Fuzzy Equational Logic

Radim Bˇelohlávek, Vilém Vychodil Fuzzy Equational Logic Studies in Fuzziness and Soft Computing, Volume 186 Editor-in...

Author: Radim Belohlávek | Vilem Vychodil

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Radim Bˇelohlávek, Vilém Vychodil Fuzzy Equational Logic

Studies in Fuzziness and Soft Computing, Volume 186 Editor-in-chief Prof. Janusz Kacprzyk Systems Research Institute Polish Academy of Sciences ul. Newelska 6 01-447 Warsaw Poland E-mail: [email protected] Further volumes of this series can be found on our homepage: springeronline.com

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Vol. 171. James J. Buckley Simulating Fuzzy Systems, 2005 ISBN 3-540-24116-7 Vol. 172. Patricia Melin, Oscar Castillo Hybrid Intelligent Systems for Pattern Recognition Using Soft Computing, 2005 ISBN 3-540-24121-3 Vol. 173. Bogdan Gabrys, Kauko Leiviskä, Jens Strackeljan (Eds.) Do Smart Adaptive Systems Exist?, 2005 ISBN 3-540-24077-2 Vol. 174. Mircea Negoita, Daniel Neagu, Vasile Palade Computational Intelligence: Engineering of Hybrid Systems, 2005 ISBN 3-540-23219-2 Vol. 175. Anna Maria Gil-Lafuente Fuzzy Logic in Financial Analysis, 2005 ISBN 3-540-23213-3 Vol. 176. Udo Seiffert, Lakhmi C. Jain, Patric Schweizer (Eds.) Bioinformatics Using Computational Intelligence Paradigms, 2005 ISBN 3-540-22901-9 Vol. 177. Lipo Wang (Ed.) Support Vector Machines: Theory and Applications, 2005 ISBN 3-540-24388-7

Vol. 180. Wesley Chu, Tsau Young Lin (Eds.) Foundations and Advances in Data Mining, 2005 ISBN 3-540-25057-3 Vol. 181. Nadia Nedjah, Luiza de Macedo Mourelle Fuzzy Systems Engineering, 2005 ISBN 3-540-25322-X Vol. 182. John N. Mordeson, Kiran R. Bhutani, Azriel Rosenfeld Fuzzy Group Theory, 2005 ISBN 3-540-25072-7 Vol. 183. Larry Bull, Tim Kovacs (Eds.) Foundations of Learning Classifier Systems, 2005 ISBN 3-540-25073-5 Vol. 184. Barry G. Silverman, Ashlesha Jain, Ajita Ichalkaranje, Lakhmi C. Jain (Eds.) Intelligent Paradigms for Healthcare Enterprises, 2005 ISBN 3-540-22903-5 Vol. 185. Dr. Spiros Sirmakessis (Ed.) Knowledge Mining, 2005 ISBN 3-540-25070-0 Vol. 186. Radim Bˇelohlávek, Vilém Vychodil Fuzzy Equational Logic, 2005 ISBN 3-540-26254-7

Radim Bˇelohlávek Vilém Vychodil

Fuzzy Equational Logic

ABC

Radim Bˇelohlávek

Vilém Vychodil

Department of Computer Science Palacky University Tomkova 40 CZ-779 00 Olomouc Czech Republic E-mail: [email protected]

Department of Computer Science Palacky University Tomkova 40 CZ-779 00 Olomouc Czech Republic E-mail: [email protected]

Library of Congress Control Number: 2005927316

ISSN print edition: 1434-9922 ISSN electronic edition: 1860-0808 ISBN-10 3-540-26254-7 Springer Berlin Heidelberg New York ISBN-13 978-3-540-26254-7 Springer 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 springeronline.com c Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands 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 authors and TechBooks using a Springer LATEX macro package Printed on acid-free paper

SPIN: 11376422

55/TechBooks

543210

To Jana Radim

To my parents Vilém

Preface

The present book deals with algebras, congruences, morphisms, reasoning about identities, classes of algebras and their axiomatizability, etc., from the point of view of fuzzy logic. We therefore deal with topics traditionally studied in universal algebra. Our approach is the following. In classical universal algebra, one works in bivalent setting. That is, one works with a two-element Boolean algebra 2 as the underlying structure of truth degrees. As an example, any two elements of an algebra either are congruent or not with respect to a given congruence relation. In our approach, we allow for more general structures of truth degrees than 2. We consider a structure of truth degrees as a parameter for which we can take any suitable structure L of truth degrees. For us, being suitable means that L is a complete residuated lattice. That is, L consists of a set L of truth degrees equipped with (suitable truth functions of) logical connectives. In this sense, we parameterize concepts and results known from classical universal algebra. If we put L = 2, we get just the concepts and results of classical universal algebra in which case the set L of truth degrees contains just two elements, namely L = {0, 1}. This is a boundary choice and we have other possibilities, e.g. L = [0, 1] (real unit interval). Returning to our example, a practical consequence is that, in general, congruence is a fuzzy relation. Two elements of an algebra may thus be congruent in degree, say, 0.8. This seems to be natural. Namely, the idea of congruence is to group together elements which are similar from a certain point of view (example from crisp setting: congruent numbers in the sense of having the same remainder when dividing by 3). Congruence in degrees thus corresponds to the fact that we may consider elements similar to degrees not necessarily equal to 0 or 1 (example: two sets might be similar in degree proportional to the extent of their overlap). Thus, allowing more general structures than the two-element Boolean algebra is not just a mathematical exercise. We thus deal with a kind of development of universal algebra and related topics from the point of view of fuzzy logic. Compared to other approaches, our approach is quite modest. Namely, equality of elements of algebras is the only component which becomes fuzzy. Although we could, we deliberately use

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neither “fuzzy subalgebras” nor “fuzzy functions”. Our “picture of the world” is the following. Classical algebras are structures of predicate logic with a language containing function symbols and a symbol of equality as a single relation symbol. We are thus interested in structures of predicate fuzzy logic with a language containing function symbols and just one relation symbol – a symbol of equality. The function symbols are interpreted by ordinary functions and the symbol of equality is interpreted by a fuzzy equality relation. We require all functions to be compatible with the fuzzy equality relation. As a consequence, if the fuzzy equality is understood as a similarity relation, our structures can be seen as sets equipped with “functions mapping similar to similar”. We call them algebras with fuzzy equality and in our setting, they play the same role as universal algebras in classical setting. From the point of view of fuzzy logic, our framework fits into that of Pavelka’s abstract fuzzy logic. We fix a particular structure L of truth degrees and develop our fuzzy logic. This gives a whole family of fuzzy logics parameterized by L. Picking a particular L, we deal with formulas, L-fuzzy sets of formulas, degrees of truth from L, and degrees of provability from L. Our formulas can be seen as generalized (since we sometimes use infinite conjunctions) formulas of predicate fuzzy logic with truth constants in language. For convenience, however, we use a particular way of writing our formulas. As a result, our formulas can be translated into formulas of predicate fuzzy logic with truth constants but are shorter. We prefer shorter formulas although not directly fitting into the framework of predicate fuzzy logic but this is a matter of taste and the reader may choose the other way. We develop a small fragment of fuzzy logic – we deal with a restricted language and restricted formulas. From this point of view, we contribute to already existing examples of restricted systems of fuzzy logic. On the one hand, they have less expressive power. On the other hand, they are more powerful than the more expressive systems since they obey desirable properties for a larger class of structures of truth degrees. For instance, compared to first-order fuzzy logic with evaluated syntax which is not syntactico-semantically complete for structures of truth degrees other than standard L ukasiewicz algebra (and its isomorphic copies), we have completeness also for other structures of truth degrees. Developing special fragments of fuzzy logic for various application domains is interesting and important: with a small fragment tailored for a particular application domain, one can hope to be able to go farther than with a full-fledged fuzzy logic. Although we do not think that our fragment will enjoy practical applications, we claim that it shows the effect of being able to go farther within a smaller system. The book is organized as follows. Chapter 1 contains preliminary notions and results. Chapter 2 deals with algebras with fuzzy equalities. Chapters 3 and 4 develop equational logic and Horn logic in fuzzy setting, their proof systems, completeness results, and results concerning definability by means of particular formulas which involve identities. At the close of each chapter is a section “Bibliographical Remarks” which contains references to

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IX

bibliographical items and also comments on alternative approaches. The book contains a list of references, a table of notation, and an index of terms. We gratefully acknowledge support by grant no. B1137301 of the Grant Agency of the Czech Academy of Sciences and institutional support, research plan MSM 6198959214. April 2005

Radim Bˇelohl´ avek Vilém Vychodil

Contents

1

Introduction to Fuzzy Sets and Fuzzy Logic . . . . . . . . . . . . . . . 1 1.1 Sets and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Structures of Truth Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 Pavelka-Style Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2

Algebras with Fuzzy Equalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.1 Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.2 Subalgebras, Congruences, and Morphisms . . . . . . . . . . . . . . . . . . 66 2.3 Direct and Subdirect Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.4 Terms, Term L-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 2.5 Direct Unions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.6 Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 2.7 Reduced Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.8 Class Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.9 Related Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 2.10 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

3

Fuzzy Equational Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.1 Syntax and Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3.2 Completeness of Fuzzy Equational Logic . . . . . . . . . . . . . . . . . . . . 146 3.3 Varieties, Free L-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.4 Equational Classes of L-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 158 3.5 Properties of Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 3.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

4

Fuzzy Horn Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.1 Syntax and Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.2 Semantic Entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 4.3 Completeness of Fuzzy Horn Logic . . . . . . . . . . . . . . . . . . . . . . . . . 204

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4.4 4.5 4.6 4.7 4.8 4.9

Fuzzy Equational Logic Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . 221 Implicationally Defined Classes of L-Algebras . . . . . . . . . . . . . . . 222 Sur-Reflections and Sur-Reflective Classes . . . . . . . . . . . . . . . . . . 235 Semivarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245 Quasivarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 Table of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

1 Introduction to Fuzzy Sets and Fuzzy Logic

This chapter surveys preliminary notions and results that we use in subsequent chapters. We pay more attention to concepts which might not be well known. Well-known notions and results are just recalled.

1.1 Sets and Structures Sets, Relations, Mappings We assume that the reader is familiar with the notion of a set as used in the intuitive set theory, i.e. as a collection of objects. Each object either belongs or does not belong to a given set. Objects which belong to a given set are also called its elements. We write a = b to denote that a and b are identical. The fact that an object a belongs to a set A is denoted by a ∈ A, the opposite by a ∈ A. Given a property ϕ, we denote by {a | ϕ(a )} the set of all objects having ϕ. The empty set (a set containing no elements) is denoted by ∅. A set A is a subset of a set B (denoted by A ⊆ B) if each element of A belongs to B. We write A ⊂ B if A ⊆ B but A = B. Given sets A and B, the intersection of A and B (denoted by A∩B) is the set containing the elements which belong to both A and B, the union of A and B (denoted by A ∪ B) is the set containing the elements which belong to A or to B (or both), the difference between A and B (denoted by A − B) is the set containing the elements belonging to A but not to B. In some situations, only elements of some given set U are taken into account. U is called a universe of discourse (universal set, universe). If U is a universe and A ⊆ U , then the complement of A (w.r.t. U , denoted by A) is the set U − A. An ordered n-tuple of objects u1 , . . . , un is denoted by u1 , . . . , un . We have u1 , . . . , un = v1 , . . . , vm iff n = m and u1 = v1 , . . . , un = vn . A direct product (or Cartesian product) of sets U1 , . . . , Un is the set U1 × · · · × Un = {u1 , . . . , un | u1 ∈ U1 , . . . , un ∈ Un }, i.e. the set of all n-tuples with ui from Ui . If U1 = · · · = Un = U , we denote the direct product by U n . R. Bˇ elohl´ avek and V. Vychodil: Fuzzy Equational Logic, StudFuzz 186, 1–58 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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A relation between sets U1 , . . . , Un is any subset r of the direct product of these sets, i.e. r ⊆ U1 × · · · × Un . The number n is called the arity of r (r is said to be n-ary). For n = 1, 2, 3, . . . we use the terms unary, binary, ternary, . . . If U1 = · · · = Un = U , r is called a relation in U (or, on U ). The inverse relation of a binary relation r ⊆ U × V is a relation r−1 ⊆ V × U defined by r−1 = {v , u | u , v ∈ r}. For r ⊆ U × V and A ⊆ U we put r(A) = {v | u , v ∈ r for some u ∈ A}. r({u }) is also denoted by [u ]r . A mapping (or function) of a set U into a set V is a binary relation f between U and V such that for each u ∈ U there is v ∈ V such that u , v ∈ f , and if u , v1 ∈ f and u , v2 ∈ f then v1 = v2 . A mapping f may be thought of as representing an assignment of elements of V to elements of U . We write f : U → V if f is a mapping of U to V . If u , v ∈ f for a mapping f , we write f (u ) = v . The set of all mappings of U to V is denoted by V U . A mapping f : V n → V is also called an n-ary operation in V . A mapping f is called injective (or injection) if u1 = u2 implies f (u1 ) = f (u2 ); surjective (or surjection) if for each v ∈ V there is u ∈ U such that f (u ) = v ; bijective (or bijection) if it is both injective and surjective. If f : U → V , g : V → W , then a composition of f and g is a mapping f ◦ g of U to W defined by (f ◦ g)(u ) = g(f (u )). Putting 2 = {0, 1} (there is no danger of confusion with the integer 2), 2U denotes the set of all mappings of U to {0, 1}. Mappings χ : X → {0, 1} are called characteristic functions of subsets of U . Namely, there is a natural bijection between 2U and the set of all subsets of U : each χ ∈ 2U induces a subset Aχ = {u ∈ U | χ(u ) = 1} of U ; each subset A of U induces a mapping χA : U → {0, 1} by χA (u ) = 1 if u ∈ A and χA (u ) = 0 if u ∈ A; χ is called a characteristic function of the corresponding subset Aχ ; we have AχA = A and χAχ = χ. For convenience, we usually do not distinguish between subsets of U and their characteristic functions and denote the set of all subsets of U also by 2U . If I is a set and for each i ∈ I we have a set Ui , we say that {Ui | i ∈ I} is a system (or family) of sets indexed by I. In a natural way, intersection and union is extended to a system of sets by putting i∈I Ui = {u | u ∈ {u | u ∈ Uj for some j ∈ I}. A direct product of Uj for all j ∈ I}, i∈I Ui = Q a system {Ui | i ∈ I} is a set i∈I UQi = {f : I → i∈I Ui | f (j) ∈ Uj for all j ∈ I}. In particular, if I = ∅, then i∈I Ui = {∅} (one-element set containing the empty set) since Q in this case, ∅ is the only mapping f : I → i∈I Ui . For I = {1, . . . , n}, i∈I Ui coincides in an obvious way with the notion of a direct product of U1 ,Q. . . , Un , as introduced above. Q The j-th projection (j ∈ I) of a Q direct product i∈I Ui is a mapping πj : i∈I Ui → Uj defined for any a ∈ i∈I Ui by πj (a) = a(j) (i.e. πj selects the j-th component). N, Z, Q, and R denote the set of all positive integers, the set of all integers, the set of all rationals, and the set of all reals, respectively. We put N0 = N ∪ {0} (non-negative integers). A set A is finite iff A is empty or for some n ∈ N there is a bijection f : A → {k ∈ N | 1 ≤ k ≤ n}. In the latter case, elements of A can be assigned

1.1 Sets and Structures

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natural numbers 1, . . . , n in a one-one way and so, n is called the number of elements of A. If a set A is not finite, it is called infinite. A set A is called denumerable (or countably infinite) if there is a bijection f : A → N. Cardinal and ordinal numbers are denoted by Greek letters α, . . . , κ, etc. Particularly, ω denotes the cardinality of N, i.e. the least infinite cardinal number. A set F ⊆ 2U is called a filter over U if X ∩Y ∈ F for every X, Y ∈ F , and if X ∈ F and X ⊆ Y imply Y ∈ F . A filter F is called proper if there is X ⊆ U such that X ∈ F . Filter F is called improper if ∅ ∈ F (in this case, F = 2U ). If {u } ∈ F for some u ∈ U , then F is said to be a trivial filter. If for every X ⊆ F we either have X ∈ F or U −X ∈ F , then F is said to be an ultrafilter. For any infinite universe set U , a filter F = {X ⊆ U | U − X is finite} is called a Fréchet filter over U . Equivalences and Partial Orders A binary relation r in a set U is called reflexive if for each u ∈ U we have u , u ∈ r; symmetric if for each u , v ∈ U , u , v ∈ r implies v , u ∈ r; antisymmetric if for each u , v ∈ U , u , v ∈ r and v , u ∈ r imply u = v ; transitive if for each u , v , w ∈ U , u , v ∈ r and v , w ∈ r imply u , w ∈ r. A binary relation in a set is called an equivalence if it is reflexive, symmetric, and transitive. For an equivalence E in a set U and u ∈ U , the set [u ]E = {v | u , v ∈ E} is called a class of E (given by u ). For each set U , there is a bijective correspondence between equivalences in U and so-called partitions of U . A partition of a set U is a system Π of non-empty subsets of U (called classes of Π) such that for each u ∈ U there is A ∈ Π with u ∈ A (classes cover U ), and for each A, B ∈ Π with A = B we have A ∩ B = ∅ (classes do not overlap). The correspondence: For each equivalence E in U , the set ΠE = {[u ]E | u ∈ U } is a partition of U . For each partition Π of U , the relation EΠ = {u , v | u , v ∈ A for some A ∈ Π} is an equivalence in U . Moreover, we have E = EΠE and Π = ΠEΠ . The set ΠE of all classes of E is also called a factor set of U by E and is denoted by U/E. A binary relation in a set is called a partial order if it is reflexive, antisymmetric, and transitive. If ≤ is a partial order, we write u ≤ v instead of u , v ∈ ≤, and write u < v if u ≤ v and u = v . If ≤ is a partial order in U , the pair U = U, ≤ is called a partially ordered set. A partially ordered set U = U, ≤ with finite U may be visualized using so-called Hasse diagram: Each element u ∈ U is depicted as a node labeled by “u ” in such a way that if u is covered by v , i.e. u < v and there is no w ∈ U such that u < w and w < v , then we connect the nodes of u and v , and put the node of u below the node of v . For a partially odered set U, ≤ , a function f (. . . , x, . . . ) : U n → U is called non-decreasing (isotone) in x if we have f (. . . , u , . . . ) ≤ f (. . . , v , . . . ) whenever u ≤ v ; f is called non-increasing (antitone) in x if we have f (. . . , u , . . . ) ≥ f (. . . , v , . . . ) whenever u ≤ v . Let U, ≤ be a partially ordered set and A ⊆ U . An element u ∈ A is called a minimal element of A if for each v ∈ A such that v ≤ u we have

