certain v0 , ..., vr−1 ∈ El with (v0 , v1 , ..., vr−1 , b0 , ..., bs−1 ) ∈

(17.11)

and ∀n ∈ Er ∀α1 , ..., αh−2 ∈ El \{an } : (α1 , ..., αr−2 , an , vn , b0 , ..., bs−1 ) ∈ . (17.12) By induction we show that for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek is valid: ∀t ≥ 0 : (a0 , ..., at−1 , vt , vt+1 , ..., vr−1 , b0 , ..., bs−1 ) ∈ .

(17.13)

For t = 0, (17.13) holds because of (17.11). Assume (17.13) is valid for t = n. Then, for t = n + 1 the statement (17.13) follows from this assumption, from the (l; r)-homogeneousness of (considering (17.12)) and from the total (l; r)-symmetry of , when one chooses v = vn for the tuple (a0 , ..., an , vn+1 , ..., vr−1 , b0 , ..., bs−1 ). (17.13) implies (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ for arbitrary a0 , ..., ar−1 ∈ El and b0 , ..., bs−1 ∈ Ek , contrary to ⊂ Elr × Eks . Consequently, our assumption γr = Elr × Eks is false and therefore is strongly (l; r)-homogeneous. To conclude our proof in this section, because of Lemma 17.5.15, it remains to show that if is strongly (l; r)-homogeneous the relation or a relation derivable from belongs to Bk;El [r]. Because of Lemma 17.5.10, we have only to prove conditions 5)–6) from the definition of an (l; r)-universal relation. Thus we can assume that is a strong (l; r)-homogeneous relation in the following. Lemma 17.5.16 For every b := (b0 , ..., bs−1 ) ∈ (Ek \El )s the relation εb := {(a, b) ∈ El2 | ∀a0 , ..., ar−3 ∈ El : (a0 , a1 , ..., ar−3 , a, b, b0 , ..., bs−1 ) ∈ } is an equivalence relation on El .

542


Proof. By the total (l; r)-reflexivity and total (l; r)-symmetry of , the reflexivity and symmetry of εb follows directly. To prove the transitivity of εb let {(a, b), (b, c)} ⊆ εb . Choosing v0 = α0 , v1 = α1 , ..., vr−3 = αr−3 , vr−2 = b and vr−1 = b in the definition of a strong (l; r)-homogeneous relation for the tuple (α0 , α1 , ..., αh−3 , a, c, b0 , ..., bs−1 ) we get (α0 , ...., αr−3 , a, c, b0 , ..., bs−1 ) ∈ for arbitrary α0 , ..., αr−3 ∈ El , whereby (a, c) ∈ εb . The following lemma is a direct consequence from Lemmas 17.5.10, 17.5.14, and 17.5.16: Lemma 17.5.17 If r = 2 then the strong (l; 2)-homogeneous relation beh longs to Bk;E [2]. l From the above lemma, we can assume that r≥3 in our further considerations. By the equivalence relation εb , the set El is partitioned into certain (nonempty) equivalence classes Ai [b] (i = 1, 2, ..., t[b]), from which we choose a representative αi [b]. With the help of the representative set Vb := {α1 [b], ..., αt[b] [b]}, we can define a mapping Fb : El −→ Vb by Fb (a) = αi [b] :⇐⇒ {a, αi [b]} ⊆ Ai [b]. Lemma 17.5.18 Let be strongly (l; r)-homogeneous. Then for every b := (b0 , ..., bs−1 ) ∈ (Ek \El )s \ιsk is valid: (a) (a0 , ...., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ (Fb (a0 ), ..., Fb (ar−1 ), b0 , ..., bs−1 ) ∈ ; (b) ((∀a0 , ..., ar−3 ∈ Vb : Vb ) =⇒ a = b.

(a0 , ..., ar−3 , a, b, b0 , ..., bs−1 ) ∈ ) ∧ {a, b} ⊆

Proof. (a): First we show that the equivalence (a, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ (b, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ (17.14) is valid for (a, b) ∈ εb and for arbitrary a0 , ..., ar−1 ∈ El . If (a, a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ , then (b, a1 , ...., ar−1 , b0 , ..., bs−1 ) ∈ follows from the strong (l; r)-homogeneousness of , choosing v0 = v1 = .... = vr−1 = a. Since εb is symmetric, we have also proven “⇐=” of (17.14). Now (a) follows from (ai , Fb (ai )) ∈ εb (i ∈ Er ), (17.14), Lemma 17.5.16 and of the total (l; r)-symmetry of :


543

(a0 , ...., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ ⇐⇒ (Fb (a0 ), a1 , ..., ar−1 , b0 , ..., bs−1 ) ∈ (Fb (a0 ), Fb (a1 ), a2 , ..., ar−1 , b0 , ..., bs−1 ) ∈ ⇐⇒ .. . (Fb (a0 ), ..., Fb (ar−1 ), b0 , ..., bs−1 ) ∈ .

⇐⇒

(b) is a conclusion from the definitions of εb and Vb and from (a). Now put ξb := Fb (b ) := {(Fb (a0 ), ..., Fb (ar−1 )) | (a0 , ..., ar−1 , b0 , ..., bs−1 ) ∈ }. By Lemma 17.5.18, (a) b is a homomorphic inverse image of this relation; i.e., it holds that b = {(a0 , ..., ar−1 ) ∈ Elr × Eks | (Fb (a0 ), ..., Fb (ar−1 )) ∈ ξb }. Because of = Elr ×Eks there is at least a b ∈ (Ek \El )s \ιsk with Elr ×{b} ⊆ . Thus w.l.o.g. we can assume Vb = Et[b] and (0, 1, ..., r − 1, b0 , ..., bs−1 ) ∈ (Vbr × {b})\ξb . To prove that has the properties 5) and (only for s ≥ 2) 6) from the definition of an (l; r)-universal relation, we consider the -derivable q-ary graphic Gq (P ol) := χq ∪ {g(κ1 , ..., κq ) | g ∈ (P ol)q }, where χq := (κ1 , ..., κq ) is the q-ary abscissa over Ek (in matrix form, see Section 2.7) and q ∈ N. Let m1 , ..., mi , n1 , ..., nj be the numbers of those rows of χri ·sj for which Ai,j := prm1 ,...,mi ,n1 ,...,nj χri ·sj is a matrix form of the relation Eri × {b0 , ..., bs−1 }j (r ≤ i ≤ l; s ≤ j ≤ k − l). Further, let µi,j := prm1 ,...,mi ,n1 ,...,nj Gri ·sj (P ol). Lemma 17.5.19 If is strongly (l; r)-homogeneous and (0, 1, 2, ..., r−1, b0 , ..., bs−1 ) ∈ , then µi,j = Eli × Ekj for all i, j with r ≤ i ≤ l and j = 1, if s = 1, and s ≤ j ≤ k − l. Proof. First we remark that = (Elr × Eks )\{(a0 , ..., ar−1 , b0 , ..., bs−1 ) | {a0 , ..., ar−1 } = Er }

544


h because of ∈ Ck;E [r]. Furthermore, if (β0 , ..., βr−1 ) ∈ , every function g l with Im(g) = {β0 , ..., βh−1 } belongs to P ol. Suppose the lemma is false for (i, j) = (r, s). Then we can form an h-ary relation with ⊂ ⊂ Elr × Eks with the help of the rr · ss -ary graphic of P ol by projection, contradicting (V I). Since all functions of P ol with at most ri · sj essential variables belong to i+1 j i j+1 (P ol)r ·s or (P ol)r ·s , it is easy to show by induction that the following holds:

∀i ∈ {r, ..., l − 1} : µi,j = Eli × Ekj =⇒ ιki+j+1 [i + 1] ⊆ µi+1,j and ∀i ∈ {s, ..., k − l − 1} : µi,j = Eli × Ekj =⇒ ιki+j+1 [i] ⊆ µi,j+1 , if s > 1. Consequently, our lemma follows from Lemma 17.5.11.

Let νs :=

rl if s = 1, rl · sk−l if s ≥ 2.

Because of Lemma 17.5.19, we can find a νs -ary function fb ∈ P ol for s ≥ 2 with den properties Im(fb ) = Vb ∪ (Ek \El ) and

⎛

⎞ Fb (0) ⎜ Fb (1) ⎟ ⎜ ⎟ ⎜ ⎟ ... ⎜ ⎟ ⎜ Fb (l − 1) ⎟ ⎜ ⎟. fb (Al,k−l ) = ⎜ ⎟ l ⎜ ⎟ ⎜ l+1 ⎟ ⎜ ⎟ ⎝ ⎠ ... k−1

In the case that s = 1, an analogous statement is valid for the matrix Al,1 : ⎞ ⎛ Fb (0) ⎜ Fb (1) ⎟ ⎟ ⎜ ⎟. ... fb (Al,1 ) = ⎜ ⎟ ⎜ ⎝ Fb (l − 1) ⎠ b0 Therefore, fulfills condition 6) of the definition of an (l; r)-universal relation. Let w0 , ..., ws−1 be certain rows of Al,k−l or Al,1 with f (wi ) = bi (i = 0, ..., s − 1).


545

Next we give some properties of the function fb on νs -ary tuples of Erνs , from h [r]. which will follow that ∈ Bk;E l We also give elements z of Erνs in the form (z[1], z[2], ..., z[νs ]) and, for every z ∈ Erνs , denote z t,a

(t ∈ {1, 2, ..., νs }, a ∈ Er )

an element of Erνs , which is defined by a, if i = t, z t,a [i] := z[i] otherwise (i ∈ {1, 2, ..., νs }). Lemma 17.5.20 Let t ∈ {1, 2, ..., νs }, z ∈ Erνs and (fb (z t,0 ), ..., fb (z t,r−1 )) ∈ ξb . Then (a) fb (z t,0 ) = ... = fb (z t,r−1 ), (b) ∀w ∈ Erνs : fb (wt,0 ) = ... = fb (wt,r−1 ). Proof. (a): For proof, it is sufficient to show that w.l.o.g. fb (z t,r−2 ) = fb (z t,r−1 ). First we prove ∀β0 , ..., βr−3 ∈ Erνs : (fb (β0 ), ..., fb (βr−3 ), fb (z t,r−2 ), fb (z t,r−1 )) ∈ ξb . (17.15) If {β0 [t], β1 [t], ..., βr−3 [t]} = {0, 1, ..., r − 3}, then (17.15) holds, since in this case all columns of the matrix ⎞ ⎛ β0 ⎜ β1 ⎟ ⎟ ⎜ ⎜ ... ⎟ ⎟ ⎜ ⎜ βr−3 ⎟ ⎜ t,r−2 ⎟ ⎟ B := ⎜ ⎜ z t,r−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w0 ⎟ ⎟ ⎜ ⎝ ... ⎠ ws−1 belong to ιhk [r] and fb preserves the relation . Let w.l.o.g. βi [t] = i for i ∈ Er−2 . Substituting the i-th row of B by z t,j , where i = j, we obtain a matrix Bi,j the columns of which belong to ιhk [r], whereby fb (Bi,j ) ∈ ξb × {b} holds. Consequently, by the strong (l; r)-homogeneousness of , choosing vj = fb (z t,j ) (j ∈ Er ) for the tuple (fb (β0 ), ..., fb (βr−3 ), fb (z t,r−2 ), fb (z t,r−1 ), b0 , ..., bs−1 ), we get (17.15). Since {fb (x) | x ∈ Erνs } = Vb , there are certain β0 , ..., βr−3 ∈ Erνs with fb (βi ) = ai (i ∈ Er−2 ) for arbitrary a0 , ..., ar−3 ∈ Vb . Consequently, by (17.15) we have

546


∀a0 , ..., ar−2 ∈ Vb : (a0 , ..., ar−3 , fb (z t,r−2 ), fb (z t,r−1 ), b0 , ..., bs−1 ) ∈ ξb × {b}. Because of Lemma 17.5.18, (b), this implies fb (z t,r−2 ) = fb (z t,r−1 ). (b): Because of (a), it is sufficient to show that (fb (wt,0 ), ...., fb (wt,r−1 )) ∈ ξb

(17.16)

holds. (17.15) implies (fb (wt,0 ), ..., fb (wt,r−3 ), fb (z t,r−2 ), fb (z t,r−1 )) ∈ ξb . Now it is easy to check that any exchange of an i-th row in ⎛ ⎞ wt,0 ⎜ ... ⎟ ⎜ t,r−1 ⎟ ⎜w ⎟ ⎜ ⎟ ⎜ w0 ⎟ ⎜ ⎟ ⎝ ... ⎠ ws−1 by wt,j for i ∈ Er−3 or by z t,j for i ∈ {r − 2, r − 1} and i = j gives a matrix M, whose columns belong to ιhk [r] and it holds f (M) ∈ ξb × {b}. Hence, by the strong (l; r)-homogeneousness of , choosing v0 = fb (wt,0 ), ..., vr−3 = fb (wt,r−3 ), vr−2 = fb (z t,r−2 ), vr−1 = fb (z t,r−1 ), we get (17.16). Lemma 17.5.21 For the function fb , there are certain digits t1 , ..., tm[b] that are exactly the essential digits of (fb )|Erνs (restriction of fb to Erνs ); i.e., it holds that ∀z, w ∈ Erνs : fb (z) = fb (w) ⇐⇒ ∀i ∈ {t1 , ..., tm[b] } : z[i] = w[i]. (17.17) Furthermore, f has the properties |Im(fb )| = rm[b] and ∀r1 , ..., rνs ∈ Err × {b} : ({rt1 , ..., rtm[b] } ⊆ ιhk [r] =⇒ fb (r1 , ..., rνs ) ∈ ) (17.18) Proof. For every t ∈ {1, 2, ..., νs }, we have either fb (z t,0 ) = fb (z t,1 ) = ... = fb (z t,r−1 ) for every z ∈ Erνs or (fb (z t,0 ), fb (z t,1 ), ..., fb (z t,r−1 ), b0 , ..., bs−1 ) ∈


547

for all z ∈ Erνs by Lemma 17.5.20. Let T := {t1 , ..., tm[b] } be the set of all t ∈ {1, 2, ..., νs }, for which (fb (z t,0 ), ..., fb (z t,r−1 ), b0 , ..., bs−1 ) ∈ holds. Now we will show that the digits ti ∈ T of fb have the properties of Lemma 17.5.21. First let fb (z) = fb (w) for certain z, w ∈ Erνs and assume there exists an i ∈ T with α := z[i] = w[i]. Then the columns of the matrix ⎛ i,0 ⎞ z ⎜ ... ⎟ ⎜ i,α−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w ⎟ ⎜ i,α+1 ⎟ ⎟ ⎜z ⎟ A := ⎜ ⎜ ... ⎟ ⎜ i,r−1 ⎟ ⎟ ⎜z ⎟ ⎜ ⎜ w0 ⎟ ⎟ ⎜ ⎝ ... ⎠ ws−1 belong to ιhk [r], whereby fb (A) ∈ ξb × {b} holds. Because of fb (w) = fb (z) = fb (z i,α ) follows (fb (z i,0 ), ..., fb (z i,r−1 ), b0 , ..., bs−1 ) ∈ ξb , contrary to i ∈ T and the definition of T . Therefore “=⇒” in (17.17) holds. Let now z[i] = w[i] for every i ∈ T and w.l.o.g. T = {1, 2, ..., m[b]}. fb (w) = fb (z) is proven if we can show that fb (un−1 ) = fb (un )

(17.19)

holds for every tuple un := (z[1], ..., z[n], w[n + 1], w[n + 2], ..., w[νs ]) and all n ∈ {m + 1, m + 2, ..., νs }, since fb (um ) = fb (w) and fb (uνs ) = fb (z). n,j By n > m[b] and the definition of T , we have fb (un,i n−1 ) = fb (un−1 ) for arbin,w[n] n,z[n] trary i, j ∈ Er . As un−1 = un−1 and un−1 = un , we have (17.19), whereby (17.17) is proven. From (17.17) and {fb (x) | x ∈ Erνs } = Vb follows |Vb | = rm[b] . Finally we prove (17.18). Assume (17.18) is false. Then there are certain z0 , z1 , ..., zr−1 ∈ Erνs with (fb (z0 ), fb (z1 ), ..., fb (zr−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} and (w.l.o.g.) zi [1] = i for every i ∈ Er and 1 ∈ T . Let w, z ∈ Erνs be arbitrary. Then each column of the matrix

548


⎛

Ci,j

w1,0 ⎜ ... ⎜ 1,j−1 ⎜w ⎜ 1,i ⎜ z ⎜ 1,j+1 ⎜w := ⎜ ⎜ ... ⎜ 1,r−1 ⎜w ⎜ ⎜ w0 ⎜ ⎝ ... ws−1

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

belongs to ιhk [r] for i = j. Thus fb (Ci,j ) ∈ ξb × {b}. By choosing vi = fb (z 1,i ) for the tuple (fb (w1,0 ), ..., fb (w1,r−1 ), b0 , ..., bs−1 ) and i ∈ Er , (fb (w1,0 ), ..., fb (w1,r−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} follows from this and the strong (l; r)-homogeneousness of , contrary to the definition of T and 1 ∈ T . Lemma 17.5.22 Let be strongly (l; r)-homogeneous, b ∈ (Ek \El )s \ιsk and Elr × {b} ⊆ . Then for certain m[b] ∈ N there is a bijective mapping ϕb from Erm[b] onto Vb with ξb = ϕb (ξm[b] ) := {(ϕb (a0 ), ..., ϕb (ar−1 )) | (a0 , ..., ar−1 ) ∈ ξm[b] } (ξm[b] denotes an r–ary elementary relation) and it holds that −1 b = {(a0 , ..., ar−1 ) ∈ Elr | (ϕ−1 b (F (a0 )), ..., ϕb (F (ar−1 ))) ∈ ξm[b] }; h i.e., ∈ Bk;E [r]. l

Proof. Subsequently, we denote m[b] with m. Because of Lemma 17.5.21, a bijective mapping from Erm onto Vb is defined by ϕb (a(m−1) hm−1 + a(m−2) hm−2 + ... + a(1) h + a(0) ) = fb (z) :⇐⇒ (z[t1 ], z[t2 ], ..., z[tm ]) = (a(m−1) , a(m−2) , ..., a(0) ), where a ∈ Erm . Let now (a0 , ..., ar−1 ) ∈ ξm . Then there are certain z0 , ..., zr−1 ∈ Erνs with the following properties: (z0 [i], ..., zr−1 [i]) ∈ ιrr for every i ∈ {1, 2, ..., νs } and (ϕb (a0 ), ϕb (a1 ), ..., ϕb (ar−1 )) = (fb (z0 ), fb (z1 ), ..., fb (zr−1 )). Then, because of fb ∈ P ol, we have (ϕb (a0 ), ..., ϕb (ar−1 ), b0 , ..., bs−1 ) ∈ ξb × {b} and therefore ϕb (ξm ) ⊆ ξb . Suppose there exists

17.6 Classes Describable by Relations of Rmax (Pl ) ∪ Rmax (Pk )

549

(a0 , ..., ar−1 ) ∈ ξb \ϕb (ξm ). Then, by definition of ϕb , there exist certain z0 , ..., zr−1 ∈ Erνs and an i ∈ {t1 , ..., tm } with (fb (z0 ), ..., fb (zr−1 )) = (a0 , ..., ar−1 ) and (z0 [i], ..., zr−1 [i]) ∈ ιrr . But this is contrary to (17.18). Thus ξb = ϕb (ξm ). The remaining assertions follow from Lemma 17.5.18, (a) and the definition of ξb . Summing up, we obtain in Case 2.2, h h ∈ Ck;E [r] ∪ Bk;E . l l

17.6 Classes Describable by Relations of Rmax(Pl) ∪ Rmax(Pk) The aim of this section is to prove the following: Theorem 17.6.1 Let be an h-ary relation of Rmax (Pl ) ∪ Rmax (Pk ). Then P olk × El is a maximal class of P olk El if and only if ∈ Rmax (Pl ) ∪ Uk;El ∪ Ck;El . With the help of Sections 5.2 and 5.4, one can easily prove the following lemma: Lemma 17.6.2 It holds: (a) ∀ ∈ Rmax (Pk )\Rmax (Pl ) : prEl (P olk × El ) = Pl =⇒ P olk × El is not maximal in P olk El . (b) ∀, ∈ Rmax (Pk ) : P olk = P olk ∧ prEl P olk × El = prEl P olk × El = Pl =⇒ P olk × El = P olk × El . Lemma 17.6.3 For every relation ∈ Mk ∪ Sk ∪ Lk , there is a certain -derivable relation which belongs to Rmax (Pl ) ∪ Ck;El . Proof. Let be an arbitrary h-ary relation of Mk ∪ Sk ∪ Lk . Then the following relations are -derivable: := ∩ Elh and El := pr1,...,h−1 (∆(El × )). We distinguish two cases for : Case 1: ∈ Mk ∪ Sk . Then the following three cases are possible for the relation : Case 1.1: = ∅. This case is possible only for ∈ Sk and the relation El ( = {x ∈ Ek | ∃a ∈ El : (a, x) ∈ } ) belongs to C1k;El , since (by = ∅) (a, b) ∈ for all a ∈ El implies b ∈ El .

550


Case 1.2: ∅ ⊂ ⊂ ι2l or ι2l ⊂ ⊂ El2 . Then, ∈ Rl \Dl , and because of Chapter 6, one can derive a relation of Rmax (Pl ) from the relations of { } ∪ Dl . Case 1.3: = ι2l . This case is possible only for ∈ Mk . Then, all elements of El are incomparable in respect to the partial order relation , and both the smallest element o and the greatest element e of Ek (in respect to ) belong to the set Ek \El . Since e ∈ El and o ∈ El , we have El ⊆ El and El ⊂ El ⊂ Ek for the above defined relation El , obviously. Consequently, El ∈ C1k;El and our assertion also holds in Case 1.3. Case 2: ∈ Lk . In this case, k = pm (p prime, m ≥ 1) and = {(a, b, c, d) ∈ Ek4 | a + b = c + d}, where (Ek ; +) is an elementar Abelean p-group. W.l.o.g. (see Lemma 5.2.4.2) we can assume that the element 0 ∈ El is the neutral element of (Ek ; +). Then the relation is not El -diagonal because of {(a, 0, 0, a), (a, 0, a, 0), (0, a, 0, a), (0, a, a, 0) | a ∈ El } ⊆ ⊆ El4 and (0, 1, 1, 1) ∈ . Consequently, by Chapter 6, one can derive a relation of Rmax (Pl ) from relations of {} ∪ Dl . Lemma 17.6.4 For every ∈ Bhk (3 ≤ h ≤ k) the clone P olk × El is not maximal in P olk El . Proof. Let ∈ Bhk . We consider the -derivable relations r := ∩ (Elr × Ekh−r ) (0 ≤ r ≤ h). Since ιhk ⊆ and h ≥ 3, the sets ιhl and {(x1 , ..., xh ) ∈ Ekh | |{x1 , ..., xh }| ≤ 2} are subsets of . Consequently, the relations r do not have any double rows for every r ∈ {0, 1, ..., h}, and it holds that ∀i ∈ Eh : pri r = pri Elr × Ekh−r . Therefore, r ∈ Invk P olk El implies r = Elr × Ekh−r . Let r∗ be the smallest number, for which is valid r∗ ∈ Invk P olk El and r ∈ Invk P olk El for all r < r∗ . The following cases are possible: Case 1: r∗ = 0. Because of the above remarks, we have 1 = El × Ekh−1 ⊆ , whereby every element c ∈ El is a central element of the relation , which cannot, however, be for the relation ∈ Bk . Case 2: r∗ = 1. In this case, the relation 1 is totally (l; 1)-reflexive, totally (l; 1)-symmetric, weakly (l; 1)-central and different from El × Ekh−1 . In addition, we can assume that for every (a2 , ..., ah ) ∈ Ekh−1 there exists a c ∈ El with (c, a2 , ..., ah ) ∈ 1 .


551

Namely, if this is false, we can derive a relation of Ck by forming pr1,...,h−1 1 , whereby P olk ×El is not a maximal class of P olk El because of Lemma 17.6.2. Let γt := {(a1 , ..., at ) ∈ El × Ekt−1 | ∃c ∈ El : ∀i2 , ..., ih ∈ {1, 2, ..., t} : (c, ai2 , ..., aih ) ∈ 1 }. This relation has the property (γt = El ×Ekt−1 ∧ t ≥ h) =⇒ γt+1 = El ×Ekt ∨ (∃ ∈ Ck : γt ) (17.20) as one can prove: Because of γt = El × Ekt−1 and the above properties of 1 , the relation γt+1 is strongly (l; 1)-reflexive, strongly (l; 1)-symmetric and := pr1,...,t+1 γt+1 = Ekt , then γt+1 ∈ Ck . weakly (l; 1)-central. Thus, if γt+1 t t If γt+1 = Ek , then for every (a2 , ..., at+1 ) ∈ Ek there is an α ∈ El with (α, a2 , ..., at+1 ) ∈ γt+1 and we can show γt+1 = El × Ekt , as follows: Let (a1 , a2 , ..., at+1 ) ∈ El × Ekt . Then there exists an α ∈ El with (α, a2 , ..., at+1 ) ∈ γt+1 . Because of the definition of γt+1 , this implies ∃c ∈ El ∀i2 , ..., ih ∈ {2, ..., t + 1} : (c, ai2 , ...aih ) ∈ 1 . In addition, we have ∀i2 , ..., ih ∈ {1, 2, ..., t+1} : (∃q ∈ {2, ..., h} : iq = 1) =⇒ (c, ai2 , ..., aih ) ∈ 1 , since 1 is a weakly (l; 1)-central relation. Therefore (a1 , a2 , ..., at+1 ) ∈ γt+1 and γt+1 = El × Ekt is proven, i.e., (17.20) holds. Analogous to the just carried out considerations or to the proof of Lemma 17.5.12, one can show γh = El × Ekh−1 . From that and from (17.20), it follows that a relation of Ck is -derivable. Consequently, P olk × El is not maximal in P olk El . Case 3: r∗ ≥ 2. We consider the (proper subclass) A := P olk r∗ of P olk El , where P olk ×El ⊆ A holds, since r∗ is -derivable. Next we show that P olk × El = A (and with that, our assertion) through the construction of a function f ∈ A, which does not preserve. For this purpose, let T be a matrix, which is a matrix representation of the relation \Elh . Then, the |T |-ary function f defined by f (T ) = α for certain α ∈ Ekh \ and |T | f (x) = 0 for the remaining tuples x ∈ Ek preserves r∗ , but does not preserve . Lemma 17.6.5 Let ∈ Uk . Then the class P olk × El is P olk El -maximal if and only if ∈ Uk;El . Proof. Let [a] := {x ∈ Ek | (a, x) ∈ }. If ∈Uk;El , then we have either := ∩ El2 ∈ {ι2l , El2 } (and therefore ∈ Ul ) or a∈El [a] ∈ {El , Ek }. In the first case a relation of Ul ⊆ Rmax (Pl ) is -derivable. In the second case, the -derivable relation El := pr1 (∆(El × )) = {x ∈ Ek | ∃a ∈ El : (a, x) ∈ } belongs to C1k;El . Our assertion results from that and from Lemma 17.3.3.

552


Lemma 17.6.6 Let ∈ Ck . Then, P olk × El is P olk El -maximal if and only if ∈ C1l ∪ Ck;El . Proof. If is a unary relation and ∈ C1l ∪C1k;El , then the -derivable relation ∩ El belongs to C1l ∪ C1k;El . Therefore, our assertion for unary relation follows from Lemmas 17.3.1 and 17.3.2. Let now ∈ Chk with 2 ≤ h ≤ k − 1 and let P olk × El be P olk El -maximal, where it is not possible to describe the class P olk × El with the aid of a relation of Rl \Dl . Because of Lemmas 17.3.5, and 17.3.6–17.3.8 it is sufficient proof to show that ∈ Ck;El is valid. Obviously, ∩ El2 ∈ {ι2l , El2 } if h = 2 and Elh ⊆ if h ≥ 3 result from our assumptions about the relation . Then the following cases are possible: Case 1: h = 2 and ∩ El2 = ι2l . Then every central element of belongs to Ek \El . We form 1 := ∩ (El × Ek ). This relation has the following properties: 1 ∩ El2 = ι2l , pr0 1 = El , 1 ⊂ El × Ek , and ∃c ∈ Ek \El : El × {c} ⊆ 1 {c} ∪ El ⊆ pr1 1 . If pr1 1 = Ek , then one can derive a relation of C1k;El from 1 and therefore from . Consequently, the -derivable relation 1 belongs to Zk;El . If there are a, c ∈ Ek \El and b ∈ El with {(a, b), (b, c)} ⊆ and (a, c) ∈ , then the binary function f defined by ⎧ (x, y) = (a, b), ⎨ a if (x, y) = (b, c), f (x, y) := c if ⎩ b otherwise, preserves 1 , but does not belong to P olk ×El . Since, by assumption, P olk × El is P olk El -maximal, ∀a, c ∈ Ek ∀b ∈ El : {(a, b), (b, c)} ⊆ =⇒ (a, c) ∈ results. Hence ∈ ZC2k;El and our lemma follows from Lemmas 17.3.5 and 17.3.6. Case 2: h ≥ 2 and Elh ⊆ . Case 2.1: has a central element c which belongs to El . For the -derivable relation := ∩ (El × Ekh−1 ) the following two cases are possible: Case 2.1.1: = El × Ekh−1 . Then, ∈ Invk P olk El and it holds


553

P olk × El ⊂ P olk ⊂ P olk El , since one can easily check that the h-ary function f with ⎛ ⎞ ⎛ ⎞ c l l ... l a1 ⎜ l c l ... l ⎟ ⎜ a2 ⎟ ⎟ ⎜ ⎟ f⎜ ⎝ . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ l l l ... c ah and f (x) = c for the remaining tuples x of Ekh preserves the relation , however f does not preserve the relation . Case 2.1.2: = El × Ekh−1 . In this case, every element of El is a central element of and belongs to Ck;El . Case 2.2: Every central element of belongs to Ek \El . If 1 = El2 × Ekh−2 then the -derivable relation 1 := ∩ (El2 × Ekh−2 ) is not an invariant of P olk El and P olk × El ⊂ P olk 1 ⊂ P olk El , holds, since the h-ary function f with ⎛ ⎛ ⎞ ⎞ c 0 0 ... 0 a1 ⎜ l c 0 ... 0 ⎟ ⎜ a2 ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . . . . . . . . . . . ⎠ := ⎝ ... ⎠ ∈ l 0 0 ... c ah (c is a central element von ) f (x) := 0 for x ∈ Elh and f (x) := c for the remaining tuples x belongs to P olk 1 \P olk . Therefore, we can assume that El2 × Ekh−2 ⊆ or {(a1 , ..., ah ) ∈ Ekh | ∃i = j : {ai , aj } ⊆ El } ⊆ .