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u = v ; a maximal element of A if for each v ∈ A such that u ≤ v we have u = v ; a least element of A if for each v ∈ A we have u ≤ v ; a greatest element of A if for each v ∈ A we have v ≤ u . If there exists a least (greatest) element of U , it is denoted by 0 (1). A lower cone of A is a set L(A) denoted by L(A) = {u ∈ U | u ≤ v for all v ∈ A}, an upper cone of A is a set U(A) denoted by U(A) = {u ∈ U | v ≤ u for all v ∈ A}. An element u ∈ L(A) is called a lower bound of A, an element u ∈ U(A) is called an upper bound of A. If L(A) has agreatest element u , then u is called an infimum of A in U, ≤ , denoted by A or inf A. Dually, if U(A) has a least element u , then u is called a supremum of A inU, ≤ , denoted by A or sup A. Note that if the least element 0 exists then U = 0 = ∅ by definition (dually, if 1 exists then ∅ = 1 = U ). A partially ordered set U, ≤ is said to be lattice ordered (or a lattice) if infimum and supremum exist for any two-element (and hence for any nonempty finite) subset of U . U, ≤ is said to be completely lattice ordered (or a complete lattice) if infimum and supremum exist for any subset of U . A partially ordered set U, ≤ is said to be linearly ordered (or a chain) if for every u , v ∈ U we have u ≤ v or v ≤ u , i.e. every two elements are comparable. In this case, ≤ is called a linear order. Each chain is a lattice with inf{u , v } = min{u , v } and sup{u , v } = max{u , v } (where min{u , v } and max{u , v } denote the smaller and the bigger one of u and v , respectively). If U, ≤ is a lattice ordered set, is called compact if for an element u ∈ U u ≤ A then there is a finite each A ⊆ U we have that if A exists and B ⊆ A such that u ≤ B still holds. A lattice is called compactly generated if each of its elements is a supremum of compact elements. An algebraic lattice is a lattice which is both complete and compactly generated. A lattice is called Noetherian if it is complete and every its non-empty subset has at least one maximal element. Therefore, in a Noetherian chain, each non-empty subset has a greatest element. If U, ≤ is a Noetherian lattice then for any A ⊆ U there is some finite F ⊆ A such that sup A = sup F . An element u of a lattice ordered set U, ≤ is called -irreducible if whenever u = A for a non-empty A ⊆ U , then u ∈ A; u is called -irreducible if whenever u = A for a non-empty A ⊆ U , then u ∈ A. We assume Zorn lemma which says that if each chain A of a partially ordered set U (i.e. a subset A ⊆ U with pairwise comparable elements) has an upper bound (i.e. U(A) = ∅) then for each u ∈ U there is a maximal element v of U such that u ≤ v . Example 1.1. (1) The genuine order ≤ on the set R of all real numbers is a partial order. Moreover, R, ≤ is a chain and thus a lattice (which is not complete). (2) For any set U , 2U , ⊆ is a complete lattice (which is not linear iff ⊆ U | i ∈ I} = U has at least two elements) with inf{A i i∈I Ai and sup{Ai ⊆ U | i ∈ I} = i∈I Ai .

1.1 Sets and Structures

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1 g c

b

e

a

f d

0 Fig. 1.1. Hasse diagram of a lattice from Example 1.1 (4)

(3) More generally, if S ⊆ 2U is a non-empty system of subsets of U that is closed w.r.t. arbitrary intersections (i.e. i∈I Ai ∈ S whenever Ai ∈ S) then S, ⊆ is a complete lattice where with intersections and infima coincide suprema are given by supi∈I Ai = {A ∈ S | i∈I Ai ⊆ A}. A system S of subsets of U that is closed w.r.t. arbitrary intersections is called a closure system in U . For a closure system S and a subset A ⊆ U , the set [A]S = {B ∈ S | A ⊆ B} is called the closure of A; it is the least (w.r.t. ⊆) subset of U that belongs to S and contains A. (4) Figure 1.1 shows a Hasse diagram of a nine-element lattice U, ≤ with U = {0, a, . . . , g, 1}. We can see from the diagram that, for instance, a ≤ c, a ≤ g, a and d are not comparable, sup{a, d} = e, inf{b, c} = inf{b, c, e, g} = a, etc. Let U = U, ≤U be a complete lattice. A mapping c : U → U is called a closure operator in U if (i) u ≤U c(u ); (ii) u1 ≤U u2 implies c(u1 ) ≤U c(u2 ); (iii) c(u ) = c(c(u )) (for any u , ui ∈ U ). If (i) is replaced by c(u ) ≤U u , c is called an interior operator in U. A subset S of U is called a closure system in U if S is closed under arbitrary infima. For a closure operator c in U, the set Sc = {u ∈ U | c(u ) = u } of all fixed points of c is a closure system in U. Conversely, for a closure system S in U, a mapping cS : U → U defined by c(u ) = {v ∈ S | u ≤ v } is a closure operator in U. Moreover, we have cSc = c and ScS = S. Let U = U, ≤U and V = V, ≤V be complete lattices. A pair ↑ , ↓ of mappings ↑ : U → V and ↓ : V → U is said to form a Galois connection between U and V if (i) u1 ≤U u2 implies u2↑ ≤V u1↑ ; (ii) v1 ≤V v2 implies ↓↑ v2↓ ≤U v1↓ ; (iii) u ≤U u ↑↓ , v , vi ∈ V ). For ; (iv) v ≤V v (for u , ui↑↓∈ U↑◦↓ ↑ ↓ = and ↓↑ = ↓◦↑ a Galois connection , , the composed mappings are closure operators in U and V, respectively. Furthermore, u ↑ = u ↑↓↑ and v ↓ = v ↓↑↓ . For U, ≤U = 2X , ⊆ and V, ≤V = 2Y , ⊆ , one obtains from the above notions the notions of a closure operator in a set X, a closure system in X, cf. also Example 1.1 (3), and that of a Galois connection between sets X and Y .

6


Algebras and Structures A structure is a set equipped with relations and functions with prescribed arities. Arities of relations and functions are given by a so-called type. Formally, a type is a triplet R, F, σ consisting of a set R (of symbols of relations), a set F (of symbols of functions) with R ∩ F = ∅, and a mapping σ of R ∪ F into a set N0 of non-negative integers. For s ∈ R ∪ F , σ(s) is called a symbol s. A structure of type R, F, σ is a triplet the arity of M = M, RM , F M where M is a non-empty set (universe of M, support of M), RM = {rM ⊆ M σ(r) | r ∈ R} is a set of relations rM in M corresponding to symbols r ∈ R, and F M = {f M : M σ(r) → M | f ∈ F } is a set of functions f M in M corresponding to symbols f ∈ F . Each rM ∈ RM is σ(r)-ary and each f M ∈ F M is σ(f )-ary. If there is no danger of confusion, we omit superscripts and write just r and f for rM and f M . An algebra is a structure M of some type R, F, σ with R = ∅ (no relation symbols). In that case, we write only F, σ and M = M, F M . If M = M, F M is M M and F = an algebra {f1 , . . . , fn }, we write only M, f1 , . . . , fn instead of M M M, {f1 , . . . , fn } , and say that M is of type σ(f1 ), . . . , σ(fn ) . Example 1.2. (1) Let R = {r}, F = ∅, and σ(r) = 2. A structure M = M, RM , F M of type R, F, σ consists of a non-empty set M , a set RM = {rM } consisting of a single binary relation rM in M , and an empty set F M . That is, a structure of type R, F, σ consists of a set M equipped with a single binary relation rM . In particular, a partially ordered set may be considered a structure of the above type R, F, σ . (2) A monoid is an algebra of type F, σ with F = {◦, e}, σ(◦) = 2, and σ(e) = 0, satisfying conditions of associativity and neutral element. That is, a monoid is an algebra M = M, ◦M , eM where M is a non-empty set, ◦M is a binary operation in M , and eM is a (constant) element of M (0ary operations are constants) such that (a ◦M b ) ◦M c = a ◦M (b ◦M c ) (associativity) and a ◦M eM = eM ◦M a (eM is a neutral element) for each a , b , c ∈ M . N, +, 0 (positive integers with addition), [0, 1], ·, 1 (real unit interval with multiplication), [0, 1], min, 1 (real unit interval with minimum) are examples of monoids. (3) We introduced lattices as partially ordered sets in which infima and suprema exist for arbitrary two-element subsets. Alternatively, lattices can be described as algebras. Call a lattice an algebra M = M, ∧, ∨ of type 2, 2 , i.e. ∧ and ∨ are binary operations in M , satisfying

u ∨v =v ∨u u ∨ (v ∨ w ) = (u ∨ v ) ∨ w u ∨ (u ∧ v ) = u

u ∧v =v ∧u u ∧ (v ∧ w ) = (u ∧ v ) ∧ w u ∧ (u ∨ v ) = u

(commutativity) , (associativity) , (absorption) ,

for each u , v , w ∈ M . If U = U, ≤ is a lattice ordered set, put u ∧ v = inf{u , v } and u ∨ v = sup{u , v }. Then U, ∧, ∨ is a lattice. Conversely, if U = U, ∧, ∨ is a lattice, put u ≤ v iff u ∧ v = u (or, which is equivalent, iff

1.2 Structures of Truth Degrees

7

u ∨ v = v ). Then U, ≤ is a lattice ordered set such that inf{u , v } = u ∧ v and sup{u , v } = u ∨ v . Moreover, these constructions are mutually inverse. A bounded lattice is an algebra U, ∧, ∨, 0, 1 where U, ∧, ∨ is a lattice and 0 and 1 satisfy 0 ∨ u = u and 1 ∧ u = u for each u ∈ U . This means that 0 and 1 are the least and the greatest element in the corresponding lattice ordered set. We now recall basic structural notions related to algebras. If M = M, F M is an algebra, a subset N of M is a subuniverse of M if it is closed under any f M ∈ F M , i.e. if for any f ∈ F , n = σ(f ), and every a1 , . . . , an ∈ M , we have f M (a1 , . . . , an ) ∈ N whenever a1 , . . . , an ∈ N . Any subuniverse N = ∅ equipped with restrictions of operations of M to N is again analgebra of the same type as M; it is called a subalgebra of M. If M = M, F M is an algebra, a binary relation θ in M is said to be compatible (with operations of M) if for each f ∈ F with σ(f ) = n and every a1 , b1 , . . . , an , bn ∈ M , we have that if a1 , b1 ∈ θ, . . . , an , bn ∈ θ then f M (a1 , . . . , an ), f M (b1 , . . . , bn ) ∈ θ. If θ is, moreover, an equivalence, it is called a congruence on M. Con(M) denotes the set of all congruence relations on M. Given an algebra M = M, F M and θ ∈ Con(M), a factor algebra of M by θ is an algebra M/θ = a congruence M/θ, F M/θ defined by putting f M/θ ([a1 ]θ , . . . , [an ]θ ) = [f M (a1 , . . . , an )]θ for each n-ary f ∈ F . Let M and N be algebras of the same type F, σ . A mapping h : M → N is called a morphism (or homomorphism) if for each f ∈ F with σ(f ) = n and every a1 , . . . , an ∈ M , we have h(f M (a1 , . . . , an )) = f N (h(a1 ), . . . , h(an )). In such a case we write h : M → N. A morphism h is called a monomorphism (epimorphism, isomorphism) if h is injective (surjective, bijective). If there is an isomorphism of M to N, we write M ∼ = N. | i ∈ I} of algebras of the same type F, σ an A direct Q product of asystem {M iQ Q Q Q i∈I Mi i∈I Mi on with operations f M , F M algebra i∈IQMi = i i i∈I i∈I defined by f i∈I Mi (a1 , . . . , an )(i) = f Mi (a1 (i), . . . , an (i)).

1.2 Structures of Truth Degrees Fuzzy Logic, Graded Truth, and Why Complete Residuated Lattices The main idea of fuzzy logic is that of graded truth. In classical logic, each proposition is either assigned truth degree 1 (true) or truth degree 0 (false). Classical logic is bivalent. In fuzzy logic, each proposition is assigned a truth degree taken from some scale L of truth degrees. Scale L is partially ordered and contains 0 and 1 as its boundary elements. That is, L is equipped with a partial order ≤ and we have 0 ≤ a ≤ 1 for each a ∈ L. Elements a from L are called truth degrees. If propositions ϕ and ψ are assigned truth degrees a and b, which we denote by ||ϕ|| = a and ||ψ|| = b, then a ≤ b means that we consider ϕ less true than ψ. That is why we want L to be partially ordered.

8


For instance, if ϕ and ψ denote “Bern is a large city” and “New York is a large city”, we might assign ||ϕ|| = 0.3 and ||ψ|| = 1 to express the fact that we consider Bern somewhat large but consider New York large without any reservations. It is in this sense that fuzzy logic is a logic of graded truth. A favorite example of L is the real unit interval [0, 1] but L need not be ordered linearly. For instance, a truth degree of “person x is financially well” might be assigned a two-dimensional truth degree a, b ∈ [0, 1] × [0, 1] with a and b being truth degrees of “investment of x is high” and “liquidity of x’s assets is high”. Notice that from this point of view, bivalent logic is a special case of fuzzy logic – just take L = {0, 1} with 0 ≤ 1. Scale L needs to be equipped with (truth functions of) logical connectives. That is, as in classical logic, we need binary functions ⊗ : L × L → L of conjunction, → : L × L → L of implication, etc. Then we get a structure L = L, ≤, . . . , ⊗, →, . . . of truth degrees. Like classical logic, fuzzy logic is truth-functional. Truth-functionality means that a truth degree of a composed proposition is computed from truth degrees of its constituent propositions using logical connectives. For instance, if propositions ϕ and ψ are assigned truth degrees ||ϕ|| and ||ψ||, then the truth degree of proposition ϕ o ψ (“ϕ and ψ”) is ||ϕ|| ⊗ ||ψ||, i.e. ⊗ applied to ||ϕ|| and ||ψ||. Contrary to classical logic, logical connectives are not uniquely given in fuzzy logic. That is, even if we fix a scale L, there is no “the right” conjunction ⊗ for L. The reason for this: In classical logic, conjunction is given by how “and” is used in natural language. We consider “ϕ and ψ” true if and only if both ϕ and ψ are true. Whence the definition of conjunction in classical logic (1 ⊗ 1 = 1, 1 ⊗ 0 = 0 ⊗ 1 = 0 ⊗ 0 = 0). Intuition behind the usage of “and” in natural language fails us if more truth degrees come into play. If ||ϕ|| = 0.7 and ||ψ|| = 0.8, what is the truth degree of “ϕ and ψ”, i.e. what is 0.7 ⊗ 0.8? There is no obvious answer. In fuzzy logic, rather than picking one particular connective, it is useful to postulate desirable properties of connectives and consider as a “good connective” any one satisfying the properties. For instance, a desirable property of conjunction is monotony – the more true the propositions, the more true their conjunction. This translates to requiring a1 ⊗ b1 ≤ a2 ⊗ b2 whenever a1 ≤ a2 and b1 ≤ b2 . In the following, we proceed this way to justify a particular structure of truth degrees. The structure will be called a complete residuated lattice and will play the role of a basic structure of truth degrees in our investigation. Our justification of complete residuated lattices as suitable structures of truth degrees is due to Goguen [45]. We already agreed on a set L of truth degrees equipped with a partial order ≤ and bounded by 0 and 1. Furthermore, we want L to have arbitrary infima and suprema, i.e. we want L, ≤, 0, 1 to be a complete residuated lattice. Why infima and suprema? Let X be a collection of persons of our interest and let ϕ(x) denote a proposition “person x is tall”. The truth degree of this proposition is ||ϕ(x)||. What then is the truth degree of “there is a person x from X who is tall”? We argue that a good choice is the supremum of ||ϕ(x)||


9

over x ∈ X, i.e. ||“there is a person x from X who is tall”|| = x∈X ||ϕ(x)||. Intuitively, evaluating “there is a person x from X who is tall” means going through x from X and take the best, i.e. the highest, ||ϕ(x)|| and this is exactly what supremum does. That is why we need suprema in L. The need of infima can be justified dually (as a truth degree of propositions of the type “each person x from X is tall”). We now turn to (truth function of) conjunction. We denote it by ⊗. Basic requirements are that L, ⊗, 1 is a commutative monoid. This means that ⊗ is a binary operation on L which is commutative, associative, and 1 is a neutral element of ⊗. Commutativity of ⊗ means that we have a ⊗ b = b ⊗ a for each a, b ∈ L. If a and b are truth degrees of propositions ϕ and ψ, respectively, then commutativity of ⊗ means ||ϕ o ψ|| = ||ϕ|| ⊗ ||ψ|| = ||ψ|| ⊗ ||ϕ|| = ||ψ o ϕ||, i.e. the truth degree of “ϕ and ψ” equals the truth degree of “ψ and ϕ”. This is an intuitively desirable property justifying commutativity. Associativity of ⊗ means a ⊗ (b ⊗ c) = (a ⊗ b) ⊗ c for each a, b, c ∈ L. Associativity results from requiring that the truth degree of “ϕ and (ψ and χ)” equals the truth degree “(ϕ and ψ) and χ” which we consider a desirable property as well. That 1 is a neutral element of ⊗ says a ⊗ 1 = a for each a ∈ L. This results from the following requirement. If ψ is a proposition which is fully true, i.e. ||ψ|| = 1, then for any proposition ϕ we have ||ϕ o ψ|| = ||ϕ|| ⊗ ||ψ|| = ||ϕ|| ⊗ 1 = ||ϕ||, i.e. the truth degree of a conjunction of ϕ with a fully true proposition equals the truth degree of ϕ. We held this property desirable as well. We proceed by (truth function of) implication which we denote by →. Several desirable properties of implication follow from a simple condition called adjointness. Adjointness itself says that for each a, b, c ∈ L we have a⊗b ≤ c if and only if a ≤ b → c. We show that adjointness follows from how modus ponens should behave in fuzzy setting. Recall first that in classical logic, modus ponens is an inference rule saying: if ϕ is valid and ϕ ⇒ ψ (“ϕ implies ψ”) is valid then we may infer that ψ is valid. An appropriate formulation of modus ponens in fuzzy setting is the following: if ϕ is valid in degree at least a and ϕ ⇒ ψ is valid in degree at least b then we may infer that ψ is valid in degree at least a ⊗ b. Observe first that if “a formula is valid” is understood as a shorthand for “a formula is valid in degree 1” then the formulation of modus ponens in fuzzy setting is equivalent to that in classical setting. This follows by a moment’s reflection since in classical logic we have only 0 and 1 as truth degrees and any formula is always valid in degree at least 0. Using modus ponens, we get a ⊗ b as a lower estimation of degree of validity of ψ. Now, we want modus ponens to satisfy two points: it should be sound and, at the same time, it should yield the highest possible estimation of validity of ψ. Soundness: The requirement of soundness says that when evaluating formulas, if the truth degree of ϕ is at least a (a ≤ ||ϕ||) and the truth degree of ϕ ⇒ ψ is at least b (b ≤ ||ϕ ⇒ ψ||) then the truth degree of ψ is at least as high as the degree obtained by modus ponens (a ⊗ b ≤ ||ψ||). In words, from lower estimations of ϕ and ϕ ⇒ ψ, modus ponens yields a lower estimation of ψ. This ensures that we do not get more than the actual truth degree of ψ.

10


Particularly, if ||ϕ|| = a and ||ψ|| = c, then since ||ϕ ⇒ ψ|| = a → c, soundness says that b ≤ a → c implies a ⊗ b ≤ c. Highest possible estimation of validity of ψ: Let ||ϕ|| = a and ||ψ|| = c. Then ||ϕ ⇒ ψ|| = a → c and since we require soundness, a ⊗ (a → c) needs to be a lower estimation of c, i.e. a ⊗ (a → c) ≤ c. Since ⊗ is non-decreasing, the lower estimation a ⊗ (a → c) is the higher, the higher a → c. Since we want a ⊗ (a → c) to be the highest possible estimation of c, a → c needs to be the highest degree for which a ⊗ (a → c) ≤ c. That is, if b is any truth degree for which a ⊗ b ≤ c then b ≤ a → c. In other words, the requirement of highest estimation of validity of ψ yields that a ⊗ b ≤ c implies b ≤ a → c. Putting the two conditions together we get that ⊗ and → should satisfy a ⊗ b ≤ c if and only if b ≤ a → c for any truth degrees a, b, c ∈ L. A structure consisting of a set L of truth degrees, a partial order ≤ bounded by 0 and 1, and functions ⊗ and → which satisfy the above conditions is called a complete residuated lattice and plays the role of a basic structure of truth degrees in our investigation. This structure will be furthermore equipped with a unary operation ∗ called a truth stresser. A truth stresser ∗ serves as (truth function of) unary connective “very true” and will be introduced later. Note that we might impose additional conditions on the structure of truth degrees if desirable for our purpose. In such a case, we restrict our attention only to structures satisfying additional conditions. For instance, we might want conjunction ⊗ to be idempotent, i.e. to satisfy a ⊗ a = a for any a ∈ L. Or, in the light of the 7 ± 2 phenomenon well-known from psychology, we might restrict our attention to structures with up to 7 ± 2 elements. Recall according to Miller’s 7 ± 2 phenomenon [65], humans are able to assign degrees to elements of some universe in a consistent manner if the scale L of truth degrees contains up to 7 ± 2 elements. With more than 7 ± 2 degrees, the assignment becomes inconsistent. For instance, when the assignment is performed twice, the same person may assign different truth degrees to the same element of universe. Complete Residuated Lattices with Truth Stressers We first introduce residuated lattices, their examples, and their basic properties. Then we add truth stressers, their examples, and properties. Definition 1.3. A complete residuated lattice is an algebra L = L, ∧, ∨, ⊗, →, 0, 1 where (i) L, ∧, ∨, 0, 1 is a complete lattice with the least element 0 and the greatest element 1, (ii) L, ⊗, 1 is a commutative monoid, i.e. ⊗ is associative, commutative, and a ⊗ 1 = a for each a ∈ L,


11

(iii) ⊗ and → satisfy adjointness, i.e. a⊗b≤c

iff

a≤b→c

(1.1)

for each a, b, c ∈ L (≤ denotes the lattice ordering). Remark 1.4. (1) If we weaken (i) so that it requires L, ∧, ∨, 0, 1 to be a bounded lattice, not necessarily a complete one, L is called a residutated lattice. (2) The operations ⊗ and → are called multiplication and residuum, respectively, ⊗, → is called an adjoint pair. For a, b ∈ L, a → b is called the residuum of b by a. For a given ⊗, there is at most one operation → satisfying (1.1). For if → is another one, we get a ≤ b → c iff a ⊗ b ≤ c iff a ≤ b → c for any a, from which it clearly follows b → c = b → c. Similarly, → uniquely determines ⊗. n (3) Since ⊗ is commutative and associative, we may write i=1 ai to denote any ⊗-product of a1 , . . . , an . For instance, a1 ⊗(a2 ⊗a3 ), (a1 ⊗a2 )⊗a3 , a2 ⊗ (a1 ⊗ a3 ), . . . , (a3 ⊗ a2 ) ⊗ a1 , all have the same value and this value is 3 denoted by i=1 ai . Most important examples of complete residuated lattices are those with the universe L being a real unit interval [0, 1] or its subchain. Example 1.5. Take L = [0, 1] (the interval of all reals between 0 and 1). The natural ordering of [0, 1] makes L, ∧, ∨, 0, 1 a complete lattice where a ∧ b = min(a, b), a ∨ b = max(a, b). Each of the following pairs of operations makes L, ∧, ∨, ⊗, →, 0, 1 a complete residuated lattice: L ukasiewicz operations:

a ⊗ b = max(a + b − 1, 0) , a → b = min(1 − a + b, 1) ,

(1.2)

a ⊗ b = min(a, b) , G¨ odel operations: 1 if a ≤ b , a→b = b otherwise ,

(1.3)

a⊗b = a·b, 1 if a ≤ b , a→b = b otherwise . a

(1.4)

Goguen (product) operations:

The corresponding algebras L are called standard L ukasiewicz algebra, standard G¨ odel algebra, and standard Goguen (product) algebra on [0, 1]. Example 1.6. Take L = {a0 , a1 , . . . , an } with an order given by 0 = a0 < · · · < an = 1. We have ai ∧ aj = amin(i,j) and ai ∨ aj = amax(i,j) . The following two pairs of operations determine a complete residuated lattice structure on L: L ukasiewicz operations:

ak ⊗ al = amax(k+l−n,0) , ak → al = amin(n−k+l,n) ,

(1.5)

12


G¨ odel operations:

ak ⊗ al = amin(k,l) , 1 if k ≤ l , ak → al = al otherwise .

(1.6)

If {a0 , . . . , an } ⊆ [0, 1] and ai = ni , then all the operations are restrictions of L ukasiewicz operations on [0, 1] to {a0 , . . . , an }, i.e. {a0 , . . . , an } is a subuniverse of the standard L ukasiewicz algebra. If {a0 , . . . , an } ⊆ [0, 1] and a0 = 0, odel operations on an = 1, then all the operations are restrictions of the G¨ odel [0, 1] to {a0 , . . . , an }, i.e. {a0 , . . . , an } is a subuniverse of the standard G¨ algebra. Remark 1.7. If L = {0, 1} then both of the structures from Example 1.6 yield the same, namely, the two-element Boolean algebra, i.e. the structure of truth degrees of classical two-valued logic. We denote this algebra by 2 and write 2 = {0, 1}. There is no other residuated lattice structure on {0, 1} (easy to check). Anything we establish for a general complete residuated lattice L (definitions, theorems, etc.), has its special “instance” for L = 2. This is a basic way of generalization from ordinary (bivalent) setting to fuzzy setting in our approach. For instance, if we prove TheoremL true for an arbitrary complete residuated lattice L and the instance Theorem2 of TheoremL for L = 2 is known from ordinary setting, we say that TheoremL is a generalization of Theorem2 to fuzzy setting. We will see particular examples later on. Example 1.8. (1) Residuated lattices from Example 1.5 are particular cases of those induced by so-called left-continuous t-norms (see Definition 1.27 and the subsequent paragraphs for details). A t-norm is a binary operation ⊗ on [0, 1] which is associative, commutative, has 1 as its neutral element, and is nondecreasing. A t-norm ⊗ is left-continuous if limn→∞ (an ⊗b) = (limn→∞ an )⊗b for any non-decreasing sequence {an ∈ [0, 1] | n = 1, 2, 3, . . . }. For a leftcontinuous t-norm ⊗, put a → b = sup{c | a ⊗ c ≤ b} .

(1.7)

Then [0, 1], min, max, ⊗, →, 0, 1 is a complete residuated lattice. It is easy to see that each of the three operations ⊗ from Example 1.5 is even a continuous t-norm and that the corresponding residua are obtained by (1.7). (2) Residuated lattices on L = {a0 , a1 , . . . , an } from Example 1.6 are particular cases of the following one. Take I = {i1 = 0, . . . , im = n} ⊆ {0, . . . , n} with i0 < · · · < im and define amax(k+l−ij+1 ,ij ) if k, l ∈ [ij , ij+1 ] ak ⊗ al = otherwise amin(k,l)  1 if k ≤ l    amin(ij+1 −k+l,ij+1 ) if k > l and ak → al = k, l ∈ [ij , ij+1 ]    otherwise . al Then L, ∧, ∨, ⊗, →, 0, 1 is a complete residuated lattice. For I = {0, 1} we get (1.5) and for I = {0, 1, . . . , n} we get (1.6).


1 g c

b

e

a

f d

0

⊗ 0 a b c d e f g 1

0 0 0 0 0 0 0 0 0 0

a 0 a a a 0 a 0 a a

b 0 a a a 0 a 0 a b

c 0 a a a 0 a 0 a c

d 0 0 0 0 0 0 0 0 d

e 0 a a a 0 a 0 a e

f 0 0 0 0 0 0 0 0 f

g 0 a a a 0 a 0 a g

1 0 a b c d e f g 1

→ 0 a b c d e f g 1

0 1 f f f g f g f 0

a 1 1 g g g g g g a

b 1 1 1 g g g g g b

c 1 1 g 1 g g g g c

d 1 f f f 1 f g f d

e 1 1 g g 1 1 g g e

13

f 1 f f f 1 f 1 f f

g 1 1 1 1 1 1 1 1 g

1 1 1 1 1 1 1 1 1 1

Fig. 1.2. Non-linear residuated lattice from Example 1.9.

Example 1.9. Figure 1.2 shows a non-linear residuated lattice L with elements 0, a, . . . , g, 1. The lattice part L, ≤ of L is shown by its Hasse diagram (left). The adjoint operations ⊗ and → are shown by their tables (middle and right). The following are important operations derived in terms of basic operations of residuated lattices. Definition 1.10. For a complete residuated lattice L we define a ↔ b = (a → b) ∧ (b → a) , ¬a = a → 0 , a0 = 1 and an+1 = an ⊗ a ,

(1.8) (1.9) (1.10)

for each a, b ∈ L and a non-negative integer n. The operations ↔, ¬, and are called biresiduum, negation, and n-th power in L.

n

One can easily check that for the biresidua corresponding to standard L ukasiewicz, G¨ odel, and product algebras on [0, 1] we have a ↔ b = 1 − |a − b| , 1 for a = b a↔b= min(a, b) otherwise ,

1 for a = b = 0 a ↔ b = min(a,b) otherwise , max(a,b) respectively. For negations of standard L ukasiewicz, Gödel, and product algebras on [0, 1] we have ¬a = 1 − a for L ukasiewicz and

¬a =

1 for a = 0 0 for a > 0

for both G¨ odel and product. For powers of standard L ukasiewicz, G¨ odel, and product algebras on [0, 1] we have

14


an = max(0, 1 − n(1 − a)) , an = a , an = the usual n-th power of a , respectively. Several important structures of truth degrees are just residuated lattices satisfying additional conditions. Some of these are introduced by the following definition. Definition 1.11. A Heyting algebra is a residuated lattice satisfying a⊗b = a ∧ b. A BL-algebra is a residuated lattice which satisfies a ∧ b = a ⊗ (a → b) (divisibility) and (a → b) ∨ (b → a) = 1 (prelinearity). An MV-algebra is a BL-algebra satisfying a = ¬¬a (law of double negation). A Π-algebra (product algebra) is a BL-algebra satisfying (c → 0) → 0 ≤ ((a ⊗ c) → (b ⊗ c)) → (a → b) and a ∧ (a → 0) = 0. A G-algebra (G¨ odel algebra) is a BL-algebra which satisfies a ⊗ a = a (idempotency). A Boolean algebra is a residuated lattice which is both a Heyting algebra and an MV-algebra. Remark 1.12. (1) Consider residuated lattices from Examples 1.5 and 1.6. Those with L ukasiewicz structure are MV-algebras, those with G¨ odel structure are G-algebras, and that with product structure is a Π-algebra. (2) Recall that a Boolean algebra is usually defined as a bounded lattice which is distributive (i.e., satisfies a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)) and complemented (i.e., a unary operation exists such that a ∧ a = 0 and a ∨ a = 1 for each a). In such a case, putting a ⊗ b = a ∧ b and a → b = a ∨ b, we get a Boolean algebra in the sense of Definition 1.11. Conversely, having a Boolean algebra in the sense of Definition 1.11, we get a Boolean algebra according to the usual definition if we put a = a → 0. (3) If L is a Heyting algebra, a → b is also called a relative pseudocomplement of a in b. We now turn our attention to truth stressers – particular unary operations on complete residuated lattices. Definition 1.13. Let L = L, ∧, ∨, ⊗, →, 0, 1 be a complete residuated lattice. A unary operation ∗ : L → L satisfying 1∗ = 1 , a∗ ≤ a ,

(1.11) (1.12)

(a → b) ≤ a∗ → b∗ ,

(1.13)

∗

for every a, b ∈ L, is called a truth stresser for L. The algebra L, ∧, ∨, ⊗, →, ∗ , 0, 1 is then called a complete residuated lattice with truth stresser.


Example 1.14. (1) Let

∗

15

be the identity on L, i.e. a∗ = a ,

(1.14)

∗

a ∈ L. Then is a truth stresser for L. (2) Let ∗ be defined by 1 if a = 1 , a∗ = 0 otherwise .