(17.21)

We consider the relation h := {(a0 , ..., ah−1 ) | ∃u ∈ El : ∀i ∈ Eh : (a0 , ..., ai−1 , u, ai+1 , ..., ah−1 ) ∈ }, which is totally symmetric. W.l.o.g. we can assume that for arbitrary a := (a1 , a3 , ..., ah ) ∈ Ekh−1 there is a ua ∈ El with (ua , a1 , a3 , ..., ah ) ∈ . Namely, if this is not valid, we have that the -derivable relation := pr1,...,h−1 ((El × Ekh−1 ) ∩ ) ; i.e., El is the set of all central elements of . belongs to Ch−1 k Because of symmetry of we have in addition (a1 , ua , a3 , ..., ah ) ∈ . Then,

554


with the total reflexivity of , (a1 , a1 , a3 , ..., ah ) ∈ h holds. Thus h is totally reflexive. Next we show that each element u ∈ El is a central element of h : Let (α, a2 , ..., ah ) ∈ El × Ekh−1 be arbitrary. Then, as already shown, there is a u ∈ El with (u, a2 , ..., ah ) ∈ . In addition, because of (17.21), we have (α, a2 , ..., ai−1 , u, ai+1 , ..., ah ) ∈ for each i ∈ {2, ..., h}. By definition of h , this implies (α, a2 , ..., ah ) ∈ h . Therefore, h is either a central relation with central elements of El or h = Ekh holds. Because of Lemma 17.6.2, we must continue to examine the case h = Ekh . We form the relation t := {(a1 , ..., at ) | ∃u ∈ El : ∀{i1 , ..., ih−1 } ⊆ {1, ..., t} : (u, ai1 , ..., aih−1 ) ∈ } (h + 1 ≤ t ≤ k). Then, t−1 = Ekt−1 implies ιtk ⊆ t (h + 1 ≤ t ≤ k). Therefore, we have either t = Ekt or t ∈ Invk P olk El . Since does not have any central elements of El , there is a tuple (α1 , α2 , ..., αh ) ∈ (El × Ekh−1 )\. Then the h-ary function g with ⎞ ⎛ ⎞ ⎛ c 0 0 ... 0 α1 ⎜ l c 0 ... 0 ⎟ ⎜ α2 ⎟ ⎟ ⎟ ⎜ g⎜ ⎝ . . . . . . . . . . . ⎠ := ⎝ ... ⎠ ∈ αh l 0 0 ... c (c is a central element of ) g(x) := α1 otherwise, does not preserve . However, g preserves the relation t for t ≥ h + 1. Thus P olk × El is P olk El -maximal only if t = Ekt holds for all t ∈ {h, ..., k}. It results, however, from k = Ekk that has a certain central element of El , which we had excluded in Case 2.2. Therefore, this case cannot occur. A direct conclusion from Lemmas 17.6.2–17.6.6 is Theorem 17.6.1, whereby the theorems of Section 17.2 are also proven.

18 Further Submaximal Classes of Pk

As already indicated in Chapter 14, there are few results about submaximal classes for arbitrary k. Supplementary to Chapters 16 and 17, all submaximal classes of a maximal class of the type S are described in this chapter. It is then shown how one can prove the special case k ∈ P of this general description. The rest of this chapter deals with submaximal classes of Pk which lie below a maximal class of the type U. In Section 18.2, one can find some maximal classes of P olk , where ∈ Uk is arbitrary. Then, in Section 18.3, the list from Section 18.2 is completed to the list of all maximal classes of P olk for 2 = Ek−1 ∪ {(k − 1, k − 1)}.

18.1 The Maximal Classes of P olks for s ∈ Sk In this section, p always denotes a prime number. Further, let k := p · l with l ∈ N. Denote s a fixed point free permutation on the set Ek , whose cycles have the same length p. Furthermore, let s := {(x, s(x)) | x ∈ Ek }. By Chapter 5 S := P olk s is a maximal class of type S. For a description of the maximal classes of S, we need the following concepts and definitions. Let θs := {(x, y) ∈ Ek2 | ∃i ∈ {0, 1, ..., p − 1} : y = si (x)}. It is easy to see that θs is an equivalence relation on Ek . An h-ary relation γ ∈ Rk is called θs -closed, if ∀(x1 , ..., xh ) ∈ γ ∀(y1 , ..., yh ) ∈ Rkh : (∀i ∈ {1, ..., h} : (xi , yi ) ∈ θs ) =⇒ (y1 , ..., yh ) ∈ γ.

556


Obviously, a θs -closed relation ⊆ Ekh is the homomorphic inverse image of an h-ary relation on the factor set (quotient set) (Ek )/θs in respect to the mapping Ek −→ (Ek )/θs , x → x/θs . It is easy to check that the following statements are valid: – An equivalence relation is θs -closed if and only if θs ⊆ . – A central relation is θs -closed if and only if it is the homomorphic inverse image of a central relation defined on (Ek )/θs . – An h-regular relation , which is defined as in Section 5.2.1 1 , is θs -closed if and only if θs ⊆ ϑ0 ∩ ϑ1 ∩ ... ∩ ϑm−1 . An equivalence relation ε on Ek is called transversal to s, if s ∈ P olk ε and ε ∩ θs = ι2k ; i.e., the permutation s maps each equivalence class (block) of ε onto another equivalence class of ε. / µ whenever A unary relation µ on Ek is called transversal to s, if si (x) ∈ i ∈ {1, ..., p − 1} and x ∈ Ek . Let q, r be prime numbers and let n be the least positive integer with q n = 1 (mod r). Moreover, denote GF (q n ) a field with the order q n .2 Let G(q, r) be the set of all mappings of the form GF (q n ) −→ GF (q n ), x → a · x + b with a, b ∈ GF (q n ) and ar = 1. It is easy to see that the algebra G(q, r) := (G(q, r); 2) is a group. P. P. Palfy proved the following fact (see [Ros-S 85]): Lemma 18.1.1 (without proof ) A finite group G has a maximal subgroup of order p (p prime) if and only if G is isomorphic to one of the groups listed below: (1) an Abelean group of order p · q, where q ∈ P; (2) G(p, q), where q ∈ P and p = 1 (mod q); (3) G(p, q), where q ∈ P \ {p}. Now we define permutation groups with the aid of groups of Lemma 18.1.1. For a group G := (G; ◦) whose order divides k = |G|·r, we consider a partition of Ek into |G|-element blocks A1 , ..., Ar and select arbitrary bijections ϕi : Ai −→ G, 1 2

See also Lemma 5.2.6.3. As generally known, there is (up to isomorphism) exactly one finite field of order qn .

18.1 The Maximal Classes of P olk s for s ∈ Sk

557

1 ≤ i ≤ r. Then, with the help of the mappings ϕi , for every g ∈ G one can define a permutation πg on Ek as follows: ∀i ∈ {1, ..., r} ∀x ∈ Ai : πg (x) := ϕ−1 (g ◦ (ϕi (x))). It is easy to check that the set of all permutations of the form πg with g ∈ G together with the operation 2 forms a group, which will be called a semiregular representation of the group G. ´ Szendrei. The following theorem was found by I. G. Rosenberg and A. Theorem 18.1.2 (Rosenberg-Szendrei Theorem; [Ros-S 85]; without proof ) The following list describes all maximal classes of S := P olk s : (1) P olk {(x, π(x)) | π ∈ G}, where G is a semiregular representation of a group, which belongs to the ones described in Lemma 18.1.1, (1)–(3) and for which s ∈ G is valid; (2) S ∩ P olk λG , where λG ∈ Lk , G := (Ek , ⊕) is an elementary Abelean p-group (see Section 5.4) and there exists an element c ∈ Ek with s(x) = x ⊕ c; (3) S ∩ P olk ε, where ε ∈ Uk is either θs -closed or transversal to s. (4) S ∩ P olk γ, where γ ∈ Ck and γ is either θs -closed or a nonempty unary relation transversal to s; (5) S ∩ P olk , where ∈ Bk and is θs -closed. There are also certain k for which the description of the maximal classes of S is simple, as the following theorem shows: Theorem 18.1.3 ([Sze 84], [Lau 84b]) Let k = p ∈ P and s ∈ Sp . Then S := P olp s has exactly two maximal classes. One maximal class has the type (2) and the other maximal has the type (4) from Theorem 18.1.2. Choosing s(x) := x + 1 (mod p), one can describe these maximal classes as follows: S ∩ P olp {0} and S ∩ P olp λ with λ := {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod p)}. The above theorem is a special case of the following theorem: Theorem 18.1.4 ([Lau 84b]) Let k ∈ N, s(x) := x + 1 (mod k), S := P olk {(x, s(x)) | x ∈ Ek } 3 and let Tk be the set of all divisors ∈ N of k. Then the following list gives all maximal classes of S: 3

S is a maximal clone of Pk iff k ∈ P. For k ∈ P the set S is a subclone of a certain clone of the type U.

558


(1) S ∩ P olk γr , where r ∈ Tk \ {1} and γr := {x ∈ Ek | r divides x}; (2) S ∩ P olk t , if k ∈ P, t ∈ Tk \ {1, k} and t := {(x, y) ∈ Ek | r divides (x − y)}; (3) S ∩ Lk , if k ∈ P and Lk := P olk {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod k)} = n n i=1 ai · xi (mod k)}. n≥1 {f ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + Proof. We prove the above theorem with the aid of Theorem 6.1 and Theorem 8.3.2. Let α be the mapping defined in Theorem 8.3.2. First we prove that the classes described in (1)–(3) are S-maximal. For this purpose, let r ∈ Tk \{1}, t ∈ Tk \{1, k} and A ∈ {P olk γr , P olk t , Lk } be arbitrary. By Chapter 5, the classes P olk γr , P olk t and Lk are maximal in Pk . Consequently, because of Theorem 8.3.2, (a), we have to show that α(S ∩ A) = A holds. Obviously, because of c0 ∈ A, we have α(S ∩ A) ⊆ A. For the proof of A ⊆ α(S ∩ A), we show that every function f n ∈ S with α(f ) ∈ A belongs to A. We distinguish three cases: Case 1: A = P olk γr . Let a1 , ..., an ∈ γr and f n ∈ S with α(f ) ∈ P olk γr be arbitrary. By definition of γr , bi := ai − a1 (mod k) is an element of γr for every i ∈ {1, ..., n}. Thus, because of α(f ) ∈ P olk γr , we have b := f (0, b2 , ..., bn ) ∈ γr and therefore b + a1 (mod k) = f (a1 , ..., an ) ∈ γr . Consequently, f preserves the relation γr , and α(S ∩ P olk γr ) = P olk γr is valid. Case 2: A = P olk t . Let (a1 , b1 ), ..., (an , bn ) ∈ t and f n ∈ S with α(f ) ∈ P olk t be arbitrary. Set ci := ai − a1 (mod k) and di := bi − b1 (mod k) for i = 1, ..., n. Then (ci , di ) ∈ t . Further, because of α(f ) ∈ P olk t , we have u 0 c2 ... cn ∈ t . := f 0 d2 ... dn v Since f ∈ S,

f

a1 a2 ... an b1 b2 ... bn

=

u + a1 (mod k) v + b1 (mod k)

∈ t

results. Consequently, f preserves the relation t and α(S ∩ P olk t ) = P olk t holds. Case 3: A = Lk . It is easy to check that S ∩ Lk = n≥1 {f n ∈ Pk | ∃a0 , ..., an ∈ Ek : f (x) = a0 + n i=1 ai · xi (mod k) ∧ a1 + ... + an = 1 (mod k)}. This implies α(S ∩ Lk ) = Lk . Now we come to the completeness proof. For this purpose, we choose an arbitrary

18.1 The Maximal Classes of P olk s for s ∈ Sk

559

subset M of S with M ⊆ A for every class A defined in (1)–(3). Our theorem is proven if we can show that [M ] = S holds. S 1 = {s, s2 , ..., sk−1 = e11 } ⊆ [M ] follows from the following considerations: Because of M ⊆ S ∩ P olk γk , there is a certain function f n ∈ M with a := f (0, 0, ..., 0) = 0. Consequently, ∆n−1 f = sa belongs to [M ]. If a and k are relatively prime, then [sa ]1 = S 1 . If a and k are not relatively prime, there is a t ∈ Tk \ {1} with [sa ]1 = {sx | x ∈ γt }. Because of M ⊆ S ∩ P olk γt , a certain function gtm with b := gt (a1 , ..., am ) ∈ γt for certain a1 , ..., am ∈ γt belongs to M . Consequently, we have gt (sa1 , ..., sam ) = sb ∈ [M ]. Further, there exists a t ∈ Tk \ {t} with [sa , sb ]1 = {sx | x ∈ γt }. If t = 1, then S 1 ⊆ [M ]. If t = 1, there is a function gt ∈ [M ], which does not preserve γt . Analogous to the above, one can form a function sc ∈ S 1 \ [sa , sb ] as a superposition over {gt , sa , sb }, and two cases are possible. The iteration of the construction above shows S 1 ⊆ [M ]. Because of Theorem 8.3.2, [M ] = S is proven if we can show that α([M ]) = Pk holds. By Theorem 6.1, α([M ]) = Pk is proven if one can show that for every relation ∈ Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk a function f ∈ [M ] that does not preserve the relation exists. Since α(sa ) = a and sa ∈ [M ] for a ∈ Ek , the constants 0, 1, ..., k − 1 belong to α([M ]). Further, [M ] ⊆ α([M ]) by Theorem 8.3.2, (b). Because of {a, sa | a ∈ Ek } ⊆ α([M ]) we have ∀ ∈ C1k ∪ Sk ∪ Mk : α([M ]) ⊆ P olk .

(18.1)

Let ∈ Chk with 2 ≤ h ≤ k−1, let c be a central element of and (a1 , ..., ah ) ∈ Ekh \. Then the function sa1 −c ∈ [M ] does not preserve because of ⎞ ⎛ ⎛ ⎞ a1 c ⎜ a2 − a1 + c (mod k) ⎟ ⎜ a2 ⎟ ⎟ ⎜ ⎜ ⎟ sa1 −c ⎜ ⎟ = ⎜ . ⎟. .. . ⎝ ⎠ ⎝ . ⎠ . ah ah − a1 + c (mod k) Consequently, we have ∀ ∈

k−1

Chk : α([M ]) ⊆ P olk .

(18.2)

h=2

Next we determine all such relations ∈ Uk that are preserved from the permutation s. In the following, let be an arbitrary equivalence relation on Ek and U := {x ∈ Ek | (0, x) ∈ }. Suppose, s ∈ P olk . Then because of (sk−a (a), sk−a (b)) = (0, b − a) and (sa (0), sa (b − a)) = (a, b), we have ∀(a, b) ∈ Ek2 : (a, b) ∈ ⇐⇒ (0, b − a) ∈ ⇐⇒ b − a ∈ U.

(18.3)

With the help of (18.3) and with the properties of equivalence relations, one can easily prove that U is a subgroup of the cyclic group (Ek ; + (mod k)). As is generally known, every subgroup of a cyclic group with k elements can be described with the aid of a divider of k. Hence, there exists a t with t ∈ Tk and U = {x ∈ Ek | t|x}. By (18.3), this implies = t . Consequently, if k ∈ P, s preserves only trivial equivalence relations and, if k ∈ P, s preserves only equivalence relations, defined in (2). Therefore, by our assumptions of M , we have

560

18 Further Submaximal Classes of Pk ∀ ∈ Uk : α([M ]) ⊆ P olk .

(18.4)

Because of |Im(f )| = k for all f ∈ S, S 1 = L1p and M ⊆ S ∩ Lp for p ∈ P and, if k ∈ P, S 1 = (P olk t )1 and M ⊆ S ∩ P olk t for t ∈ Tk \ {1, k} α([M ]) ⊆ P olk ιkk

(18.5)

is obviously valid. Next, let ∈ Bhk be arbitrary with 3 ≤ h ≤ k − 1. Then by definition there is a surjective mapping q : Ek −→ Ehm with m ≥ 1 and (a1 , ..., ah ) ∈ ⇐⇒ (q(a1 ), ..., q(ah )) ∈ ξm , where ξm is an h-ary elementary relation on Ehm (see Chapter 5). If q is a bijective mapping, then with the aid of Theorem 5.2.6.1, one can prove that s does not preserve the relation . If q is not bijective, then the mapping equivalence σ := {(x, y) ∈ Ek2 | q(x) = q(y)} belongs to Uk . We have shown that a function fσn , which does not preserve the relation σ, belongs to α([M ]) ∩ S. Therefore, there are tuples (a1 , b1 ), ..., (an , bn ) ∈ σ with (c, d) := (fσ (a1 , ..., an ), fσ (b1 , ..., bn )) ∈ σ. Because of ιhh ∩ Ehhm ⊆ ξm , we have {(ai , bi , x3 , ..., xh ) | x3 , ..., xh ∈ Ek } ⊆ for all i = 1, ..., n. The definition of and the choice of the elements c and d imply the existence of certain u3 , ..., uh ∈ Ek with (c, d, u3 , ..., uh ) ∈ . Im(fσ ) = Ek holds because of fσ ∈ S. Consequently, there exist certain (ai , bi , ci3 , ..., cin ) ∈ , i = 1, ..., n, with ⎛ ⎞ ⎛ ⎞ c a 1 a2 . . . a n ⎜ b1 b2 . . . b n ⎟ ⎜ d ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ fσ ⎜ ⎜ c13 c23 . . . cn3 ⎟ = ⎜ u3 ⎟ . ⎝ ................ ⎠ ⎝ ... ⎠ c1h c2h . . . cnh uh Hence, fσ does not preserve the relation ∈ Bhk and ∀ ∈

k−1

Bhk : α([M ]) ⊆ P olk

(18.6)

h=3

is valid. It still remains to be proven that ∀λ ∈ Lk : α([M ]) ⊆ P olk λ.

(18.7)

For this purpose let λ ∈ Lk . Then there exist p ∈ P, m ∈ N with k = pm and an elementary Abelean p-group (Ek ; ⊕) with the property λ := {(a, b, c, d) ∈ Ek4 | a ⊕ b = c ⊕ d}. W.l.o.g.4 let 0 be the neutral element of the group (Ek ; ⊕). If k ∈ P, then either S 1 ⊆ P olk λ or λ = {(a, b, c, d) ∈ Ek4 | a + b = c + d (mod p)}. If k ∈ P, (18.6) results from that and then (with the aid of (18.1), (18.2), (18.4)–(18.6) and Theorems 8.3.2 and 6.1) our theorem. Let k ∈ P in the following. 4

See Section 5.2.4

18.2 Some Maximal Classes of a Maximal Class of Type U

561

To prove (18.7), we show first that there is a relation γ ∈ Uk ∪ Mk ∪ Ck ∪ Bk with S ∩ P olk λ ⊆ S ∩ P olk γ. It is easy to check that the following three relations are invariants of S ∩ P olk λ: λ1 := {(a1 , ..., ap ) ∈ Ekp | a1 ⊕ a2 ⊕ ... ⊕ ap = 0}, λ2 := {(a1 , ..., ap , b1 , ..., bp ) ∈ Ek2·p | a1 ⊕ a2 ⊕ ... ⊕ ap = b1 ⊕ b2 ⊕ ... ⊕ bp } and, where + denotes the addition modulo k, := {(i, i + 1, i + 2, ..., i + p − 1) | i ∈ Ek }. For i ∈ Ek we set: ti := (i ⊕ (i + 1) ⊕ (i + 2) ⊕ ... ⊕ (i + p − 1), where + is the addition modulo k. Because of k = p the equations t0 = t1 = ... = tk−1 are not possible. If there is a j with tj = 0 (this is possible only for p = 2), then pr1 ( ∩λ1 ) ∈ C1k . If ti = 0 for all i ∈ Ek , there are certain r, s with tr = ts and r = s, whereby the relation := pr1,p+1 (( × ) ∩ λ2 ) is a binary reflexive non-diagonal relation. A conclusion of Chapter 6 is the fact that a relation γ ∈ Uk ∪ Mk ∪ Ck ∪ Bk is derivable from . Consequently, we obtain from the assumption α([M ]) ⊆ P olk λ a contradiction to the statements (18.1), (18.2), and (18.4)–(18.6). Our assumption was therefore false, and (18.7) is valid. Thus, the set α([M ]) is no subset of an arbitrary maximal class of Pk . Hence, α([M ]) = Pk is valid. This and Theorem 8.3.2 imply [M ] = S.

18.2 Some Maximal Classes of a Maximal Class of Type U Let be an arbitrary relation of Uk , where Ai (i ∈ Et ) denote the equivalence classes of this relation. In addition, a/ denotes the equivalence class of , which contains a ∈ Ek . One checks the following lemma easily (see Table 18.1):5 5

See also Lemma 1.4.6.

562


Lemma 18.2.1 Let Ai (i ∈ Et ) be a partition the set Ek and ai ∈ Ai (i ∈ Et ). Furthermore let y for ∃i ∈ Et : x = ai and y ∈ Ai , x y := x otherwise. Then, an arbitrary function f n ∈ Pk is a superposition over the functions z, gf , fi (i ∈ Et ), defined by z(x, y) := x y, gf (x1 , ..., xn ) := ai ⇐⇒ f (x1 , ..., xn ) ∈ Ai , f (x1 , ..., xn ) for f (x1 , ..., xn ) ∈ Ai , fi (x1 , ..., xn ) := otherwise ai (i ∈ Et ), and a representation of f is given by f (x) = ((...((gf (x) f0 (x)) f1 (x)) ...) ft−1 (x)).

(18.8)

Table 18.1

x f (x) gf (x) fi (x) gf f0 (gf f0 ) f1 .. }= a0 }= ai }= f (x) }= f (x) . }∈ A0 .. }= a1 }= ai }= a1 }= f (x) . }∈ A1 .. }= a2 }= ai }= a2 }= a2 . }∈ A2 .. .. .. .. .. .. . . . . . . .. }= ai }= f (x) }= ai }= ai . }∈ Ai .. .. .. .. .. .. . . . . . . .. }= at−1 . }∈ At−1 }= at−1 }= ai }= at−1 Remark

(18.8) is also valid, if one defines y for ∃i ∈ Et : {x, y} ⊆ Ai , x y := x otherwise

... ... ... ... .. . ... .. . ...

(18.9)

and if gf fulfills the following condition: ∀x ∈ Ekn : (gf (x), f (x)) ∈ .

Let f n ∈ Pk be arbitrary and let (18.8) be a representation of f . Then f ∈ P olk if and only if gf ∈ P ol , since the functions fi (i ∈ Et ) and


563

the function z preserve . Obviously, every function of the form gf (∈ P ol ) is unambiguously characterized through its values on {ai | i ∈ Et }n . Consequently, ({gf | f ∈ P ol }; ζ, τ, ∆, ∇, ) is isomorphic to (Pt ; ζ, τ, ∆, ∇, ). Let q a unary function of Pk defined by ∀i ∈ Et : q(x) = ai :⇐⇒ x ∈ Ai . Lemma 18.2.2 follows from the above considerations: Lemma 18.2.2 ∀f n ∈ Pk : f ∈ P ol ⇐⇒ ∃ hn ∈ P{a0 ...,at−1 } : gf (x) = h(q(x1 ), ..., q(xn )). 2 Since [Pk2 ] = Pk and [Pk,A ] = Pk,A , the fact

ord P olk = 2 follows from Lemmas 18.2.1 and 18.2.2 (see also proof of Theorem 11.2.2). In analog mode to a corresponding statement for Pk , the following lemma results from that then: Lemma 18.2.3 For every proper subclone T of P olk there is a maximal clone of P olk , which contains T . P olk has only finite many maximal clones. 2 For every maximal class M of P olk , there is a relation ψ ∈ Rkk with M = P olk ψ. W.l.o.g. let |A0 | ≥ |A1 | ≥ ... ≥ |At−1 | and A0 := {0, 1, ..., l − 1},

l≥2

in the following. For the purpose of defining a homomorphism αϕ : Pk −→ Pt we need the mapping ϕ defined by ϕ : Ek −→ Et , ∀i ∈ Et ∀x ∈ Ai : ϕ(x) := i. Obviously, αϕ : Pk −→ Pt , f → f αϕ ∀x1 , ..., xn ∈ Ek : f αϕ (ϕ(x1 ), ..., ϕ(xn )) := ϕ(f (x1 , ..., xn )) is a homomorphism. Next we describe maximal clones of

564


U := P olk which contain all unary functions of U or which have the form P olk ∩ P olk , where P olk is Pk -maximal. Some of these clones can be described in the form P olk βi for certain indices i. Put λ := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ {0, 1}}, ιhi := {(a1 , ..., ah ) ∈ Eih | |{a1 , ..., ah }| ≤ h − 1}, −1 ϕ (λ) if t = 2, β1 := ϕ−1 (ιtt ) if t ≥ 3, λ := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | ∃i : |Ai | = 2 ∧ {a, b} ⊆ Ai }∪ {(x, x, x, x) | ∃j : Aj = {x}}, λ if l = 2, β2 := ι if l ≥ 3. For the purpose of describing a third type of maximal clones of U , we first define the following relation for i1 , ..., ir ∈ N: γi1 ,...,ir := {(x1,1 , x1,2 , ..., x1,i1 , x2,1 , x2,2 , ..., x2,i2 , ..., xr,1 , xr,2 , ..., xr,ir ) ∈ Eki1 +...+ir | ∀j ∈ {1, ..., r} ∃q : {xj,1 , xj,2 , ..., xj,ij } ⊆ Aq }. Then we can define the relations βi1 ,...,ir , as follows: x := (x1,1 , x1,2 , ..., x1,i1 , x2,1 , x2,2 , ..., x2,i2 , ..., xr,1 , xr,2 , ..., xr,ir ) ∈ βi1 ,...,ir :⇐⇒ x ∈ γi1 ,...,ir ∧ (∃s ∈ {i1 , ..., ir } : |{xs,1 , xs,2 , ..., xs,ij }| ≤ ij − 1). For the case |A2 | = ... = |At | = 1 we need the following relations: σr,s := {(x1 , ..., xr , y1 , ..., ys ) ∈ Ekr+s | x1 = ... = xr ∨ (x1 , ..., xr ) ∈ ιrl ∨ ({x1 , ..., xr } = El =⇒ |{x1 , ..., xr , y1 , ..., ys }| ≤ r + s − 1} (2 ≤ r ≤ l, 1 ≤ s ≤ k − l). Lemma 18.2.4 Let ∅ ⊂ σ ⊂ Ek . Then P olk σ ∩ P olk is a maximal clone of U := P olk if and only if the following condition is valid: (∃I ⊂ {0, 1, ..., t − 1} : σ = Ai ) ∨ (∀j ∈ {0, 1, ..., t − 1} : σ ∩ Aj = ∅). i∈I

(18.10)


565

Proof. “=⇒”: Let P olk σ ∩ P olk be U -maximal and let σ := {y ∈ Ek | ∃x ∈ Ek : (x, x, y) ∈ σ × }. The following two cases are possible: Case 1: ∅ ⊂ σ ⊂ Ek . In this case, we have U ∩ P olk σ ⊆ U ∩ P olk σ ⊂ U and σ has the form ∃I ⊆ {1, 2, ..., t} : σ =

Ai ,

i∈I

since

a ∈ σ =⇒ (∀b ∈ a/ : b ∈ σ )

holds (a/ := {x ∈ Ek | (a, x) ∈ }). Case 2: σ = Ek . (Such a case occurs, for example, if k = 3, := {(0, 0), (1, 1), (2, 2), (0, 1), (1, 0)} and σ := {0, 2}.) It is easy to check that σ fulfills the condition ∀j ∈ {0, 1, ..., t − 1} : σ ∩ Aj = ∅ in this case. “⇐=”: Let f n ∈ U \P olk σ, where σ has the property (18.10). Then there are a1 , ..., an ∈ σ with f (a1 , ..., an ) = a ∈ Ek \σ. Since the constant cs with s ∈ σ belongs to U ∩ P olk σ, we have ca ∈ [(U ∩ P olk σ) ∪ {f }]. Let now g m ∈ U be arbitrary. We show that g ∈ [(U ∩ P olk σ) ∪ {f }]. For this purpose, we consider a function hm+1 ∈ U with hg (x1 , ..., xm , a) = g g(x1 , ..., xm ). Our assertion is proven, ifwe can show that there is a such function hg in the set U ∩ P olk σ. If σ = i∈I Ai for a certain I ⊂ Et , we set g(x1 , ..., xm ), if xm+1 ∈ a/, hg (x1 , ..., xm+1 ) := s otherwise, where s ∈ σ. Obviously, hg ∈ U ∩ P olk σ. Let σ ∩ Aj = ∅ for all j ∈ Et . Then one must choose hg (a1 , ..., am , am+1 ) ∈ (b/) ∩ σ, if hg (a1 , ..., am , am+1 ) = b was chosen and (ai , ai ) ∈ for every i ∈ {1, ..., m+1} is valid. For the remaining tuples x, one can set e.g. hg (x) = s, where s ∈ σ, so that hg ∈ U ∩ P olk σ. Lemma 18.2.5 (1) Let γ ⊆ Eth and let P olt γ Pt -maximal. Furthermore, let −1 αϕ (γ) := {(a1 , ..., ah ) ∈ Ekh | (αϕ (a1 ), ..., αϕ (ah )) ∈ γ}. −1 Then P olk αϕ (γ) is U -maximal.

566


(2) P olk β1 is U -maximal. (3) Let A be a subset of U with U 1 ⊆ A and A ⊆ P olk β1 . Then ∀f ∈ Pt ∃g ∈ [A] : αϕ (g) = f.

(18.11)

−1 Proof. (1): Let f ∈ U \P olk αϕ (γ) be arbitrary. Then we have obvious −1 [αϕ (P olk αϕ (γ) ∪ {f })] = Pt . Consequently, −1 (γ) : αϕ (g) = h. ∀h ∈ Pt ∃g ∈ P olk αϕ

(18.12)

−1 For arbitrary functions q ∈ U it follows from the definition of P olk αϕ (γ): −1 (γ) ⇐⇒ αϕ (q) ∈ P olt γ. q ∈ P olk αϕ

(18.13)

Case 1: {c0 , c1 , ..., ct−1 } ⊂ P olt γ. With the help of (18.13) and the fact that the constant functions c0 , ..., ct−1 of Pt belong to P olt γ, one gets: {} ∪

t−1

−1 Pk,Ai ⊆ P olk αϕ (γ).

(18.14)

i=0 −1 Then (with the help of Lemma 18.2.1), (18.12), and (18.14) imply [P olk αϕ (γ)∪ −1 {f }] = U , i.e., P olk αϕ (γ) is U -maximal. Case 2: γ ⊂ Et . In this case, our assertion follows from Lemma 18.2.4. Case 3: γ ∈ St . Because of (18.12), one can find a unary function o with αϕ (o) = c0 in −1 (γ) ∪ {f })]. Since a unary function p with [αϕ (P olk αϕ

∀x ∈ A0 : p(x) = 0 −1 belongs to P olk αϕ (γ), the constant c0 = p o is a superposition over −1 P olk αϕ (γ) ∪ {f }. Let g ∈ U m be arbitrary. Then there is an (m + 1)-ary function hg with hg (0, x1 , ..., xm ) = g(x1 , ..., xm ) −1 −1 in P olk αϕ (γ), whereby g = hg c0 ∈ [P olk αϕ (γ)∪{f }] holds. Consequently, −1 P olk αϕ (γ) is U -maximal.

(2) is a special case of (1). (3): A conclusion of Rosenberg’s Theorem is [αϕ (A)] = Pt , i.e., (18.11) holds.