(1.15)

Then ∗ , called a globalization in [83], is a truth stresser for L. (3) Denote the two above truth stressers by ∗1 (identity) and ∗2 (globalization). Trivially, for each truth stresser ∗ , if a < 1 then a∗2 = 0 ≤ a∗ ≤ a = a∗1 , and 1 = 1∗1 = 1∗ = 1∗2 . Therefore, truth stressers are bounded by ∗1 and ∗2 . Remark 1.15. (1) If L is a complete residuated lattice and ∗ a truth stresser for L then the resulting complete residuated lattice with truth stresser, i.e. the algebra L, ∧, ∨, ⊗, →, ∗ , 0, 1 , will be denoted by L∗ . (2) A truth stresser ∗ may be thought of as a (truth function of) unary connective “very true”. That is, if ϕ is a proposition then a truth degree ∗ of “ϕ is very true” is ||ϕ|| . This interpretation of truth stressers is due to Hájek [51]. In particular, identity may be thought of as connective “· · · is true”. Globalization may be thought of as connective “· · · is fully true”. As we will see, a ≤ b is equivalent to a → b = 1. Therefore, ||ϕ|| ≤ ||ψ|| means that ||ϕiψ|| = 1 (truth degree of “ϕ implies ψ” is 1). Taking this into account, we can see that conditions (1.11)–(1.13) are natural properties of truth function of “very true”. Namely, (1.11) ensures that if ϕ is true in degree 1 then “ϕ is very true” has degree 1 (succinctly: if ϕ is fully true then ϕ is very true). Equation (1.12) says that if ϕ is very true then ϕ is true. Finally, using adjointness, ∗ (1.13) is equivalent to a∗ ⊗ (a → b) ≤ b∗ , and so it says that if ϕ is very true and if it is very true that ϕ implies ψ then ψ is very true. ∗ (3) Note that for technical reasons H´ ajek requires also (a ∨ b) ≤ a∗ ∨ b∗ ∗ ∗ in [51]. Note also that a truth stresser satisfying (a ∨ b) ≤ a∗ ∨ b∗ and a∗ ∨ ¬a∗ = 1 is just a truth function of a unary connective satisfying Baaz’s axioms for from [4]. We will see later that if L is a linearly ordered residuated lattice then ∗ is a truth stresser for L satisfying the two additional conditions iff ∗ is globalization. (4) It is an obvious but useful fact that if we prove a theorem true for a particular complete residuated lattice L, then this theorem remains true for any L∗ resulting from L by an arbitrary truth stresser ∗ for L. We continue with further examples of truth stressers. Example 1.16. (1) For any L, c1 , . . . , ck ∈ L, and non-negative integers n1 , . . . , nk , let 1 if a = 1 , (1.16) a∗ = k ni c ⊗ a if a 0 there exists δ > 0 such that for each a ∈ [0, c] such that |a − c| < δ we have |g(a) − g(c)| < ε. g is called left-continuous (in [0, 1]) if g is left-continuous at c for each c ∈ [0, 1]. We will need the following lemma.


23

Lemma 1.30. A function g : [0, 1] → [0, 1] is left-continuous iff we have (1.50) lim g(an ) = g lim an n→∞

n→∞

for any non-decreasing sequence {an ∈ [0, 1] | n = 1, 2, 3, . . . }. Proof. We make use of the following fact well-known from basic calculus. Claim. g is left-continuous at c iff limn→∞ an = c implies limn→∞ g(an ) = g(c) for each sequence {an | n = 1, 2, . . . } of elements an ∈ [0, c]. “⇒”: Take any non-decreasing sequence {an }n = {an ∈ [0, 1] | n = 1, 2, . . . }. Since {an }n is non-decreasing and bounded, it has a limit, say limn→∞ an = c. Since an ≤ c, left-continuity of g and Claim give limn→∞ g(an ) = g(limn→∞ an ). “⇐”: Consider any sequence {an }n of elements an ∈ [0, c] such that limn→∞ an = c. By Claim, we need to show limn→∞ g(an ) = g(c). Suppose by contradiction that limn→∞ g(an ) = g(c) is not the case. By definition of a limit we have that (*) there exists ε > 0 such that for each n ∈ N there is m ≥ n such that |g(am ) − g(c)| ≥ ε. Using (*) for n = 1, there exists m1 ≥ n such that |g(am1 ) − g(c)| ≥ ε; denote this am1 by b1 . Using (*) for n = m1 + 1, there exists m2 ≥ n such that |g(am2 ) − g(c)| ≥ ε; denote this am2 by b2 . Repeating this over and over, we get a subsequence {bn }n of {an }n . Since {bn }n is bounded (namely, bn ≤ c for each n), it contains a non-decreasing subsequence {cn }n of elements cn ∈ [0, c]. Notice that for each cn we have |g(cn ) − g(c)| ≥ ε .

(1.51)

As {cn }n is a subsequence of {an }n , we have limn→∞ cn = limn→∞ an = c. Applying assumption (1.50) to {cn }n we get limn→∞ g(cn ) = g(limn→∞ cn ) = g(c) which is impossible due to (1.51). The proof is complete. Lemma 1.30 has the following corollary relating left-continuity of a t-norm ⊗ to left-continuity of ⊗ as a function of a real variable. Corollary 1.31. For a t-norm ⊗, the following claims are equivalent. (i) ⊗ is a left-continuous t-norm. (ii) for each b ∈ [0, 1], function g(x) = x ⊗ b is left-continuous (in the sense of Definition 1.29). Recall now that a function f (x, y) : [0, 1] × [0, 1] → [0, 1] is called nondecreasing (non-increasing) in x if for any a1 , a2 , b ∈ [0, 1], a1 ≤ a2 implies f (a1 , b) ≤ f (a2 , b) (f (a1 , b) ≥ f (a2 , b)). The notions non-decreasing (nonincreasing) in y are defined analogously. The following lemma translates algebraic properties of ⊗ and → on [0, 1] in terms of infima and suprema to their continuity properties known from basic calculus. Lemma 1.32. Let a function f (x, y) : [0, 1] × [0, 1] → [0, 1] be non-decreasing in x. Then f is left-continuous in x iff for any {aj | j ∈ J} ⊆ [0, 1] and each b ∈ [0, 1], we have

24


f ( j∈J aj , b) = j∈J f (aj , b) .

(1.52)

f is right-continuous in x iff for any {aj | j ∈ J} ⊆ [0, 1] and each b ∈ [0, 1], we have f ( j∈J aj , b) = j∈J f (aj , b) . (1.53) Therefore, f is continuous in x iff both (1.52) and (1.53) hold true. We obtain analogous statements if one replaces “in x” by “in y”; a dual statement holds true for f non-increasing. Proof. We prove the assertion concerning the left-continuity of a non-decreasing function. The case of right-continuity is symmetric, the case of continuity is clear, and for non-increasing f , the proof is symmetric. Let fbe both non-decreasing and left-continuous in x. We verify (1.52). For a = {a Either a = ({aj | j ∈ J} − j | j ∈ J}, there are two possibilities. {a}) or a > ({aj | j ∈ J} − {a}). If a = ({aj | j ∈ J} − {a}) then for each n ∈ N there exists some aj(n) ∈ {aj | j ∈ J} − {a} such that a − aj(n) < n1 . Clearly, we may safely assume aj(n) ≤ aj(n+1) . Then we have limn→∞ aj(n) = a and aj(n) < a (n ∈ N). A moment’s reflection shows that f ( j∈J aj , b) = f ( lim aj(n) , b) = lim f (aj(n) , b) = n→∞ n→∞ = n∈N f (aj(n) , b) ≤ j∈J f (aj , b) by the definition of left-continuity in x. If a > ({aj | j ∈ J} − {a}) then 1 there is some n ∈ N such that a − ak >n for each ak ∈ {aj | j∈ J} − {a}. But then a = al for some l ∈ J,hence f ( j∈J aj , b) = f (al , b) ≤ j∈J f (aj , b). The converse inequality, i.e. f ( j∈J aj , b) ≥ j∈J f (aj , b), holds since f is nondecreasing in x. Conversely, assume (1.52). By Lemma 1.30, to show that f is leftcontinuous in x it suffices to show that for any a ∈ [0, 1] and any nondecreasing sequence {an ∈ [0, 1] | n = 1, 2, . . . } such that limn→∞ an = a = f (limn→∞ an , b). Using (1.52),this readily follows we have limn→∞ f (an , b) from limn→∞ f (an , b) = n∈N f (an , b) and limn→∞ an = n∈N an . Theorem 1.33 (left-continuous t-norms and residuated lattices). A binary operation ⊗ in [0, 1] is a left-continuous t-norm iff a structure [0, 1], min, max, ⊗, →, 0, 1 with → defined by a → b = {c | a ⊗ c ≤ b} , (1.54) is a residuated lattice. Proof. By Corollary 1.31 iff and Lemma 1.32, ⊗ is a left-continuous t-norm we have a ⊗ b = (a ⊗ b ). By Theorem 1.22, we have a ⊗ b = i i i i∈I i∈I i∈I i∈I (a⊗bi ) iff (1.54) makes [0, 1], min, max, ⊗, →, 0, 1 a complete residuated lattice.


25

Example 1.34. (1) An example of a left-continuous t-norm which is not continuous is “nilpotent minimum” [40, 60] T0 defined by min(x, y) for x + y > 1 T0 (x, y) = 0 for x + y ≤ 1 . (2) An example of a t-norm that is even not left-continuous is the “drastic product” [60] defined by x · y for max(x, y) = 1 Td (x, y) = 0 otherwise . (3) A moment’s reflection shows that for each t-norm T we have Td (x, y) ≤ T (x, y) ≤ min(x, y) for every x, y ∈ [0, 1]. The following property of left-continuous t-norms will be used in the sequel. Lemma 1.35. For a left-continuous t-norm ⊗, let a, b ∈ [0, 1]. (i) If a ≥ b ⊗ c for each c ∈ [0, 1) then a ≥ b. (ii) If a < b then there is c ∈ [0, 1) such that a < b ⊗ c. Proof. (i): If a ≥ b ⊗ c for each c = 1 then a ≥ {b ⊗ c | c = 1} = b ⊗ {c | c = 1} = b ⊗ 1 = b. (ii) is a restatement of (i). We now turn our attention to continuous t-norms and the corresponding residuated lattices. Definition 1.36. A t-norm ⊗ is called continuous if it is continuous as a real function of two variables. Lemma 1.37. Let f : [0, 1]2 → [0, 1] be a function such that for each a ∈ [0, 1], f (x, a) : [0, 1] → [0, 1] and f (a, x) : [0, 1] → [0, 1] are continuous. Then f is continuous. Proof. We have to show that for every a, b ∈ [0, 1], for each ε > 0 there is some δ > 0 such that |f (x, y) − f (a, b)| < ε for every x ∈ (a − δ, a + δ), y ∈ (b − δ, b + δ) (if, e.g., x = 1 then “x ∈ (a − δ, a + δ)” is to be replaced by “x ∈ (a − δ, a]”; similarly for the other boundary cases). Since for any A, B ∈ [0, 1], both f (A, x) and f (x, B) are continuous functions on a compact set (closed interval), they are uniformly continuous. Therefore, there are δ1 , δ2 > 0 such that |f (x1 , B) − f (x2 , B)| < 2ε and |f (A, y1 ) − f (A, y2 )| < 2ε for any x1 , x2 , y1 , y2 ∈ [0, 1] such that |x1 −x2 | < δ1 , |y1 −y2 | < δ2 . Put δ = min(δ1 , δ2 ) and take any x ∈ (a − δ, a + δ), y ∈ (b − δ, b + δ). We have |f (x, y) − f (a, b)| = |f (x, y) − f (a, y) + f (a, y) − f (a, b)| ≤ ≤ |f (x, y) − f (a, y)| + |f (a, y) − f (a, b)| < completing the proof.

ε 2

+

ε 2

=ε

26


Due to Lemma 1.37, asserting that a t-norm is continuous (as a real function of two arguments) is tantamount to asserting that it is continuous in both the first and the second variable. Corollary 1.31 and Lemma 1.32 yield the following corollary. Corollary 1.38. A left-continuous t-norm ⊗ is continuous iff it satisfies a ⊗ i∈I bi = i∈I (a ⊗ bi ) .

Lemma 1.39. A binary operation ⊗ in [0, 1] is a continuous t-norm iff the structure [0, 1], min, max, ⊗, →, 0, 1 with → defined by (1.54) is a divisible residuated lattice. Proof. First, we show that a residuated lattice L satisfies divisibility iff for each a ≤ b there exists c such that a = b ⊗ c. Denote this alternate condition by (D). On the one hand, if L satisfies divisibility and a ≤ b, we have a = b ∧ a = b ⊗ (b → a), i.e. one can take c = b → a in (D). On the other hand, if L satisfies (D), then for any a, b there exists c such that a ∧ b = a ⊗ c. By adjointness, c ≤ a → (a ∧ b) ≤ a → b. Therefore, a ∧ b = a ⊗ c ≤ a ⊗ (a → b). Since a ⊗ (a → b) ≤ a ∧ b is always the case, (D) implies divisibility. Now, let ⊗ be continuous. We verify (D). Let a, b ∈ [0, 1], a ≤ b and consider a function f (z) : [0, 1] → [0, b] defined by f (z) = b ⊗ z. Since f is a non-decreasing function and since 0 ≤ a ≤ b, continuity yields some c ∈ [0, 1] such that a = f (c), i.e. a = b ⊗ c. Conversely, let (D) hold. For an arbitrary b ∈ [0, 1], let f (z) = b ⊗ z. By (D), f is a surjection from [0, 1] onto [0, b] and since it is non-decreasing, it is continuous. The proof is complete. Theorem 1.40 (continuous t-norms and residuated lattices). A binary operation ⊗ in [0, 1] is a continuous t-norm iff the structure [0, 1], min, max, ⊗, →, 0, 1 with → defined by (1.54) is a BL-algebra. Proof. The assertion follows from Lemma 1.39 since prelinearity is always satisfied in a linearly ordered residuated lattice. The following definition summarizes further properties of t-norms which we will need. Definition 1.41. A t-norm ⊗ is called (i) Archimedean iff for each a, b ∈ (0, 1) there exists n ∈ N such that an < b, (ii) strictly monotone iff for each a > 0 and b < c we have a ⊗ b < a ⊗ c, (iii) strict iff it is continuous and strictly monotone. Continuous Archimedean t-norms can be characterized by a simple condition, see e.g. [47, 60]. Lemma 1.42. A continuous t-norm ⊗ is Archimedean iff we have a ⊗ a < a for each a ∈ (0, 1).


27

Furthermore, continuous Archimedean t-norms can be represented using so-called additive generators, see e.g. [47, 60]. Theorem 1.43 (additive generators). A mapping ⊗ : [0, 1]2 → [0, 1] is a continuous Archimedean t-norm iff there is a continuous additive generator f such that x ⊗ y = f (−1) (f (x) + f (y)) , i.e. f is a strictly decreasing continuous mapping f : [0, 1] → [0, ∞] with f (1) = 0 and f (−1) is the pseudoinverse of f defined by f (−1) (x) = f −1 (x) if x ≤ f (0) and f (−1) (x) = 0 otherwise. Example 1.44. L ukasiewicz as well as product t-norms are both continuous and Archimedean. f (x) = 1 − x and f (−1) (x) = max(1 − x, 0) are an additive generator and its pseudoinverse of L ukasiewicz t-norm. f (x) = − log(x) and f (−1) (x) = e−x are an additive generator and its pseudoinverse of product t-norm. Min is continuous but not Archimedean. The following theorem presents a well-known representation of continuous t-norms by ordinal sums. Theorem 1.45 (Mostert-Shields representation). Let ⊗ be a continuous t-norm. Then the corresponding complete residuated lattice L on [0, 1] is isomorphic to an ordinal sum i∈I Li of complete residuated lattices such that ukasiewicz I is the set of all idempotents of ⊗, and each Li is a standard L algebra on [0, 1], a standard product algebra on [0, 1], or a one-element algebra. Proof. The proof is technically involved, see e.g. [47, 49, 60].

Remark 1.46. (1) The following claim is easy to check. Let I ⊆ [0, 1] be a closed set (w.r.t. standard metric d on [0, 1] given by d(x, y) = |x − y|). Consider the operation + w.r.t. I, ≤ with ≤ being a natural ordering of I (see Definition 1.23). Let Li be a one-element algebra if i = i+ , and let Li be a standard L ukasiewicz algebra on [0, 1] or a standard product algebra on [0, 1] if i < i+ . Then there is a continuous ⊗ t-norm such that the corresponding complete residuated lattice on [0, 1] is isomorphic to the ordinal sum i∈I Li . (2) Theorem 1.45 and (1) give a precise meaning of the claim that continuous t-norms are just ordinal sums of L ukasiewicz t-norms and product t-norms. Properties of Truth Stressers Theorem 1.47 (basic properties of truth stressers). Each truth stresser on a complete residuated lattice satisfies 0∗ = 0 , monotony, i.e. a ≤ b implies a∗ ≤ b∗ , ∗ a∗ ⊗ b∗ ≤ (a ⊗ b) .

(1.55) (1.56) (1.57)

28


Proof. (1.55): By (1.12), 0∗ ≤ 0, i.e. 0∗ = 0. (1.56): If a ≤ b then a → b = 1. Therefore, (1.11) and (1.13) imply 1 = 1∗ = ∗ (a → b) ≤ a∗ → b∗ from which we get a∗ ≤ b∗ . ∗ (1.57): By (1.22), a ≤ b → (a ⊗ b). Using (1.56) we get a∗ ≤ (b → (a ⊗ b)) . ∗ ∗ ∗ ∗ ∗ Furthermore, (b → (a ⊗ b)) ≤ b → (a ⊗ b) by (1.13). Therefore, a ≤ b → ∗ ∗ (a ⊗ b) from which we get a∗ ⊗ b∗ ≤ (a ⊗ b) by adjointness. Remark 1.48. Note that (1.12),(1.56), and(1.17) say that ∗ operator on L, ≤ . Therefore, i∈I ai ∗ = ( i∈I ai ∗ ) .

∗

is an interior

We are now ready to prove that (1.16) defines a truth stresser. Lemma 1.49. For each complete residuated lattice, truth stresser.

∗

defined by (1.16) is a

Proof. (1.11) and (1.12) are obvious. For (1.13), distinguish the following cases. First, if a = 1, then (1.13) becomes b∗ ≤ b∗ which is true. Second, ∗ if 1 = a ≤ b, then a → b = 1 and so (1.13) says 1 = (a → b) ≤ a∗ → b∗ which ∗ ∗ ∗ ∗ is equivalent to a ≤ b . Now observe that a ≤ b is true both for b = 1 (since a∗ ≤ 1 = b∗ ) as well as for b < 1 (by monotony of ⊗). Third, if 1 = a ≤ b = 1, k k k then (1.13) becomes i=1 ci ⊗ (a → b)ni ≤ i=1 ci ⊗ ani → i=1 ci ⊗ bni . By adjointness, this inequality k is true iff for each p, q = 1, . . . , k we have cp ⊗ (a → b)np ⊗ cq ⊗ anq ≤ i=1 ci ⊗ bni which is true. Indeed, for np ≤ nq , cp ⊗ (a → b)np ⊗ cq ⊗ anq ≤ cp ⊗ (a → b)np ⊗ cq ⊗ anp ≤ k ≤ cp ⊗ cq ⊗ anp ⊗ (a → b)np ≤ cp ⊗ cq ⊗ bnp ≤ cp ⊗ bnp ≤ i=1 ci ⊗ bni ,

and analogously for np > nq .

Theorem 1.50 (implicational truth stressers). Each implicational truth stresser on a complete residuated lattice satisfies ∗

(a ⊗ b) = a∗ ⊗ b∗ . ∗

(1.58) ∗

Proof. Due to (1.57), it is enough to see (a ⊗ b) ≤ a∗ ⊗b∗ . Since (a ⊗ b) ≤ a∗ ∗ ∗ ∗ and (a ⊗ b) ≤ b∗ , (1.18) and monotony of ∗ yield (a ⊗ b) = (a ⊗ b) ⊗ ∗ (a ⊗ b) ≤ a∗ ⊗ b∗ . Remark 1.51. Looking at (1.58) one can ask under what conditions L∗ satisfies ∗ (a → b) = a∗ → b∗ . It turns out that this condition is too restrictive. Indeed, ∗ if ∗ is an implicational truth stresser satisfying (a → b) = a∗ → b∗ , we get ∗ a ≤ (a → a∗ ) → a∗ ≤ (a → a∗ ) → a∗ = (a∗ → a∗∗ ) → a∗ = 1 → a∗ = a∗ , i.e. a ≤ a∗ which means a = a∗ . Therefore, if ∗ is an implicational truth stresser ∗ satisfying (a → b) = a∗ → b∗ , ∗ must be identity. The following lemma shows how to obtain implicational truth stressers in BL-chains (i.e. linearly ordered BL-algebras) which satisfy certain additional condition.


Lemma 1.52. Let L be a complete BL-chain satisfying a ⊗ i∈I bi = i∈I (a ⊗ bi ) . Then a unary operation

∗

defined by a∗ =

n∈N0

29

(1.59)

an

(1.60) ∗

for all a ∈ L is an implicational truth stresser. Moreover, a is the greatest idempotent less or equal a. Proof. (1.11) is obvious. It remains to check conditions (1.12), (1.13), (1.17)– (1.19). (1.12): For n = 1 we have a1 ≤ a, thus a∗ = n∈N0 an ≤ a. (1.13): From (1.44) and (1.35) it follows ∗ n (a → b) = n∈N0 (a → b) ≤ n∈N0 (an → bn ) ≤ ≤ n∈N0 an → n∈N0 bn = a∗ → b∗ . (1.17): Due to (1.59), we have m n m a∗∗ = m∈N0 = m∈N0 i=1 ni ∈N0 ani = n∈N0 a m m = m∈N0 n1 ,...,nm ∈N0 i=1 ani = m∈N0 n1 ,...,nm ∈N0 a i=1 ni = = m∈N0 n∈N0 an = n∈N0 an = a∗ . 1.18: Analogously as for (1.17), we have n m a ⊗ a∗ = ⊗ = m,n∈N0 an+m = k∈N0 ak = a∗ . n∈N0 a m∈N0 a n (1.19): We have to show i∈I n∈N0 ani = n∈N0 i∈I ai . The “≥”-part n is evident since for each i ∈ I we have n∈N0 ani ≥ n∈N0 i∈I ai . For the “≤”-part, observe that since ani ≤ ai1 ⊗ · · · ⊗ ain for ai = min {ai1 , . . . , ain } holds for each n (recall that L is supposed to be a chain), we have n n n . i∈I n∈N0 ai ≤ i∈I ai ≤ i1 ,...,in ∈I (ai1 ⊗ · · · ⊗ ain ) = i∈I ai ∗

Hence, the required inequality follows immediately. Altogether, ∗ is an implicational truth stresser for a complete BL-chain satisfying (1.59). It remains to show that a∗ is the greatest idempotent which is less or equal toa. Indeed, let b ∈ L be an idempotent such that b ≤ a. Then b = n∈N0 bn ≤ n∈N0 an = a∗ . Remark 1.53. (1) Since each BL-algebra satisfies a ⊗ (b ∧ c) = (a ⊗ b) ∧ (a ⊗ c) [49], every finite BL-chain satisfies (1.59). (2) If L = [0, 1] , max, min, ⊗, →, 0, 1 is a residuated lattice with ⊗ being a continuous t-norm, then L is a BL-algebra and (1.59) is a consequence of right-continuity of ⊗. Condition (1.21) of Horn truth stressers follows from two simpler conditions.

30


Lemma 1.54. If L∗ is a complete residuated lattice with a truth stresser ∗ such that a → i∈I bi ≤ i∈I (a → bi ) for every non-empty index set I , (1.61) ∗ ∗ ≤ i∈I bi for every index set I , (1.62) i∈I bi then L∗ satisfies (1.21). Proof. Applying monotony of ∗ to (1.61) and using 1.62 we get ∗ ∗ ∗ a → i∈I bi ≤ i∈I (a → bi ) ≤ i∈I (a → bi ) ,

which is the non-trivial inequality of (1.21).

Remark 1.55. (1) Observe that the converse inequality of (1.62) is always sat isfied. Indeed, for every i ∈ I, we have bi ≤ i∈I bi , so by monotony of ∗ we ∗ ∗ have b∗i ≤ i∈I bi . Thus, i∈I b∗i ≤ i∈I bi . (2) Condition (1.62) does not hold for a general implicational truth stresser. For instance, for L being a four-element Boolean algebra L = {0, a, b, 1}, ∨, ∧, ⊗, →, 0, 1 (i.e. a, b ∈ L are incomparable), globalization ∗ ∗ is an implicational truth stresser. We have a ∨ b = 1, i.e. (a ∨ b) = 1. On the ∗ ∗ ∗ ∗ other hand a = b = 0, thus a ∨ b = 0. In each residuated lattice, prelinearity is equivalent to a → (b ∨ c) = (a → b) ∨ (a → c) [10, Theorem 2.34]. Therefore, we have the following consequence of Lemma 1.54. Corollary 1.56. Let L be a finite residuated lattice, which satisfies the condition of prelinearity. Then every implicational truth stresser for L satisfying (1.20) and (1.62) is a Horn truth stresser. We now present two examples of implicational truth stressers satisfying (1.20) and (1.62). Example 1.57. (1) Let L be any complete Heyting algebra (i.e. a complete residuated lattice, where ⊗ = ∧, the residuum is then a relative pseudocomplement). The implicational truth stresser ∗ defined by a∗ = a, for all a ∈ L, satisfies (1.20) and (1.62) trivially. (2) Let L be an arbitrary complete residuated lattice equipped with globalization. Evidently, ∗ satisfies (1.20) trivially because 0 ⊗ a = 0 ∧ a, and 1 ⊗ a = 1 ∧ a for all a ∈ L. Furthermore, ∗ satisfies (1.62) iff 1 (the greatest element of L) is ∨-irreducible, i.e. iff for every family {bi < 1 | i ∈ I} we have i∈I bi < 1. Globalization on chains can be characterized by an additional identity. Lemma 1.58 (globalization on chains). Let L be a linearly ordered complete residuated lattice. A truth stresser ∗ is a globalization for L iff it satisfies a∗ ∨ ¬a∗ = 1 .

(1.63)


31

Proof. Evidently, globalization satisfies (1.63). Let ∗ satisfy (1.63). We have 1∗ = 1 due to (1.11). If a < 1 then a∗ ≤ a < 1. Since L is linearly ordered, we have max(a∗ , ¬a∗ ) = a∗ ∨ ¬a∗ = 1, and so ¬a∗ = 1 from which we get a∗ = 0 (indeed, if ¬a∗ = 1 then 1 ≤ ¬a∗ , i.e. 1 ≤ a∗ → 0, whence a∗ ≤ 0 by adjointness). Therefore, ∗ is globalization. Remark 1.59. A characterization of truth stressers which are globalizations for general complete residuated lattices by means of identities or inequalities is not possible. First, note that each inequality can be equivalently replaced by identity. Namely, x ≤ y can be replaced by x ∨ y = y. Now, if L∗11 and L∗22 are two complete residuated lattices with globalizations ∗ 1 and ∗ 2 , their direct product L∗ = L∗11 × L∗22 is a complete residuated lattice with a truth stresser ∗ which is not a globalization since for 0, 1 ∈ L1 × L2 we have ∗ 0, 1 = 0∗1 , 1∗2 = 0, 1 = 0, 0 although 0, 1 < 1, 1 . On the other hand, L∗11 × L∗22 satisfies each identity satisfied by both L∗11 and L∗22 . This shows that globalizations cannot be characterized by identities. Truth Stressers on Ordinal Sums We now present some ways to define truth stressers on ordinal sums of complete residuated lattices. Theorem 1.60. Let I, ≤ be a finite chain with the least element 0 and the greatest element 1. Let {L∗ii | i ∈ I} be a family of complete residuated lattices Li = Li , ∨i , ∧i , ⊗i , →i , 0i , 1i with implicational truth stressers ∗i satisfying (1.20) and (1.62). Suppose {Li | i ∈ I} can be ordinally added. Then the op eration ∗ on i∈I Li defined by a∗ = a∗i for a ∈ Li is an implicational truth stresser on i∈I Li satisfying (1.20) and (1.62). Proof. Clearly, conditions (1.11), (1.12), (1.17), and (1.18) follow directly from the definition. Denote the operations on i∈I Li by ∧, ∨, ⊗, →, 0, 1. (1.13): We consider three separate cases. If a ≤ b then clearly a∗ ≤ b∗ , ∗ ∗ i.e. (a → b) = 1 = a∗ → b∗ . If a → b = b, then (a → b) = b∗ ≤ a∗ → b∗ . Finally, when a b and a → b = a →i b, i.e. a, b ∈ Li , then we have ∗ ∗ (a → b) = (a →i b) = (a →i b)∗i ≤ a∗i →i b∗i ≤ a∗ → b∗ . (1.19): Take {aj | j ∈ J}. Let k = min {i ∈ I | there is j ∈ J such that aj ∈ Li }. For J = {j | aj ∈ Lk }, ∗ k we have j∈J aj = j∈J aj . Thus, j∈J a∗j = j∈J a∗j k = = j∈J aj ∗ a . j∈J j (1.20): If a, b∗ ∈ Li , the claim is trivial. If b∗ ∈ Li and a ∈ Li , we have a ⊗ b∗ = a ∧ b∗ by definition. 1.62: For {bj | j ∈ J}, take l = max{i ∈ I | there is j ∈ J such that bj ∈ ∗ ∗l L }, put J = {j | bj ∈ Ll }, and observe that = = j∈J bj j∈J bj i ∗l ∗ b = b . j j∈J j j∈J

32

1 Introduction to Fuzzy Sets and Fuzzy Logic 11 = 1 11

L1 1i = 0 1 1i

Li 0i 10 10

L0 00 = 0

Fig. 1.7. Ordinal sum of {Li ⊕ 2 | i ∈ I}

Lemma 1.61. For any complete residuated lattice L, globalization defined on L ⊕ 2 (see Example 1.26) is an implicational truth stresser satisfying (1.20) and (1.62). Proof. As seen in Example 1.19, globalization is an implicational truth stresser for any complete residuated lattice, thus in particular for L ⊕ 2. The above-described structures Li ⊕2 can be ordinally added, yielding new structures with non-trivial implicational truth stressers satisfying (1.20) and (1.62). This construction is illustrated in Fig. 1.7 and is justified by the next assertion which is a direct consequence of Theorem 1.60 and Lemma 1.61. Corollary 1.62. Let I, ≤ be a finite chain with the least element 0 and the greatest element 1. Let {Li | i ∈ I} be a family of complete residuated lattices Li = Li , ∨i , ∧i , ⊗i , →i , 0i , 1i , where L i ∩ Lj = ∅ for all i, j ∈ I, i = j. More be Li ⊕ 2. = L , ∨ , ∧ , ⊗ , → , 0 , 1 For the complete residuated over, let L i i i i i i i i lattice i∈I Li we define a unary operation ∗ on i∈I Li by 1i for a = 1i , ∗ (1.64) a = for a ∈ Li , a = 1i . 0i Then ∗ is an implicational truth stresser for i∈I Li satisfying (1.20) and (1.62).

1.3 Fuzzy Sets Fuzzy Sets and Fuzzy Relations The concepts of a fuzzy set and a fuzzy relation are the basic ones in fuzzy set theory and its applications. Fuzzy sets and relations are mathematical models

1.3 Fuzzy Sets

33

of vaguely delineated collections of objects and relationships between objects. Unlike the case of an ordinary set, an element may belong to a fuzzy set in a degree different from 0 and 1. In general, the degree is taken from a suitable scale L of truth degrees. We assume that L is a support set of some complete residuated lattice L. For instance, if L = [0, 1], an element u may belong to a fuzzy set A in degree, say 0.9. We denote this fact by A(u ) = 0.9. This may be interpreted as “u almost belongs to A” since 0.9 is a high truth degree. A(u ) = 1 means that u fully belongs to A, A(u ) = 0 means that u does not belong to A at all. For example, we might say that a person who is 183 cm tall belongs to a fuzzy set of tall people in degree 0.9 while a person 190 cm tall belongs to that fuzzy set in degree 1. From this point of view, ordinary sets correspond to fuzzy sets over the scale L = {0, 1} – each element either belongs (i.e., belongs in degree 1) or does not belong (i.e., belongs in degree 0) to an ordinary set. Fuzzy sets are mathematical objects. The aim of this section is to introduce basic notions and facts concerning fuzzy sets. Definition 1.63. An L-set (or fuzzy set with truth degrees in L) in a universe set U is a mapping A : U → L. If A is an L-set in U , A(u ) ∈ L is interpreted as the truth degree of “u belongs to A”. Here and in the sequel, truth degrees (i.e. the elements of L) will be denoted by a, b, c, . . . while elements of universes of L-sets will be denoted by a , b , c , . . . , u , v , . . . , both possibly with indices. Remark 1.64. (1) 2-sets coincide with characteristic functions of ordinary sets. This is a basic way the concept of an L-set generalizes the concept of an ordinary set. (2) Note that a characteristic function χA of an ordinary set A ⊆ U also can be seen as an L-set (for any L) such that for each element u we have χA (u ) ∈ {0, 1}. Example 1.65. (1) Let X = {circle, square, hexagon}, L be the standard G¨ odel algebra on [0, 1]. Let A : X → [0, 1] be given by A(circle) = 1, A(square) = 0, A(hexagon) = 0.5. Then A is an L-set in X. A may be thought of as representing the concept of “being circle-shaped” in the universe X. (2) Let X be the set of all real numbers, L be the standard L ukasiewicz algebra on [0, 1]. Define an L-set A ∈ LX by   x − 4 for 4 ≤ x ≤ 5 A(x) = 6 − x for 5 < x ≤ 6  0 otherwise . Then A is a fuzzy set that represents the concept “approximately 5”. A is depicted in Fig. 1.8. (3) Both medical doctors (experts) and their patients (non-experts) understand and use the term “normal blood pressure”. The left part of Fig. 1.9 shows a fuzzy set which represents the meaning of this term. If one would like

34


1

1

2

3

4

5

6

7

Fig. 1.8. Fuzzy set representing “approximately 5”

to represent the meaning of “normal blood pressure” by an ordinary set, one would have to come up with a set like the one depicted by its membership function in the right part of Fig. 1.9. That is, one would have to pick some interval around a prototypical value of a normal blood pressure. Everybody can see that this is not appropriate – a very small difference in blood pressure can cause a transition from being normal to not being normal. This is in contradiction to how experts and non-experts understand “normal blood pressure”. 1

1

0

0 100 120 140

100 120 140

Fig. 1.9. Normal blood pressure as a fuzzy set (left) and as an ordinary set (right)

Note that A : U → L is sometimes called a membership function of a fuzzy set. We do not follow this approach since it implies a distinction between a fuzzy set and its membership function, and this causes unnecessary complications. For us, a fuzzy set is a function A : U → L (which might be occasionally called a membership function). For every L-set A : U → L, we define an ordinary set Supp(A) by Supp(A) = {u | A(u ) > 0}. Supp(A) is called a support set of A. An L-set A in U is said to be finite if Supp(A) is a finite set. A finite L-set A in U such that Supp(A) = {u1 , . . . , un } is also denoted by A = {A(u1 )/u1 , . . . , A(un )/un }. An L-set A : U → L such that Supp(A) = {u } is called a singleton. For every Lset A : U → L and a ∈ L, we define an ordinary set aA by aA = {u | A(u ) ≥ a}. a A is called an a-cut of A.