567

Lemma 18.2.6 For some i ∈ Et let |Ai | ≥ 2. Furthermore, let γ ⊆ Ahi be a relation that describes a maximal clone P olAi γ in PAi , where all constant functions of PAi belong to P olAi γ. Set t−1

γ := γ ∪

Ahj .

j=1, j=i

Then the clone P olk γ is U -maximal. Proof. W.l.o.g. let Ai := Er with r ≥ 2 and γ ∈ Ur ∪ Mr ∪ Lr ∪ Br ∪

r−1

Chr .

h=2

Obviously, P olk γ ⊂ U and {, c0 , c1 , ..., ck−1 } ∪ U0 ∪

t−1

Pk,Aj ⊆ P olk γ .

j=1, j=i

Elements of P olk γ are also all n-ary functions qg with g(x) if x ∈ Ern , qg (x) := 0 otherwise, for arbitrary g ∈ P olr γ. Moreover, all unary functions pd0 ,...,dr−1 defined by pd0 ,...,dr−1 (x) := dj ⇐⇒ x ∈ Aj (j = 0, 1, ..., t − 1) belong to P olk γ for all d0 , d1 , ..., dt−1 ∈ Ai = Er . Let f ∈ U \P olk γ be arbitrary. To prove [{f } ∪ P olk γ ] = U , we have only to show that Pk,r ⊆ [{f } ∪ P olk γ ] is valid because of Lemma 18.2.2 and because of the above considerations. Obviously, there are certain (a1j , a2j , ..., ahj ) ∈ γ for j = 1, ..., m and certain (b1l , b2l , ..., bhl ) ∈ γ \γ for l = 1, ...n with ⎞ ⎛ ⎞ ⎛ α1 a11 a12 ... a1m b11 b12 ... b1n ⎜ a21 a22 ... a2m b21 b22 ... b2n ⎟ ⎜ α2 ⎟ h ⎟ ⎜ ⎟ f⎜ ⎝ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎠ = ⎝ ... ⎠ ∈ Er \γ. ah1 ah2 ... ahm bh1 bh2 ... bhn αh By choosing certain m-ary functions f1 ..., fn from the m-ary function f ∈ Pk,r by means of

t−1 j=1, j=i

Pk,Aj one receives

f (x1 , ..., xm ) := qe11 (f (x1 , ..., xm , f1 (x1 , ..., xm ), ...., fn (x1 , ..., xm ))), 1

where

568


⎞ ⎛ α1 a11 a12 ... a1m ⎜ ⎟ ⎜ α2 ⎜ a21 a22 ... a2m ⎟ =⎜ f ⎝ . . . . . . . . . . . . . . . . ⎠ ⎝ ... ah1 ah2 ... ahm αh ⎛

⎞ ⎟ ⎟ ∈ Erh \γ ⎠

holds, as a superposition over P olk γ ∪{f }. With the help of the completeness criterion for Pk,r (see Theorem 12.4.3 or the proof of the following lemma) one can easy prove that {f } ∪ {qg | g ∈ Pr } ∪ {pd0 ,...,dt−1 | d0 , ..., dr−1 ∈ Er } is a generating system for Pk,r , whereby Pk,r ⊆ [{f } ∪ P olk γ ] was shown. With that, a generating system for the clone U was proven in {f } ∪ P olk γ . Therefore, P olk γ is U -maximal. Lemma 18.2.7 (1) P olk β2 is U -maximal. (2) Let A be a subset of U with U 1 ⊆ A and A ⊆ P olk β2 . Furthermore, A fulfills the condition (18.11). Then ∀i ∈ {0, 1, ..., t − 1} : Pk,Ai ⊆ [A].

(18.15)

Proof. (1): For proof, we use the completeness criterion for Pk,s from Chapter 12: Let pr : Pk,s −→ Ps be defined by pr(f n ) = g m :⇐⇒ n = m ∧ ∀x ∈ Esn : f (x) = g(x). With the aid of the mapping pr, one can describe the following maximal classes of Pk,s : pr−1 M := {f ∈ Pk,s | pr(f ) ∈ M }, where M is an arbitrary maximal clone of Ps . Furthermore, let ζi,t := {(x, x) | x ∈ Es } ∪ {(i, t)} for all i, t with i < t ≤ k − 1 and s ≤ t. Then, for all A ⊆ Pk,s is valid: [A] = Pk,s ⇐⇒ A is no subset of every set of the form pr−1 M (where M is a maximal clone of Ps ) and A does not preserve every relation of the form ζi,t . 1 (This means that the set Pk,s ∪ {f }, where f ∈ Pk,s and pr(f ) ∈ L for s = 2 s and pr(f ) ∈ P ols ιs for s ≥ 3, is complete in Pk,s .) Obviously, U 1 ⊂ P olk β2 . Let f ∈ U \P olk β2 be arbitrary. Then a function g ∈ Pk,l with pr(g) ∈ L for l = 2 and with pr(g) ∈ P oll ιll for l ≥ 3 is a superposition over U 1 ∪ {f }. Since U 1 ⊆ P olk β2 in addition, we have

Pk,l ⊆ [P olk β2 ∪ {f }]


569

because of the completeness criterion, and also ∀i ∈ Et : Pk,Ai ⊆ [P olk β2 ∪ {f }]. It is easy to check that the function defined in the remark on Lemma 18.2.1 preserves the relation β2 . Moreover, we have ∀h ∈ Pt ∃g ∈ P olk β2 : αϕ (g) = h. Consequently, the clone [P olk β2 ∪ {f }] contains a generating system for U , whereby (1) is proven. (2) follows from the proof for (1). Lemma 18.2.8 Let |A1 | = ... = |At−1 | = 1 (i.e., A0 = El is the only equivalence class of that contains at least two elements). Furthermore, let U0 := {f ∈ U | |Im(f ) ∩ El | ≤ 1}, U1,α := {f ∈ U | Im(f ) ⊆ El ∪ {α}}, U1 := α∈Ek \El U1,α . Then (a) ∀f ∈ U \P olk σl,1 : U1 ⊆ [{f } ∪ U 1 ∪ Pk,l ∪ U0 ]; (b) P olk σl,1 is U -maximal. Proof. The definition of σr,s , which one can find before Lemma 18.2.4, implies σl,1 = {(x1 , ..., xl , y) ∈ Ekl+1 | x1 = ... = xl ∨ {x1 , ..., xl+1 } ⊆ El ∨ ({x1 , ..., xl } ∈ ιll } (a): Let f ∈ U \P olk σl,1 be arbitrary and M := {f } ∪ U 1 ∪ Pk,l ∪ U0 . Then, an n-ary function f1 with the property ⎞ ⎛ ⎞ ⎛ 0 0 0 a13 a14 · · · a1n ⎜ 0 1 a23 a24 · · · a2n ⎟ ⎜ 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 2 a33 a34 · · · a3n ⎟ ⎜ 2 ⎟ ⎟, ⎜ ⎜ ⎟ f1 ⎜ ⎟ ⎟=⎜ ⎜ ........................ ⎟ ⎜ ··· ⎟ ⎝ 0 l − 1 al3 al4 · · · aln ⎠ ⎝ l − 1 ⎠ l l 0 b 3 b4 · · · bn where (a1i , a2i , ..., ali ) (i = 3, 4, ..., n) are certain tuples of ιll and b3 , ..., bn are certain elements of Ek , is a superposition over M . The above property of f1 and f1 ∈ U implies that f1 has only values of El on tuples of Eln . Next we consider the function

570


f1 := (f1 c0 )|El ∈ Pl , that is, we consider the restriction of f1 (c0 (x1 ), x2 ..., xn ) on (x1 ..., xn ) ∈ Eln . Then the following cases are possible: Case 1: x2 is the only essential variable of f1 . In this case, we have ⎛

0 0 0 0 ··· 0 ⎜0 1 0 0 ··· 0 ⎜ ⎜0 2 0 0 ··· 0 f1 ⎜ ⎜ ..................... ⎜ ⎝ 0 l − 1 0 0 ··· 0 l 0 b 3 b4 · · · bn

⎞ 0 ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎟ ⎟=⎜ ⎟ ⎜ ··· ⎟ , ⎟ ⎟ ⎜ ⎠ ⎝l−1⎠ l ⎞

⎛

and for every α ∈ Ek \El , a certain binary function fα with the property ⎞ ⎞ ⎛ ⎛ 0 0 0 ⎜0 1 ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜0 2 ⎟ ⎜ 2 ⎟ ⎟ ⎜ ⎟ ⎜ fα ⎜ ⎟ ⎟=⎜ ⎜ ....... ⎟ ⎜ ··· ⎟ ⎝0 l−1⎠ ⎝l−1⎠ α l 0 is a superposition over M . Let g be an arbitrary m-ary function of U1,α with α ∈ Ek \El . The m-ary functions g1 ∈ U0 and g2 ∈ Pk,l defined by 0, if g(x) ∈ El , g1 (x) := l otherwise,

and g2 (x) :=

= f (x), if g(x) ∈ El , 0 otherwise,

belong to M , whereby g ∈ [M ] holds because of g(x) = fα (g1 (x), g2 (x)). Consequently, we have U1,α ⊆ [M ] and (since U 1 ⊆ M ) also U1 ⊆ [M ], i.e., our assertion (a) is proven in Case 1. Case 2: x2 and xi for certain i ∈ {3, .., n} are essential variables of f1 . In this case, the Fundamental Lemma of Jablonskij (see Theorem 1.4.4) implies the existence of some ci , di , eji ∈ El (i = 2, ..., n; j = 4, ..., l) with


⎛

0 c2 c3 c4 · · · cn ⎜ 0 c2 d3 d4 · · · dn ⎜ ⎜ 0 d2 d3 d4 · · · dn ⎜ f1 ⎜ ⎜ 0 e42 e43 e44 · · · e4n ⎜ ...................... ⎜ ⎝ 0 el2 el3 el4 · · · eln l 0 b3 b4 · · · bn

⎞

⎛ α0 ⎟ ⎟ ⎜ α1 ⎟ ⎜ ⎟ ⎜ α2 ⎟=⎜ ⎟ ⎜ ··· ⎟ ⎜ ⎟ ⎝ αl−1 ⎠ l

571

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and {α0 , ..., αl−1 } = El . First we show that Uα1 ,α2 ,l := {f ∈ U | Im(f ) ⊆ {α1 , α2 , l} } ⊆ [M ]

(18.16)

holds. For this purpose, we use the property ⎞ ⎛ ⎞ ⎛ 0 c2 d3 d4 · · · dn α1 f1 ⎝ 0 d2 d3 d4 · · · dn ⎠ = ⎝ α2 ⎠ . l 0 b 3 b 4 · · · bn l Let h be an arbitrary m-ary function of Uα1 ,α2 ,l . Then the m-ary functions h1 , h2 , ..., hn defined by 0, if h(x) = α1 , h1 (x) := (∈ U0 ), l otherwise ⎧ h(x) = α1 , ⎨ c2 , if h(x) = α2 , (∈ Pk,l ), h2 (x) := d2 , if ⎩ 0 otherwise di , if h(x) ∈ {α1 , α2 } hi (x) := (∈ U0 ), bi otherwise (i = 3, ..., n), belong to [M ], and we have: h(x) = f1 (h1 (x), h2 (x), ..., hn (x)), whereby (18.16) is proven. Since U 1 ⊆ M , (18.16) implies {f ∈ U | Im(f ) ⊆ {a, b, c} } ⊆ [M ]. a,b∈El , c∈Ek \El

Then, analogously to the above and ⎛ 0 c2 c3 c4 ⎜ 0 c2 d3 d4 f1 ⎜ ⎝ 0 d2 d3 d4 l 0 b 3 b4

with the aid of ⎞ ⎛ · · · cn α0 ⎜ α1 · · · dn ⎟ ⎟=⎜ · · · dn ⎠ ⎝ α2 · · · bn l

⎞ ⎟ ⎟, ⎠

572


one can show that {f ∈ U | Im(f ) ⊆ {α0 , α1 , α2 , l} } ⊆ [M ] is valid. Consequently, because of U 1 ⊂ M , we have {f ∈ U | Im(f ) ⊆ {β0 , β1 , β2 , γ} } ⊆ [M ]. β0 ,β1 ,β2 ∈El , γ∈Ek \El

Then (a) results by means of induction from what was shown till now. (b): It is easy to see that the binary function z defined by x if x ∈ El , z(x, y) := y otherwise, preserves the relation σl,1 . Moreover, we obviously have: U 1 ∪ Pk,l ∪ U0 ⊆ P olk σl,1 .

(18.17)

Let f ∈ U \P olk σl,1 be arbitrary. Then, because of (a) and (18.17), U1 ⊆ [{f } ∪ P olk σl,1 ] holds. Let q be an arbitrary m-ary function of U . Then, one can form q as a superposition over z and the m-ary functions q1 , q2 defined by q(x), if q(x) ∈ El , q1 (x) := (∈ U1 ), l otherwise

and q2 (x) :=

0, if q(x) ∈ El , (∈ U0 ), q(x) otherwise

as follows: q(x) = z(q1 (x), q2 (x)). From that and from what was shown till now [{f } ∪ P olk σl,1 ] = U , through which (b) is proven. Theorem 18.2.9 Let k = l + 1. Then U has exactly three maximal clones that have U 1 as a subset. These clones are P olk β1 , P olk β2 , P olk σl,1 .

(18.18)

Proof. The U -maximality of the given clones was proven in Lemmas 18.2.5, 18.2.7, and 18.2.8. Let A be an arbitrary subset of U which has U 1 as a subset and which is no subset of the clones given in (18.18). To prove our theorem, we have to show that [A] = U . Lemmas 18.2.5 and 18.2.7 and U 1 ⊆ U imply U0 ∪ Pk,l ⊆ A. Consequently, with the aid of Lemma 18.2.8, (a), we have that U1 = U is a subset of [A].

18.3 The Maximal Classes of P olk (Ek2-1 ∪ {(k-1, k-1)})

573

18.3 The Maximal Classes of P olk(Ek2-1 ∪ {(k-1, k-1)}) In this section let k ≥ 3, 2 := Ek−1 ∪ {(k − 1, k − 1)}

and U := P olk . The sets U0 := {f ∈ U | k − 1 ∈ Im(f ) ∧ |Im(f )| ≤ 2} and Ua,b,k−1 := {f ∈ U | Im(f ) ⊆ {a, b, k − 1} }. are subsets of U . 18.3.1 Definitions of the U -Maximal Classes For k = 3, the clone U has exactly 13 maximal classes, which were given in Chapter 13, Theorem 13.1.8. In generalization of these classes (for a = 0 and b = 1), we will define six types of classes from which in the following two sections it is shown that they are the only maximal classes of the clone U . Type I (homomorphic inverse images of the maximal classes of P2 ): With the help of the mapping ϕ : Ek −→ E2 , ∀x ∈ Ek−1 : ϕ(x) := 0, ϕ(k − 1) := 1. −1 one can define a homomorphic inverse image αϕ (γ) of a relation γ ⊆ E2h as follows: −1 αϕ (γ) := {(x1 , ..., xh ) ∈ Ekh | (ϕ(x1 ), ϕ(x2 ), ..., ϕ(xh )) ∈ γ}.

In the case that γ describes a maximal class of P2 , we obtain the following maximal classes of U : −1 U ∩ P olk αϕ ({0}), −1 U ∩ P olk αϕ ({1}), 0 1 −1 U ∩ P olk αϕ , 1 0 0 0 1 −1 , P olk αϕ ⎛0 1 1 0 0 1 1 0 ⎜ −1 ⎜ 0 1 1 0 1 P olk αϕ ⎝ 1 0 0 1 1 1 1 0 0 0

1 0 0 1

0 0 0 0

⎞ 1 1⎟ ⎟. 1⎠ 1

574


For k = 3, these are the classes with the numbers (2), (1), (8), (9), and (13) of Theorem 13.1.8. Type II (U -maximal classes described by certain unary relations): Let σ be a subset of Ek , which has one of the following two properties: 1) 2)

σ = Ek−1 , {k − 1} ⊆ σ ⊂ Ek .

Then U ∩ P olk σ is maximal in U (see Lemma 18.2.4). For k = 3 there are exactly four maximal clones that are describable in this manner. Two of these classes were already recorded by means of type I. Type III (U -maximal classes that are determined by the maximal h classes of Pk−1 ; first possibility): Let γ ⊆ Ek−1 , 2 ≤ h ≤ k − 1, be an h-ary relation, which describes by means of P olk−1 γ a maximal class of Pk−1 and which contains all constant functions of Pk−1 . Then P olk (γ ∪ {(k − 1, k − 1, ..., k − 1)}) is U -maximal. For k = 3 there are exactly two U -maximal clones of type III: 0 1 2 0 P ol3 , 0 1 2 1 ⎛ ⎞ 0 0 0 1 1 0 1 1 2 ⎜0 0 1 1 0 1 0 1 2⎟ ⎟ P ol3 ⎜ ⎝ 0 1 0 0 1 1 0 1 2 ⎠. 0 1 1 0 0 0 1 1 2 The U -maximality of the classes of type III follows from Lemma 18.2.6. Type IV (U -maximal classes that are determined by the maximal classes of Pk−1 ; second possibility): This type of U -maximal class occurs only for k ≥ 4, since for k = 3 the conditions (a) and (b) mentioned below h , 2 ≤ h ≤ k − 1, an h-ary relation with the cannot be met. Denote γ ⊆ Ek−1 following properties: (a) P olk−1 γ is a maximal class of Pk−1 ; (b) γ is totally reflexive and totally symmetric, i.e., we can assume γ ∈ Uk−1 ∪

k−2 h=2

Moreover, let

Chk−1 ∪

k−1 h=3

Bhk−1 .


575

h+1 γ := Ek−1 ∪ {(x, x, ..., x, y) | x, y ∈ Ek }

∪{(x1 , ..., xh , k − 1) | (x1 , ..., xh ) ∈ γ}. Then P olk γ is U -maximal by Lemma 18.3.2.4. Example Let k = 4 and 0 1 2 3 0 3 γ := . 0 1 2 3 3 0 It is well-known that P ol3 γ is a maximal class of P3 and that γ is a reflexive and symmetric relation, whereby the relation ⎛ ⎞ 0 1 2 3 0 3 γ := E33 ∪ {(x, x, y) | x, y ∈ E4 } ∪ ⎝ 0 1 2 3 3 0 ⎠ 3 3 3 3 3 3 describes a maximal class of U . Type V (U -maximal classes that are described by certain binary 2 central relations): Denote τ a binary central relation ⊆ Ek−1 (i.e., a binary reflexive and symmetric relation, which has at least a central element c ∈ Ek−1 2 with {(c, x) | x ∈ Ek−1 } ⊆ τ and which is different from Ek−1 .) If τ fulfills the two following conditions (a) ⊆ τ , (b) ∃c ∈ Ek−1 : c is central element of τ , then U ∩ P olk τ is U -maximal by Lemma 18.3.2.9. Examples For k = 3 there are exactly two maximal classes of this type: 0 1 2 0 1 0 2 U ∩ P ol3 0 1 2 1 0 2 0 0 1 2 1 0 1 2 U ∩ P ol3 . 0 1 2 0 1 2 1 Type VI (Two U -maximal classes that are described by ternary relations): For A ∈ {Ek−1 , {k − 1}} let 2 σA := (Ek−1 × A) ∪ {(x, x, y) | x, y ∈ Ek }.

The class P olk σEk−1 is U -maximal by Lemma 18.3.2.5 and the class P olk σ{k−1} is U -maximal by Lemma 18.3.2.7. Example For k = 3 these classes have the number (10) and (11) in Theorem 13.3.8.

576


18.3.2 Proof of the U -Maximality of the Classes Defined in 18.3.1 The U -maximality of a class of the type I, II or III follows from Lemmas 18.2.4–18.2.6. Lemma 18.3.2.1 Let M be a subset of U which has the two following properties: (1) The set {f n ∈ Pk−1 | ∃g n ∈ M ∩ P olk {k − 1} ∩ P olk Ek−1 : n ∀x ∈ Ek−1 : g(x) = f (x)} is complete in Pk−1 . (2) There are two different elements a, b ∈ Ek−1 with Ua,b,k−1 := {f ∈ U | Im(f ) ⊆ {a, b, k − 1}} ⊆ [M ]. Then M is complete in U . Proof. Because of (1) a (k − 1)-ary function p with the property ⎞ ⎞ ⎛ ⎛ 0 b a a ... a a a ⎟ ⎜ ⎜ a b a ... a a a ⎟ ⎟ ⎜ 1 ⎟ ⎜ ⎟ ⎜ 2 ⎟ ⎜ a a b ... a a a ⎟ ⎟ ⎜ ⎜ ⎜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎟ ⎜ ... ⎟ ⎟ ⎜ ⎟ ⎜ =⎜ p⎜ ⎟ a a ... b a a ⎟ ⎟ ⎜k−4⎟ ⎜ a ⎟ ⎜k−3⎟ ⎜ a a a ... a b a ⎟ ⎟ ⎜ ⎜ ⎝ a a a ... a a b ⎠ ⎝k−2⎠ k−1 k − 1 k − 1 k − 1 ... k − 1 k − 1 k − 1 belongs to [M ]. Then one can form an arbitrary n-ary function u ∈ U as a superposition over {p} ∪ Ua,b,k−1 ⊆ [M ] as follows: The n-ary functions u0 , u1 , ..., uk−2 defined by ⎧ if u(x) = i, ⎨b if u(x) ∈ Ek−1 \{i}, ui (x) := a ⎩ k − 1 if u(x) = k − 1 (i = 0, 1, ..., k − 2) belong to Ua,b,k−1 . Then u = p(u0 , u1 , ..., uk−2 ) ∈ [M ], whereby [M ] = U is proven. Therefore, M is complete in U . Lemma 18.3.2.2 Let M be a subset of U , which has the two following properties:


577

(1) The set n : g(x) = f (x)} {f n ∈ Pk−1 | ∃g n ∈ M ∩ P olk Ek−1 : ∀x ∈ Ek−1

is complete in Pk−1 . (2) There are two different elements a, b ∈ Ek−1 with Pk,{a,b} := {f ∈ Pk | Im(f ) ⊆ {a, b} } ⊆ [M ]. Then Pk,k−1 ⊆ [M ]. Proof. The proof is similar to the proof of Lemma 18.3.2.1: Because of (1) a (k − 1)-ary function p with the property ⎞ ⎞ ⎛ ⎛ 0 b a a ... a a a ⎜ a b a ... a a a ⎟ ⎜ 1 ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ a a b ... a a a ⎟ ⎜ 2 ⎟ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ p⎜ ⎜ . . . . . . . . . . . . . . . . ⎟ = ⎜ ... ⎟ ⎜ a a a ... b a a ⎟ ⎜ k − 4 ⎟ ⎟ ⎟ ⎜ ⎜ ⎝ a a a ... a b a ⎠ ⎝ k − 3 ⎠ k−2 a a a ... a a b belongs to [M ]. Then one can form an arbitrary n-ary function u ∈ Pk,k−1 as a superposition over {p} ∪ Pk,{a,b} ⊆ [M ], as follows: The n-ary functions u0 , u1 , ..., uk−2 defined by b if u(x) = i, ui (x) := a if u(x) ∈ Ek−1 \{i} (i = 0, 1, ..., k − 2) belong to Pk,{a,b} . Then u = p(u0 , u1 , ..., uk−2 ) ∈ [M ], whereby Pk,k−1 ⊆ [M ] is proven.

2 2 Lemma 18.3.2.3 For each f ∈ U \P olk (ιk−1 k−1 ) and for each (a, b) ∈ Ek−1 \ιk−1 the set {f } ∪ U 1 ∪ Pk,k−1 ∪ Ua,b,k−1

is complete in U .

Proof. Let f ∈ U \P olk (ιk−1 k−1 ) be arbitrary. Then, w.l.o.g. there are certain k (a1i , a2i , ..., aki ) ∈ Ek−1 (i = 1, ..., m) and (b1j , b2j , ..., bk−1,j ) ∈ ιk−1 k−1 (j = 1, ..., n) with ⎞ ⎛ ⎞ ⎛ 0 a11 a12 ... a1m b11 b12 ... b1n ⎜ ⎟ ⎜ a21 a22 ... a2m b21 b22 ... b2n ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎜ ⎜ ⎟ f ⎜ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ⎟ = ⎜ ... ⎟ ⎟. ⎝ ak−1,1 ak−2,2 ... ak−1,m bk−1,1 bk−1,2 ... bk−1,n ⎠ ⎝ k − 2 ⎠ k−1 ak1 ak2 ... akm k − 1 k − 1 ... k − 1

578


By replacing of some variables of f through certain functions of Pk,k−1 one

can form an n-ary function f ∈ U \P olk (ιk−1 k−1 ) with f (k−1, k−1, ..., k−1) = k − 1. Consequently, the set {f } ∪ {g ∈ U 1 | g(k − 1) = k − 1} fulfills the assumption (1) of Lemma 18.3.2.1. Then with the help of Lemma 18.3.2.1, one can prove [{f } ∪ U 1 ∪ Pk,k−1 ∪ Ua,b,k−1 ] = U . k−2 Lemma 18.3.2.4 Let γ ∈ Uk−1 ∪ Bk−1 ∪ k=2 Chk−1 be an h-ary relation. Then P olk γ is U -maximal. Proof. Clearly, P olk γ is a proper subset of U . For each n-ary function g ∈ P olk−1 γ we define an n-ary function fg ∈ Pk by ⎧ n x ∈ Ek−1 , ⎨ g(x) if x = (k − 1, k − 1, ..., k − 1), fg (x) := k − 1 if ⎩ 0 otherwise. It is easy to check that fg belongs to P olk γ . Moreover, we have Pk,k−1 ⊆ P olk γ and there are two different elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ P olk γ . More precisely: If γ ∈ Uk−1 ∪ C2k−1 one chooses the elements a and b so that (a, b) ∈ γ\ι2k−1 . If γ has an arity h ≥ 3, then one can choose the elements a and b arbitrary. Now let f ∈ U \P olk γ be arbitrary. Then, w.l.o.g. there are certain tuh+1 (i = 1, ..., n) and (b1j , b2j , ..., bhj ) ∈ γ ples (a1i , a2i , ..., ah+1,i ) ∈ Ek−1 (j = 1, ..., m) with ⎛ ⎞ ⎛ ⎞ a1 a11 a12 ... a1n b11 b12 ... b1m ⎜ ⎜ a21 ⎟ a22 ... a2n b21 b22 ... b2m ⎟ ⎜ ⎟ ⎜ a2 ⎟ ⎜ ⎜ ⎟ ⎟ · f⎜ ⎟ = ⎜ ... ⎟ ⎝ ⎝ ah1 ⎠ ah2 ... ahn bh1 bh2 ... bhm ah ⎠ ah+1,1 ah+1,2 ... ah+1,n k − 1 k − 1 ... k − 1 k−1 h and (a1 , ..., ah ) ∈ Ek−1 \γ. By replacing some variables of f through certain functions of Pk,k−1 ⊆ P olk γ one can form a function f ∈ [{f } ∪ P olk γ ], which preserves the relations {k − 1} and Ek−1 but does not preserve the relation γ. Then, the maximality of P olk−1 γ in Pk−1 implies that the set {f } ∪ {fg | g ∈ P olk γ} fulfills the condition (1) of Lemma 18.3.2.1. Since we have already proven that P olk γ fulfills condition (2) of Lemma 18.3.2.1, the set {f } ∪ P olk γ is complete in U . Hence P olk γ is U -maximal.

Lemma 18.3.2.5 Let 3 σEk−1 := Ek−1 ∪ {(a, a, b) | k − 1 ∈ {a, b} ⊆ Ek }.

Then


579

(1) For every function f ∈ U \P olk σEk−1 there are two different elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ [{f } ∪ Pk,k−1 ∪ U0 ]. (2) P olk σEk−1 is U -maximal. Proof. (1): Let f ∈ U \P olk σEk−1 be arbitrary. W.l.o.g., we can assume that there are certain (a1j , a2j , a3j ) ∈ 3 3 (j = 1, 2, ..., n) and certain (bi , bi , ci ) ∈ Ek3 \Ek−1 (i = 1, 2, ..., m) with Ek−1 the properties ⎞ ⎛ ⎛ ⎞ a a11 ... a1n b1 ... bm f ⎝ a21 ... a2n b1 ... bm ⎠ = ⎝ b ⎠ , a31 ... a3n c1 ... cm k−1 a = b and {a, b} ⊆ Ek−1 . We choose some functions fi ∈ Pk,k−1 and gj ∈ U0 with the properties ⎛ ⎞ ⎛ ⎞ 0 0 a1i fi ⎝ 0 1 ⎠ = ⎝ a2i ⎠ a3i k−1 0 (i = 1, ..., n) and

⎛

⎞ ⎛ ⎞ 0 0 bj gj ⎝ 0 1 ⎠ = ⎝ bj ⎠ cj k−1 0

(j = 1, ..., m). Thus the binary function f with f := f (f1 , f2 , ..., fn , g1 , ..., gm ) belongs to [{f } ∪ Pk,k−1 ∪ U0 ] and it holds that ⎛ ⎞ ⎛ ⎞ 0 0 a f ⎝ 0 1 ⎠ = ⎝ b ⎠ . k−1 0 k−1 Next we show that every function of Ua,b,k−1 := {q ∈ U | Im(q) ⊆ {a, b, k−1}} is a superposition over {f } ∪ U0 ∪ Pk,k−1 : Let q t ∈ Ua,b,k−1 be arbitrary. Then the t-ary function q1 with 0 if q(x) ∈ {a, b}, q1 (x) := k − 1 otherwise, belongs to U0 and the t-ary function q2 with 1 if q(x) = b, q2 (x) := 0 otherwise, belongs to Pk,k−1 . Because of

580


q = f (q1 , q2 ) ∈ [{f } ∪ Pk,k−1 ∪ U0 ] we have Ua,b,k−1 ⊆ [{f } ∪ Pk,k−1 ∪ U0 ]. (2): Clearly, U = P olk σEk−1 . Since from σEk−1 the relation is derivable, P olk σEk−1 ⊂ U holds. Let f ∈ U \P olk σEk−1 be arbitrary. Since Pk,k−1 ∪U0 ⊆ P olk σEk−1 , above statement (1) implies that Ua,b,k−1 ⊆ [{f } ∪ P olk σEk−1 ] is valid for two certain elements a, b ∈ Ek−1 , whereby {f } ∪ P olk σEk−1 fulfills the condition (2) of Lemma 18.3.2.1. Obviously, each n-ary function p ∈ U with ∀x ∈ Ekn : p(x) = k − 1 ⇐⇒ x = (k − 1, k − 1, ..., k − 1) is a function of P olk σEk−1 . Consequently, {f } ∪ P olk σEk−1 also fulfills the condition (1) of Lemma 18.3.2.1. Therefore, the U -maximality of P olk σEk−1 follows from Lemma 18.3.2.1. Lemma 18.3.2.6 Denote max ∈ Pk the binary function that is defined with respect to the order 0 < 1 < 2 < ...k − 1 as usual. Then, the set U := {max} ∪ Pk,k−1 ∪ U0 is complete in U . Proof. Obviously, Pk,k−1 ∪ U0 ⊆ U . Since the function max has the property max(x, y) = k − 1 ⇐⇒ k − 1 ∈ {x, y}, max belongs to U . Consequently, U is a subset of U . Let f n ∈ U be arbitrary. Then, the n-ary functions f1 and f2 defined by f (x), if f (x) ∈ Ek−1 , f1 (x) := 0 otherwise,

and f2 (x) :=

0, if f (x) ∈ Ek−1 , k − 1 otherwise,

belong to U . Hence, the U -completeness of U follows from f = max(f1 , f2 ). Lemma 18.3.2.7 Let 2 × {k − 1}) ∪ {(a, a, b) | {a, b} ⊆ Ek }. σ{k−1} := (Ek−1

Then: (1) For every function f ∈ U \P olk σ{k−1} and every set M ⊆ U with [prEk−1 M ] = Pk−1 , where prEk−1 M := {g n ∈ Pk−1 | ∃g1 ∈ M ∩ P olk Ek−1 : n : g(x) = g1 (x) }, ∀x ∈ Ek−1 is valid:

Pk,k−1 ⊆ [{f } ∪ U0 ∪ M ∪ (U 1 ∩ P olk {k − 1})].


581

(2) P olk σ{k−1} is maximal in U . Proof. (1): Let f ∈ U \P olk σ{k−1} be arbitrary. W.l.o.g. we can assume that there are certain ai , bi ∈ Ek−1 (i = 1, 2, ..., r), cj ∈ Ek (j = 1, 2, ..., s) and dl , el ∈ Ek−1 (l = 1, ..., t) with the property ⎛ ⎞ ⎛ ⎞ a1 ... ar k − 1 ... k − 1 d1 ... dt a f ⎝ b1 ... br k − 1 ... k − 1 d1 ... dt ⎠ = ⎝ b ⎠ , k − 1 ... k − 1 c1 ... cs e1 ... et c where a = b and {a, b, c} ⊆ Ek−1 . Then, because ⎛ a1 ... ar k − 1 ... k − 1 d1 f ⎝ b1 ... br k − 1 ... k − 1 d1 k − 1 ... k − 1 c1 ... cs d1

of f ∈ U , we have ⎞ ⎛ ⎞ ... dt a ... dt ⎠ = ⎝ b ⎠ ... dt c

for a certain c ∈ Ek−1 . W.l.o.g. let c = a. We choose some unary functions fi ∈ U 1 ∩ P olk {k − 1} (i = 0, 1, ..., r) and gj ∈ U0 with the properties 0 if x = a, f0 (x) = 1 if x ∈ Ek−1 \{a}, ⎞ ⎛ ⎞ ⎛ ai 0 fi ⎝ 1 ⎠ = ⎝ bi ⎠ k−1 k−1 (i = 1, ..., r) and

⎛

⎞ ⎛ ⎞ 0 k−1 gj ⎝ 1 ⎠ = ⎝ k − 1 ⎠ k−1 cj

(j = 1, ..., s). Therefore, the unary function f with f := f0 (f (f1 , f2 , ..., fr , g1 , ..., gs , cd1 , ..., cdt )) belongs to [{f } ∪ (U 1 ∩ P olk {k − 1}) ∪ U0 ], where f ∈ Pk,2 (because of f ∈ U and f (k − 1) = k − 1) and ⎛ ⎞ ⎛ ⎞ 0 0 f ⎝ 1 ⎠ = ⎝ 1 ⎠ k−1 1 are valid. Now it is possible to prove that Pk,2 ⊆ [{f } ∪ U0 ∪ M ] holds with

582


the help of the following completeness criterion for Pk,2 (see Theorem 12.4.3): A subset T ⊆ Pk,2 is a generating system for Pk,2 if and only if T fulfills the following two conditions: (i) The set pr{0,1} T := {f n ∈ P2 | ∃f1n ∈ T : ∀x ∈ E2n : f (x) = f1 (x)} is a generating system for P2 . (ii) For each a ∈ Ek−1 and each b ∈ Ek \{0, 1} with a < b it is valid: 0 1 a . T ⊆ Pk,2 ∩ P olk 0 1 b Clearly, the set pr{0,1} {f g | g ∈ M } fulfills the above condition (i) because of our assumption prEk−1 M = Pk−1 . For arbitrary a ∈ Ek−1 , the unary function qa with ⎧ if x = a, ⎨0 x = k − 1, qa (x) := k − 1 if ⎩ 1 otherwise, belongs to U 1 ∩ P olk {k − 1}. Then, for arbitrary a ∈ Ek−1 and b ∈ Ek \{0, 1} with a < b we have a 0 (f qa ) = , b 1 whereby {f } ∪ (U 1 ∩ P olk {k − 1}) also fulfills the conditions of (ii). Thus by the completeness theorem for Pk,2 , Pk,2 ⊆ [{f } ∪ U0 ∪ M ∪ (U 1 ∩ P olk {k − 1})] is valid. This and Lemma 18.3.2.2 imply our assertion (1). (2): Clearly, P olk σ{k−1} = U . Further, it is easy to prove that is derivable from σ{k−1} . Hence P olk σ{k−1} ⊂ U . Let f ∈ U \P olk σ{k−1} be arbitrary. Obviously, U0 ⊆ P olk σ{k−1} and each n-ary function p ∈ U (n ∈ N) with the property n ∀x ∈ Ekn : p(x) = k − 1 ⇐⇒ x ∈ Ekn \Ek−1

belongs to P olk σ{k−1} . In particular, max ∈ P olk σ{k−1} . Then [{f } ∪ P olk σ{k−1} ] = U follows from statement (1) and Lemma 18.3.2.6. Lemma 18.3.2.8 For certain t ∈ {1, 2, ..., k − 2} we set C := {t, t + 1, ..., k − 2}. Furthermore let 2 α := {(x, x) | x ∈ Ek } ∪ Ek−1 ∪ {(x, y), (y, x) | x ∈ Ek , y ∈ C},


583

i.e., α is a binary central relation, where C is the set of all central elements of α and α has the property Ek2 \α = {(u, k − 1), (k − 1, u) | u ∈ Et }. Then max ∈ U ∩ P olk α. Proof. Because of Lemma 18.3.2.6, we have max ∈ U . Suppose max does not preserve the relation α. Then there are certain (a, b), (c, d) ∈ α and a u ∈ Et with a c u max = . b d k−1 W.l.o.g. we can assume that a = u and d = k − 1, i.e., c ≤ u and b ≤ k − 1. Then, because of u ∈ Et we have however c ∈ Et , contrary to (c, d) = (c, k − 1) ∈ α. Lemma 18.3.2.9 Let α ∈ C2k be a binary central relation with ⊆ α and the property that at least an element of Ek−1 is a central element of α. Then U ∩ P olk α is U -maximal. Proof. Because of ⊆ α and ∈ Dk , element k − 1 cannot be a central element of α. Therefore, w.l.o.g. we can assume that the set C := {t, t + 1, ..., k − 2} for a certain t ∈ {1, 2, ..., k − 2} is the set of all central elements of α, whereby we have 2 ∪ {(x, y), (y, x) | x ∈ Ek , y ∈ C} α = {(x, x) | x ∈ Ek } ∪ Ek−1

and

Ek2 \α = {(u, k − 1), (k − 1, u) | u ∈ Et }.

Notice that, with that, k − 2 is a central element of α. Let f n ∈ U \P olk α be arbitrary. Then there are certain (a1 , b1 ), ..., (an , bn ) ∈ α with a1 a2 ... an a f = b1 b2 ... bn k−1 and (a, k − 1) ∈ α. Since the functions tr,s (i = 1, ..., n) with (r, s) ∈ α and qa (a ∈ Ek−1 ) with r f¨ u x ∈ Ek−1 , tr,s (x) := s if x = k − 1, and

⎧ f¨ u x = a, ⎨b qa,b (x) := k − 2 if x ∈ Ek−1 \{a}, ⎩ k − 1 if x = k − 1,

for arbitrary b ∈ Ek−1 belong to U ∩ P olk α, the functions

584


tb,k−1 = qa,b (f (ta1 ,b1 , ..., tan ,bn )) and tk−1,b = qa,b (f (tb1 ,a1 , ..., tbn ,an )) are superpositions over {f } ∪ (U ∩ P olk α) for all b ∈ Ek−1 . Consequently, U0 ⊆ [{f } ∪ (U ∩ P olk α)]. Moreover, max ∈ U ∩ P olk α (see Lemma 18.3.2.8) and (because of ⊆ α) Pk,k−1 ⊆ P olk α. Then the U -completeness of {f } ∪ (U ∩ P olk α) follows from Lemma 18.3.2.6. 18.3.3 Proof of the Completeness Criterion for U Let M be an arbitrary subset of U , which is not contained in any class from type I–VI. We show that U = [M ] results from this assumption when we prove the generating system from Lemma 18.3.2.3 in [M ]. Since M is no subset of a class of type I, it results from the completeness theorem for P2 that the following is valid: ∀g m ∈ P2 ∃g1m ∈ [M ] : αϕ (g1 ) = g.

(18.19)

In particular, (18.19) implies: ck−1 ∈ [M ].

(18.20)

With the aid of the functions of M , which do not belong to the clones of type II, one sees from (18.20) that the constant functions of Pk belong to [M ]: c0 , c1 , ..., ck−1 ∈ [M ].

(18.21)

Because of (18.19), there is a unary function in [M ] that is an inverse image of the function g ∈ [M ] with x g(x) 0 1 . 1 0 Let g be this function with x 0 1 2 . . k−2 k−1 where a is a certain element of Ek−1 . The relation

g (x) k−1 k−1 k−1 , . . k−1 a


585

2 Ek−1 ∪ {(k − 1, k − 1), (a, k − 1), (k − 1, a)}

describes a certain class of the type V. Thus there is a function q ∈ M that does not preserve this relation. W.l.o.g. we can assume b a1 a2 ... an k − 1 a k − 1 = , q k−1 b1 b2 ... bn k − 1 k − 1 a where a1 , ..., an , b1 , ..., bn are certain elements of Ek−1 . Because of q ∈ U , we have also b a1 a2 ... an k − 1 a k − 1 = . q k−1 a1 a2 ... an k − 1 k − 1 a Consequently, the functions tb,k−1 (x) := q(ca1 (x), ..., ca2 (x), ck−1 (x), g (g (x)), g (x)) and

tk−1,b (x) := q(ca1 (x), ..., ca2 (x), ck−1 (x), g (x), g (g (x)))

belong to [M ], where x tb,k−1 (x) 0 b 1 b 2 b . . . . k−2 b k−1 k−1

tk−1,b k−1 k−1 k−1 . . . k−1 b

With the aid of further functions of M , which do not belong to the clones of type V, in analog mode, we see that all functions of the form α if x ∈ Ek−1 , tα,β (x) := β if x = k − 1, belong to [M ] for all α, β ∈ Ek−1 with k − 1 ∈ {α, β}. Next we show that U0 := {f ∈ U | |Im(f )| ≤ 2 ∧ k − 1 ∈ Im(f )} ⊆ [M ]

(18.22)

holds. For this purpose, let f n be an arbitrary function of U0 with Im(f ) = {a, k−1}. Because of (18.19) there is an n-ary function f ∈ [M ] with the property ∀x ∈ Ekn : f (x) = k − 1 ⇐⇒ f (x) = k − 1. With the aid of the above function ta,k−1 ∈ [M ], one can prove that

586


ta,k−1 f = f ∈ [M ]. Therefore, and because of (18.21), (18.22) is valid. If one forms superpositions over the constants and the functions of M , which do not belong to the clones of type III, one obtains functions gγm ∈ [M ] with x

gγ (x)

m ∈ Ek−1

}= gγ ∈ P olk−1 γ

otherwise

certain values

for all γ with the properties: (a) P olk−1 γ is maximal in Pk−1 and (b) c0 , ..., ck−1 ∈ P olk−1 γ. Hence, because of (18.21), there are functions gγ ∈ [M ] that do not fulfill (b). This and the completeness criterion for Pk−1 imply that the following function belongs to [M ]: x

y

s(x, y)

2 ∈ Ek−1

}= max(x, y) + 1 (mod k − 1)

k−1 k−1 otherwise

α certain values

We distinguish two cases for α := s(k − 1, k − 1): Case 1: α ∈ Ek−1 . By s ∈ [M ] the functions s := ∆s and

s := (s )k−1 ,

(s (x) = s(x, x))

(s (x) = s (s (s (...s (x)...))) ) k−1

belong to [M ]. Let β := s (k − 1) and γ ∈ Ek−1 \{β}. With the help of the completeness criterion for Pk,k−1 , one can easily prove that the set M1 := {s s, tβ,γ = s tk−1,γ } ⊆ [M ] is a generating system for Pk,k−1 . The completeness criterion for Pk,k−1 is a special case of Theorem 12.4.3 and says: An arbitrary subset T ⊆ Pk,k−1 is a generating system for Pk,k−1 , if and only if T fulfills the following two conditions:


587

(i) The set n pr T := {f n) ∈ Pk−1 | ∃f1n ∈ T : ∀x ∈ Ek−1 : f (x) = f1 (x)}

is a generating system for Pk−1 . (ii) For all a ∈ Ek : T ⊆ Pk,k−1 ∩ P olk

0 1 ... k − 2 a 0 1 ... k − 2 k − 1

.

Because of s s ∈ Pk,k−1 , ∀x, y ∈ Ek−1 : (s s)(x, y) = max(x, y) + 1 (mod k − 1) and the fact that max(x, y)+1 is a Sheffer function for Pk−1 fulfills (i). Because of a β tβ,γ = k−1 γ

6

the set T := M1

T := M1 also fulfills the condition (ii). Thus Pk,k−1 ⊆ [M ] was proven in Case 1. Then Lemma 18.3.2.5 implies the existence of two different elements a, b ∈ Ek−1 with Ua,b,k−1 ⊆ [M ].

(18.23)

Except for the unary function e ∈ U \U0 with x e(x) 0 1 . ∈ Ek−1 . k−2 k−1 k−1 we have proven that the other unary functions of U belong to [M ]. Next, with the aid of the completeness theorem for Pk−1 , we show that the functions of the form e are superpositions over {c0 , c1 , ..., ck−1 } ∪ U0 ∪ Pk,k−1 ⊆ [M ] and functions of M , which do not belong to the clones of type IV. 6

See Theorem 7.1.5.

588


k−2 Let γ ∈ Uk−1 ∪Bk−1 ∪ h=1 Ck−1 be an h-ary relation; i.e., γ is a totally reflexive and totally symmetric relation that describes the maximal class P olk−1 γ of Pk−1 . Then, the relation h+1 ∪ {(x, x, ..., x, y) | x, y ∈ Ek } γ := Ek−1

∪{(x1 , ..., xh , k − 1) | (x1 , ..., xh ) ∈ γ} describes a maximal class of U of type IV. If one replaces the variables of a function fγ ∈ M \P olk γ by certain functions of the set {c0 , c1 , ..., ck−1 } ∪ U0 ∪ Pk,k−1 ⊆ [M ], one receives an n-ary function fγ ∈ [M ] with fγ (x)

x n ∈ Ek−1

*

= fγ (x) ∈ P olk−1 γ

(18.24)

k − 1 k − 1 ... k − 1 k−1 otherwise certain values k−2 for all γ ∈ Uk−1 ∪Bk−1 ∪ h=1 Ck−1 . Since function ta,k−1 belongs to U0 ⊆ [M ] for every a ∈ Ek−1 , there are also functions fγ ∈ [M ] with the property (18.24) for all γ ∈ Sk−1 ∪ C1k−1 . Because of (18.23), there is a function fγ ∈ [M ] with the property (18.24) for every γ ∈ Mk−1 ∪ Lk−1 . With the help of the completeness criterion for Pk−1 and the fact that the function fγ ∈ [M ] k−2 preserves the relation {k −1} for every γ ∈ Mk−1 ∪Sk−1 ∪Uk−1 ∪ h=1 Ck−1 ∪ k−1 1 1 h=3 Bk−1 , one can prove that U ∩ P olk {k − 1} ⊆ [M ]. Consequently, U ⊆ [M ]. To summarize, we have proven Pk,k−1 ∪ U0 ∪ Ua,b,k−1 ∪ U 1 ⊆ [M ] in Case

1. Since a function of U \P olk (ιk−1 k−1 ) belongs to M , this and Lemma 18.3.2.3 implies [M ] = U . Case 2: α = k − 1. Since function s preserves {k − 1} in this case and s is a Sheffer function for 2 Pk−1 on the tuples of Ek−1 , every unary function e with e(k − 1) = k − 1 is a superposition over M . Furthermore, [prEk−1 M ] = Pk−1 . Since we have proven that U0 ⊆ [M ] holds, the first statement of Lemma 18.3.2.7 implies Pk,k−1 ⊆ [M ]. Therefore, there exists a function of [M ] that fulfills the assumption of Case 1. Consequently, [M ] = U in Case 2, and we have proven the following theorem: Theorem 18.3.3.1 (Completeness Theorem for U ) (1) The clones defined in Section 18.3.1 are the only maximal classes of U . (2) An arbitrary set M ⊆ U is U -complete if and only if there exists no class of the type I, II, ... or V I that has M as a subset.

19 Minimal Classes and Minimal Clones of Pk

In this chapter we deal with classes (of Lk ), which are either direct predecessors of the empty set (so-called minimal classes) or which are direct predecessors of the set of all projections (so-called minimal clones). It will turn out that it is not heavy to determine the minimal classes. However, for the minimal clones only partial results can be given.

19.1 Minimal Classes A subclass A of Pk is called a minimal class, if no subclass B = ∅ with B ⊂ A exists. Because of [∆n−1 f ] ⊂ [f ] for every function f ∈ Pk \[Pk1 ], every minimal classes has the order 1. If k = 2 then [c0 ], [c1 ] and [e11 ] are the only minimal classes. For arbitrary k one can easy check that Jk is a minimal class and, for every other minimal class A, there exists a unary function f ∈ Pk1 (k − 1) with the property f f = f and [f ] = A. Then a minimal class of P6 is e.g. x f (x) 0 1 1 1 2 1 3 3 4 4 5 4 By generalizing this example, one obtains the following theorem which is a conclusion from [P¨ os-K 79], Section 4.4.

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Theorem 19.1.1 Pk contains exactly 1+

k−1 r=1

k · rk−r r

minimal classes. These are class Jk and the classes of the form [f ] with f ∈ Pk1 (k − 1) and f f = f . One can describe a function f ∈ Pk1 (k − 1) with the property f f = f as follows: Let θ be an equivalence relation on Ek , which is different from ι2k (= κ0 ) and has exactly r equivalence classes. Further, denote a := (a1 , ..., ar ) an r-tuple with elements, which in pairs are not θ-equivalent. For such a θ und such an a, one can define the mapping faθ ∈ Pk by faθ (x) = ai :⇐⇒ x ∈ [ai ]θ . Then, for a function f ∈ Pk1 (k − 1), f f = f holds, if and only if there is an equivalence relation θ and a tuple a of the above form with f = faθ .

19.2 The Five Types of Minimal Clones A clone A ⊆ Pk is called minimal in Pk , iff Jk is a maximal clone of A, i.e., it holds (19.1) ∀f ∈ A \ Jk : [Jk ∪ {f }] = A. Therefore, a basis of a minimal clone has at most two elements. Furthermore, it is easy to see that the following implication holds: (A is a minimal clone with ord A ≥ 2) =⇒ ∀f ∈ A \ Jk : A = [f ].

(19.2)

From Chapter 3 it follows directly: Theorem 19.2.1 P2 has exactly 7 minimal clones. These are the clones I ∪ C0 = [e11 , c0 ], I ∪ C1 = [e11 , c1 ], I = [e11 ], K = [∧], D = [∨], L ∩ S ∩ T0 = [r] and [S ∩ M ∩ T0 ] = [h32 ]. One has solved the problem of describing minimal clones of the form [f n ] ⊆ Pk for arbitrary k thus far only for n = 1.

Theorem 19.2.2 ([P¨ os-K 79], [Har 74]; without proof ) Pk has exactly k−1 r=1

k−2 k k · rk−r + · r r r=0

p·t=k−r,p∈P,t∈N

pt

(k − r)! · t! · (p − 1)

19.2 The Five Types of Minimal Clones

591

minimal clones of the order 1. A minimal clone A of the order 1 has either the form A = Jk ∪ [f ], where [f ] is a minimal class different to Jk (see Theorem 19.1.1), or has the form A = [s], where s ∈ Pk1 [k] \ {e11 } is a permutation for which there is a prime number p with sp = e11 . The following theorem supplies the existence of at most finite many minimal clones (for a fixed k) and a coarse division of the minimal clones. Theorem 19.2.3 (Rosenberg’s Classification of the Minimal Clones; [Ros 82]) For every k ∈ N \ {1} there is only finite many minimal clones. If A = [Jk ∪ {f n }] is an arbitrary minimal clone of Pk , where [An−1 ] = Jk , then this clone fulfills one of the following five conditions: (1) n = 1 and A is described in Theorem 19.2.2. (2) n = 2 and f is idempotent, i.e., f (x, x) = x holds for arbitrary x ∈ Ek . (3) n = 3 and ∀x, y ∈ Ek : f (x, x, y) = f (x, y, x) = f (y, x, x) = y,

(19.3)

i.e., f is a so-called ternary minority function. A minority function g 3 of Pk generates a minimal clone if and only if g(x, y, z) = x ⊕ y ⊕ z and (Ek ; ⊕) is an elementary 2-group. (4) n = 3 and ∀x, y ∈ Ek : f (x, x, y) = f (x, y, x) = f (y, x, x) = x, (5)

(19.4)

i.e., f is a so-called ternary majority function. n ∈ {3, 4, ..., k} and f is a semiprojection, i.e., there exists an i ∈ {1, ..., n} with f (a1 , ..., an ) = ai for every tuple (a1 , ..., an ) ∈ Ek with |{x1 , ..., xn }| ≤ n − 1.

1

Proof. Let A be an arbitrary minimal clone of the order n ∈ N. Because of Theorem 19.2.2 and by (19.2), we can assume w.l.o.g. that n ≥ 2 and that A = [f n ] for certain f ∈ Pk . Obviously, because of ord f ≥ 2 and the minimality of the clone A, it holds: ∀i ∈ {1, ..., n − 1} : ∆i f ∈ Jk

(19.5)

In particular this implies f (x, x, ..., x) = x, i.e., f is idempotent. Thus, (2) is proven for n = 2. Because of (19.5) we obtain for n = 3 that 1

´ The following statement is also mentioned Swierczkowski Lemma in the literature. ´ S. Swierczkowski published in [Swi 60] a theorem from which statement (5) of Theorem 19.2.3 results.

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{f (x, x, y), f (x, y, x), f (y, x, x)} ⊆ {x, y}. Consequently, the following eight cases are possible: 1 f (x, x, y) = x f (x, y, x) = x f (y, x, x) = x

2 x x y

3 x y x

Case 4 5 x y y x y x

6 y x y

7 y y x

8 y y y

In Case 1, the function f fulfills the condition (19.4). In Cases 2, 3, and 5 the function f is a semiprojection, i.e., f fulfills the condition (5). Case 8 gives the condition (19.3). The remaining cases 4, 6, and 7 cannot occur for a function f which generates a minimal clone. This can be shown as follows: Suppose it holds Case 6; i.e., we have f (x, x, y) = f (y, x, x) = y and f (x, y, x) = x. Then the function g(x, y, z) := f (x, f (x, y, z), z) is a majority function. Since every ternary superposition t ∈ Jk over the majority function f is a majority function, too 2 , this is a contradiction to the minimality of clone A. In analog mode, the other cases can be led to a contradiction. Let now n = 4. Then, by assumption [A3 ] = Jk . Therefore, in particular ∆f ∈ Jk , i.e., we have f (x1 , x1 , x3 , x4 ) ∈ {x1 , x3 , x4 }. Then the following two cases are possible: Case 1: f (x1 , x1 , x3 , x4 ) = x1 . Then, because of [A3 ] = Jk it holds: xa := f (x1 , x2 , x1 , x4 ) ∈ {x1 , x2 , x4 }, xb := f (x1 , x2 , x3 , x1 ) ∈ {x1 , x2 , x3 }, xc := f (x1 , x2 , x2 , x4 ) ∈ {x1 , x2 , x4 }, xd := f (x1 , x2 , x3 , x2 ) ∈ {x1 , x2 , x3 }, xe := f (x1 , x2 , x3 , x3 ) ∈ {x1 , x2 , x3 }. If one puts x1 = x2 in the above equations and if one compares this with f (x1 , x1 , x3 , x4 ) = x1 , it follows {a, b, c, d, e} ⊆ {1, 2}. Since there exists a t ∈ {1, 2} with f (x1 , x2 , x1 , x1 ) = xt in addition, one obtain from the above equations: a = b = c = d = e = t. Thus f is a semiprojection in Case 1. 2

Use induction on the number of occurrences of f in t; see e.g. [Sza 83b] or [Qua 95].


593

Case 2: There exists an i ∈ {3, 4} with f (x1 , x1 , x3 , x4 ) = xi . If i = 3 then f (x1 , x2 , x3 , x3 ) = x3 . Consequently, we can continue analogously to the first case with the proof. If i = 4, then we have f (x1 , x2 , x3 , x1 ) = x1 and one can show (as in the first case) that f is a semiprojection. In analog mode to the case n = 4, one can examine the case n ≥ 5. It remains to show that a minority function g 3 , which generates a minimal clone, fulfills condition (4). One can show this most easily with the aid of the ´ Szendrei in [Sze 87]: following statement, proven by A. Let k ≥ 2 and d ∈ Pk be a ternary function, which fulfills the Mal’tsevcondition d(x, y, y) = d(y, y, x) = x. Then [d] is a minimal clone of Pk if and only if there exists an elementary Abelean p-group (Ek ; ⊕) with d(x, y, z) = x ) y ⊕ z. 3 Since a minority function g fulfills the Mal’tsev-condition according to definition, the function g has the form given in the above statement. However, the function g(x, y, z) = x ) y ⊕ z is a minority function iff p = 2.

Theorem 19.2.4 ([Pal 86]) For every t ∈ {1, 2, ..., k} there is a minimal clone of the order t. Proof. For t = 1 one can find examples for minimal clones of the order 1 in Theorem 19.2.1. In [P¨ os-K 79] one finds proof that the binary function f with f (x, y) = x ◦ y, where the operation ◦ fulfills the equations x ◦ x = x (idempotent law), (x ◦ y) ◦ z = x ◦ (y ◦ z) (associative law), (x ◦ y) ◦ x = x ◦ y (absorption law), generates a minimal clone. Let t ∈ {3, ..., k − 1}. Denote a1 , ..., at+1 pairwise distinct elements of Ek . The t-ary semiprojection f is defined by at+1 if x1 = a1 ∧ {x2 , ..., xt } = {a2 , ..., at }, f (x1 , ..., xt ) := x1 otherwise. In [Pal 86] it was proven that [f ] is a minimal clone of the order t. Put x1 if |{x1 , ..., xk }| ≤ k − 1, g(x1 , ..., xk ) := x2 otherwise. Then it is easy to see that [g] is a minimal clone of the order k. Of the many results found in the literature on minimal clones, only some are given without proof. 3

One finds this statement also proven in [Qua 95].

594


Theorem 19.2.5 ([Csa 83b]; without proof ) P3 has exactly 84 minimal clones. One obtains every one of these clones by using an inner automorphism of P3 onto exactly one of the 24 following clones (under that 4 of the order 1, 12 of the order 2 and 8 of the order 3): [j2 ], [u2 ], [s2 ], [s4 ] (see Table 15.1), [bi ] and [mj ], where i ∈ {1, 2, ..., 12}, bi idempotent, j ∈ {1, 2, ..., 8}, mj majority function and x 0 1 0 2 1 2

y 1 0 2 0 2 1

b1 0 0 0 0 0 0

b2 0 0 0 2 0 2

x 0 0 1 1 2 2

y 1 2 0 2 0 1

z 2 1 2 0 1 0

b3 0 0 0 0 1 1

b4 0 0 0 0 1 2

b5 0 0 0 2 1 1

b6 0 0 0 2 1 2

b7 0 0 0 2 2 2

b8 0 1 0 2 0 0

b9 b10 b11 b12 0 0 0 2 1 2 0 2 0 0 2 1 2 1 2 1 0 1 1 0 2 2 1 0

m1 m 2 m 3 m 4 m 5 m 6 m 7 m 8 0 0 2 0 0 0 0 1 0 1 1 0 0 0 0 2 0 1 2 0 0 0 2 0 0 0 0 0 0 2 2 2 0 0 1 0 2 2 1 0 0 1 0 0 2 2 1 1

Note that the cardinality of the set of all subclasses of P3 , which contain a fixed minimal clone of the order 1, can be found in [Pan-V 2000]. It was proven by B. Szczepara in his 210-page long Ph.D. thesis that there are exactly 2182 binary minimal clones for k = 4 (see [Szc 95]). All minimal clones, which are generated by majority functions of P4 , were determined by T. Waldhauser in [Wal 2000]. In [Lev-P 96] one can find all minimal clones C of the order 2 with |C 2 | ∈ {3, 4, 6}. Furthermore, one can find examples for minimal clones C of the order 2 with |C 2 | = 2t + 2 (t ≥ 1) or |C 2 | = 3t + 2 (t ≥ 2) in this paper. Up to isomorphic functions, all binary commutative functions, which generate minimal clones of the order 2, were determined in [Kea-S 99]. In [P¨ os-K 79] it was proven that Pk is the smallest clone, which contains all minimal clones of Pk . Further, it was proven that Jk is the intersection of all maximal clones of Pk . The following problem results from that: How many clones M1 , ..., Mt does one need so that at least t minimal (or maximal) t [ i=1 Mi ] = Pk (or i=1 Mi = Jk ) is valid, respectively? In [Zsa 92] it was proven that t ≤ 3 holds. The solution t = 2 for the above problem can be found in [Czé-H-K-P-S 2001]. A necessary and sufficient condition for [f, g] = Pk , if [f ] and [g] are minimal clones, can be found in [Ros-M 2001]. The answer 3 for the corresponding problems during the investigation of partial minimal (or maximal) clones was proven in [Had-M-R 2002].


595

One finds supplements to this chapter in the survey articles [Qua 95] and [Csa 2002]. In the next chapter, we deal with partial functions, for which we will handle similar problems as in the previous chapters. The next theorem shows that the problem of determining minimal partial clones can be reduced to the problem of determining minimal clones. Theorem 19.2.6 ([B¨ or-H-P 91]; without proof ) k . Then C is either a minimal clone of Pk or Let C be a partial clone of P C is generated by a partial n-ary projection with a nontrivial totally reflexive and totally symmetric domain D(e) ⊂ Ekn for certain n ≤ k. Denote t(k) the number of all minimal clones of Pk and m(k) the number of k . Then all partial minimal clones of P m(k) = t(k) +

k

k (2( i ) − 1).

i=1

In particular, we have m(2) = 11 and m(3) = 99, where t(2) = 7 and t(3) = 84 (see Theorems 19.2.1 and 19.2.5).

20 Partial Function Algebras

In Part I, Chapter 1, we introduced the concept of partial operation over a set A. By choosing A = Ek and replacing the concept of “operation” with the concept of “function”, we get the concept “partial function” over Ek . One k of can then introduce certain modified Mal’tsev-operations over the set P all partial functions on Ek . Then the set Pk together with these operations k ; τ, ζ, ∆, ∇, ), which can forms a so-called (full) partial function algebra (P be examined similar to the function algebra (Pk ; τ, ζ, ∆, ∇, ). The choice of results on partial function in this chapter focuses on questions that were already treated for Pk in the previous chapter. After a composition of some basic concepts in Section 20.1, Section 20.2 shows k is isomorphic to a certain sublatthat the lattice of all partial clones of P tice of the lattice of all clones of Pk+1 . Thus, one gets many properties of the partial clones from the properties of the clones that were already found. To find certain partial clones, however, with the aid of the above-mentioned isomorphism, is not possible, because of the absence of results on clones. One k with the help could, for example, not solve the completeness problem for P of isomorphism. In Section 20.3, we show how one can describe partial clones by relations. Sections 20.4 and 20.5 deal with the maximal partial clones with whose aid, analogously to Pk , one can solve the completeness problem of the partial 3 has exactly 58 maximal 2 has exactly 8 and P logic. We will prove that P partial clones. In Section 20.5 one can find the complete list of all maxik for arbitrary k ∈ N, which was found by L. Haddad and mal clones of P I. G. Rosenberg. The list is given without proof. In Section 20.6, we determine the descriptive relations of the maximal clones of Pk that are also k . In addition, a surdescriptive relations of the maximal partial clones of P vey those papers that deal with determining the orders of the maximal partial clones. Section 20.7 deals with determining the cardinality of the set k | C = [C] ∧ C ∩ Pk = A}, where A is an arbitrary maximal I(A) := {C ⊆ P clone of Pk . We prove, that, if A has the type U, S or C, I(A) is a finite set.