1.3 Fuzzy Sets

35

The set of all L-sets in U is denoted by LU or LU . An empty L-set in U , denoted by ∅U , is a mapping ∅U : U → L where ∅U (u ) = 0 for all u ∈ U . If the universe of discourse U is clear from context, we denote ∅U by ∅. A full L-set in U , denoted by 1U , is a mapping 1U : U → L, where 1U (u ) = 1 for every u ∈ U . If there is no danger of confusion, we write U instead of 1U . An L-set A in U is called crisp if A(u ) ∈ {0, 1} for each u ∈ U . Crisp L-sets in U correspond in an obvious way to ordinary subsets of U (crisp L-sets are the characteristic functions of ordinary subsets). Therefore, we will sometimes identify crisp L-sets with the corresponding ordinary subsets. An n-ary L-relation (or fuzzy relation with truth degrees in L) in a universe set U is an L-set in the universe set U n . That is, a binary L-relation R in U is a mapping R : U × U → L. A binary L-relation R : U × U → L is called a restriction of R : U × U → L, if R (u , v ) > 0 implies R (u , v ) = R(u , v ) for every elements u , v ∈ U . If a restriction R of R is a finite L-relation, it is called a finite restriction of R. For a binary L-relation R, we define an inverse L-relation R−1 of R by putting R−1 (u , v ) = R(v , u ) for all u , v ∈ U . For binary L-relations R1 , R2 in U , the ◦-composition of R1 and R2 is a binary L-relation on U defined by (R1 ◦ R2 )(u , v ) = w ∈ U (R1 (u , w ) ⊗ R2 (w , v )) . (1.65) for all u , v ∈ U . For L-sets A and B in U we define S(A, B) = u ∈ U A(u ) → B(u ) , A ≈ B = u ∈ U A(u ) ↔ B(u ) .

(1.66) (1.67)

S(A, B) is called a degree of subsethood of A in B, and A ≈ B is called a degree of equality of A and B. Note that S(A, B) can be thought of as a truth degree of “for each u ∈ U : if u belongs to A then u belongs to B.” Likewise, A ≈ B can be thought of as a truth degree of “for each u ∈ U : u belongs to A iff u belongs to B.” Example 1.66. For standard L ukasiewicz, Gödel, and product structures on [0, 1], we have S(A, B) = inf{1 − A(u ) + B(u ) | u ∈ U, A(u ) > B(u )} (Lukasiewicz) , S(A, B) = inf{B(u ) | u ∈ U, A(u ) > B(u )} (G¨ odel) , S(A, B) = inf{B(u )/A(u ) | u ∈ U, A(u ) > B(u )}

(product) ,

and A ≈ B = inf{1 − |A(u ) − B(u )| | u ∈ U } (Lukasiewicz) , A ≈ B = inf{min(A(u ), B(u )) | u ∈ U, A(u ) = B(u )} (G¨ odel) ,  0 if for some u ∈ U :     A(u ) = 0 = B(u ) or A≈B= (Goguen) . u ) = 0 = B(u ) A(    A( u ) B( u )  inf u ∈U min , otherwise B(u )

A(u )

36


From the properties of residuated lattices it follows that A ≈ B = S(A, B) ∧ S(B, A). Furthermore, we write A ⊆ B (A is a subset of B) if S(A, B) = 1. Since S(A, B) = 1 means u ∈U (A(u ) → B(u )) = 1, S(A, B) = 1 is equivalent to A(u ) → B(u ) = 1 for each u ∈ U which is, due to (1.23), equivalent to A(u ) ≤ B(u ) for each u ∈ U . That is, A ⊆ B means that for each u ∈ U we have A(u ) ≤ B(u ). Common operations with L-sets which generalize the ordinary operations with sets result by componentwise extension of operations on L. That is, any operation o (possibly infinitary) on L induces in a componentwise manner an operation O on LU by putting O(A, . . . )(u ) = o(A(u ), . . . ) for arbitrary L-sets A, . . . and arbitrary u ∈ U . Arguments of O are L-sets in U and the result of O applied to A, . . . is an L-set O(A, . . . ) in U to which an element u ∈ U belongs to a degree obtained by applying o to A(u ), . . . In particular, for L-sets A, B, Ai (i ∈ I) in U we define (A ∧ B)(u ) = A(u ) ∧ B(u ) , (A ∨ B)(u ) = A(u ) ∨ B(u ) , i∈I Ai (u ) = i∈I Ai (u ) , i∈I Ai (u ) = i∈I Ai (u ) ,

etc. Following commonusage we denote A ∧ B, A ∨ B, i∈I Ai , and i∈I Ai also by A ∩ B, A ∪ B, i∈I Ai , and i∈I Ai , respectively. The componentwise extensions make LU = LU , ∧, ∨,Q⊗, →, ∅U , 1U a complete residuated lattice. Namely, LU is a direct product u ∈U Lu of complete residuated lattices Lu = L. Theorem 1.67. For any complete residuated lattice L and any set U = ∅, LU isa complete residuated lattice where infima and suprema are given by and , respectively, and the corresponding lattice order is ⊆. Moreover, all identities and inequalities satisfied by L are valid also in LU . The following proposition summarizes the basic properties of the degree of subsethood. Theorem 1.68. For L-sets A, B, and C in U , we have S(A, A) = 1 ,

(1.68)

S(A, B) ⊗ S(B, C) ≤ S(A, C) , S(A, B) = 1 iff A ⊆ B ,

(1.69) (1.70)

S(∅, A) = 1 and S(A, U ) = 1 , S(A, i∈I Bi ) = i∈I S(A, Bi ) , S( i∈I Ai , B) = i∈I S(Ai , B) ,

(1.71) (1.72) (1.73)

S(A, A ∩ B) = S(A, B) = S(A ∪ B, B) .

(1.74)

Proof. We will use Theorem 1.67 and the fact that if au ≤ bu (au , bu ∈ L, u ∈ U ) then u ∈U au ≤ u ∈U bu .

1.3 Fuzzy Sets

37

(1.68): S(A, A) = u ∈U (A(u )→ A(u )) = u ∈U 1 = 1. (1.69): S(A, B) ⊗ S(B, C) = u ∈U (A(u ) → B(u )) ⊗ v ∈U (B(v ) → C(v )) ≤ ((A( u ) → B( u )) ⊗ (B( v ) → C( v ))) ≤ u ∈U v ∈U u ∈U ((A(u ) → B(u )) ⊗ (B(u ) → C(u ))) ≤ u ∈U (A(u ) → C(u )) = S(A, C). (1.70): If S(A, B) = 1 iff for each u ∈ U we have A(u ) → B(u ) = 1 iff for each u ∈ U we have A(u ) ≤ B(u ) iff A ⊆ B. → A(u )) = 1 and (1.71): S(∅, A) = u ∈U (0 S(A, U ) = u∈U (A(u ) → 1) = 1. (1.72): S(A, i∈I Bi ) = u ∈U (A(u ) → i∈I Bi (u )) = u ∈U i∈I (A(u ) → Bi (u )) = i∈I u ∈U (A(u ) → Bi (u )) = i∈I S(A, Bi ). Ai , B) = u ∈U ( i∈I Ai ( u ) → B(u )) = u ∈U i∈I (Ai (u ) → (1.73): S( i∈I B(u )) = i∈I u ∈U (Ai (u ) → B(u )) = i∈I S(Ai , B) (1.74): By (1.68), (1.72), and (1.73), S(A, A ∩ B) = S(A, A) ∧ S(A, B) = 1 ∧ S(A, B) = S(A, B), and S(A ∪ B, B) = S(A, B) ∧ S(B, B) = S(A, B) ∧ 1 = S(A, B). Analogous assertions can be proved for the degree of equality. Notice how the assertions of Theorem 1.68 can be described verbally. For instance, take (1.69). If L = 2 (bivalent case), (1.69) expresses exactly that if a set A is a subset of B, and B is a subset of C, then A is a subset of C. For a general L, the meaning of (1.69) is the same, only interpreted in many-valued setting. The same is true of other assertions. Fuzzy Equivalences and Fuzzy Equalities The concept of a fuzzy equivalence relation results by carrying over the concept of an ordinary equivalence relation to fuzzy setting. Fuzzy equivalences belong to the most studied fuzzy relations. The reason for this is that a fuzzy equivalence can be interpreted as similarity – a degree to which two object are equivalent is understood as a degree of their similarity. Similarity is one of crucial phenomena accompanying human reasoning and perception. With the concept of fuzzy equivalence, fuzzy logic provides a simple model of similarity. Definition 1.69. An L-equivalence (fuzzy equivalence) relation θ on a set U is a mapping θ : U × U → L satisfying θ(u , u ) = 1 ,

(1.75)

θ(u , v ) = θ(v , u ) , θ(u , v ) ⊗ θ(v , w ) ≤ θ(u , w ) ,

(1.76) (1.77)

for every u , v , w ∈ U . An L-equivalence θ on U where θ(u , v ) = 1

implies

u=v

(1.78)

will be called an L-equality (fuzzy equality). Remark 1.70. (1) There are various other terms used for L-equivalences: the most common are similarity (or fuzzy similarity) and indistinguishability.

38


These terms, however, suggest an epistemic interpretation that may be disputable. (2) Conditions (1.75), (1.76), and (1.77) are called reflexivity, symmetry, and transitivity, respectively. These conditions generalize their bivalent counterparts. Indeed, consider L = 2. Then (1.75) and (1.76) obviously express reflexivity and symmetry (in bivalent sense). Equation (1.77) implies that if u , v and v , w are in θ (i.e. θ(u , v ) = 1 and θ(v , w ) = 1) then 1 = 1 ⊗ 1 = θ(u , v ) ⊗ θ(v , w ) ≤ θ(u , v ). Therefore, θ(u , w ) = 1, i.e. u , w is in θ, i.e. θ is transitive (in bivalent sense). Conversely, let θ be transitive (in bivalent sense). There are two possibilities: Either θ(u , v ) = 0 or θ(v , w ) = 0 and then (1.77) is true; or both θ(u , v ) = 1 and θ(v , w ) = 1 in which case u , v and v , w are in θ, and so also u , w is in θ, i.e. θ(u , w ) = 1, i.e. (1.77) holds true. (3) Equations (1.75)–(1.77) are equivalent to saying that first-order logical formulas which express reflexivity, symmetry, and transitivity, are true in degree 1. Note that denoting conjunction and implication by o and i, the formulas are (∀x)(x ∼ x) , (∀x, y)(x ∼ y i y ∼ x) , (∀x, y, z)(((x ∼ y) o (y ∼ z)) i x ∼ z) . (4) There has been a lot of debates of whether reflexivity, symmetry, and transitivity as described verbally are appropriate properties of similarity, see e.g. [82, 86]. By verbal description we mean e.g. “if u and v are similar then v and u are similar” in case of similarity. Reflexivity seems to be mostly agreed upon. There is an argument against symmetry. It is being pointed out that “u is similar to v ” is not the same as “v is similar to u ” (e.g. u being a father and v being his son). However, if one is interested in relation describing “u and v are similar to each other”, symmetry seems to be an obvious property. Transitivity of similarity has been a point of disagreement. One usually argues against transitivity as follows. If similarity were transitive then any two colors would be similar. For we may suppose that two colors with sufficiently close wave lengths are similar. Now, for any two colors u and v we may find a chain u = u1 , u2 , . . . , un = v , of colors such that ui and ui+1 are similar. Using transitivity, u and v are similar. On the other hand, if the transitivity condition is formulated verbally (i.e. “if u and v are similar, and if v and w are similar then u and w are similar”), it seems acceptable. The solution to this puzzle lies in the fact that similarity, by its nature, is a graded (fuzzy) notion. If we look at the meaning of transitivity in fuzzy setting, we find it quite natural. For example, if θ(u , v ) = 0.8 (u and v are similar in degree 0.8) and θ(v , w ) = 0.8 (v and w are similar in degree 0.8) then u and w have to be similar at least in degree 0.8 ⊗ 0.8. Thus, in case of the product conjunction, transitivity forces θ(u , w ) ≥ 0.8 ⊗ 0.8 = 0.64 which seems to reflect intuition in a reasonable way.

1.3 Fuzzy Sets

39

(5) L-equivalences will usually be denoted by θ. L-equalities will usually be denoted by ≈ and we will use an infix notation, i.e. we write u ≈ v instead of ≈ (u , v ). In order to facilitate reading, we use also (u ≈ v ) instead of u ≈ v . Example 1.71. (1) Figure 1.10 describes an L-equivalence relation θ in U = {u1 , u2 , u3 , u4 } with L being the standard L ukasiewicz algebra on [0, 1]. That

Fig. 1.10. L-equivalence θ

is, θ(u1 , u1 ) = 1, θ(u1 , u2 ) = 0.9, θ(u1 , u3 ) = 0.8, etc. Note that θ is not an L-equality. Indeed, θ(u1 , u4 ) = 1 but u1 = u4 . On the other hand, restriction of θ to U − {u4 } is an L-equality on {u1 , u2 , u3 }. (2) If L is the standard G¨ odel algebra on [0, 1] (i.e. a⊗b = min(a, b)), θ from Fig. 1.10 is not an L-equivalence. Indeed, 0.9 ⊗ 0.8 = θ(u2 , u1 ) ⊗ θ(u1 , u3 ) = min(0.9, 0.8) ≤ θ(u2 , u3 ) = 0.7, and so θ is not transitive. We now list basic properties of fuzzy equivalences. Theorem 1.72. Let θ be an L-equivalence in U . (i) For each a ∈ L, a θ is a reflexive and symmetric relation. If a ⊗ a = a then a θ is an equivalence relation. (ii) 1 θ is an equivalence relation. If θ is an L-equality then 1 θ is the identity in U . Proof. (i): Reflexivity and symmetry of a θ is obvious. Let a⊗a = a. If u , v ∈ θ and v , w ∈ a θ then a ≤ θ(u , v ) and a ≤ θ(v , w ) and thus a = a ⊗ a ≤ θ(u , v ) ⊗ θ(v , w ) ≤ θ(u , w ), i.e. u , w ∈ a θ and so a θ is also transitive. (ii): Since 1 ⊗ 1 = 1, the fact that 1 θ is an L-equivalence follows from (i). If θ is an L-equality then 1 θ is identity in U by definition.

a

∗

∗

∗

In classical setting, equivalence relations correspond uniquely to partitions, see Sect. 1.1. We are going to show that an analogous correspondence is available also in fuzzy setting. For an L-equivalence θ in U and u ∈ U , denote by [u ]θ an L-set in U defined by (1.79) [u ]θ (v ) = θ(u , v ) for any v ∈ U . We call [u ]θ a class of θ given by u . Therefore, a degree to which v belongs to a class given by u is the degree to which u and v are equivalent.

40


Definition 1.73. An L-partition of a set U is a set Π of L-sets in U such that (i) for each u ∈ U there is A ∈ Π such that A(u ) = 1; (ii) for each A ∈ Π there is u ∈ U such that A(u ) = 1; (iii) for every A, B ∈ Π, u ∈ U we have A(u ) ⊗ B(u ) ≤ (A ≈ B). Remark 1.74. (1) For L = 2, the concept of an L-partition coincides with the concept of a partition. (i) says that Π covers U ; (ii) says that each A ∈ Π is non-empty; (iii) says that if u belongs to A and B then A and B are equal. (2) Since A ≈ B denotes the equality degree defined by (1.67), (iii) implies that if A(u ) = 1 and B(u ) = 1 for some A, B ∈ Π, then A = B. Theorem 1.75 (fuzzy equivalences and partitions). Let θ be an Lequivalence in U , Π be an L-partition of U . Define a binary L-relation θΠ in U by θΠ (u , v ) = Au (v ) where Au ∈ Π is such that Au (u ) = 1, and put Πθ = {[u ]θ | u ∈ U }. Then (1) Πθ is an L-partition of U ; (2) θΠ is an L-equivalence in U ; and (3) θ = θΠθ and Π = ΠθΠ . Proof. (1): We verify that Πθ is an L-partition of U . By reflexivity of θ, [u ]θ (u ) = 1 verifying (i) and (ii). To verify (iii) we have to show that for every u , v , w ∈ U we have [u ]θ (w ) ⊗ [v ]θ (w ) ≤ ([u ]θ ≈ [v ]θ ) which holds iff for each z ∈ U we have both [u ]θ (w ) ⊗ [v ]θ (w ) ≤ [u ]θ (z ) → [v ]θ (z ) and [u ]θ (w ) ⊗ [v ]θ (w ) ≤ [v ]θ (z ) → [u ]θ (z ) . Due to symmetry of both of the cases we check only the first inequality which is equivalent to [u ]θ (z ) ⊗ [u ]θ (w ) ⊗ [v ]θ (w ) ≤ [v ]θ (z ) , i.e. to θ(z , u ) ⊗ θ(u , w ) ⊗ θ(w , v ) ≤ θ(z , v ) , which is true by transitivity of θ. Therefore, (1) is proved. (2): We verify that θΠ is an L-equivalence in U . First, notice that θΠ is defined correctly. Indeed, by Remark 1.74 (2), there is just one Au ∈ Π such that Au (u ) = 1. Reflexivity of θΠ is obvious. Symmetry: We have to show that if A(u ) = 1 and B(v ) = 1 for u , v ∈ U and A, B ∈ Π, then A(v ) = B(u ). By definition we have A(v ) ⊗ B(v ) ≤ (A ≈ B), hence A(v ) = A(v ) ⊗ 1 = A(v ) ⊗ B(v ) ≤ A(u ) → B(u ) = 1 → B(u ) = B(u ), i.e. A(v ) ≤ B(u ). Similarly, one gets B(u ) ≤ A(v ), proving the required equality. Transitivity: We have to show Au (v ) ⊗ Av (w ) ≤ Au (w ), i.e. Au (v ) ≤ Av (w ) → Au (w ).