598


On the other hand, the set I(A) has the cardinality of continuum, if A has the type L or B. For the type M we can give only partial results. Section 20.8 gives a survey on the cardinalities of the sets I(A), where A is an arbitrary subclass of P2 . In last section, we determine the congruences on the maximal partial clones. It k has exactly 4 congruences, whereas a maximal is proven particularly that P partial clone has exactly 4, 8, or 10 pairwise distinct congruences.

20.1 Basic Concepts Let A be nonempty sets and let T be a proper subset of An . For an arbitrary mapping f from T into A and for (a1 , ..., an ) ∈ An \T , one can use the notation f (a1 , ..., an ) = ∞ with ∞ ∈ A, to indicate that f (a1 , ..., an ) is not defined. Then D(f ) := {(x1 , ..., xn ) ∈ An | f (a1 , ..., an ) = ∞} is the domain of f . In the following, we use these notations for A = Ek and for functions which are defined over subsets of Ekn . More exact: For a fixated k ∈ N \ {1} let k := Ek ∪ {∞}. E An n-ary mapping f of the form k f : Ekn −→ E is called (n-ary) partial function. and the set of all n-ary partial functions let n k . P Furthermore, we put

k := P

n

k . P

n≥1

By the above definition, the functions of Pk are also partial functions. To k \ Pk , we call these the distinguish these functions from functions of the set P total functions of Pk . n n k , if g(a) ∈ {f (a), ∞} k is called subfunction of f ∈ P A function g ∈ P holds for all a ∈ Ekn . In the following, the notation g ⊆p f is used to indicated the fact that g is a subfunction of f . We take the notations introduced for total functions and we set

20.1 Basic Concepts

599

cn∞ (x1 , ..., xn ) := ∞, i.e., cn∞ is the notation for the n-ary function with empty domain, and C∞ := {cn∞ | n ∈ N}. The reduction of a function f n ∈ Pk to T ⊂ Ekn is a function defined by f (x), if x ∈ T, g(x) := ∞ otherwise, which is also denoted with f|T . k . Now we consider the operations over P n m For arbitrary f , g ∈ Pk we define the unary Mal’tsev-operations analogous to Chapter 1 and the binary operation as follows: (ζf )(x1 , x2 , ..., xn ) := f (x2 , x3 , ..., xn , x1 ), (τ f )(x1 , x2 , ..., xn ) := f (x2 , x1 , x3 , ..., xn ), (∆f )(x1 , x2 , ..., xn−1 ) := f (x1 , x1 , x2 , ..., xn−1 ) for n ≥ 2, ζf = τ f = ∆f = f for n = 1, (∇f )(x1 , x2 , ..., xn+1 ) := f (x2 , x3 , ..., xn+1 ) and (f ∗ g)(x1 , ..., xm+n−1 ) := f (g(x1 , ..., xm ), xm+1 , ...., xm+n−1 ) if g(x1 , ..., xm ) ∈ Ek , ∞ otherwise. We adopt the concepts (such as closure, closed set, clone ...) and the notations (such as [...]) coupled with the Mal’tsev-operations from Chapter 1. If distinctions are required, we complete these concepts with the word “partial”. For k , which contains example,“partial clone” instead of “clone” if a closed set of P the set Jk of all projections of Pk , is meant. A partial clone C is called strong if it contains all subfunctions of its functions. If C is a clone of Pk , then let k | ∃g ∈ Pk : g|D(f ) = f }. Str(C) := {f ∈ P It is easy to check that Str(C) is a partial strong clone. As we see in Section 20.3, partial clones, like the clones of Pk , are describable k ) suitably chosen. with the help of relations (on E At first we want to show, however, how one can embed the lattice of the k into the lattice of the subclones of Pk+1 isomorphically. subclones of P

600


20.2 One-Point Extension In this section, let A := Ek and B := Ek ∪ {∞}. The following mappings establish relations between P A := Pk and PB (isomorphic to Pk+1 ): n n + : P A −→ PB , f → f+ , n n − : PB −→ P A , g → g− n where f+ is the so-called extended function defined by f (x) if x ∈ D(f ), f+ (x) := ∞ otherwise, n and g− , the so-called restricted function, is defined by g(x) if x ∈ Ekn ∧ g(x) ∈ Ek , g− (x) := ∞ otherwise.

Furthermore, for subsets F ⊆ PA and G ⊆ PB we put: F+ := {f+ | f ∈ F } and G− := {g− | g ∈ G}. To distinguish the projections from PA of the projections from PB , we use the notation eni,X with X ∈ {A, B}, i.e., ∀x1 , ..., xn ∈ X : eni,X (x1 , ..., xn ) := xi (i ∈ {1, ..., n}) holds. Let JX be the set of all projections of the form eni,X . One can prove that, if f ∈ P A and D(f ) = ∅, ∇f+ ∈ F+ holds. In the following, the clones (⊆ PB ), which are formed by closing certain subsets of PB , are important to further considerations: k )+ ]. H := [(Jk )+ ] and U := [(P The next Lemma gives properties of the clones defined above. Lemma 20.2.1 ([Ros 88]) (1) A function q t ∈ PB belongs to U , if and only if the following two statements hold: (a) q preserves ∞ and (b) for every essential place i of q and for arbitrary b1 , ..., bn ∈ B it holds q(b1 , ..., bi−1 , ∞, bi+1 , ...,t ) = ∞. (2) H is a minimal clone of PB .

20.2 One-Point Extension

601

Proof. (1): Denote U the set of all functions of PB , which fulfills conditions (a) and (b) above. At first we show that U is a clone. The equation U = U from which the assertion (1) immediately follows is then proven. Let f n , g m ∈ U be arbitrary. Then obviously τ f, ζf, ∇f ∈ U and ∆f ∈ U for n = 1. Let n > 1. If ∆ f depends on the first place essentially, then f depends at least on the places 1 and 2 essentially. Consequently, we have (∆ f )(∞, x3 , ..., xn ) = f (∞, ∞, x3 , ..., xn ) = ∞ for all x3 , ..., xn ∈ B. If ∆f depends on the i-th place with 2 ≤ i ≤ n − 1 essentially, then f depends on the (i + 1)-th place essentially and (∆ f )(x1 , x3 , ..., xi−1 , ∞, xi+1 , ..., xn ) = f (x1 , x1 , x3 , ..., xi−1 , ∞, xi+1 , ..., xn ) =∞ holds for all x1 , x3 , .., , , xn ∈ B. Consequently, the function ∆ f also belongs to U in the case n > 1. To prove f g ∈ U , we put hm+n−1 := f g. Obviously h fulfills the condition (a). Let h be dependent of the i-th place and let x := (x1 , ..., xm+n−1 ) ∈ B m+n−1 be arbitrary, where xi = ∞. Then the following two cases are possible: Case 1: i ∈ {1, ..., m}. In this case, the first place of f and the i-th place of g is essential. Thus by assumption, we have g(x1 , ..., xm ) = ∞ and h(x) = f (∞, xm+1 , ..., xm+n−1 ) = ∞. Case 2: i ∈ {m + 1, ..., m + n − 1}. Since in this case f depends on the (i − m + 1)-th place essentially, h(x) = ∞ holds. It also holds that h = f g ∈ U . Thus U is a closed set. Since the function e21,B fulfills conditions (a) and (b), U is a clone. It is easy to check that (PA )+ ⊆ U and U ⊆ U hold. For the missing proof of U ⊆ U , we consider an arbitrary function q t ∈ U with at least an essential place. Using the operations ζ, τ, ∆ we get from q a function q1 ∈ U that depends on all its places essentially and which belongs to (PA )+ . Since we can get again the function q by using the operations ∇, ζ, τ from q1 , we have q ∈ [(PA )+ ] = U . Thus U = U and (1) is proven. (2): It is easy to check that JB ⊂ [(e21,A )+ ] = H holds. To prove the maximality of JB in H we put b := |B| and := {(x1 , ..., xb ) ∈ B b | |{x1 , ..., xb }| = b}. Since (e21,A )+ ∈ P olB , we have H ⊆ P olB . Let q t ∈ H \JB be arbitrary. Since no constants belong to P olB , the function q has at least an essential place i. Suppose i is the only one essential place of q. Then we have q(x1 , ..., xi−1 , xi , xi+1 , ..., xn ) = g(xi , ..., xi , ..., xi ) = xi , i.e., q t = eti,B , in contradiction to q ∈ JB . Thus the function q has also at least two essential places. Using the operations ζ, τ, ∆ we get from q a binary

602


function g, which depends on both places essentially, preserves all elements of B, and belongs to H ⊂ U ; i.e., for all x ∈ B it holds: g(x, x) = x, g(x, ∞) = g(∞, x) = ∞, g(∞, ∞) = ∞. Consequently, g = (e21,A )+ and therefore H ⊆ [g] ⊆ H. Thus JB is a maximal clone of H. Lemma 20.2.2 For arbitrary f n , g m ∈ U holds: (1) α(f− ) = (α f )− for every operation α ∈ {ζ, τ, ∆, ∇}; (2) f− g− = h− , where hm+n−1 (x1 , ..., xm+n−1 ) := (e21 )+ (f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), g(x1 , ..., xm )); (3) f− g− ⊆p (f g)− ; (4) f− g− = (f g)− , if the first place of f is essential; (5) f− g− = (f g)− , if g− ∈ Pk . Proof. (1): For α ∈ {ζ, τ, δ} the assertion is obvious. For arbitrary (x1 , ..., xn+1 ) ∈ A is by definition (∇f )(x1 , ..., xn+1 ) = f (x2 , ..., xn+1 ). Furthermore (x1 , ..., xn+1 ) ∈ D((∇f )− ) holds iff (x2 , ..., xn+1 ) ∈ D(f− ). Consequently, we have (∇f )− (x1 , ..., xn+1 ) = ((∇f )− )(x2 , ..., xn+1 ) for arbitrary (x2 , ..., xn+1 ) ∈ D(f− ). (b2): Let x ∈ Am+n−1 be arbitrary. It is easy to check that h(x) = ∞ iff (f g)(x) = ∞. If h(x) = ∞, then h(x) = (f g)(x). Consequently, D(h− ) = {(x1 , ..., xm+n−1 ) ∈ Am+n−1 | (x1 , ..., xm ) ∈ D(g− ) ∧ (g− (x1 , ..., xm ), xm+1 , ..., xn+m−1 ) ∈ D(f− )} = D(f− g− ). Thus, for x := (x1 , ..., xm+n−1 ) ∈ D(h− ) we have g(x1 , ..., xm ) = g− (x1 , ..., xm ) and h(x) = (f g)(x) ∈ A. Hence h− = f− g− . (3): Let x := (x1 , ..., xm+n−1 ) ∈ D(f− g− ) be arbitrary. Then (x1 , ..., xm ) ∈ D(g− ) and (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ). Thus (f g)(x) = ∞. Consequently, x belongs to D((f g)− ) and we have (f− g− )(x) = f (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) = (f g)(x), i.e., f− g− ⊆p (f g)− . (4): Let the first place of f be essential and let x := (x1 , ..., xm+n−1 ) ∈ D((f g)− ) be arbitrary. Because of f ∈ U it follows that g(x1 , ..., xm ) = ∞. Thus (x1 , ..., xm ) ∈ D(g− ) and (g(x1 , ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ). Hence x ∈ D(f− g− ) and by (3) we have f− g− = (f g)− . (5): Let g ∈ PA and let x := (x1 , ..., xm+n−1 ) ∈ Am+n−1 be arbitrary. Then, (x1 , ..., xm ) ∈ D(g− ) and, because of x ∈ D(f− g− ) ⇐⇒ (g(x − 1, ..., xm ), xm+1 , ..., xm+n−1 ) ∈ D(f− ) ⇐⇒ x ∈ D((f g)− ), the assertion follows with the help of (c).

20.2 One-Point Extension

603

Lemma 20.2.3 For arbitrary f, g ∈ P A it holds: (1) α(f+ ) = (α f )+ for every operation α ∈ {ζ, τ, ∆}; (2) ∇(f+ ) = (∇f )+ , if f ∈ C∞ ; (3) f+ g+ = (f g)+ . Proof. (1) and (3) are immediate conclusions from the definition of the mapping +. (2) is easy to check. Lemma 20.2.4 Let G ⊆ U ⊆ PB be a clone and let F ⊆ P A be a partial clone with G− ⊆ F . Then ([G])− ⊆ F. (20.1) Proof. Let f, g ∈ G be arbitrary. Then by assumption the function f− and g− belong to F . Because of Lemma 20.2.2, (1) it follows that (αf )− = α(f− ) ∈ F for every α ∈ {ζ, τ, ∆, ∇}. Let h := f g ∈ G. If the first place of f is essential, then h− = (f g)− = f− g− ∈ F holds by Lemma 20.2.2, (4). If the first place of f is fictitious, then m m m hm+n−1 = f n g m = f n em 1,B . Since (e1,B )− = e1,A , we have h− = f− e1,A by Lemma 20.2.2, (5). Theorem 20.2.5 ([Ros 88], [B¨ or-P 90], [B¨ or 97]) (1) For every partial clone F ⊆ P A we have F = ([F+ ])− . (2) For every clone G ⊆ PB with H ⊆ G the set G− is a partial clone of P A with the property G = [(G− )]+ . (3) The mapping ϕ : LB (H; U ) −→ LA (JA ; P A ), G → G−

(20.2)

+ is a lattice isomorphism between the lattices LB (H, U) and LA (JA ; P A ), −1 where ϕ (F ) = [F+ ] holds for every F ∈ LA (JA ; PA ). Proof. (1): Because of f = (f+ )− for every f ∈ PA we have F ⊆ ([F ]+ )− . It follows from (F+ )− ⊆ F and Lemma 20.2.4 that ([F+ ])− ⊆ F . (2): Let rn and sm be two arbitrary functions of G− . Then there are two total functions un and v m of G with u− = r and v− = s. Because of Lemma 20.2.1, (a) we have αr = α(u− ) = (αu)− ∈ G− for all α ∈ {ζ, τ, ∆, ∇}. Furthermore, by Lemma 20.2.1, (b) the following is valid: (r s)(x1 , ..., xm+n−1 ) = (u− v)(x1 , ..., xm+n−1 ) = ((e21,A )+ (u(v(x1 , ..., xm ), xm+1 , ..., xm+n−1 ), v(x1 , ..., xm )))− . Thus r s ∈ G− . Consequently, G− is a partial clone. Let hn1 ∈ G ⊆ U be arbitrary. By using the operations ζ, τ, ∆ one can form

604


from h1 a function hm 2 , which depends on all its places essentially. Then, for the function h2 and for arbitrary j ∈ {1, ..., m} and arbitrary x1 , ..., xm ∈ B, it holds: h2 (x1 , ..., xj−1 , ∞, xj+1 , ...xm ) = ∞, i.e., h2 = ((h2 )−)+ ∈ (G− )+ . With the help of the operations ζ, τ, ∆, ∇, one can form the function h1 from the function h2 . Consequently, hn1 ∈ [(G− )+ ] and G ⊆ [(G− )+ ] hold. Let ht3 ∈ G− be arbitrary. Then there is a function ht4 ∈ G with (h4 )− = h3 . t Because of H ⊆ G we have (et+1 1,A )+ ∈ G. Then, for the function h5 with t h5 (x1 , ..., xt ) := (et+1 1,A )+ (h4 (x1 , ..., xt ), x1 , ..., xt )

it follows: h5 ∈ G and h5 = (h3 )+ . Thus (G− )+ ⊆ G. (3): Let G1 and G2 be two different clones of PB with H ⊆ G1 ⊂ G2 ⊆ U . By (1) (G1 )− and (G2 )− are partial clones of LA [JA , PA ]. Obviously, (G1 )− ⊆ (G2 )− . Suppose, (G1 )− = (G2 )− . Then, by (2) it follows that G1 = [((G1 )− )+ ] = [((G2 )− )+ ] = G2 , in contradiction to G1 = G2 . Consequently, the mapping ϕ is an order-preserving bijective mapping from LB (H; U ) into LA (JA ; P A ). Let F ∈ LA [JA , PA ] be arbitrary. Then G := [F+ ] is a clone with H ⊆ G ⊆ U , since if e21,A ∈ F , then (e21,A )+ ∈ G and F+ ⊆ (PA )+ ⊆ U also hold. Further, we have ϕ(G) = ([F+ ])− = F by (1). Consequently, ϕ is surjective and ϕ−1 (F ) = [F+ ].

20.3 Description of Partial Clones by Relations h

k , h-ary relations (i.e., subsets of E k ), h ≥ 1, To describe closed subsets of P are suitable. We often write the elements of relations in the form of columns and we often give a relation in the form of a matrix, the columns of which are the elements of the relation. h k let The set of all h-ary relations over E k R and we put k := R

h

h

k . R

h≥1

k preserves an h-ary relation over E k , iff We say that a function f ∈ P for all r1 , r2 , ..., rn ∈ with ri := (r1i , r2i , ..., rhi ), i = 1, 2, ..., n, it holds ⎛ ⎞ f (r11 , r12 , ..., r1n ) ⎜ f (r21 , r22 , ..., r2n ) ⎟ ⎜ ⎟ f (r1 , ..., rn ) := ⎜ ⎟ ∈ , .. ⎝ ⎠ . f (rh1 , rh2 , ..., rhn )

20.3 Description of Partial Clones by Relations

605

n

k \E h . where f (a) = ∞ is defined for all a ∈ E k Let pP olk h

k . Furtherk that preserve the relation ⊆ E be the set of all functions of P more, h

k \E h )). pP OLk := pP olk ( ∪ (E k The following Lemma is easy to check. Lemma 20.3.1 For every relation ∈ Rkh is valid: (1) pP olk ρ and pP OLk ρ are partial clones. (2) Str(P olk ) ⊆ pP OLk . h

k , the following conditions are equivLemma 20.3.2 For each relation ∈ R alent: (1) pP olk is a partial clone; (2) e21 ∈ pP olk ; (3) If (a1 , . . . , ah ), (b1 , . . . , bh ) ∈ and (c1 , . . . , ch ) is defined by ai if bi ∈ Ek , ci := ∞ if bi = ∞, (i = 1, ..., n), then (c1 , . . . , ch ) ∈ holds. Proof. Obviously, (1) ⇐⇒ (2) holds by definition. (2) ⇐⇒ (3) follows from ⎛ ⎞ ⎛ ⎞ a1 b1 c1 . .. ⎠ ⎝ .. ⎠ = . e21 ⎝ .. . . ah bh ch The following two lemmas give important properties that are needed to determine the maximal partial clones. 1 We need the following concept for the wording of the lemmas: The relation ∈ Rkh is called irredundant, iff it fulfills the following two conditions: 1) for all i, j with 1 ≤ i < j ≤ h, there is a tuple (a1 , ..., ah ) ∈ with ai = aj ; 2) No i ∈ {1, ..., h} exists, such that (a1 , ..., ah ) ∈ implies (a1 , ..., ai−1 , x, ai+1 , ..., ah ) ∈ for all x ∈ Ek . We say that for i = 1, ..., n the relations χi ⊆ {1, ..., t}hi cover the set {1, ..., t}, if for every x ∈ {1, ..., t} there exists an i ∈ {1, ..., n} such that x ∈ {a1 , ..., ahi } holds for at least a tuple (a1 , ..., ahi ) ∈ χi . 1

As one can gather from Sections 20.4 and 20.5, the maximal partial clones are all strong clones, with an exception.

606


k be a strong partial Lemma 20.3.3 ([Rom 81], without proof ) Let C ⊆ P of irredundant relations clone. Then there is a certain nonempty set M ⊆ R k with C = ∈M pP OLk . Lemma 20.3.4 (Representation Lemma of B. A. Romov, [Rom 81]; without proof ) Let αi ⊆ Ekhi for i = 1, ..., n and let β ⊆ Ekt be an irredundant relation. Then n pP OLk αi ⊆ pP OLk β i=1

if and only if there are certain (help-)relations χi ⊆ {1, ..., t}hi for i = 1, ..., n that cover the set {1, ..., t} and for which β = {(b1 , ..., bt ) ∈ Ekt | ∀j ∈ {1, ..., n} ∀(ij1 , ..., ijhj ) ∈ χj : (bij , ..., bij ) ∈ αj } 1

hj

holds.

2 and P 3 20.4 The Maximal Partial Classes of P Lemma 20.4.1 Let f be an n-ary function of P3 , which essentially depends on at least two variables (w.l.o.g. let x1 and x2 be the essential variables). Furthermore, let {a, b, c} = E3 and δI := {(a0 , a1 , a2 ) ∈ E33 , | ∀α, β ∈ I : aα = aβ }, I ⊆ Ek . Then 3 3 ∪ δ{1,2} : f (r1 , ..., rn ) ∈ E33 \ι33 ; (a) Im(f ) = E3 ⇒ ∃ r1 , ..., rn ∈ δ{0,1} 3 3 3 3 ∪ δ{b,c} : f (r1 , ..., rn ) ∈ δ{a,c} \δ{0,1,2} ; (b) |Im(f )| = 2 ⇒ ∃ r1 , ..., rn ∈ δ{a,b} 1 (c) Im(f ) = E3 ⇒ [{f } ∪ { g ∈ P3 | g(a) = g(b) ∨ g(b) = g(c) }] = P3 . Proof. (a) is a special case of the “fundamental lemma of Jablonskij” and (b) is an easy conclusion from this lemma (see Theorem 1.4.4). (c): W.l.o.g. let a = 0, b = 1 and c = 2. Obviously, then, we have { g ∈ P31 | g(0) = g(1) ∨ g(1) = g(2) } = {c0 , c1 , c2 , jα , uα , vα | α ∈ {0, 2, 3, 5}}. It is easy to check that this set of unary functions is not a subset of maximal classes of type M, U, S, C and L. Since Im(f ) = E3 and f ∈ P3 \[P31 ] hold, f does not preserve by (a) the relation ι33 . Thus (c) follows from the completeness criterion for P3 . k \(Pk ∪ [{c∞ }]) it holds [Pk ∪ {g}] = P k . Lemma 20.4.2 (a) For every g ∈ P k that contains Pk . (b) Pk ∪ [{c∞ }] is the only maximal class of P k \(Pk ∪ [{c∞ }]). Since the function g := e2 ∗ g has Proof. (a): Let g m ∈ P 2 k . Consequently, k + 1 different values, we can assume w.l.o.g. Im(g) = E

2 and P 3 20.4 The Maximal Partial Classes of P

607

k ). there are k + 1 tuples ai := (ai1 , ai2 , ..., aim ) ∈ Ekm with g(ai ) = i (i ∈ E n Let f be an arbitrary function of Pk . Independently from f , one can define the following functions fj (j = 1, 2, ..., m): fj (b1 , ..., bn ) = aij :⇐⇒ f (b1 , ..., bn ) = i k ). Thus we have f (x) = g(f1 (x), ..., fm (x)). Hence (b1 , ..., bn ∈ Ek ; i ∈ E f ∈ [Pk ∪ {g}]. (b) follows directly from (a). 3 The next four lemmas deal with the maximality of certain subclasses of P (or Pk ) in P3 (or Pk ), respectively. The following statement (a) was proven in [Bur 67] (see also Chapter 4) and (b) of the following lemma was proven in [Rom 80] (or in [Lau 77], [Lau 88]). Lemma 20.4.3 Let 1 := {(a, a, b, b), (a, b, a, b) | a, b ∈ Ek }, 2 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek }, i := {(a1 , a2 , ..., ai ) ∈ Eki | |{a1 , ..., ai }| ≤ i − 1 } (i = 3, ..., k). Then (a) The classes P olk i (i = 1, 2, ..., k) are the only proper subclasses of Pk that contain Pk1 . Furthermore, it holds that [Pk1 ] = P olk 1 ⊂ P olk 2 ⊂ ... ⊂ P olk k−1 ⊂ P olk k ⊂ Pk . k and the (b) The classes pP OLk i (i = 1, 2, ..., k) are maximal classes of P 1 only maximal classes of Pk that contain Pk . Lemma 20.4.4 Let := δ ∪ σ, where δ denotes a certain h-ary diagonal relation, which is different from Ekh , and σ fulfills the condition ∅ = σ ⊆ {(a1 , ..., ah ) ∈ Ekh | |{a1 , ..., ah }| = h }. There exists to every a := (a1 , ..., ah ) ∈ σ a certain equivalence relation εa on Ek with the following two properties: (1) For every i ∈ {1, ..., h} there exists exactly an equivalence class of εa which contains ai . (2) To every b ∈ one can find in pP OLk a unary function ga,b with ga,b (a) = b and ga,b (x) = ga,b (y) for all (x, y) ∈ εa . k . Then pP OLk is a maximal class of P k . Let f ∈ P k \pP OLk . Since has the Proof. Obviously, pP OLk = P properties (1) and (2), one gets a certain unary function hr with hr (r) ∈ Ekh \ as a superposition over unary functions of pP OLk and f for every r ∈ σ. Let σ = {r1 , ..., rm }. One can find in pP OLk certain functions, which arbitrary k on rows of the form values have of E (x1 , x2 , gr1 (x1 ), gr2 (x1 ), ..., grm (x1 ), gr1 (x2 ), gr2 (x2 ), ..., grm (x2 ))

608


and otherwise only have the value ∞. Consequently, arbitrary functions of Pk2 are superpositions over {f } ∪ pP OLk . Hence (by well-known properties of Pk ) it follows that Pk ⊆ [{f } ∪ pP OLk ]. Since pP OLk obtains functions with exactly k + 1 different values, it follows from Lemma 20.4.2, (a) that k holds. Hence pP OLk is a maximal class of P k . [{f } ∪ pP OLk ] = P Table 20.1 i 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37

39

τi {0} {2} {0, 2} 0 1 1 2 0 2 2 0 0 0 1 2 0 1 2 2 0 1 1 1 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 1 2 ⎛0 ⎞ 0 ⎝1⎠ ⎛2 ⎞ 0 2 ⎝1 1⎠ 2 0

i 2 4 6 8 10 12 14

2 1 0 1 1 2 0 2 0 1 0 2 1 0 0 1 0 1 0 1

16

18

20

2 0 0 2 1 2 2 1 0 2 0 2 1 0

2 1 1 2 1 2 2 1

22 24 26 28 30 32 34 36 38

40

τi {1} {0, 1} {1, 2} 0 2 0 1 1 0 1 2 2 1 0 2 1 1 0 1 0 1 0 2 0 2 1 2 0 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 2 2 ⎛0 1 ⎞ 0 0 ⎝1 2⎠ ⎛2 1⎞ 0 1 ⎝1 0⎠ 2 2

2 0 2 1 0 2 0 1 1 2 0 1 0 1 2 0 0 1 0 1 0 2

1 0 2 1 2 1 2 0 1 2 2 1 1 0 2 0

2 0 0 2 1 2

2 0 2 1


i τ⎛i 41 ⎝ ⎛ 43 ⎝ ⎛ 45 ⎝ ⎛ 47 ⎝

i 49 50 51 52 53 54 55 56 57

0 1 2 0 0 0 0 0 0 0 0 0

1 2 0 1 1 1 1 1 1 1 1 1

2 0 1 2 2 2 2 2 2 2 2 2

Table 20.2 i τ⎛i ⎞ 0 0 ⎠ 42 ⎝ 1 2 ⎛2 1 ⎞ 0 0 1 1⎠ 44 ⎝ 0 1 2 ⎞ ⎛0 1 0 2 0 1 1 1⎠ 46 ⎝ 0 1 2 0 ⎞ ⎛0 1 0 1 2 0 1 1 2 0 ⎠ 48 ⎝ 0 1 2 0 1 0 1

1 0 2 2 2 2 2 2 2 2 2 2

1 2 0 0 1 2 0 1 2 0 1 2

2 0 1 0 2 1 1 0 2 0 2 1

609

⎞ 2 1⎠ 0 ⎞ ⎠ ⎞ ⎠ ⎞ 1 1 2 2 0 2 0 1⎠ 2 0 1 0

Table 20.3 τi { (0, 1, 2), (a, a, b) | a, b ∈ E3 } { (0, 1, 2), (a, b, a) | a, b ∈ E3 } { (0, 1, 2), (b, a, a) | a, b ∈ E3 } { (0, 1, 2), (1, 0, 2), (a, a, b) | a, b ∈ E3 } { (0, 1, 2), (2, 1, 0), (a, b, a) | a, b ∈ E3 } { (0, 1, 2), (0, 2, 1), (b, a, a) | a, b ∈ E3 } { (a, a, b, b), (a, b, a, b) | a, b ∈ E3 } { (a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E3 } { (a, b, c) ∈ E33 | |{a, b, c}| ≤ 2 }

Lemma 20.4.5 (1) For every ∈ {{0}, {1}, {(0, 1)}, {(0, 1), (1, 0)}, {(0, 0), (0, 1), (1, 1)}, λ2 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E2 }, G2 ([P21 ])}, the set pP Ol2 is 2 . a maximal class of P (2) The classes pP OL3 τi (i ∈ {1, 2, ..., 57}, see Tables 20.1 - 20.3) are maximal classes of P3 . Proof. (1): The maximality of pP OL2 for ∈ {λ2 , G2 ([P21 ])} was proven in Lemma 20.4.3. 2 : Let f n ∈ Next we show that pP OL2 {(0, 1)} is a maximal class of P n−1 1 P2 \ pP OL2 {(0, 1)} be arbitrary. Then ∆ f ∈ {c0 , c1 , e1 }. We distinguish two cases: Case 1: ca ∈ [f ] for certain a ∈ E2 . The unary function g with g(a) = a and g(a) = ∞ belongs to pP OL2 {(0, 1)}, whereby {c0 , c1 } ⊂ [{f } ∪ pP OL2 {(0, 1)}]. With the aid of Theorem 3.2.4.1, it is easy to prove that P2 = [(T0 ∩ T1 ) ∪ {c0 , c1 }] ⊆ [{f } ∪ pP OL2 {(0, 1)}].

610


Further, we have pP OL2 {(0, 1)} ⊆ P2 ∪ [c∞ ]. Therefore, by Lemma 20.4.2, 2 holds, i.e., pP OL2 {(0, 1)} is a maximal class of [{f } ∪ pP OL2 {(0, 1)}] = P P2 . Case 2: e11 ∈ [f ]. In this case, by Theorem 3.2.4.1, P2 = [(T0 ∩T1 )∪{e11 }] ⊆ [{f }∪pP OL2 {(0, 1)}], whereby Case 2 is reducible to Case 1. The maximality of the other classes pP OL2 is easy to check. (2): For i ∈ {1, 2, ..., 54} one can easily prove the lemma with the help of Lemma 20.4.4. If i ∈ {55, 56, 57}, the above statement follows from Lemma 20.4.3. Theorem 20.4.6 ([Fre 66]) 2 has exactly 8 maximal classes: P pP OL2 σi , where σ0 := {0}, σ1 := {1}, σ2 := {(0, 1)}, σ3 := {(0, 1), (1, 0)}, σ4 := {(0, 0), (0, 1), (1, 1)}, σ5 := G2 ([P21 ]) = {(a, a, b, b), (a, b, a, b) | a, b ∈ E2 } and σ6 := {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ E2 }, and P2 ∪ [c∞ ] = pP ol2 (Ek2 ∪ {(∞, ∞)}). Proof. Because of Lemmas 20.4.2 and 20.4.5, we must show that each subset 2 , which fulfills A ⊆ pP OL3 τi for all i ∈ {0, 1, 2, ..., 6} and A ⊆ P2 ∪[c∞ ], A of P 2 . is complete in P Let A ⊆ P2 be an arbitrary set, for which there are f0 , ..., f6 , f7 ∈ A with fi ∈ pP OL2 σi (i = 0, 1, ..., 6) and f7 ∈ P2 ∪ [c∞ ]. 2 defined by g(x) := f2 (x, x, ..., x). Then g ∈ [A] and g ∈ {c0 , c1 , e1 }. Let g ∈ P 1 Consequently, since c1 ∈ [f0 , c0 ], c0 ∈ [f1 , c1 ], {c0 , c1 , e11 } ⊂ [f3 , e11 ] and e11 ∈ [c0 , c1 , f4 ], we have P21 ⊂ [A]. Thus there exists a function h ∈ [{f5 }∪P21 ] ⊆ [A] with h ∈ P2 \ [P21 ]. We distinguish two cases: Case 1: h is non-linear. In this case, by Theorem 3.2.4.1, we have [P21 ∪ {h}] = P2 ⊆ [A]. Because of 2 . A ⊆ P2 ∪ [c∞ ] and Lemma 20.4.2, this implies [A] = P 1 Case 2: h ∈ [P2 ] is a linear function. Then it is easy to check that all binary linear functions belong to [A], whereby (by f6 ∈ A) a non-linear function of P2 is a superposition over A. Thus Case 2 is reducible to Case 1. Theorem 20.4.7 ([Lau 77], [Rom 80]) 3 has exactly 58 maximal classes. These classes are the sets pP OL3 τi (i = P 1, 2, 3, ..., 57), where τi are given in Tables 20.1–20.3, and the set P2 ∪ [c∞ ].


611

Proof.2 Because of Lemmas 20.4.2 and 20.4.5, we must show that each subset M

3 , which fulfills M ⊆ pP OL3 τi for all i ∈ {1, 2, ..., 57} and M ⊆ P2 ∪ [c∞ ], is of P 3 . complete in P 3 be an arbitrary set that fulfills M ⊆ P2 ∪ [c∞ ] and M ⊆ pP OL3 τi for Let M ⊆ P all i ∈ {1, 2, ..., 57}). Consequently, there are functions fini ∈ M \ pP OL3 τi for all i = 1, 2, ..., 57) and f58 ∈ M \ (P2 ∪ [c∞ ]). If τi = (σ1 σ2 ... σmi ) (i ∈ {1, 2, ..., 57}) is an hi -ary relation, then we can assume w.l.o.g. ni = mi and fi (σ1 , σ2 , ..., σmi ) ∈ E3hi \τi . As already mentioned, we must show that [M ] = P3 . First we prove {c0 , c1 , c2 } ⊆ [M ].

(20.3)

The function f37 is unary and belongs to P31 \{s1 }. Then the following three cases are possible: Case 1: f37 = ca (a ∈ E3 ). Obviously, we have {c0 , c1 , c2 } ⊆ [{f37 , f1 , f2 , ..., f6 }] in this case. Case 2: |Im(f37 )| = 2. W.l.o.g. let Im(f37 ) = {0, 1}, i.e., f37 ∈ {j0 , j1 , ..., j5 }. Because of j2 ∗ j2 = c0 , j0 ∗ j0 = j5 , j3 ∗ j3 = c1 , j4 ∗ j4 = j1 and Case 1 it is sufficient to assume that f37 ∈ {j1 , j5 }. 2.1: f37 = j1 . We form f7 := f7 ∗ j1 ∈ {c0 , c1 , c2 , j4 , u1 , u4 , v1 , v4 }. Because of Case 1, u1 ∗ u1 = c0 , v1 ∗ v1 = v4 and u4 ∗ u4 = c2 , we can confine ourselves to f7 ∈ {j4 , v4 }. 2.1.1: f7 = j4 . Putting the functions f37 (= j1 ) and f7 (= j4 ) into f10 provides a unary function (x) := f10 (j1 (x), j4 (x)) with f10 ∈ {c0 , c1 , c2 , u1 , u4 , v1 , v4 }. Because of u1 ∗ u1 = f10 = v4 . It c0 , u4 ∗ u4 = c2 and v1 ∗ v1 = v4 we still have to examine the case f10 (x) := f17 (j1 (x), j4 (x), v1 (x), v4 (x)), then f17 ∈ holds v4 ∗ j4 = v1 . If we form f17 {c0 , c1 , c2 , u1 , u4 } holds and f17 ∗ f17 is a constant function. With that we have reduced the Case 2.1.1 to the first case. 2.1.2: f7 = v4 . (x) := f14 (j1 (x), v4 (x)) belongs to {c0 , c1 , c2 , j4 , u1 , u4 , v1 }. Because The function f14 of u1 ∗ u1 = c0 , u4 ∗ u4 = c2 , j1 ∗ v1 = j4 and by Case 2.1.1, we reduce Case 2.1.2 to the first case. 2.2: f37 = j5 . In this case, one can proceed to Case 2.1 analogously using the function f16 instead of f17 and f13 instead of f14 . Case 3: |Im(f37 )| = 3 and f37 = s1 . With the help of functions f38 , ..., f42 , this case is reduced at first to Case 1 or 2 and, therefore, to Case 1. Consequently, the constant functions belong to [M ]. Let f43 (x) := f43 (c0 (x), c1 (x), c2 (x), x). This function belongs to P31 ∩ [M ] and has is a permutation, then with the help of functions at least two different values. If f43 2

The definitions of the following unary functions of P3 are in Chapter 15, Table 15.1.

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f44 , ..., f48 , we can form a function with 2-element range. Thus w.l.o.g. we can ) = E2 in the following; i.e., f43 ∈ {j0 , j1 , ..., j5 }. assume Im(f43 Let Ma,b := {f ∈ P31 | f (a) = f (b)}. Next ∃ a, b ∈ E3 : a = b ∧ Ma,b ⊂ [M ]

(20.4)

shall be proven. Because of j0 ∗ j0 = j5 , j4 ∗ j4 = j1 and for reasons of the duality, ∈ {j1 , j2 }. we can assume that f43 Case 1: f43 = j1 . ∈ {j4 , u1 , u4 , v1 , v4 } as a superposition over the constant functions, One receives f19 j1 and f19 . Because of v1 ∗ v1 = v4 and f22 (c0 , c1 , c2 , j1 , j4 ) ∈ {u1 , u4 , v1 , v4 } we can ∈ {u1 , u4 , v4 }. assume f19 = u1 . 1.1: f19 (x) := f25 (c0 (x), c1 (x), c2 (x), j1 (x), u1 (x)) In this case, we can form a function f25 ∈ {j4 , u4 , v4 }. with f25 ∈ {j4 , u4 , v1 , v4 }. Further, we can assume w.l.o.g. f25 1.1.1: f25 = j4 . := f34 (c0 , c1 , c2 , j1 , j4 , It holds that u1 ∗j4 = u4 . Consequently, the unary function f34 u1 , u4 ) ∈ {v1 , v4 } is a superposition over M . Thus by v1 ∗ v1 = v4 and v4 ∗ j4 = v1 it holds: {c0 , c1 , c2 , j1 , j4 , u1 , u4 , v1 , v4 } = {f ∈ P31 | f (0) = f (2)} ⊂ [M ]. = u4 . 1.1.2: f25 := f23 (c0 , c1 , c2 , u1 , u4 ) we have f23 ∈ {j1 , j4 , v1 , v4 }. Because For the function f23 of v1 ∗ v1 = v4 and j4 ∗ j4 = j1 , we can assume f23 ∈ {j1 , v4 }. = j1 . 1.1.2.1: f23 When one substitutes u1 instead of u4 and u4 instead of u1 into f23 , one receives = j4 instead of f23 . Now one can continue the proof as in Case 1.1.1. f23 1.1.2.2: f23 = v4 . When one substitutes u1 instead of u4 and u4 instead of u1 into f23 , one receives the = v1 instead of f23 . Further, we have f36 := f36 (c0 , c1 , c2 , u1 , u4 , v1 , v4 ) ∈ function f23 = j1 , we substitute u1 instead of u4 , u4 instead of u1 , v1 instead of {j1 , j4 }. If f36 = j4 instead of f36 . v4 and v4 instead of v1 into f36 and we receive the function f36 Thus Case 1.1.2.2 was also reduced to Case 1.1.1. = v4 . 1.1.3: f13 := f31 (c0 , c1 , c2 , j1 , u1 , v4 ) ∈ {j4 , u4 , v1 } is a superposition over M . BeThen, f31 cause of j1 ∗ v1 = j4 , this case is reducible to Cases 1.1.1 and 1.1.2. = u4 . 1.2: f19 := f28 (c0 , c1 , c2 , j1 , u4 ) ∈ {j4 , u1 , v1 , v4 } or w.l.o.g. (by v1 ∗ v1 = v4 We can form f28 and u4 ∗ j4 = u1 ) f28 ∈ {u1 , v4 }. 1.2.1: f28 = u1 . In this case, one can continue the proof as in Case 1.1. = v4 . 1.2.2: f28 := f32 (c0 , c1 , c2 , j1 , v4 , u4 ) ∈ {j4 , u1 , v1 }. Because of v4 ∗ j4 = v1 Here we have f32 and u4 ∗ v1 = u1 we can assume f32 = u1 . Thus Case 1.2 is reducible to Case 1.1. 1.3: f19 = v4 . := f26 (c0 , c1 , c2 , j1 , v4 ). Then, f26 ∈ {j4 , u1 , u4 , v1 } or w.l.o.g. (by We form f26 v4 ∗ j4 = v1 ) f26 ∈ {u1 , u4 , v1 }. ∈ {u1 , u4 }. 1.3.1: f26 Continuation of the proof as in Cases 1.1 or 1.2.


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1.3.2: f26 = v1 . := f35 (c0 , c1 , c2 , j1 , It holds that j1 ∗v1 = j4 . Consequently, we can form a function f35 j4 , v1 , v4 ) ∈ {u1 , u4 }; i.e., Case 1.3.2 is reducible to Cases 1.1 and 1.2. Hence, we have proven (20.4) in Case 1. = j2 . Case 2: f43 Then f19 := f19 (c0 , c1 , c2 , j2 ) ∈ {j3 , u2 , u3 , v2 , v3 }. Because of u3 ∗ u3 = u2 and ∈ {j3 , u2 , v2 }. v3 ∗ v3 = v2 let w.l.o.g. f19 = j3 . 2.1: f19 := f22 (c0 , c1 , c2 , j2 , j3 ) and f22 := f22 (c0 , c1 , c2 , j3 , j2 ). Then {f22 , f22 } ∈ Set f22 {{u2 , u3 }, {v2 , v3 }} and {u2 , u3 , v2 , v3 } ⊆ [{f22 , f22 , j2 , j3 , f34 , f35 }]. Thus M0,1 ⊆ [M ] is proven. = u2 . 2.2: f19 := f25 (c0 , c1 , c2 , j2 , u2 ) ∈ {j3 , u3 , v2 , v3 }. Because of j2 ∗ u3 = j3 , Here we have f25 = v2 . Since f33 (c0 , c1 , c2 , j2 , u2 , v2 ) ∈ v3 ∗ v3 = v2 and Case 2.1 we can assume f25 {j3 , u3 , v3 }, j2 ∗ u3 = j3 and j2 ∗ v3 = j3 , Case 2.2 is reducible to Case 2.1. = v2 . 2.3: f19 Analogously to 2.2.

Thus (20.4) is proven. Therefore, w.l.o.g. we can assume M0,1 := { f ∈ P31 | f (0) = f (1) } = {c0 , c1 , c2 , j2 , j3 , u2 , u3 , v2 , v3 } ⊂ [M ].

(20.5)

By substituting the functions of M01 and identifying variables in f49 , one can form ∈ P31 with f49 (0) = f49 (1) and f49 = s1 . a function f49 Case 1: |Im(f49 )| = 2. In this case it is easy to check that M0,1 ∪ Ma,b ⊂ [M ] for a certain (a, b) ∈ {(0, 2), (1, 2)}. W.l.o.g. let (a, b) = (1, 2). Further, by (20.4) w.l.o.g. we can assume that, for certain α, β, γ, δ ∈ E3 , ⎛ ⎞ α γ ⎜α δ ⎟ 4 ⎟ f55 ⎜ ⎝ β γ ⎠ ∈ E3 \τ55 β δ holds. Then we can form the function f55 (x, y) := f55 (x), g2 (y)) (g1 g1 ∈ [M ], where 0 0 α and g2 are certain functions of M0,1 ∪ M1,2 with g1 = = and g2 1 1 β γ ∈ P3 \[P31 ]. . Obviously, f55 δ If |Im(f55 )| = 3, then it follows from Lemma 20.4.1, (c) that P3 ⊆ [M ]. This implies 3 with the help of Lemma 20.4.2, (a) and f58 ∈ P 3 \(P3 ∪ [{c∞ }]). [M ] = P Let )| = 2 (20.6) |Im(f55

be in the following. Because of Lemma 20.4.1, (b) there are certain a1 , a2 , a3 , b1 , b2 , b3 ∈ E3 with ⎛ ⎞ ⎛ ⎞ a 1 a2 a3 f55 ⎝ b1 a2 ⎠ = ⎝ b3 ⎠ , a1 b 2 b3

614


where a3 = b3 . By substituting functions of M0,1 ∪ M1,2 (⊂ [M ]) into f55 , one can form a unary function f55 with the property: M0,2 ⊆ [{f55 } ∪ M0,1 ∪ M1,2 ]. Then, with the help of function f57 ∈ M and Lemma 20.4.1, (a), it follows P31 ⊂ [M ]. By Lemma 20.4.3, (a) we have then P ol3 τ56 ⊆ [M ]. With the help of functions f56 and f57 of M , it is easy to prove that all functions of P3 are superpositions on M . Because of f58 ∈ M and Lemma 20.4.2, (a), this implies [M ] = P3 . )| = 3. Case 2: |Im(f49 In this case, f49 is a permutation = s1 . If f49 = s3 , then, because of (20.5), one can form a certain unary function g with g(0) = g(1) as a superposition over M and one continues to be able to use the proof as in the first case. = s3 , then it is possible to form a unary function h ∈ [M0,1 ∪ {f52 }](⊂ [M ]), If f49 which is either a permutation ∈ {s1 , s3 } or h is a function with |Im(h)| = 2 and h ∈ M0,1 . Consequently, one can completely reduce the second case to the first case.

Next, some remarks on partial Sheffer functions: k is called Sheffer iff [f ] = P k . A partial function f ∈ P In [Had-R 91] all partial Sheffer functions for k = 2 and all binary Sheffer function for k = 3 were described. Further, the statement 2 ⇐⇒ [f ] = P (∀A ∈ {pP OL2 {0}, pP OL2 {1}, pP OL2 {(0, 1), (1, 0)}, P2 ∪ [c∞ ]} : f ∈ A) was proven. In [Had-L 2006] one finds the proof of the following criterion: 3 ⇐⇒ [f ] = P (∀A ∈ {pP OL3 τi | i ∈ {1, 2, 3, 4, 5, 6, 10, 11, 12, 16, 17, 18, 19, 20, 21, 22, 23, 24, 41, 42, 47, 48, 55, 56, 57}} ∪ {P2 ∪ [c∞ ]} : f ∈ A), where the relations τi are defined in Tables 20.1–20.3. In [Had-L 2006] one finds also the proof that it is not possible to reduce the conditions from the above criteria; i.e., these conditions are independent of each other.

k 20.5 The Completeness Criterion for P In analog mode to the sixth chapter, one can show that a completeness critek . Subsequently, k can be found using the maximal partial classes of P rion for P the maximal partial classes are described in a form found by L. Haddad and I. G. Rosenberg. The following definitions are needed: Definitions Let Eqh be the set of all equivalence relations over {1, ..., h}. An h-ary relation ⊆ Ekh is called • areflexive, if ∩δε = ∅ for every ε ∈ Eqh , ε = ι2h , i.e., for all (x1 , . . . , xh ) ∈ we have xi = xj for all 1 ≤ i < j ≤ h.

k 20.5 The Completeness Criterion for P

615

• quasi-diagonal, if = σ ∪ δε , where σ is a nonempty areflexive relation, ε ∈ Eqh \ {ι2h } holds, and further = Ek2 for h = 2. Furthermore 1 := = 2 := :=

{(a, a, b, b), (a, b, a, b) | a, b ∈ Ek } δ{1,2},{3,4} ∪ δ{1,3},{2,4} , {(a, a, b, b), (a, b, a, b), (a, b, b, a) | a, b ∈ Ek } δ{1,2},{3,4} ∪ δ{1,3},{2,4} ∪ δ{1,4},{2,3} .

In the following, denote an h-ary relation of the form =σ∪( (δε )), ε∈F

where σ is an areflexive h-ary relation and F ⊂ Eqh . Let Gσ : = {π ∈ Sh | σ ∩ σ (π) = ∅}, where Sh denotes the set of all permutations over the set {1, ..., h} and σ (π) := {(aπ(1) , ..., aπ(h) ) | (a1 , ..., ah ) ∈ σ}. The model of is the h-ary relation M () : = {(π(1), . . . , π(h)) | π ∈ Gσ } ( {(x1 , . . . , xh ) ∈ {1, . . . , h}h | ε∈F

(i, j) ∈ ε ⇒ xi = xj })

on the set {1, . . . , h}.3 Suppose h, F , and σ fulfill exactly one of the following five conditions: i) h ≥ 2, F = ∅ and σ = ∅, i.e., is a nonempty h-ary areflexive relation; ii) h ≥ 2, F = {ε}, where ε = ι2h , σ = ∅ and σ ∪ δε = Ek2 , i.e., is a trivial quasi-diagonale h-ary relation; iii) h = 4 and F = {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}, {{1, 4}, {2, 3}}}, i.e., = σ ∪ 2 , where σ is an areflexive 4-ary relation (the empty set is possible); iv) h = 4 and F = {{{1, 2}, {3, 4}}, {{1, 3}, {2, 4}}}, i.e., = σ ∪ 1 , where σ is an areflexive 4-ary relation (= ∅ is possible); v) h = 2, h ≤ k, F = {i, j} and = Ekh , i.e., is a totally reflexive 1≤ij≤h

and totally symmetric not trivial relation.4 We say that is coherent, iff 3 4

In some papers the set Eh is elected instead of {1..., h} in defining the model. For h = 1 we have ∅ ⊂ ⊂ Ek .

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(1) Gσ = {π ∈ Sh | σ (π) = σ} and π(ε) := {(π(x), π(y)) | (x, y) ∈ ε} = ε for all π ∈ Gσ , if fulfills either the above condition i) or ii), Gσ = {π ∈ Sh | σ (π) = σ} ∪ {π ∈ Sh | π(F ) = F }, if fulfills iii) or iv), Gσ = {π ∈ Sh | σ (π) = σ} = Sh , if fulfills the condition v), and (2) for every nonempty subset σ of σ there exists a relational homomorphism γ : Ek → {1, . . . , h} of σ in M (), such that (γ(i1 ), . . . , γ(ih )) = (1, . . . , h) for at least an h-tuple (i1 , . . . , ih ) ∈ σ . Theorem 20.5.1 (Haddad-Rosenberg Theorem; [Had-R 89], [Had-R 92]; without proof) k there is a maximal parLet k ≥ 2. For every proper partial subclass A of P k , then either tial clone that contains A. If C is a maximal partial clone of P 2 C = Pk ∪ {f ∈ Pk | D(f ) = ∅} (= pP olk Ek ∪ {(∞, ∞)}) or C = pP OLk , where is one of the following relations: (1) an h-ary not trivial totally reflexive and totally symmetric relation with 1 ≤ h ≤ k; (2) an h-ary areflexive or quasidiagonal relation with h ≥ 2, which is coherent; (3) a quaternary relation 2 or 1 ; (4) a quaternary coherent relation σ ∪ τi , where i ∈ {1, 2} and σ = ∅ is a quaternary areflexive relation. max be the set of all relations (∈ Rk ) given in the above theorem. Then Let R the following theorem is a consequence of the above: k ; [Had-R 92]) Theorem 20.5.2 (Completeness Criterion for P k if and only if C ⊆ pP OLk for all ∈ R max and k . Then [C] = P Let C ⊆ P C ⊆ Pk ∪ {f ∈ Pk | D(f ) = ∅}.

20.6 Some Properties of the Maximal Partial Clones k of P We show first that each maximal clone of Pk is a subset of exactly a k . Then we specify the relations ∈ Rmax := maximal partial clone of P Mk ∪ Uk ∪ Sk ∪ Lk ∪ Ck ∪ Bk , with the property that pP OLk is a maximal k . partial clone of P A survey of the orders of the maximal partial clones forms the end of this section.

k 20.6 Some Properties of the Maximal Partial Clones of P

617

Next to the notations from the fifth chapter, we still need the following notations for certain relation sets: Let Pk,p be the set of all fixed-point-free permutations on Ek whose cycles have the same prim number length p. Let Sk,p := { {(x, s(x)) | x ∈ Ek } | s ∈ Pk,p } and let M k be the set of all ∈ Mk with the property that (Ek ; ) is a lattice. Theorem 20.6.1 For every maximal clone C ⊆ Pk there is exactly one maxk with C ∩ Pk = C. imal partial clone C ⊆ P k . Put Proof. Let C be a partial clone of P 2 := C n≥1 {f ∈ Pk | ∀f1 , . . . , fn ∈ C ∩ Pk :

(f (f1 , . . . , fn ) ∈ Pk =⇒ f (f1 , . . . , fn ) ∈ C) }. It was already shown Notice that C ∩ Pk2 = ∅. As C is a partial clone, C ⊆ C. is a partial clone with the following property: in [Fre 66] that C = P k . k =⇒ C C = P Consequently, if C is a maximal partial clone, then C = C.

(20.7)

Let M be a maximal clone of Pk and let C1 , C2 be two maximal partial clones )1 and C2 = C )2 . From M = Pk of Pk that contain M . Then, by (20.7) C1 = C 2 and [Pk ] = Pk (see Theorem 1.4.2), we have M 2 = (C1 ∩ Pk )2 = (C2 ∩ P2 )2 ⊂ Pk2 . )2 and by (20.7), C1 = C2 . we have C )1 = C By definition of C, Theorem 20.6.2 ([Had-L 2000]; without proof ) (1) Let M := Ck ∪ Mk ∪ Sk ∪ Uk ∪ Lk ∪ Bk and M1 := Sk \Sk,2 . For every k . ∈ M \ (Lk ∪ M1 ) the set pP OLk is a maximal partial clone of P (2) For s ∈ Pk,p with p ≥ 3 let so := {(x, s(x)) | x ∈ Ek } s2 := {(x, s(x), s2 (x), . . . , sp−1 (x)) | x ∈ Ek }. Then pP OLk s2 is a maximal partial clone that properly contains the partial clone pP OLk s0 .

618


We need the following notations for the statement still missing on classes of type L. Let p ∈ P, m ∈ N and W := Epm . As shown in Section 5.2.4, a maximal class of Ppm of type L is isomorphic to the set LW . The set LW one can also describe in the form P olW λ with λ := {(a, b, c, d) ∈ (Epm )4 | a ⊕ b = c ⊕ d}, where (a1 , ..., am ) ⊕ (b1 , ..., bm ) := (a1 + b1 , ..., am + bm ) and + is the addition modulo p. With the help of ⊕ and (α(a1 , ..., am ) := (α·a1 (mod p), ..., α·am (mod p)) for α ∈ Ep and (a1 , ..., am ) ∈ W ) we can define the following p-ary relation over W : λp := {(a, a ⊕ b, a ⊕ 2 b, ..., a ⊕ (p − 1) b) | a, b ∈ W }. Theorem 20.6.3 ([Had-L 2000]; without proof ) + For p ∈ P the partial clone pP OLW λp is a maximal clone of P W that properly contains the partial clone pP OLW λ. Now, we come to some theorems that deal with the finite generating of partial clones. 2 has partial subclasses, which are not It was shown already in [Fre 66] that P finitely generating. In [Lau 88], it was proven that there are maximal partial 2 that are not finitely generating. These are exactly the partial clones of P clones pP OL2 with ∈ {λ2 , G2 ([P21 ])} (see also [Bör-H 97]). The order of 2 agrees with the order of each finite generating maximal partial clone C of P C ∩ P2 . We need the following concept for the following criterion about the finite generating of strong partial clones: Let r > 0 and let C be a clone of Pk . Then C is called r-separable, if for every n > 0 and every b ∈ Ekn there are certain n-ary functions g1 , ..., gr ∈ C such that the mapping g : Ekn −→ Ekr with g(a) := (g1 (a), ..., gr (a)) has the property g −1 (g(b)) = {b} for all b ∈ Ekn . Theorem 20.6.4 ([B¨ or-H 97]; without proof ) Let C ⊆ Pk be a clone. Then the partial clone Str(C) is finitely generated if and only if C is finitely generated and there is an r ∈ N such that C is r-separable. If C r-separable and finitely generated, then ord(Str(C)) ≤ max{ord(C), r}.

Theorem 20.6.5 ([Noz-L 97], [Had-L 2000]; without proof ) Let k ≥ 3. Then (1) For every ∈ Mk ∪ Uk ∪ C1k ∪ C2k it holds ord(pP OLk ) = 2.

20.7 Intervals of Partial Clones That Contain a Maximal Clone

619

(2) For every ∈ Chk with 3 ≤ h ≤ k − 1 it holds ord(pP OLk ) ≤ h. (3) For every ∈ Mk ∪ Bk it holds ord(Str(P ol )) = 2. (4) ord(pP OLk s2 ) = 2 (see Theorem 20.6.2).

20.7 Intervals of Partial Clones That Contain a Maximal Clone Since there are many results about subclasses of Pk , it is obvious to classify k after Pk ∩ C. For an arbitrary subclass A of Pk : the subclasses C of P k | [F ] = F ∧ F ∩ Pk = A}. I(A) := {F ⊆ P This section aims to obtain cardinality statements over the sets I(A) for the maximal clones A of Pk . First, however, some general properties of the set I(A): Lemma 20.7.1 ([Str 97]; without proof ) Let A ⊆ Pk be a clone. Then (1) I(A) contains the partial clones A ∪ C∞ and Str(A) as well all partial clones of the form pP OLk with P olk = A. (2) If A is finitely generated with ord A = n, then pP OLk Gn (A) is the greatest element in the lattice (I(A); ⊆). Now we obtain from (2) of the above lemma: Theorem 20.7.2 If A is a finitely generated clone on Ek of the order n, then I(A) is exactly the interval [A, pP OLk Gn (A)] of the lattice of all partial k . clones of P Lemma 20.7.3 Let A = [A] ⊆ Pk . Then (1) A is a maximal subclass of A ∪ C∞ . (2) If {c10 , . . . , c1k−1 } ⊂ A then A ∪ C∞ ⊆ B for all B ∈ I(A)\{A}. Proof. (1) is clear. (2): Let B ∈ I(A)\{A}. If B = A ∪ C∞ , we do not have to prove anything. Otherwise, there is a function f n ∈ B \(A ∪ C∞ ). Let f (a1 , . . . , an ) = ∞. Now c1∞ = f (c1a1 , . . . , c1an ) ∈ B and from this one can easily verify that C∞ ⊆ B. Theorem 20.7.4 ([Had-L-R 2002; without proof ) Let ∅ = ⊂ Ek , T := P olk and k | f (n ) = {∞} }. {f n ∈ P T∞ := n≥1

Then I(T ) consists exactly of the partial clones:

620


T , T ∪ C∞ , pP olk , T ∪ T∞ , pP olk ∪ C∞ , pP olk ∪ T∞ , pPOL k . The partial clones are pairwise distinct for || > 1 whereas for || = 1 pP olk ∪ T∞ and pP OLk coincide. Their inclusions are shown in Figure 20.1. q pP OLk

q T ∪ T∞

q T ∪ C∞

q T

q pP olk ∪ T∞ q pP olk ∪ C∞ q pP olk

Fig. 20.1

Theorem 20.7.5 ([Had-L-R 2002]; without proof ) Let h ≥ 2 and ∈ Chk . Then the set I(P olk ) is a 3-element chain: P olk ⊂ (P olk ) ∪ C∞ ⊂ pP olk .