1.3 Fuzzy Sets

41

The last inequality is true since Au (v ) = Au (v ) ⊗ Av (v ) ≤ (Au ≈ Av ) ≤ Av (w ) → Au (w ). (3): We show θ = θΠθ : θΠθ (u , v ) = A(v ) for A ∈ Πθ such that A(u ) = 1. Since A = [u ]θ , we have θΠθ (u , v ) = [u ]θ (v ) = θ(u , v ). We show Π = ΠθΠ : We have A ∈ ΠθΠ iff A = [u ]θΠ . Now, for Au ∈ Π such that Au (u ) = 1 we have A(v ) = [u ]θΠ (v ) = θΠ (u , v ) = Au (v ). Thus, A = Au ∈ Π. ∗

∗

∗

Similarity of objects is often related to a collection of attributes of these objects. Two objects are considered similar if they are similar according to having some attributes. The following assertion says that fuzzy equivalences are just fuzzy relations which result by the criterion of having the same attributes. Theorem 1.76 (Leibniz equivalence). For each system S ⊆ LU of L-sets in U , the L-relation θS in U defined by θS (u , v ) = A∈S (A(u ) ↔ A(v )) (1.80) is an L-equivalence in U . For each L-equivalence θ in U , there is some S ⊆ LU such that θ = θS . Moreover, θS is an L-equality iff for every distinct u and v there is some A ∈ S such that A(u ) = A(v ). Proof. First, we show that θS isan L-equivalence. Reflexivity: Since A(u ) ↔ A(u ) = 1, we have θS (u , u ) = A∈S (A(u ) ↔ A(u )) = 1. Symmetry follows from A(u ) ↔ A(v ) = A(v ) ↔ A(u ). Transitivity: = ≤ ≤ ≤

θS (u , v ) ⊗ θS (v , w ) = A∈S (A(u ) ↔ A(v )) ⊗ B∈S (B(v ) ↔ B(w )) ≤ ((A( u ) ↔ A( v )) ⊗ (B(v ) ↔ B(w ))) ≤ A∈S B∈S ((A(u ) ↔ A(v )) ⊗ (A(v ) ↔ A(w ))) ≤ A∈S A∈S (A(u ) ↔ A(w )) = θS (u , w ) .

Conversely, let θ be an L-equivalence in U . Put S = Πθ , i.e. S = {[u ]θ | u ∈ U }. Due to Theorem 1.75, it suffices to show that θS (u , v ) = [u ]θ (v ). On the one hand, θS (u , v ) = w ∈U ([w ]θ (u ) ↔ [w ]θ (v )) ≤ [u ]θ (u ) ↔ [u ]θ (v ) = [u ]θ (v ). On the other hand, [u ]θ (v ) ≤ θS (u , v ) iff for each w ∈ U we have [u ]θ (v ) ≤ [w ]θ (u ) ↔ [w ]θ (v ). Due to symmetry, we verify only [u ]θ (v ) ≤ [w ]θ (u ) → [w ]θ (v ) which is equivalent to [w ]θ (u ) ⊗ [u ]θ (v ) ≤ [w ]θ (v ) which follows from transitivity: [w ]θ (u ) ⊗ [u ]θ (v ) = θ(w , u ) ⊗ θ(u , v ) ≤ θ(w , v ) = [w ]θ (v ). Finally, let u and v be distinct (i.e. u = v ). Then θS (u , v ) = 1 iff there is some A ∈ S such that A(u ) ↔ A(v ) = 1, i.e. A(u ) = A(v ). This proves the last claim. Remark 1.77. (1) The set S from Theorem 1.76 can be considered a set of attributes. Each A ∈ S can be thought of as a (fuzzy) attribute, A(u ) being

42


a degree to which u ∈ U has attribute A. Then θS (u , v ) is a truth degree of “for each attribute A: u has A iff v has A” or, more succinctly, a truth degree of “u and v have the same attributes (of S)”. Having the same attributes has been considered by Leibniz [3] a criterion for being the same. Therefore, (1.80) can be considered a fuzzy counterpart of Leibniz definition. (2) Equality degree ≈ defined by (1.67) is an L-relation in LU . It is easy to see that ≈ is an L-equality. This fact can be proved directly but it also follows from Theorem 1.76. Namely, having a set V , put U = LV , S = {Av ∈ LU | v ∈ V, Av (B) = B(v) for B ∈ U }. Then by Theorem 1.76, θS is just ≈ defined by (1.67) and it is an L-equality in LV . ∗

∗

∗

Next, we recall a relationship between fuzzy equivalences and generalized pseudometrics. A generalized pseudometric on a non-empty set U is a mapping δ : U × U → [0, ∞] satisfying δ(u , u ) = 0 , δ(u , v ) = δ(v , u ) , δ(u , w ) ≤ δ(u , v ) + δ(v , w ) . If, in addition to the first condition, we require that δ(u , v ) = 0 implies u = v , we speak of a generalized metric. A metric (pseudometric) is a generalized metric (pseudometric) δ such that δ(u , v ) ∈ [0, ∞), i.e. δ(u , v ) < ∞, for each u , v ∈ U . Note that a generalized (pseudo)metric is sometimes called simply a (pseudo)metric. Theorem 1.78 (fuzzy equivalences vs. pseudometrics). Let ⊗ be a continuous Archimedean t-norm with an additive generator f , L be a residuated lattice on [0, 1] given by ⊗, ≈ be an L-equivalence on X, δ be a generalized pseudometric on U . Then (i) δ≈ : [0, 1]2 → [0, ∞] defined by δ≈ (u , v ) = f (u ≈ v )

(1.81)

is a generalized pseudometric which is a generalized metric iff ≈ is an L-equality; (ii) ≈δ : [0, 1]2 → [0, 1] defined by (u ≈δ v ) = f (−1) (δ(u , v ))

(1.82)

is an L-equivalence on U which is an L-equality iff δ is a generalized metric; (iii) ≈ equals ≈δ≈ and if δ(U, U ) ⊆ [0, f (0)] then δ equals δ≈δ . Proof. (i): δ≈ (u , u ) = 0 and δ≈ (u , v ) = δ≈ (v , u ) follow from (u ≈ u ) = 1, (u ≈ v ) = (v ≈ u ), and f (1) = 0. Triangle inequality for δ≈ can be obtained as follows: Transitivity of ≈ yields f (−1) (f (u ≈ v ) + f (v ≈ w )) ≤ (u ≈ w ).

1.3 Fuzzy Sets

43

Since f is decreasing, we have f (u ≈ w ) ≤ f (f (−1) (f (u ≈ v ) + f (v ≈ w ))). Now, there are two possibilities: either f (u ≈ v ) + f (v ≈ w ) > f (0) and then f (u ≈ w ) ≤ f (f (−1) (f (u ≈ v ) + f (v ≈ w ))) = f (0) < f (u ≈ v ) + f (v ≈ w ), or f (u ≈ v ) + f (v ≈ w ) ≤ f (0) and then f (u ≈ w ) ≤ f (f (−1) (f (u ≈ v ) + f (v ≈ w ))) = f (u ≈ v ) + f (v ≈ w ). In both of the cases we have f (u ≈ w ) ≤ f (u ≈ v ) + f (v ≈ w ) which means δ≈ (u , w ) ≤ δ≈ (u , v ) + δ≈ (v , w ), i.e. the required triangle inequality. That δ≈ is a metric metric iff ≈ is an L-equality follows easily from the fact that f is strictly decreasing. (ii): (u ≈δ u ) = 1 and (u ≈δ v ) = (v ≈δ u ) follow from δ(u , u ) = 0, δ(u , v ) = δ(v , u ), and f (−1) (0) = 1. Transitivity of ≈δ : Note first that f (−1) (u ) = f −1 (min(f (0), u )), f (u ≈δ v ) = min(f (0), δ(u , v )), and that f −1 is decreasing. Using triangle inequality for δ, we have (u ≈δ w ) = f (−1) (δ(u , w )) = f −1 (min(f (0), δ(u , w ))) ≥ ≥ f −1 (min(f (0), δ(u , v ) + δ(v , w ))) = = f −1 (min(f (0), f (u ≈δ v ) + f (v ≈δ w ))) = (u ≈ v ) ⊗ (v ≈ w ) , verifying transitivity of ≈δ . Since f (−1) is strictly decreasing, ≈δ is an Lequality iff δ is a generalized metric. (iii): (u ≈δ≈ v ) = f (−1) (δ≈ (u , v )) = f (−1) (f (u ≈ v )) = (u ≈ v ). If δ(u , v ) ≤ f (0) then δ≈δ (u , v ) = f (u ≈δ v ) = f (f (−1) (δ(u , v ))) = f (f −1 (min(f (0), δ(u , v )))) = f (f −1 (δ(u , v ))) = δ(u , v ). Sets with Fuzzy Equalities In the ordinary setting (of naive set theory), a set U is always considered to be available with the identity idU in U , though idU is mostly not explicitly mentioned. Intuitively, idU provides trivial information about which elements are distinct and which are not and this information is prior to the concept of a set. Nevertheless, identity is used in formulation of various properties. For instance, the condition of antisymmetry of a partial order ≤ in U says that for each u , v ∈ U , if u ≤ v and v ≤ u then u equals v , i.e. u , v ∈ idU . The concept of a fuzzy equality is an extension of the concept of ordinary identity. First, an L-equality ≈ in U carries information about the ordinary identity in U since, due to (1.75) and (1.78), we have u = v iff (u ≈ v ) = 1. Second, ≈ can be thought of as carrying information about similarity of elements of U since we may have (u ≈ v ) > 0 even if u = v . Note, however, that we can always have an ordinary identity in fuzzy setting as well since an L-relation ≈ in U defined by 1 for u = v , (u ≈ v ) = 0 for u = v , is an L-equality. That is, an L-equality in U can be thought of as providing information about a kind of underlying similarity of elements of U such that being similar in degree 1 means being equal. The underlying similarity needs

44


to be taken into account when considering additional functions and relations in U . Taking the underlying L-equality ≈ into account means that additional functions and (fuzzy) relations in U should be, in a sense, compatible with ≈. For functions, compatibility means that for any n-ary function f in U , if u1 is similar to v1 and · · · and un is similar to vn , then f (u1 , . . . , un ) is similar to f (v1 , . . . , vn ). In words, similar arguments are mapped to similar results. For (fuzzy) relations, compatibility means that for any n-ary (fuzzy) relation r in U , if u1 is similar to v1 and · · · and un is similar to vn , and if u1 , . . . , un are in r, then v1 , . . . , vn are in r. In words, similar arguments cannot be distinguished by being or being not in a relation. Therefore, in analogy to ordinary setting, where we start by a set U and consider functions and relations in U , in fuzzy setting, a natural situation is the following. We start by a set U equipped by a fuzzy equality relation ≈. Fuzzy equality ≈ imposes the above-described compatibility constraints on additionally considered functions and (fuzzy) relations in U . This leads to the following definitions. Definition 1.79. A set with L-equality (set with fuzzy equality ) is a pair U, ≈ where ≈ is an L-equality in U . Compatibility of functions and relations with an L-equality is captured by the following definition. Definition 1.80. An n-ary function f : U n → U in U is said to be compatible with an L-equality ≈ in U if for arbitrary u1 , v1 , . . . , un , vn ∈ U we have (u1 ≈ v1 ) ⊗ · · · ⊗ (un ≈ vn ) ≤ (f (u1 , . . . , un ) ≈ f (v1 , . . . , vn )) .

(1.83)

n

An n-ary L-relation r ∈ LU in U is said to be compatible with an L-equality ≈ in U if for arbitrary u1 , v1 , . . . , un , vn ∈ U we have (u1 ≈ v1 ) ⊗ · · · ⊗ (un ≈ vn ) ⊗ r(u1 , . . . , un ) ≤ r(v1 , . . . , vn ) .

(1.84)

Remark 1.81. (1) The above compatibility conditions make sense for arbitrary binary L-relation ≈ in U . Particularly, it is often used for ≈ being an Lequivalence relation. (2) Verbal descriptions of compatibility conditions are the well-known equality axioms (sometimes called congruence axioms or compatibility axioms). For functions, compatibility condition says “if u1 and v1 are in ≈ and · · · un and vn are in ≈ then f (u1 , . . . , un ) and f (v1 , . . . , vn ) are in ≈”. For fuzzy relations, compatibility condition says “if u1 and v1 are in ≈ and · · · un and vn are in ≈, and u1 , . . . , un are in r then v1 , . . . , vn are in r”. It is almost immediate that (1.83) and (1.84) are satisfied iff the first-order formulas expressing the equality axioms are true in degree 1. (3) Compatibility of fuzzy relations has useful equivalent formulations. Using adjointness, (1.83) is equivalent to (u1 ≈ v1 ) ⊗ · · · ⊗ (un ≈ vn ) ≤ r(u1 , . . . , un ) → r(v1 , . . . , vn ) ,

1.3 Fuzzy Sets

45

which is due to symmetry of ≈ equivalent to (u1 ≈ v1 ) ⊗ · · · ⊗ (un ≈ vn ) ≤ r(u1 , . . . , un ) ↔ r(v1 , . . . , vn ) . (4) If ≈ is a (characteristic function of) ordinary identity, both conditions (1.83) and (1.84) are satisfied for free. (5) For an arbitrary n-ary L-relation r in U , the least L-relation C≈ (r) containing r which is compatible with ≈ is given by C≈ (r)(u1 , . . . , un ) = v1 ,...,vn ∈U r(v1 , . . . , vn ) ⊗ (u1 ≈ v1 ) ⊗ · · · ⊗ (un ≈ vn ) . C≈ (r) is called the ≈-closure (extensional closure) of r. C≈ is a particular n closure operator in LU , see in [12, 59]. Lemma 1.82. An L-equivalence θ is compatible with an L-equality ≈ on U iff ≈ ⊆ θ. Proof. Suppose θ is compatible with ≈, i.e. θ(a , b ) ⊗ (a ≈ a ) ⊗ (b ≈ b ) ≤ θ(a , b ) ,

(1.85)

for all a , a , b , b ∈ M . 1.85 and reflexivity of θ and ≈M yield (a ≈ b ) = θ(a , a ) ⊗ (a ≈ a ) ⊗ (a ≈ b ) ≤ θ(a , b ). Conversely, if ≈ ⊆ θ, we get θ(a , b ) ⊗ (a ≈ a ) ⊗ (b ≈ b ) ≤ θ(a , b ) ⊗ θ(a , a ) ⊗ θ(b , b ) ≤ θ(a , b ), by transitivity and symmetry of θ. Given two sets with fuzzy equalities, U, ≈U and V, ≈V , it might be desirable to consider a mapping h of U to V . Such a mapping h should again be compatible with the underlying fuzzy equality. In terms of the following definition, h should be an ≈-morphism. Definition 1.83. For sets with L-equalities U, ≈U and V, ≈V , a mapping h : U → V is called an ≈-morphism if

u1 ≈U u2 ≤ h(u1 ) ≈V h(u2 )

(1.86)

for all u1 , u2 ∈ U . For a mapping h : U → V denote by θh a binary L-relation on U for which we have θh (u1 , u2 ) = h(u1 ) ≈V h(u2 ). θh is called a kernel of the ≈-morphism h. U V Remark 1.84. (1) If h : U → V is an ≈-morphism of U, ≈ to V, ≈ , we U V usually denote this fact explicitly by h : U, ≈ → V, ≈ . (2) Condition (1.86) says “if u1 and u2 are in ≈U then h(u1 ) and h(u2 ) are in ≈V ”. Since we interpret ≈U and ≈V as underlying similarities, (1.86) in fact says “if u1 and u2 are similar then h(u1 ) and h(u2 ) are similar”, i.e. an ≈-morphism is required to map similar elements to similar ones. (3) If ≈U is a (characteristic function of) ordinary identity then (1.86) is satisfied for free. The following assertions summarize basic properties of ≈-morphisms.