Theorem 20.7.6 ([Had-L-R 2002]; without proof ) Let ∈ Uk , X := P olk and 1 := ∪ {(∞, ∞)}. Then I(X) is a 4-element chain: X ⊂ X ∪ C∞ ⊂ pP olk 1 ⊂ P OLk . For clones M of type M, there are only partial results over I(M ). The following is one of these results: Theorem 20.7.7 ([Had-L-R 2002]; without proof ) Let ≤ ∈ Mk . Set M := P olk ≤, k }, ≤0 := ≤ ∪ {(∞, x) | x ∈ E k }, ≤1 := ≤ ∪ {(x, ∞) | x ∈ E 3 2 := {(x, y, z) ∈ Ek | x ≤ y ≤ z} ∪ {(∞, x, y), (x, y, ∞) | x, y ∈ Ek , x ≤ y}∪ ({∞} × Ek × {∞}), Mi := pP olk (≤i ) (i = 0, 1) and M2 := pP olk 2 . Then


621

{M, M ∪ C∞ , M0 , M1 , M2 , Str(M )} is the set of all partial clones from I(M ) included in Str(M ). Their inclusions are shown in Figure 20.2. Str(M ) q q M 2 @ @ @q M 1

M0 q @ @

@q M ∪ C ∞ q M

Fig. 20.2

Because of Str(P olk ≤) = P olk ≤ ⇐⇒ ≤∈ M k , Theorem 20.7.8 follows. Theorem 20.7.8 For every ∈ M k is |I(P olk )| = 6. Next we prove that I(A) is a finite set if A is a maximal class of type S. For this, we need some notations: Let p be a prime factor of k and s ∈ Pk,p a fixed-point-free permutation on Ek comprising of cycles of the same length p. Set so := {(x, s(x)) | x ∈ Ek }, S := P olk so , s2 := {(x, s(x), . . . , sp−1 (x)) | x ∈ Ek } and Smax := pP OLk s2 . By Lemma 20.6.2, we have that Smax is the (unique) maximal partial clone containing the maximal clone P olk so . As usual, the powers of s are defined recursively by setting s0 (x) := x and si+1 (x) := s(si (x)) for all x ∈ Ek . To describe functions of Smax , we define the following relations on Ekn : x =s y :⇐⇒ (∃i ∈ {0, 1, ..., p − 1} : si (x) = y), where si (x) = si ((x1 , . . . , xn )) := (si (x1 ), . . . , si (xn )). The relation =s is an equivalence relation of Ekn with rn := k n /p equivalence classes (or blocks) which we denote by U1 , . . . , Urn . Fix vi ∈ Ui (i = 1, . . . , rn ) and set Vn := {v1 , v2 , . . . , vrn }. Since for all f n ∈ S, b ∈ Ekn and i ∈ Ep

622


f (si (b)) = si (f (b)), each function f n ∈ S is fully determined by its values on Vn . Set p

k \E p | ∃α ∈ Ek ∀i ∈ Ep (ai = ∞ =⇒ ai = si (α)}. γ := {(a0 , . . . , ap−1 ) ∈ E k For i = 1, 2, . . . , p let the p-ary relation γi consist of all (a0 , a1 , . . . , ap−1 ) ∈ γ with exactly i coordinates ∞. Furthermore, set γ0 := s2 . For every I ⊆ Ep let τI := {(a0 , a1 , . . . , ap−1 ) ∈ γ | (∀i ∈ I : ai = ∞) ∧ (∀j ∈ Ep : \I aj = ∞)}. For every function f n ∈ Smax set χ (f ) := {(f (a), f (s(a)), . . . , f (sp−1 (a))) | a ∈ Vn } and χ(f ) := {(f (a), f (s(a)), . . . , f (sp−1 (a))) | a ∈ kn }. k with χ (f ) ⊆ χ(f ) and Notice that χ (f ) and χ(f ) are p-ary relations on E p

k \E p ) ∪ s2 , χ(f ) ⊆ (E k

(20.8)

and hence, in general, χ(f ) is not a subrelation of s2 . It is easy to check that χ(f ) = {(ai , ai+1 , . . . , ap−1 , a0 , a1 , . . . , ai−1 ) | (a0 , a1 , ..., ap−1 ) ∈ χ (f ), i ∈ Ep }. (20.9) p p 2 Moreover, for R ⊆ (Ek \Ek ) ∪ s and α ∈ {χ , χ}, set α−1 (R) := {g ∈ Smax | α(g) ⊆ R}. Then it holds that α−1 (R) = {g ∈ Smax | ∀ r1 , . . . , rn ∈ γ0 g+ (r1 , . . . , rn ) ∈ R}. We start with Lemma 20.7.9 Let f n ∈ Smax and I, I ⊆ Ep . Then (1) Gf := {g n ∈ Str (S) | D(g) = D(f )} ⊆ [S ∪ {f }], (2) χ−1

( χ (f ) ) ⊆ [S ∪ {f }],


623

(3) (α0 , α1 , . . . , αp−1 ) ∈ χ (f ) =⇒ χ−1

( {(α1 , α2 , . . . , αp−1 , α0 )} ∪ χ (f )) ⊆ [S ∪ {f }], (4) there is a function g ∈ [S ∪ {f }] with χ (g) = χ(f ), (5) χ−1 ( χ(f ) ) ⊆ [S ∪ {f }], (6) χ(f ) ∩ τI = ∅ =⇒ χ−1 (τI ∪ χ(f )) ⊆ [S ∪ {f }], (7) (χ(f ) ∩ τI = ∅ ∧ χ(f ) ∩ τI = ∅) =⇒ χ−1 (τI∪I ∪ χ(f )) ⊆ [S ∪ {f }], p (8) (p ∈ {2, 3} ∧ j ∈ {1, . . . , p} ∧ χ(f )∩γj = ∅) =⇒ χ−1 ( i=j γi ∪χ(f )) ⊆ [S ∪ {f }], 3

3

k \E 3 ∪χ(f )) ⊆ [S∪{f }]. k \(E 3 ∪γ)) = ∅) =⇒ χ−1 (E (9) (p = 3 ∧ χ(f )∩(E k k Proof. (1): Let g n ∈ Gf and g1 ∈ S with g1|D(f ) = g. Then g = e21 (g1 , f ) ∈ [S ∪ {f }]. (2): Let g m ∈ χ−1

(χ (f )) be arbitrary. Then there is for every v ∈ Vm a bv := (bv1 , bv2 , . . . , bvn ) ∈ χ (f ) with (g(v), g(s(v)), . . . , g(sp−1 (v))) = (f (bv ), f (s(bv )), . . . , f (sp−1 (bv ))). It is easy to check that the functions gim ∈ S (i = 1, 2, . . . , n) exist with gi (v) = bvi for all i ∈ {1, . . . , n} and v ∈ Vm . Then we have that g = f (g1 , . . . , gn ) and (2) hold. (3): Let a1 , . . . , aq ∈ Vn with ai = (ai1 , ai2 , . . . , ain ) (i = 1, . . . , q), χ (f ) = {(f (ai ), f (s(ai )), . . . , f (sp−1 (ai ))) | i = 1, . . . , q} and (f (aq ), f (s(aq )), . . . , f (sp−1 (aq ))) = (α0 , α1 , . . . , αp−1 ). Then there exists a t ∈ N, b1 , . . . , bq+1 ∈ Vt and functions gjt ∈ S (j = 1, 2, . . . , n) with gj (bi ) := aij (i = 1, 2, ..., q, j = 1, . . . , n) and gj (bq+1 ) := s(aqj ) (j = 1, . . . , n). Let g := f (g1 , . . . , gn ). So we have g ∈ [S ∪ {f }] and it is easy to check that χ (f ) ∪ {(α1 , α2 , . . . , αp−1 , α0 )} ⊆ χ(g). Consequently, (3) follows from (2). (4) follows from (20.9) and (3). (5) follows from (4) and (2). In the following, we can assume χ (f ) = χ(f ). (6): Let (a0 , a1 , . . . , ap−1 ) ∈ χ(f ) ∩ τI . Then there exists an α ∈ Ek with ∀i ∈ Ep (ai = ∞ =⇒ ai = si (α) .

624


Obviously, there are a t ∈ N, a function q t ∈ S with χ (q) = γ0 and a function h2 ∈ S with y if x ∈ {α, s(α), . . . , sp−1 (α)}, h(x, y) := x otherwise. Then n+t n+t h1 := h(f (en+t , en+t ..., en+t n ), q(en+1 , . . . , en+t )) ∈ [S ∪ {f }] 1 2

and it is easy to check that τI ∪ χ(f ) ⊆ χ(h1 ). Thus by (5) we have χ−1 (τI ∪ χ(f )) ⊆ χ−1 (χ(h1 )) ⊆ [S ∪ {f }]; i.e., (6) holds. (7): Let a1 , . . . , aq ∈ Vn with ai = (ai1 , ai2 , . . . , ain ) (i = 1, . . . , q), χ(f ) = {(f (ai ), f (s(ai )), . . . , f (sp−1 (ai ))) | i = 1, . . . , q}, (f (aq−1 ), f (s(aq−1 )), . . . , f (sp−1 (aq−1 ))) ∈ τI and (f (aq ), f (s(aq )), . . . , f (sp−1 (aq ))) ∈ τI . Then there exists t ∈ N, b1 , . . . , bq+1 ∈ Vt and functions gjt , htj ∈ S (j = 1, 2, . . . , n) with gj (bi ) := hj (bi ) := aij (i = 1, 2, ..., q, j = 1, . . . , n) and gj (bq+1 ) := aq−1,j , hj (bq+1 ) := aq,j (j = 1, . . . , n). Let g := e21 (f (g1 , . . . , gn ), f (h1 , . . . , hn )). Then g ∈ [S ∪{f }], and it is easy to check that χ(f ) ⊆ χ(g) and χ(g)∩τI∪I = ∅. Then, it follows from (6): χ−1 (τI∪I ∪ χ(f )) ⊆ [S ∪ {g}] ⊆ [S ∪ {f }]. (8) follows from (3) and (5)–(7). (9): Let p = 3, (a, b, ∞) ∈ χ(f ), a, b ∈ Ek and s(a) = b. Furthermore, let t ∈ N, k t /3 ≥ k 2 and {(αi , βi ) | i = 1, 2, ..., k2 } := Ek2 . Then there are c1 , . . . , ck2 ∈

:= {(a, ci ), (b, s(ci )) | i = 1, 2, . . . , k2 } with Vt with x =s y for all x, y ∈ Vt+1

⊆ Vt+1 holds, there is a x = y. Since we can choose Vt+1 such that Vt+1 t+1 ∈ S with function g ⎧ ⎨ αi if x = a, x = ci , g(x, x) := βi if x = b, x = ci , ⎩

. x if x ∈ Vt+1 \Vt+1 Then


625

n+t n+t h := g(f (en+t , . . . , en+t n ), en+1 , . . . , en+t ) ∈ [S ∪ {f }] 1

and it is easy to check that Ek2 × {∞} ∪ χ(f ) ⊆ χ(h). Then by (3) and (8) it follows that (9) holds. Theorem 20.7.10 ([Had-L-R 2002]) The set I(S) is finite. Proof. Obviously, Smax =

χ−1 (R).

(20.10)

k p \E p )∪s2 R⊆(E k

By Lemma 20.7.9, (5) we have χ−1 (χ(f )) ⊆ [S ∪ {f }]

(20.11)

for every function f n ∈ Smax . Let G := {χ(f ) | f ∈ Smax }. By (20.8) G is a finite set. Let C ∈ I(S). Obviously, H := {χ(f ) | f ∈ C} ⊆ G is finite. Thus H = {χ(1 ), . . . , χ(h )} for certain 1 , . . . h ∈ C. Furthermore, it holds that C ⊆ χ−1 (χ(1 )) ∪ . . . ∪ χ−1 (χ(h )), where, by (20.11), χ−1 (χ(1 ))∪. . .∪χ−1 (χ(h )) ⊆ [S∪{1 }]∪. . .∪[S∪{h }] ⊆ [S∪{1 , . . . , h }] ⊆ C. Thus C = [S ∪ {1 , . . . , h }]. Consequently, the partial clone C ∈ I(S) is generated from S and from not more than |G| functions of Smax \ S. Now we determine exactly the set I(S) for the cases p = 2 and p = 3. We begin with the case p = 2. Str(S) q

S1 q @ @

q S ∪S 1 2 @ @ @q S 2 @q S ∪ C ∞ q S

Fig. 20.3

626


Theorem 20.7.11 ([Had-L-R 2002]; without proof ) Let k ≥ 2, s ∈ Pk,2 and S = P olk so . Furthermore, let S1 := χ−1 (γ0 ∪ γ2 ) and S2 := S ∪ χ−1 (γ1 ∪ γ2 ). Then I(S) = {S, S ∪ C∞ , S1 , S2 , S1 ∪ S2 , Str(S)}, where Smax = pPOL k so = Str(S). The lattice (I(S)); ⊆) is given in Figure 20.3. Smax

r

Str(S) ∪ S5 r

H

HH

HH

H H r S3 ∪ S5 = S1 ∪ S3 ∪ S5

Str(S) r

r S1 ∪ S3

r S4 ∪ S5 = S 1 ∪ S4 ∪ S5

S ∪ S r 1 4 r S ∪ S5 = S1 ∪ S5 r r S1 S2 ∪ S4 H HH HH H H r S2 S4 r @ @ @ @r S ∪ C∞ r S3

rS Fig. 20.4

Theorem 20.7.12 ([Had-L-R 2002]; without proof ) Let s ∈ Pk,3 and S = Pol k so . Furthermore, let S1 := S ∪ χ−1 (γ1 ∪ γ2 ∪ γ3 ), S2 := S ∪ χ−1 (γ2 ∪ γ3 ), 3

k \ E 3 ), S3 := χ−1 (γ0 ∪ γ2 ∪ γ3 ), S4 := χ−1 (γ0 ∪ γ3 ) and S5 := χ−1 (E k where Str(S) = χ−1 (γ0 ∪ . . . ∪ γ3 ). Then

20.8 Intervals of Boolean Partial Classes

627

I(S) = {S, S ∪ C∞ , S2 , S4 , S2 ∪ S4 , S1 , S3 , S1 ∪ S4 , S1 ∪ S3 , S ∪ S5 , S4 ∪ S5 , S3 ∪ S5 , Str (S), Str (S) ∪ S5 , Smax }. The lattice (I(S); ⊆) is given in Figure 20.4. Theorem 20.7.13 ([Ale-V 94], [B¨ or-H 98], [Had-L 2003]) For every ∈ Lk ∪ Bk , the set I(P olk ) has the cardinality of continuum. Proof. For k = 2 (and therefore = {(a, b, c, d) | a + b = c + d (mod 2)}) the theorem was proven in [Ale-V 94]. If k is an arbitrary prime number power, then |I(P olk )| = c with ∈ Lk was proven in [Had-L 2003], where the proof of the general result includes the proof from [Ale-V 94]. or-H 98]. This result and Theorem 5.2.6.1 |I(P olk ιkk )| = c was shown in [B¨ were used in [Had-L 2003] to prove the remaining statements of the theorem.

20.8 Intervals of Boolean Partial Classes In continuation of the examinations of Section 20.7, one can find cardinality statements over the set I(A) for arbitrary subclasses A of P2 in the following. For this we use the notations from Chapter 3. Theorem 20.8.1 ([Ale-V 94], [Str 97b]) Let A be a subclass of P2 with A ⊆ L or A ⊆ B ∈ {I ∪ C, D ∪ C, K ∪ C, T0,∞ , T1,∞ }. Then the set I(A) has the cardinality of continuum. Proof. For A = L, the theorem was proven by V. B. Alekzeev and L. L. Voronenko in [Ale-V 94]. By easy modification of the proof from [Ale-V 94], one gets the statements of the theorem for A ∈ {L, L ∩ T0 , L ∩ T1 , L ∩ T0 ∩ T1 }. One finds proof of the remaining statements of the theorem in [Str 97b]. Theorem 20.8.2 ([Ale-V 94], [Str 97a], [Str 95], [Str 96], [Lau 2006]) Let A be a subclass of P2 with T0 ∩ T1 ∩ M ⊆ A or T0 ∩ S ⊆ A (or T1 ∩ S ⊆ A). Then I(A) is a finite set and it holds that

628


A |I(A)| P2 3 Ta (a ∈ {0, 1}) 6 M 6 S 6 T0 ∩ T1 30 Ta ∩ M (a ∈ {0, 1}) 15 T0 ∩ T1 ∩ M 101 T0 ∩ S ? < 2000 For the remaining subclasses A of Pk , i.e., M ∩ Ta ∩ Ta,∞ ⊂ A ⊆ Ta,2 for certain a ∈ E2 or A = T0 ∩ M ∩ S, it holds |I(A)| ≥ ℵ0 . 2 } and therefore |I(P2 )| = 3 follows from Proof. I(P2 ) = {P2 , P2 ∪ C∞ , P Lemma 20.4.2. The statements of the theorem over maximal classes of P2 different from L were proven by V. B. Alekzeev and L. L. Voronenko and by B. Strauch independently of each other. (see [Ale-V 94], [Str 94], [Str 97a]). Certain elements of the sets I(A) with A ∈ {T0 , T1 , S, M } were already determined in [Lau 88] 2 . On can find the description at the determination of submaximal classes of P of the elements of I(A) with A ∈ {T0 , T1 , S, M } in Theorems 20.7.4, 20.7.7, and 20.7.11. The sets I(T0 ∩ T1 ) and I(T0 ∩ M ) were determined in [Str 97a]. One can find I(M ∩ T0 ∩ T1 ) in [Str 95]. The finiteness of the set I(S ∩ T0 ∩ T1 ) was proven in [Str 96]. In [Lau 2006] it was proven that every set I(A) with A ⊆ T0,2 has infinitely many elements.

20.9 On Congruences of Partial Clones This section is a revised version of the papers [Lau-D 90] and [Lau-D 91]. For an arbitrary partial clone C, we define the following equivalence relations which – as A. I. Mal’tsev in [Mal 66] was already proving – are congruences k , are the only possible congruences (see Theorem over C and, for C = P 20.9.3): κ0 := {(f, f ) | f ∈ C} κ1 := C × C κa := {(f n , g m ) ∈ C × C | n = m} κ∞ := κ0 ∪ {(f n , g m ) ∈ C × C | D(f ) = D(g) = ∅}

20.9 On Congruences of Partial Clones

629

κ1 r @ @ κa r @r κ∞ κ0 r One is easily able to transfer many results from Chapter 9 (because of results of Section 20.2) to partial clones. Therefore, only the congruences on the maximal partial clones shall be determined. For this we need the following notations, which are introduced for relations ∈ Rkh . κ1 () := κ0 ∪ {(f m , g n ) ∈ κ1 | ∀r1 , ..., rmax(m,n) ∈ : k )h \E h }, {f (r1 , ..., rm ), g(r1 , ..., rn )} ⊆ (E k κa () := κ1 () ∩ κa , U () := {α ∈ Ek | ∀(a1 , ..., ah ) ∈ : α ∈ {a1 , ..., ah }}, µ0 () := {(f m , g m ) ∈ κa | ∀a ∈ (Ek \U ())m : f (a) = g(a)}, µ() := {(f m , g m ) ∈ κa | ∀r1 , ..., rm ∈ : k )h \E h ∨ {f (r1 , ..., rm ), g(r1 , ..., rm )} ⊆ (E k f (r1 , ..., rm ) = g(r1 , ..., rm )}. Obviously, the above relations are equivalence relations with the following Hasse-diagram: κ1 r @ @ κa r @r κ1 () ∪ µ() κa () ∪ µ() r r κ () ∪ µ0 () 1 κa () ∪ µ0 () r r κ1 () κa () r r κ∞ @ @ @r κ0 Since Ek \U () = , if is a unary relation, we have in this case µ() = µ0 (). If U () = ∅, then µ0 () = κ0 . If all constant functions and a function cn∞ belong to the partial clone pP OLk , then we have κ1 () = κ∞ and κa () ∪ µ() = κ0 . The following Lemma is easy to prove.

630


Lemma 20.9.1 k . Then the relations κ0 , κa , κ1 , κ∞ are congruences on C. (1) Let C ⊆ P k . Then the relations κa (), κa () ∪ (2) Let ⊆ Ekh and C := pP OLk ⊆ P µ0 (), κa () ∪ µ() κ1 (), κ1 () ∪ µ0 (), κ1 () ∪ µ() are congruences on C. k of the In the following, it shall be proven that a maximal partial clone of P form pP OLk has only those congruences given above. Then, dependently of , one has 4, 8 or 10 pairwise distinct congruences per partial clone pP OLk . Lemma 20.9.2 Let C be a partial clone on Ek and κ a congruence of C. Then (a) κ ∩ (Pk × Pk ) ⊆ κa =⇒ κ = κ1 ; (b) (c1∞ ∈ C ∧ κ ⊆ κa ) =⇒ κ∞ ⊆ κ. Proof. (a): Let κ∩(Pk ×Pk ) ⊆ κa . Then there are functions f m , g n ∈ C ∩Pk with m < n and (f, g) ∈ κ. Consequently, we have f11 := ∆n−2 f ∼ g12 := ∆n−2 g (κ) and

e22 = e22 f1 ∼ e33 = e22 g (κ).

Thus (enn , e11 ) ∈ κ for all n ≥ 1 and e11 h = h ∼ e22 h = et+1 t+1 (κ) for all ht ∈ C, i.e., κ = κ1 . (b): Let κ ⊆ κa and c∞ ∈ C. Then there are two functions f m , g n , m < n, in C with (f, g) ∈ κ. Thus (∆n−2 g) c1∞ = c2∞ ∼ (∆n−2 f ) c1∞ = c1∞ (κ). Consequently, c2∞ er1 = cr+1 ∼ c1∞ er1 = cr∞ (κ) for arbitrary r ∈ N. ∞ Therefore, κ∞ ⊆ κ. Theorem 20.9.3 ([Mal 66]) Let C be a clone with Str({ca | a ∈ Ek }) ⊂ C. Then C has exactly the following four congruences: κa , κa , κ∞ , κ1 . Proof. Let κ = κ0 be a congruence of C. Then the following cases are possible: Case 1: κ0 ⊂ κ ⊆ κa . In this case, there are two functions f m , g n ∈ C and an n-tuple a := (a1 , ..., an ) with f (ca1 , ..., can ) =: cα , g(ca1 , ..., can ) =: g , g ∈ {cβ , c∞ }, {α, β} ⊆ Ek , (cα , g ) ∈ κ and cα = g .


631

Since Str({ca | a ∈ Ek }) ⊂ C, there is a function h1 ∈ C with D(h) = {α} and h(α) = α. Thus we have h c1α ∼ h g = c1∞ (κ) and e11 = ∆(e22 cα ) ∼ ∆(e22 c1∞ ) = c1∞ (κ). We obtain e11 t = t ∼ cn∞ (κ) for all tn ∈ C. Thus κa ⊆ κ and κ = κa . Case 2: κ0 ⊂ κa . Since c∞ ∈ C, it follows from Lemma 20.9.2: κ∞ ⊆ κ. Suppose, κ = κ∞ . Then there are two functions f m , g n ∈ C with m = n, {f, g} ⊆ {cn∞ | n ∈ N} and (f, g) ∈ κ. W.l.o.g. we can assume that f = cm ∞ and m < n. Let a := (a1 , ..., am ) ∈ D(f ) and f (a) = α ∈ Ek . Then (...((∆((∆((∆(f ca1 )) ca2 )) ca3 )) ca4 ))... cam = f (ca1 , ..., cam ) = cα ∼ (...((∆((∆((∆(g ca1 )) ca2 )) ca3 )) ca4 ))... cam =: g1r (κ) with r := n − m + 1. W.l.o.g. let r = 2. If g1 = c2∞ , we obtain (c1α , c1∞ ) ∈ κ and analogously to Case 1, κa ⊆ κ. κa ⊆ κ and κ∞ ⊆ κ imply κ = κ1 . If g1 = c2∞ , there exists a (b1 , b2 ) ∈ Ek2 with g1 (b1 , , b2 ) ∈ Ek and thus we have c1α = c1α (τ (cα cb1 ) cb2 ) ∼ c1α (τ (g1 cb1 ) cb2 ) = c2α (κ). By Lemma 20.9.2, (a) we get κ = κ1 . Theorem 20.9.4 ([Lau-D 90]) max \{ ∈ R max | ∅ ⊂ Let C = Pk ∪ {cn∞ | n ∈ N} or C = pP OLk with ∈ R ⊂ Ek or is a coherent areflexive relation}, where Rmax denotes the set of all relations by which a maximal partial clone pP OLk is described (see 20.5). Then C has exactly the following four congruences: κ0 , κa , κ∞ , κ1 . max \{ ∈ R max | ∅ ⊂ ⊂ Ek or is a Proof. If C = pP OLk with ∈ R coherent areflexive relation}, our theorem follows from Theorem 20.9.3. It was proven in Chapter 9 (see Theorem 9.1.2) that Pk has only the congruences κ0 , κa , κ1 . From this and from Lemma 20.9.2 follows our theorem for C = Pk ∪ {cn∞ | n ∈ N}. Lemma 20.9.5 Let κ be a congruence of pP OLk . Then (a) κ0 ⊂ κ ⊆ κa =⇒ κa () ⊆ κ, (b) κ∞ ⊂ κ =⇒ κ1 () ⊆ κ, (c) κa () ⊂ κ ⊆ κ1 () =⇒ κ = κ1 (), (d) κa () ∪ µ0 () ⊂ κ ⊆ κ1 () ∪ µ0 () =⇒ κ = κ1 () ∪ µ0 (), (e) κa () ∪ µ() ⊂ κ ⊆ κ1 () ∪ µ() =⇒ κ = κ1 () ∪ µ(). Proof. (a): Let κ0 ⊂ κ ⊆ κa . Then there are functions f n , g n with (f n , g n ) ∈ κ and certain a := (a1 , ..., an ) ∈ Ekn , a, b ∈ Ek , a = b with f (a) =: a = b := g(a). Furthermore, pP OLk contains functions tα (α ∈ Ek ) and hβ,γ (β, γ ∈ Ek ) with

632


tα (x, y) :=

and hβ,γ (x) :=

y if x = α, ∞ otherwise, γ if x = β, ∞ otherwise,

where, if is areflexive or {ca | a ∈ Ek } ⊂ pP OLk holds, β and γ are arbitrary elements of Ek ; otherwise we must choose, however, β ∈ {a | (a, a, ..., a) ∈ }. Thus we obtain f (hβ,a1 (x), ..., hβ,an (x)) = hβ,a (x) ∼ g(hβ,a1 (x), ..., hβ,an (x)) = hβ,b (x) (κ) and

ta (hβ,a (x), y) = tβ (x, y) ∼ ta (hβ,b (x), y) = c2∞ (x, y) (κ).

Let um be an arbitrary function from pP OLk with the property ∀(r11 , r21 , ..., rh1 ), ..., (r1n , r2n , ..., rhn ) ∈ ∃i ∈ {1, ..., h} : (ri1 , ri2 , ..., rin ) ∈ D(u). Then the function v m defined by β if u(x) ∈ Ek , v(x) := ∞ otherwise, belongs to pP OLk . Consequently, we have tβ (v(x), u(x)) = u(x) ∼ c2∞ (v(x), u(x)) = cm ∞ (x) (κ), i.e., it holds κa () ⊆ κ. (b): For κ∞ ⊂ κ and κ ∩ κa = κ0 , we have κa () ⊆ κ because of (a), since the inclusion κa () ⊆ κ follows from κ ∩ κa = κ = κ0 and since the inclusion κa () ⊆ κ follows from κ ⊆ κ. κa () ⊆ κ and κ∞ ⊆ κ imply κ1 () ⊆ , since, if (f m , g n ) ∈ κ1 () (m = n), n n the inclusions (f m , cm ∞ ) ∈ κ and (c∞ , g ) ∈ κ follow, because of κa () ⊆ κ m n and (c∞ , c∞ ) ∈ κ by κ∞ ⊆ κ. By the transitivity of κ we get (f m , g n ) ∈ κ. If κ∞ ⊂ κ and κ ∩ κa = κ0 , there are two functions sl , tr (l = r) in pP OLk with (s, t) ∈ κ. We can assume l > r and s = cl∞ , i.e., there exists an element a := (a1 , ..., al ) with s(a) ∈ Ek . We distinguish the following two cases: Case 1: For all i ∈ {1, ..., l} it holds (ai , ..., ai ) ∈ . In this case we have {ca1 , ..., cal } ⊂ pP OLk . From (s, t) ∈ κ and κ∞ ⊆ κ the existence of two κ-kongruent constant functions with different arities follows. Therefore, by Lemma 20.9.2, (a) we get (Pk ∩ pP OLk ) × (Pk ∩ pP OLk ) ⊆ κ, a contradiction to κ ∩ κa = κ0 .


633

Case 2: There exists an i ∈ {1, ..., l} with (ai , ..., ai ) ∈ . W.l.o.g. we can assume (a1 , ..., a1 ) ∈ . Then the function h1a1 ,aj belongs to pP OLk for certain aj ∈ Ek , j = 1, ..., n. Further, we can assume a1 = al−r . (If a1 = al−r , then we can choose s := s e22 and t := t e2 and then use the operations ζ and τ to get functions s and t with (s , t ) ∈ κ, s (a) ∈ Ek , (a1 , ..., a1 ) ∈ and a1 = al−r ). Thus s1 := (....(ζ((ζ((ζs) ha1 ,a1 )) ha1 ,al−1 ha1 ,al−2 )) ...) ha1 ,al−r ) ∼ t1 := (....(ζ((ζ((ζt) ha1 ,a1 )) ha1 ,al−1 ha1 ,al−2 )) ...) ha1 ,al−r ) (κ) with s1 (x1 , ..., xl ) = s(xl−r+1 , ..., xl , ha1 ,al−r (x1 ), ..., ha1 ,al (xl−r )) and t1 (x1 , ..., xr ) = t(ha1 ,al−r+1 (x2 ), ..., ha1 ,al−1 (xr ), ha1 ,al (ha1 ,al−r (xl ))) = cr∞ (x), since ha1 ,al ha1 ,al−r = c1∞ and because of a1 = a1−r . From this and from κ∞ ⊆ κ, it follows that κ ∩ κa = κ0 , in contradiction to our assumption. Therefore (b) holds. (c): Let κa () ⊂ κ ⊆ κ1 (). Then κ ⊆ κa . By Lemma 20.9.2, (b) we have κ∞ ⊆ κ. From κ∞ ⊆ κ ⊆ κ1 () it follows that κ = κ∞ or κ = κ1 () (by (b)). Since the first case is not possible, (c) holds. (d) and (e) follow from (c). Lemma 20.9.6 Let ∅ ⊂ ⊂ Ek or let be an h-ary relation on Ek with the properties h ≥ 2 and ∀(a1 , ..., ah ) ∈ ∃t ∈ Pk ∩ pP OLk : Im(t) ⊆ {a1 , ..., ah }.

(20.12)

Then for an arbitrary congruence κ of pP OLk , (a) (κ ⊆ κa () ∪ µ() ∧ κ ⊆ κa ) =⇒ κ = κa , (b) (κ ⊆ κ1 () ∪ µ() ∧ κ ⊆ κa ) =⇒ κ = κ1 . (c) (κ ⊆ κa () ∧ κ ⊆ κa ) =⇒ κa () ∪ µ0 () ⊆ κ. Proof. (a): κ ⊆ κa ∪µ() and κ ⊆ κa imply that there are functions (f n , g n ) ∈ κ and a set R := {(rj1 , rj2 , ..., rjn ) | i = 1, ..., h} with {(r1i , r2i , ..., rhi ) | i = 1, ..., n} ⊆ , and 1) R ⊆ D(f ), R ⊆ D(g) and f (rj1 , rj2 , ..., rjn ) = g(rj1 , rj2 , ..., rjn ) for certain j ∈ {1, ..., h} or 2) R ⊆ D(f ) and R ⊆ D(g). The first case can be reduced to the second case as follows: Let e33 be a ternary function of pP OLk defined by

634


e33 (x, y, z) =

z if x = y, ∞ otherwise.

Then e33 (f (x), f (x), f (x)) = f (x) ∼ e33 (f (x), g(x), f (x)) =: g (x) (κ) with R ⊆ D(g ). Consequently, we can assume that R ⊆ D(f ) and R ⊆ D(g). Let (a1 , ..., ah ) ∈ . Then the functions t1 , ..., th with ⎛ ⎞ ⎛ ⎞ r1i a1 ⎜ a2 ⎟ ⎜ r2i ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ti ⎜ ⎜ . ⎟=⎜ . ⎟ ⎝ . ⎠ ⎝ . ⎠ ah rhi and ti (x) = ∞ for x ∈ {a1 , ..., ah } (i = 1, 2, ..., n) belong to pP OLk . Then, from (f, g) ∈ κ it follows f (t1 (x), ..., tn (x)) =: f (x) ∼ g(t1 (x), ..., tn (x)) =: g (x) (κ), where {r1 , ..., rh } ⊆ D(g ) and (f (a1 ), ..., f (ah )) ∈ for all (r1 , ..., rh ) ∈ . By Lemma 20.9.2 we have κa () ⊆ κ and thus (g , c1∞ ) ∈ κ and (f , c1∞ ) ∈ κ. (20.12), implies the existence of a function t1 ∈ Pk ∩ pP OLk with Im(t1 ) ⊆ {a1 , ..., ah }. Using this function and the fact that f = f t, we obtain another function of Pk ∩ pP OLk with (f , c1∞ ) ∈ κ. Thus (∆(e22 f ), ∆(e22 c1∞ )) = (e11 , c1∞ ) ∈ κ and κ = κa . (b): It is easy to check that κ ⊆ κ1 () ∪ µ() and κ ⊆ κa imply κ ∩ κa ⊆ κa () ∪ µ(). Hence (b) follows from (a) and κ∞ ⊆ κ. (c): Obviously, (c) holds, if U () = ∅. Thus we can assume U () = ∅. Because of Lemma 20.9.5, (a) we have to show that µ0 () ⊆ κ. Let κ be a congruence of pP OLk with κ ⊆ κa and κ ⊆ κa (). Then there are two different nary functions f, g with (f, g) ∈ κ and ρj := (r1j , r2j , ..., rhj ) ∈ (j = 1, ..., n) with (ri1 , ri2 , ..., rin ) ∈ D(f ) for all i ∈ {1, ..., h} or (ri1 , ri2 , ..., rin ) ∈ D(g) for all i ∈ {1, ..., h}. We can assume that κ = κa . Then by (a) we have κ ⊆ κa () ∪ µ(), i.e., f (r1 , ..., rn ) = g(r1 , ..., rn ) holds. Because of f = g there exists an n-tuple a := (a1 , ..., an ) ∈ Ekn with f (a) = g(a). It was already shown in proof of (a) that we can assume a ∈ D(g) and a ∈ D(f ). Let (α1 , ..., αh ) ∈ and a ∈ U (). Then the functions h1 , ..., hn defined by (hi (α1 ), ...hi (αh )) = ri , hi (a) = ai and hi (x) := ∞ otherwise belong to pP OLk . By this we have f (x) := f (h1 (x), ..., hn (x)) ∼ g(h1 (x), ..., hn (x)) =: g (x) (κ), where f (αi ) = g (αi ), f (a) = f (a1 , ..., an ) and a ∈ D(g ). Obviously, if is a unary relation or because of (20.12) (if is an areflexive relation and


635

h ≥ 2) pP OLk contains a function t ∈ Pk with Im(t) ⊆ {α1 , ..., αn }. Thus the function t defined by t(x) if x ∈ U (), t (x) := a otherwise, belongs to pP OLk and we get f t =: f ∼ g := g t (κ) and e22 (f (x), x) = e11 (x) ∼ e11 (x) := e22 (g (x), x) (κ) with x if x ∈ U (), e(x) := ∞ if x ∈ U (). Let pm be an arbitrary function of pP OLk and let p(x) if x ∈ (Ek \U ())m , pa (x) := a otherwise,

and p∞ (x) :=

p(x) if x ∈ (Ek \U ())m , ∞ otherwise.

From (e11 , e11 ) ∈ κ it follows e11 pa = pa ∼ e11 pa = p∞ (κ) and e22 (pa (x), p(x)) = p(x) ∼ e22 (p∞ (x), p(x)) = p∞ (x) (κ). Consequently, two m-ary functions p, q with p(x) = q(x) for x ∈ (Ek \U ())m are κ-kongruent, i.e., µ0 () ⊆ κ. Lemma 20.9.7 Let be an areflexive h-ary relation with h ≥ 2 and let (0, 1, ..., h − 1) ∈ . Further, for every (r1j , r2j , ..., rhj ) ∈ , j = 1, ..., n, there ∈ Pk ∩ pP OLk with are unary functions h1 , ..., hn (h1 (i), h2 (i), ..., hn (i)) = (ri1 , ri2 , ..., rin ) for all i ∈ {0, ..., h − 1}. Then for every congruence κ of pP OLk : (a) µ0 () ⊂ κ\κ1 () ⊆ µ() =⇒ µ() ⊆ κ; (b) κa () ∪ µ0 () ⊂ κ ⊆ κa () ∪ µ() =⇒ κ = κa () ∪ µ(); (c) κ1 () ∪ µ0 () ⊂ κ ⊆ κ1 () ∪ µ() =⇒ κ = κ1 () ∪ µ(). Proof. (a): Let κ be a congruence of pP OLk with µ0 () ⊂ κ\κ1 () ⊆ µ(). Then there are two κ-kongruent functions f n , g n , such that for certain h-tuple (r11 ..., rh1 ), ..., (r1n , ..., rhn ) ∈ there are an n-tuple (α1 , ..., αn ) ∈ Ekn and an element α ∈ Ek with f (α1 , ..., αn ) = α, (α1 , ..., αn ) ∈ D(g) and ⎛ ⎛ ⎞ ⎛ ⎞ ⎞ a1 r11 ... r1n r11 ... r1n ⎜ . ... . ⎟ ⎜ . ... . ⎟ ⎜ . ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ f⎜ ⎝ . ... . ⎠ = ⎝ . ⎠ = g ⎝ . ... . ⎠ . rh1 ... rhn ah rh1 ... rhn By assumption, there are functions h1 , ...., hn ∈ Pk ∩ pP OLk with {(h1 (i), ..., hn (i)) | i ∈ Eh } = {(rj1 , rj2 , ..., rjn ) | j ∈ {1, 2, ..., h}}. Let tm be an arbitrary function of pP OLk . Then, with the help of

636


Dt := ⎧ ⎛ a1 ⎪ ⎪ ⎨ ⎜ . m x ∈ Ek | ∃a1 , ..., ah : x ∈ {a1 , ..., ah } ∧ ⎜ ⎝ . ⎪ ⎪ ⎩ ah

⎫ ⎞ a1 ⎪ ⎪ ⎬ ⎜ . ⎟ ⎟ ⎟ ⊆ ∧ t⎜ ⎟∈ ⎝ . ⎠ ⎠ ⎪ ⎪ ⎭ ah ⎞

⎛

one can define the functions ti by ⎧ x ∈ Dt , ⎨ hi (t(x)) if if x∈ Dt ∧ x ∈ D(t), ti (x) := αi ⎩ ∞ otherwise (i = 1, 2, ..., n). These functions belong to pP OLk and we get u(f (t1 (x), ..., tn (x)) = t(x) ∼ t (x) = u(g(t1 (x), ..., tn (x)) (κ)

with u(x, y) :=

y if x ∈ {a1 , ..., ah , α}, ∞ otherwise,

and

t (x) :=

t(x) if x ∈ Dt , ∞ otherwise.

Thus µ() ⊆ κ. (b) and (c) follow from (a). Lemma 20.9.8 Let ⊆ Ekh be an areflexive coherent relation with h ≥ 2. Then (a) ∀(a1 , ..., ah ) ∈ ∃t ∈ Pk ∩ pP OLk : Im(t) ⊆ {a1 , ..., ah } and w.l.o.g. (b) one can assume that fulfills the assumptions of Lemma 20.9.7. Proof. By definition of (see Section 20.5) we can assume that there is a subgroup G of Sh and a surjective function ϕ : Ek −→ {0, 1, ..., h − 1} with the following properties: = {(π(0), π(1), ..., π(h − 1)) | π ∈ G }, is symmetric in respect to every π ∈ G and ∀(a0 , ..., ah−1 ) ∈ : (ϕ(a0 ), ..., ϕ(ah−1 )) ∈ . For an arbitrary a := (a0 , ..., ah−1 ) ∈ we consider the function ϕa : {0, 1, ..., h − 1} −→ {a0 , ..., ah−1 } with ϕa (i) = ai for all i ∈ {0, ..., h − 1}. Further, let ha := ϕa ϕ. Then Im(ha ) = {a0 , ..., ah−1 } and ha ∈ pP OLk , since for every b := (b0 , .., bh−1 ) ∈ there exists a permutation πb ∈ G with ϕ(b) = (ϕ(b0 ), ..., ϕ(bh−1 )). Thus we obtain ha (b) = ϕa (ϕ(b)) = ϕa ((πb (0), ..., πb (h − 1))) = (aπb (0) , ..., aπb (h−1) ) ∈ because of symmetry of with respect to every permutation π ∈ G . Consequently, (a) and (b) hold.


637

Theorem 20.9.9 ([Lau-D 91]) Let ∅ ⊂ ⊂ Ek or let ⊆ Ekh an areflexive coherent relation with h ≥ 2 and U () = ∅. Then pP OLk has exactly 8 congruences with the following congruence lattice: κ1 r @ @ κa r @r κ () ∪ µ() 1 κa () ∪ µ() r r κ1 () κa () r r κ∞ @ @ @r κ0

Proof. Since µ() = µ0 () for all with ∅ ⊂ ⊂ Ek and for all areflexive relations with U () = ∅, our theorem follows from Lemmas 20.9.2 and 20.9.5– 20.9.8. The following theorem also follows from Lemmas 20.9.2, 20.9.5–20.9.8: Theorem 20.9.10 ([Lau-D 91]) Let ⊆ Ekh be an areflexive relation with h ≥ 2 and U () = ∅. Then pP OLk has exactly 10 congruences, and the congruence lattice of pP OLk is given by

κa κa () ∪ µ() κa () ∪ µ0 () κa ()

κ1 r @ @ r @r κ () ∪ µ() 1 r κ () ∪ µ0 () r 1 κ () r r 1 r κ∞ r @ @ @r κ0

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Glossary

Ud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 ∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∨ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ¬ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 =⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ⇐⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 :=. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17 :⇐⇒ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∃ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∃! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 ∀ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 af . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 D(f, A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 D(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Im(f ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 (n) c∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 f 2g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 g1 g2 ...gr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 (A; f1 , ..., fr ) . . . . . . . . . . . . . . . . . . . . . . . . 26 (A; (fi )i∈I ) . . . . . . . . . . . . . . . . . . . . . . . . . . 26 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 f A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 fA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (D1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (D2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (M1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

(M2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (M3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (M4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (L1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (L5 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (DL1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (DL2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (B1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Pkn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 [T ]A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 [T ]F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32 [T ]f1 ,...,fr . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 [T ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 S(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (L1 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L1 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L2 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L2 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L3 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L3 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L4 a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (L4 b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (O1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

656

Glossary

(O2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (O3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (O4 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 sup Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (S1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (S2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 inf Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (I1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (I 1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (E1 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (E2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (E3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Eq(A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42 a = b (mod ) . . . . . . . . . . . . . . . . . . . . . . . 43 a ∼ b (mod ) . . . . . . . . . . . . . . . . . . . . . . . 43 ∇A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ∆A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 Π(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 A∼ = B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 ∼ = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Ker ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Con(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 NG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 ker ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 IR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 (σ, τ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 (GC1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 (GC2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 ≤δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 pr1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 pr2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 S(K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .71 H(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 P (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 I(K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 XY (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 (F, τ ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73 T (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 T(X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

T F (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 t < x1 , ..., xn > . . . . . . . . . . . . . . . . . . . . . . 76 s := t < t1 , ..., tn > . . . . . . . . . . . . . . . . . . 76 s ≈ t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 A |= s ≈ t . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 IdX (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 M od(Σ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .77 IdX (K) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 ConsX (Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . 77 T(X)/IdX (K) . . . . . . . . . . . . . . . . . . . . . . 79 FK (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (x1 , ..., xn ) . . . . . . . . . . . . . . . . . . . . . . . 79 FK (n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (x1 , x2 , ...) . . . . . . . . . . . . . . . . . . . . . . . 79 FK (ℵ0 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 FK (ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 R1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Rep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 tˆ, t ∈ P rop . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 PAn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Pkn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 F n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA,B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA (l). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 Pk (l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 PA [l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Pk [l] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 x(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 x. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .92 JA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Jk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .93 cn a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ∧ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ∨ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x + y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ⇒ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x ⇐⇒ y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 x · y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 xy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 πs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ∆t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Glossary ∇q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 PA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .96 f (g1 , ..., gn ) . . . . . . . . . . . . . . . . . . . . . . . . . 97 [F ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↓A (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↑A (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↑k (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 L↓k (F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 LA (F ; G) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 ord F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ord F = ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ιhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 δ{α,β} 3 δ{1,2,3} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 P rop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 |= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 Cons(Σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 sub . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 V ar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111 F ORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 =xk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 vA,u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Rkh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Qh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk,ε δε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δεh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126 Dkh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Dk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk;ε 1 ,...,εr δε1 ,...,εr . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 δk;

657

h δk;E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 k ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 × . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ∧ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Rk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 [Q] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 σs (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .128 prα1 ,...,αt () . . . . . . . . . . . . . . . . . . . . . . . . 128 ∆i,j () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 νi () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ∇i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ◦t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 P olk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 P ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 P olk Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Invk f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Invk A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 P oln Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Inv n A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 χk;n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 χn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 χ(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 Gn (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 ΓA (σ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 P olA Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Rphk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Rpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 α(, ). . . . . . . . . . . . . . . . . . . . . . . . . . . . .140 ∇(, ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 (, ) × (µ, µ ) . . . . . . . . . . . . . . . . . . . . . 141 E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ν1,a (, ) . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ν2,a (, ) . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Rpk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 m(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . 146 t(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 q(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 r(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 hµ (x1 , ..., xµ+1 ) . . . . . . . . . . . . . . . . . . . . 146 xσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 f δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

658

Glossary

T0,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T1,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T0,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 T1,∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Ca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 ζ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 ∆ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 ∇ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Ut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 ιhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 λk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 prE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 Mk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 ≤ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Sk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Skn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Fr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 ja (x) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Uk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 Lk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 λG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 (W ; ⊕) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Ln W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Xi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 LW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Chk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Ck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 a(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 ξm Bhk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Bk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 z(x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

gf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .175 fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Rmax (Pk ) . . . . . . . . . . . . . . . . . . . . . . . . . . 183 ∼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 M∼ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 S∼ k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 ∼ (Pk ) . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Rmax o(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 z(i) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 ch (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 b(k, h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 +o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 αi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .205 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 ξ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 µi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 z t,a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 z t,a [i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 ϕn (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 gI,J Con A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 f ∼ g (κ) . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κ1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Con1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Cona A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .235 µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 f 1 g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 κa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 µ(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 ki (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 k(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 α(κ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 r(x, y, z) . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 TA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 NA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 κ(I, U ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 qa (x, y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Glossary QA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .252 ⊕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 " . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 LM ;id . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .254 κs,t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 κU,µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 κ0, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 κZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 κc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .261 κN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 κf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 µr (k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 µr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 µr,N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 Con(k). . . . . . . . . . . . . . . . . . . . . . . . . . . . .268 κα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 ess(f ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . .270 αµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 βµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 γµ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Tn (U1 , U2 , ..., Ut ) . . . . . . . . . . . . . . . . . . . 273 κU1 ,...,Ut . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 κU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 K0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 Ka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 prl−1 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 σn,κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 πn,κ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 F (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 f α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 ar(). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .291 armax (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . 291 d(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 dim A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Q0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 Q1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 k1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 ja . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 prl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 pr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .337 pr−1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Nk (B) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 P olPk,l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

659

P ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 B a1 ,...,ar . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 Za,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 dm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 t(i ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 R(α) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Tn (Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 Eq(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 kf,I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Kf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 KM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 K0,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 Ud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 rd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 ϕ(q) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 t(q, k) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 Tα,U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 SU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Cα,U,I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Cα,U,∅ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 Qt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 LM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 ⊗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 Ta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 Ta,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 min . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404 T0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 U . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434 ϕi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .435 ϕi (A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

660

Glossary

τ (H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435 L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 λ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 La,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 pra,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 −1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 pra,b Za,b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457 Bc,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 Br . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458 M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 numf (fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . 465 num(fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 465 R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 (III). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .468 (IV ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Ji . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Ji,r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 Ui . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 A2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 A3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 I1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 Bi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Ij . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 Ai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 486 Aj . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 T2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 TQ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Mk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Uk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Sk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Pk,Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Lk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Ck;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 Nk;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500 MA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 MA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 UA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501

UA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 SA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 SA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 PA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 LA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 LA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 CA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 CA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 NA;Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 BA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501 Rmax (TQ ) . . . . . . . . . . . . . . . . . . . . . . . . . . 501 (t1 , t2 , ..., tn ) . . . . . . . . . . . . . . . . . . . . . . 515 [t]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .515 ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 Uk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Sk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 (1) Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Chk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 ZC2k;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 516 Zk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .516 Nk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 h [r] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Ck;E l Ck;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Hk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 γr,s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 h Bk;E [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518 l Bk;El [r] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Bk;El . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Rmax (Pl ) . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Rmax (P olk El ) . . . . . . . . . . . . . . . . . . . . . 519 (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 (I ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 530 εb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541 Ai [b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 αi [b] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Vb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 Fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542 ξb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .543 Ai,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 µi,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 νs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 fb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 wi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 544 z[i] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 z t,a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545 un . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547

Glossary s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 θs .n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 P ⊆p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598 f+ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 f− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 en i,X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 JX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600 k h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 R k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 R

661

pP olk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 pP OLk . . . . . . . . . . . . . . . . . . . . . . . . . . . 605 Eqh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614 max . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 616 R κ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .628 κ1 (). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .629 κa () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 U () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 µ() . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629 µ0 () . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 629

Index

A. I. Mal’tsev’s theorem, 268 adding of certain fictitious variables, 94 adding of fictitious coordinates (rows), 129 algebra, 26 semiprimal, 104 axiom of, 27 closed subset, 32 demiprimal, 104 demisemiprimal, 104 directly irreducible, 64 extension of, 31 factor, 55 finite, 26 finitely axiomatizable, 84 finitely based, 84 free, 79 fundamental operations, 26 generating system, 32 hemiprimal, 104 infinite, 26 infraprimal, 104 of finite type, 27 partial, 26 preprimal, 104 primal, 104 quasiprimal, 104 quotient, 55 semiprimal, 501 set of all subalgebras, 33 simple, 53 subalgebra, 31 type, 26, 73

universal, 26 universe, 26 algebras of same type, 27 all relation, 43 all-congruence, see congruence antiisomorphic, 59 antiisomorphism, 59 arity, see operation arity congruence, 235 atom, 112 atomic proposition, 106 automorphism, 52 inner, 285 basis, 98 block, 44 Boolean algebra, 30 Cartesian product, 64 chain, 36 characterization theorem for Shefferfunctions, 215 class, 97 B-projectable, 337 l-class, 280 inverse image, 337 maximal, 98 minimal, 589 of type B, 174 of type C, 173 of type L, 171 of type M, 165 of type U, 170 of type X, 165

664

Index

order, 98 submaximal, 98 class of algebras closed, 72 class of all models of Σ, 77 clone, 97 minimal, 590 strong, 599 closed set system, 45 closure, 97 deductive, 82 closure operator, 45 algebraic, 46 co-class, 141 co-clone, 127 co-group, 138 co-monoid, 138 complete, 98 completeness criterion for P2 , 156 for Pk , 191 for Pk,l , 352 for TQ , 501 for the class of all idempotent functions of Pk , 501 completeness problem, 117 completeness theorem for the equational logic, 84 completeness theorem of proportional logic, 110 composition, 25 general, 129 conclusion, 77 congruence, 52, 234 n-congruence, 279 congruence class, 55 fully invariant, 83 of the first kind, 235 of the second kind, 235 theorem for maximal clones, 265 trivial, 53, 234 congruence relation, see congruence congruence theorem for P2 , 237 constant, 93 countability criterion, 221 cyclical exchanging of the lines, 127 deductive closure, 82 depth of a subclass, 433

diagonal, 126 diagonale, 43 dimension, 291 direct product, 61 of classes, 397 of functions, 397 DNF, 99 domain, see operation doubling of coordinates (rows), 129 dual isomorphic, 59 duality principle of the lattice theory, 36 element central, 174 greatest, 165 inverse, 28 least, 165 neutral, 28 elementary operations, 95 embedding, 67 endomorphism, 52 equation, 76 equational class, 77 equational theory, 77 equivalence class, 43 equivalence relation, 42 equivalence class, 43 finer, 351 permutable, 62 transversal to s, 556 trivial, 43 exchange of the first two rows, 127 factor algebra, 55 factor set, 43 family of sets, 65 fictitious place of a function, 93 field, 29 floor function, 331 free algebra, 79 free generating set, 79 function n-ary on A, 91 r-th component, 168 autoduale, 167 Boolean, 93 component, 371 components of f , 359

Index constant, 93 extended, 600 linear, 171 monotone, 165 near unanimity function, 342 partial, 598 preserves the relation , 130 quasi-linear, 171 quasilinear, 456 reducible, 150 reduction, 599 restricted, 600 semiprojection, 591 subfunction, 598 total, 598 function algebra, 30 full, 96 iterative full, 96 functions κ-congruent, 234 are associated, 238 identity of, 93 fundamental group, 218 fundamental lemma of Jablonskij, 102 fundamental operations, see algebra fundamental semigroup, 218 fundamental set, 218 fuzzy logic, 116 Galois connection, 59 Galois correspondence, 59 generating set, 48 generating system, 48, 98 Gorlov’s tqheorem, 281 graphic n-te of A, 133 group, 28 Abelian, 28 additive notation, 28 commutative, 28 semiregular representation, 557 gruppoid, 27 Haddad-Rosenberg theorem, 616 Hasse diagram, 36 Hilbert-type-calculus, 107 homomorphism, 51 kernel, 52 natural, 55

665

quotient, 55 homomorphism theorem for groups, 57 for rings, 58 general, 55 hull, 45, 97 hull system, 45 I. A. Mal’tsev’s theorem, 241 ideal, 58 identification of certain variables of f , 94 identification of coordinates, 129 identity, 43, 76 inductively set system, 68 information transformer, 116 intersection, 127 inverse element, 28 inverse image homomorphic, 174 isomorphic, 39 isomorphic lattices, 39 isomorphism, 39, 52 anti-, 59 dual, 59 kernel of a group homomorphism, 57 of a homomorphism, 52 of a ring homomorphism, 58 Krasner-algebra of first kind, 138 of second kind, 138 lattice, 30 bounded, 30 complete, 42 distributive, 30 first definition, 35 isomorphic, 39 second definition, 37 sublattice, 41 with 0 and 1, 30 left unit, 241 lexicographical order, 132 limit class, 280 main theorem of the equational theory first, 81 second, 84

666

Index

majority function, 591 Mal’tsev-operations, 31, 95 mapping homomorphic, 51 isomorphic, 52 order-preserving, 39 projection-, 61 minority function, 591 module, 29 R-module, 29 over a unitary ring, 29 over the ring R, 29 modus ponens, 108 monoid, 28 neutral element, 28 normal form disjunctive, 99 normal subgroup, 56 operation n-ary partial, 25 arity, 25 domain, 25 elementary on Rk , 127 nullary, 25 range, 25 operation symbol, 73 order, 98 dual, 59 partial, 36 order diagram, 36 partition, 44 Peirce decomposition, 397 permutation of coordinates, 128 permutation of variables of f , 94 polymorphism, 130 poset, 36 antiisomorphic, 59 complete, 41 dual isomorphic, 59 Post’s theorem, 148 predecessor proper, 292 predicate, 111 preserve a relation pair, 140 preserving of a set, 97

preserving of relations, 130 product Cartesian, 127 projection, 93 onto the α1 -te, ..., αt -te coordinates, 128 onto the i-th coordinate, 127 projection mapping, see mapping proposition, 105 quotient algebra, 55 range, see operation reduct of an algebra, 104 relation (l; r)-homogeneous, 540 M -permissible, 363 θs -closed, 555 -derivable, 127, 499, 515 {ζ, τ, pr, ∧, ×}-derivable, 128 h-ary, 125 h-ary elementary, 174 h-regular, 178 h−universal, 175 (l; 2)-universal, 518 (l; r)-central, 517 (l; r)-universal, 518 areflexive, 614 central, 173 central element c ∈ El , 517 coherent, 615 derivable, 127 diagonal, 126 induced relation, 269 invariant of the function f , 130 irredundant, 605 length, 126 primitive, 197 quasidiagonal, 615 row, 126 strong, 179 strongly (l; r)-homogeneous, 540 strongly (l; 2)-homogeneous, 517 strongly homogeneous, 204 totally (l; r)-reflexive, 517 totally (l; r)-symmetric, 517 totally reflexive, 174 totally symmetric, 174 unary transversal to s, 556

Index

667

weakly (l; r)-central, 517 width, 126 relation algebra on Ek , 127 full, 127 relation degree, 291 relation pair, 140 relation pairalgebra full, 141 relation product, 129 relation set α-permissible, 350 -independent, 179 h-regular, 178 is closed, 127 minimal coarsening, 351 permissible, 350 relation-pair algebras, 141 replacement of the i-th variable of f through the function g and the changing of the denotation of variables, 95 replacement rule, 82 representation theorem for functions of PA , 98 residue class ring, 58 right zero, 241 ring, 28 ideal, 58 with unit element, 28 ring, unitary, 28 Rosenberg’s completeness criterion, 191

infimum of a subset, 37 linearly ordered, 36 of all invariants, 131 of conjunctions, 147 of constant functions of P2 , 147 of diagonal relations, 126 of disjunctions, 147 of linear functions of P2 , 147 of monotone functions of P2 , 146 of projections of P2 , 147 of self-dual functions of P2 , 146 of the h-ary relations on Ek , 126 partially ordered, 36 partition, 44 supremum of a subset, 36 totally ordered, 36 Sheffer-function, 211 Sheffer-function for P olk , 10, 307 subclass, 97 congruence on, 234 depth, 433 dimension, 291 maximal, 98 relation degree, 291 subdirect product, 66 subdirect representation, 67 subdirectly irreducible, 67 substitution rule, 82, 107 superposition operations, 94 superposition over F , 96 Slupecki-function, 211

selector, 93 semigroup, 28 commutative, 28 semilattice, 29 semiprojection, 591 semiring, 28 set CH -basis, 49 CH -independent, 49 J-closed, 274 basis, 49 closed, 46, 97 complete, 98 complete in a class, 98 finitely generated, 48 generated, 46 independent, 49

tautology, 106 term, 74 induces a term function, 75 term algebra, 74 term function, 75 theorem of Webb, 215 theorem on the orders of the maximal classes, 309 theorem over the cardinality of Lk , 221 transitive t-fold, 218 tuple h-tuple, 125 universe, see algebra valuation, 106

668

Index

variable, 74 bound, 112 essential, 93 fictitious, 93 free, 112 variety, 72

vector space over the field K, 29 zero-congruence, see congruence zigzag, 325 Zorn’s Lemma, 68

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Abhyankar, S.S. Resolution of Singularities of Embedded Algebraic Surfaces 2nd enlarged ed. 1998 Alexandrov, A.D. Convex Polyhedra 2005 Andrievskii, V.V.; Blatt, H.-P. Discrepancy of Signed Measures and Polynomial Approximation 2002 Angell, T. S.; Kirsch, A. Optimization Methods in Electromagnetic Radiation 2004 Ara, P.; Mathieu, M. Local Multipliers of C*-Algebras 2003 Armitage, D.H.; Gardiner, S.J. Classical Potential Theory 2001 Arnold, L. Random Dynamical Systems corr. 2nd printing 2003 (1st ed. 1998) Arveson, W. Noncommutative Dynamics and E-Semigroups 2003 Aubin, T. Some Nonlinear Problems in Riemannian Geometry 1998 Auslender, A.; Teboulle M. Asymptotic Cones and Functions in Optimization and Variational Inequalities 2003 Banasiak, J.; Arlotti, L. Perturbations of Positive Semigroups with Applications 2006 Bang-Jensen, J.; Gutin, G. Digraphs 2001 Baues, H.-J. Combinatorial Foundation of Homology and Homotopy 1999 Böttcher, A.; Silbermann B. Analysis of Toeplitz Operators 2nd ed. 2006 Brown, K.S. Buildings 3rd printing 2000 (1st ed. 1998) Chang, K. Methods in Nonlinear Analysis 2005 Cherry, W.; Ye, Z. Nevanlinna’s Theory of Value Distribution 2001 Ching, W.K. Iterative Methods for Queuing and Manufacturing Systems 2001 Crabb, M.C.; James, I.M. Fibrewise Homotopy Theory 1998 Chudinovich, I. Variational and Potential Methods for a Class of Linear Hyperbolic Evolutionary Processes 2005 Dineen, S. Complex Analysis on Infinite Dimensional Spaces 1999 Dugundji, J.; Granas, A. Fixed Point Theory 2003 Ebbinghaus, H.-D.; Flum J. Finite Model Theory 2006 Elstrodt, J.; Grunewald, F. Mennicke, J. Groups Acting on Hyperbolic Space 1998 Edmunds, D.E.; Evans, W.D. Hardy Operators, Function Spaces and Embeddings 2004 Engler, A.; Prestel, A. Valued Fields 2005 Fadell, E.R.; Husseini, S.Y. Geometry and Topology of Configuration Spaces 2001 Fedorov, Y.N.; Kozlov, V.V. A Memoir on Integrable Systems 2001 Flenner, H.; O’Carroll, L. Vogel, W. Joins and Intersections 1999 Gelfand, S.I.; Manin, Y.I. Methods of Homological Algebra 2nd ed. 2003 Griess, R.L. Jr. Twelve Sporadic Groups 1998 Gras, G. Class Field Theory corr. 2nd printing 2005 Hida, H. p-Adic Automorphic Forms on Shimura Varieties 2004 Ischebeck, F.; Rao, R.A. Ideals and Reality 2005 Ivrii, V. Microlocal Analysis and Precise Spectral Asymptotics 1998 Jech, T. Set Theory (3rd revised edition 2002) corr. 4th printing 2006 Jorgenson, J.; Lang, S. Spherical Inversion on SLn (R) 2001 Kanamori, A. The Higher Infinite corr. 2nd printing 2005 (2nd ed. 2003) Kanovei, V. Nonstandard Analysis, Axiomatically 2005 Khoshnevisan, D. Multiparameter Processes 2002 Koch, H. Galois Theory of p-Extensions 2002 Komornik, V. Fourier Series in Control Theory 2005 Kozlov, V.; Maz’ya, V. Differential Equations with Operator Coefficients 1999

Lau, D. Function Algebras on Finite Sets 2006 Landsman, N.P. Mathematical Topics between Classical & Quantum Mechanics 1998 Leach, J.A.; Needham, D.J. Matched Asymptotic Expansions in Reaction-Diffusion Theory 2004 Lebedev, L.P.; Vorovich, I.I. Functional Analysis in Mechanics 2002 Lemmermeyer, F. Reciprocity Laws: From Euler to Eisenstein 2000 Malle, G.; Matzat, B.H. Inverse Galois Theory 1999 Mardesic, S. Strong Shape and Homology 2000 Margulis, G.A. On Some Aspects of the Theory of Anosov Systems 2004 Miyake, T. Modular Forms 2006 Murdock, J. Normal Forms and Unfoldings for Local Dynamical Systems 2002 Narkiewicz, W. Elementary and Analytic Theory of Algebraic Numbers 3rd ed. 2004 Narkiewicz, W. The Development of Prime Number Theory 2000 Onishchik, A.L.; Sulanke, R. Projective and Cayley-Klein Geometries 2006 Parker, C.; Rowley, P. Symplectic Amalgams 2002 Peller, V. Hankel Operators and Their Applications 2003 Prestel, A.; Delzell, C.N. Positive Polynomials 2001 Puig, L. Blocks of Finite Groups 2002 Ranicki, A. High-dimensional Knot Theory 1998 Ribenboim, P. The Theory of Classical Valuations 1999 Rowe, E.G.P. Geometrical Physics in Minkowski Spacetime 2001 Rudyak, Y.B. On Thom Spectra, Orientability and Cobordism 1998 Ryan, R.A. Introduction to Tensor Products of Banach Spaces 2002 Saranen, J.; Vainikko, G. Periodic Integral and Pseudodifferential Equations with Numerical Approximation 2002 Schneider, P. Nonarchimedean Functional Analysis 2002 Serre, J-P. Complex Semisimple Lie Algebras 2001 (reprint of first ed. 1987) Serre, J-P. Galois Cohomology corr. 2nd printing 2002 (1st ed. 1997) Serre, J-P. Local Algebra 2000 Serre, J-P. Trees corr. 2nd printing 2003 (1st ed. 1980) Smirnov, E. Hausdorff Spectra in Functional Analysis 2002 Springer, T.A.; Veldkamp, F.D. Octonions, Jordan Algebras, and Exceptional Groups 2000 Sznitman, A.-S. Brownian Motion, Obstacles and Random Media 1998 Taira, K. Semigroups, Boundary Value Problems and Markov Processes 2003 Talagrand, M. The Generic Chaining 2005 Tauvel, P.; Yu, R.W.T. Lie Algebras and Algebraic Groups 2005 Tits, J.; Weiss, R.M. Moufang Polygons 2002 Uchiyama, A. Hardy Spaces on the Euclidean Space 2001 Üstünel, A.-S.; Zakai, M. Transformation of Measure on Wiener Space 2000 Vasconcelos, W. Integral Closure. Rees Algebras, Multiplicities, Algorithms 2005 Yang, Y. Solitons in Field Theory and Nonlinear Analysis 2001 Zieschang, P.-H. Theory of Association Schemes 2005

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A Course on Set Theory

A Course on Set Theory

A Course on Group Theory

A course on set theory

A Course on Set Theory

A Course on Finite Groups (Universitext)

A Course on Mathematical Logic (Universitext)

A Course on Mathematical Logic, 2nd edition

Notes on Complex Function Theory

Function theory on planar domains

A course on geometric group theory

A Short Course on Spectral Theory

A Short Course on Spectral Theory

A Short Course On Spectral Theory

A Short Course on Banach Space Theory

A short course on Banach space theory

A Short Course on Spectral Theory

A Short Course on Spectral Theory

A Short course on approximation theory

A short course on Banach space theory

A Short Course on Banach Space Theory

Abstract Analytic Function Theory And Hardy Algebras

Function Algebras on Finite Sets. A Basic Course on Many-Valued Logic and Clone Theory