46


Theorem 1.85. Let h : U → V be a mapping between U, ≈U and V, ≈V . Then h is an ≈-morphism if and only if ≈U ⊆ θh . Proof. Follows directly from Definition 1.83. Theorem 1.86. Let us have ≈-morphisms g : U, ≈U → U , ≈U and h : U , ≈U → U , ≈U . Then

(i) the composed mapping (g ◦ h) : U → U is an ≈-morphism. (ii) If g is a bijection, then g −1 is an ≈-morphism iff u1 ≈U u2 = g(u1 ) ≈U g(u2 ) for all u1 , u2 ∈ U . Proof. (i): The mapping g ◦ h, i.e. (g ◦ h)(u ) = h(g(u )) for every u ∈ U , is evidently an ≈-morphism. (ii): “⇒”: Suppose g is a bijection and g −1 is an ≈-morphism. Then u1 ≈U u2 ≤ g(u1 ) ≈U g(u2 ) ≤ g −1 (g(u1 )) ≈U g −1 (g(u2 )) = u1 ≈U u2 , which implies the requested condition u1 ≈U u2 = g(u1 ) ≈U g(u2 ). U “⇐”: If g is a bijection satisfying u1 ≈ u2 = g(u1 ) ≈U g(u2 ) for all −1 u1 , u2 ∈ U , then g is evidently an ≈-morphism. Theorem 1.87. Let h : U, ≈U → V, ≈V be an ≈-morphism. Then (i) θh is an L-equivalence compatible with ≈U ; (ii) θh is an L-equality iff h is injective. Proof. (i): Since θh (u1 , u2 ) = h(u1 ) ≈V h(u2 ) and since ≈V is an Lequivalence, θh is an L-equivalence. By Theorem 1.85, ≈U ⊆ θh , and by Lemma 1.82, θh is compatible with ≈U . (ii): “⇒”: For an L-equality θh and h(u1 ) = h(u2 ) we have θh (u1 , u2 ) = 1 which implies u1 = u2 . Thus, h is injective. “⇐”: Let h be an injective ≈-morphism. For θh (u1 , u2 ) = 1 we have h(u1 ) ≈V h(u2 ) = 1, and so h(u1 ) = h(u2 ), that is u1 = u2 , i.e. θh is an L-equality. Let us note that if u1 ≈U u2 = h(u1 ) ≈V h(u2 ) for all u1 , u2 ∈ U , then θh = ≈U , i.e. θh is an L-equality. By applying Theorem 1.87 (ii), we obtain the following corollary. Corollary 1.88. If for an ≈-morphism h : U, ≈U → V, ≈V we have u1 ≈U u2 = h(u1 ) ≈V h(u2 ) for every u1 , u2 ∈ U , then h is injective. Factor Sets by Fuzzy Equivalences One of the basic constructions in the ordinary setting is that of formation of a factor set U/θ of a set U by an equivalence relation θ in U , see Sect. 1.1. Theorem 1.75 gives a way to proceed analogously in fuzzy setting. That is, given an L-equivalence θ in U , one may define a factor set U/θ of U by θ by U/θ = Πθ , i.e. U/θ = {[u ]θ | u ∈ U }. In the light of the previous section,

1.3 Fuzzy Sets

47

however, one might ask what happens if we start by a set with fuzzy equality rather than just by a set, i.e. by U, ≈ rather than just by U . First, note that if starting by U, ≈ , i.e. a set U equipped with an Lequality ≈, we require the L-equivalence θ to be compatible with ≈, i.e. (u1 ≈ v1 ) ⊗ (u2 ≈ v2 ) ⊗ θ(u1 , u2 ) ≤ θ(v1 , v2 ) , cf. also Lemma 1.82. Second, note that Theorem 1.75 can be carried over to the more general setting when we start by U, ≈ . In more detail, one can proceed as follows. Modify the definition of an L-partition (Definition 1.73) to that of an Lpartition compatible with ≈ by requiring that each A ∈ Π be compatible with ≈. Then the corresponding modification of Theorem 1.75 remains true (now, it concerns a bijective correspondence between L-equivalences θ in U compatible with an L-equivalence ≈, and L-partitions Π of U compatible with ≈). To see this, it is sufficient to check that starting from an L-equivalence θ in U compatible with ≈, each [u ]θ is compatible with ≈, and that starting from an L-partition Π of U compatible with ≈, the induced L-equivalence θΠ is compatible with ≈. Both claims are true. Indeed, since [u ]θ (v ) ⊗ (v ≈ w ) = θ(u , v ) ⊗ (v ≈ w ) ≤ θ(u , w ) = [u ]θ (w ) , each [u ]θ is compatible w.r.t. ≈. Furthermore, compatibility of θΠ with ≈ means (u1 ≈ v1 ) ⊗ (u2 ≈ v2 ) ⊗ θΠ (u1 , u2 ) ≤ θΠ (v1 , v2 ) , i.e. (u1 ≈ v1 ) ⊗ (u2 ≈ v2 ) ⊗ Au1 (u2 ) ≤ Av1 (v2 ) , where Au1 , Au2 ∈ Π are (uniquely determined) elements of Π such that Au1 (u1 ) = 1 and Av1 (v1 ) = 1. Now, the latter inequality is true since (u1 ≈ v1 ) ⊗ (u2 ≈ v2 ) ⊗ Au1 (u2 ) ≤ ≤ (u1 ≈ v1 ) ⊗ Au1 (v2 ) = = Av1 (v1 ) ⊗ Au1 (u1 ) ⊗ (u1 ≈ v1 ) ⊗ Au1 (v2 ) ≤ ≤ Av1 (v1 ) ⊗ Au1 (v1 ) ⊗ Au1 (v2 ) ≤ Av1 (v2 ) , the last inequality being true due to condition (iii) of the definition of Lpartition. Now, since we work in the framework of sets with fuzzy equalities, a factor set U, ≈ /θ of U, ≈ by an L-equivalence θ in U compatible with ≈ should be again a set with fuzzy equality. A natural choice is to define U, ≈ /θ = U/θ, ≈U/θ where U/θ = Πθ and ≈U/θ is an L-equality in U/θ defined by ([u ]θ ≈U/θ [v ]θ ) = θ(u , v ) . Remark 1.89. In order to check that the above definition of U, ≈ /θ is sound, we need to verify that ≈U/θ is correctly defined and that it is indeed an L-equality. To see that the definition of ≈U/θ is correct, observe first the

48


following Claim: For any u , u ∈ U , [u ]θ = [u ]θ iff θ(u , u ) = 1. Indeed, if [u ]θ = [u ]θ then θ(u , u ) = [u ]θ (u ) = [u ]θ (u ) = 1; if θ(u , u ) = 1 then by symmetry and transitivity of θ, for any v ∈ U we have [u ]θ (v ) = θ(u , v ) = θ(u , u ) ⊗ θ(u , v ) ≤ θ(u , v ) = [u ]θ (v ) and similarly, [u ]θ (v ) ≤ [u ]θ (v ), whence [u ]θ = [u ]θ . Now, if [u ]θ = [u ]θ and [v ]θ = [v ]θ , we have ([u ]θ ≈U/θ [v ]θ ) = θ(u , v ) = [u ]θ (v ) = [u ]θ (v ) = θ(u , v ) = θ(v , u ) = [v ]θ (u ) = [v ]θ (u ) = θ(v , u ) = θ(u , v ) which means that ([u ]θ ≈U/θ [v ]θ ) does not depend on the particular u and v and so ≈U/θ is defined correctly. Now, the fact that θ is reflexive, symmetric, and transitive immediately implies the corresponding properties of ≈U/θ . By definition, ([u ]θ ≈U/θ [v ]θ ) = 1 means θ(u , v ) = 1, from which we get [u ]θ = [v ]θ by Claim. Therefore, ≈U/θ is an L-equality in U/θ. By the above generalization of Theorem 1.75, U/θ is a set of L-sets in U which are compatible with ≈. Moreover, θ can be reconstructed from U/θ (namely, U/θ = Πθ and θ = θΠθ ). U/θ, ≈U/θ is thus a good candidate for being a set with L-equality which results by factorization of U, ≈ by θ. But now, notice that such a definition of a factor set of U, ≈ by θ can be simplified. The point is that using U/θ, ≈U/θ as a factor set with fuzzy equality, a complete information about θ is present in both U/θ and ≈U/θ . Consider, instead of U/θ, ≈U/θ , an alternate definition of a factor set with fuzzy equality. By Theorem 1.72 (ii), 1 θ is an ordinary equivalence relation. 1 1 One may thus consider a pair U/1 θ, ≈U/ θ where ≈U/ θ is an L-relation in the ordinary factor set U/1 θ defined by ([u ]1 θ ≈U/

1

θ

[v ]1 θ ) = θ(u , v ) .

(1.87)

By Claim of Remark 1.89, [u ]θ = [u ]θ iff θ(u , u ) = 1 iff u , u ∈ 1 θ iff [u ]1 θ = [u ]1 θ . Therefore, a mapping sending [u ]θ to [u ]1 θ is a bijection between U/θ and U/1 θ. As a direct consequence of the above definitions and considerations we have the following lemma. Lemma 1.90. Let U, ≈ be a set with L-equality, θ be an L-equivalence in U compatible with ≈. Then (i) U/θ, ≈U/θ is a set with L-equality, 1 1 (ii) U/ θ, ≈U/ θ is a set with L-equality, (iii) a mapping h : U/θ → U/1 θ defined by h([u ]θ ) = [u ]1 θ is a bijective ≈ 1 morphism between U/θ, ≈U/θ and U/1 θ, ≈U/ θ for which 1 ([u ]θ ≈U/θ [v ]θ ) = ([u ]1 θ ≈U/ θ [v ]1 θ ). Lemma 1.90 thus says that we have two almost equivalent ways to define a factor set with fuzzy equality of U, ≈ by θ. Since by Claim of Remark 1.89, 1 [u ]1 θ = 1 [u ]θ , i.e. [u ]1 θ is a subset of [u ]θ . That is, U/1 θ, ≈U/ θ is more simple a concept than U/θ, ≈U/θ . Therefore the following definition.

1.3 Fuzzy Sets

49

Definition 1.91. For a set U, ≈ with L-equality and an L-equivalence θ in 1 U which is compatible with ≈, U/1 θ, ≈U/ θ is called a factor set with fuzzy equality of U, ≈ by θ. 1 Remark 1.92. To simplify notation, a factor set U/1 θ, ≈U/ θ with fuzzy equality will be denoted just by U/θ, ≈U/θ and classes [u ]1 θ just by [u ]θ . This means that, for the sake of simplicity, we sacrifice uniqueness of notation. Namely, U/θ, ≈U/θ introduced above is now redefined to denote 1 1 U/ θ, ≈U/ θ . But since the above-introduced meaning of U/θ, ≈U/θ will not be used any more, there is no danger of confusion involved here. Further Concepts from Fuzzy Sets Here we recall a concept of a particular closure operator in LU . For a truth stresser ∗ , an L∗ -closure operator (fuzzy closure operator with truth stresser) on U [14] is a mapping C : LU → LU satisfying A ⊆ C(A) , ∗ S(A, B) ≤ S(C(A), C(B)) , C(A) = C(C(A)) ,

(1.88) (1.89) (1.90)

for every L-sets A, B ∈ LU . Remark 1.93. (1) Conditions (1.88) and (1.90) are the usual conditions of extensivity and idempotency. Condition (1.89) is a particular form of graded monotony. It says “if is is very true that A is a subset of B then C(A) is a subset of C(B)” where the interpretation of “very true” is given by ∗ . (2) It is easy to see that if ∗ is globalization, (1.89) says that A ⊆ B implies C(A) ⊆ C(B). This is a commonly required condition of fuzzy closure operators, see e.g. [42]. If ∗ is identity, (1.89) says S(A, B) ≤ S(C(A), C(B)). This form of monotony is considered e.g. in [7]. This is a stronger form of monotony than the one for ∗ being globalization. The corresponding L∗ -closure operators are called L-closure operators. (3) If L = 2, the only truth stresser is identity (which coincides with globalization in this case). In this case, the concept of an L∗ -closure operator coincides with the concept of an ordinary closure operator on U , see Sect. 1.1. We will make use of the following characterization of L∗ -closure operators. Lemma 1.94. A mapping C : LU → LU is an L∗ -closure operator if it satisfies (1.88) and ∗ (1.91) S(A1 , C(A2 )) ≤ S(C(A1 ), C(A2 )) . ∗

Proof. Assume (1.88)–(1.90). We get S(A1 , C(A2 )) ≤ S(C(A1 ), C(C(A2 ))) = S(C(A1 ), C(A2 )), verifying (1.91). Conversely, assume (1.88) and (1.91). By (1.88), A2 ⊆ C(A2 ), whence S(A1 , A2 ) ≤ S(A1 , C(A2 )). Furthermore,

50

1 Introduction to Fuzzy Sets and Fuzzy Logic ∗

∗

S(A1 , A2 ) ≤ S(A1 , C(A2 )) ≤ S(C(A1 ), C(A2 )) , ∗

proving (1.89). By (1.91), 1 = S(C(A), C(A)) ≤ S(C(C(A)), C(A)), i.e. we have C(C(A)) ⊆ C(A). Since the converse inclusion holds by (1.88), we conclude (1.90).

1.4 Pavelka-Style Fuzzy Logic In this section we recall an approach developed in Pavelka’s seminal [74]. Particularly, we introduce what Pavelka calls an abstract fuzzy logic. Our main goal is to show what has become known as Pavelka-style fuzzy logic. The basic setting in logic can be seen as follows. We work with formulas, i.e. particular sequences of symbols of a particular language. From an abstract point of view, we might consider formulas just as elements of some (abstract) set Fml of all formulas. Furthermore, we want to evaluate formulas, i.e. to assign truth degrees to formulas. In a general setting, a set of all truth degrees may be supposed to form a complete lattice L (i.e., not necessarily equipped with ⊗ and →) with support L. A truth degree from L may be interpreted as a degree of truth in the sense of fuzzy logic, as a probability of a given formula (event), but any other meaningful interpretation is possible. The process of assignment of truth degrees to formulas has two “inputs” and one “output”. The first input is a formula ϕ ∈ Fml . The second input is a semantic component S in which formulas can be evaluated. The output is a truth degree ||ϕ||S ∈ L of ϕ in S. This way, a mapping E : Fml → L, i.e. an L-set E in Fml , is induced by E(ϕ) = ||ϕ||S . From an abstract point of view, we may forget about the semantic components S and see the induced mappings E as primitive concepts which describe the semantics of our logic. This way we come to the following definition. Definition 1.95. An L-semantics for a set Fml of formulas is a set S of L-sets in Fml , i.e. S ⊆ LFml . Elements E ∈ S are called evaluations. For ϕ ∈ Fml and E ∈ S, E(ϕ) is called a truth degree of ϕ in E. To retain the intended meaning of E (E as a semantic component of evaluation), we use also ||ϕ||E instead of E(ϕ). Example 1.96. (1) Propositional logic. Let Fml be a set of formulas of classical propositional logic. That is, Fml contains propositional variables p, q, . . . , and if ϕ, ψ ∈ Fml then nϕ ∈ Fml and (ϕ i ψ) ∈ Fml (n and i are symbols of negation and implication). Let L be a two-element chain, (i.e. L = {0, 1}, 0 ≤ 1). Each mapping e (called elementary evaluation) assigning truth degrees e(p), e(q), . . . from L to propositional variables p, q, . . . induces a mapping || · · · ||e assigning truth degrees ||ϕ||e from L to formulas ϕ ∈ Fml as follows: For a propositional variable p, put ||p||e = e(p); for formulas ϕ, ψ ∈ Fml , put ||nϕ||e = ¬||ϕ||e and ||ϕ i ψ||e = ||ϕ||e → ||ψ||e where ¬ and → are truth functions of classical negation and implication, respectively. Then

1.4 Pavelka-Style Fuzzy Logic

51

S = {E ∈ LFml | for some e : E(ϕ) = ||ϕ||e for each ϕ ∈ Fml } is an L-semantics. This shows how the usual semantics of ordinary propositional logic can be seen in the framework of Pavelka’s abstract approach. (2) Predicate fuzzy logic with truth constants in language. Let L be a complete residuated lattice and Fml be a set of all formulas of predicate fuzzy logic with (some) truth constants in a language. That is, we start with a language consisting of a non-empty set R of relation symbols (each r ∈ R with its arity σ(r) ∈ N0 ), a set F of function symbols (each f ∈ F with its arity σ(f ) ∈ N0 ), (object) variables x, y, . . . , symbols o, i, c, d of connectives, symbols ∀, ∃ of quantifiers, symbols a of truth constants for each a ∈ K for some K ⊆ L, and possibly auxiliary symbols. Then, in a usual way, we define terms (each variable x is a term; if t1 , . . . , tn are terms and f ∈ F is n-ary then f (t1 , . . . , tn ) is a term) and formulas (if a ∈ K then a is a formula; if t1 , . . . , tn are terms and r ∈ R is n-ary then r(t1 , . . . , tn ) is an (atomic) formula; if ϕ, ψ are formulas and x is a variable then (ϕ o ψ), (ϕ i ψ), (ϕ c ψ), (ϕ d ψ), (∀x)ϕ, (∃x)ϕ are formulas). Denote Fml the set of all such formulas. Then, we define a structure of a given language as a triplet M = M, RM , F M where M = ∅ is a universe, σ(r) RM = {rM ∈ LM | r ∈ R} is a set of L-relations corresponding to relation symbols, and F M = {f M : M σ(f ) → M | f ∈ F } is a set of (ordinary) functions corresponding to function symbols. Given a structure M and a valuation v assigning elements v(x), v(y), · · · ∈ M to variables x, y, . . . , we define evaluation of terms and evaluation of formulas. A value ||t||M,v ∈ M of term t is defined as follows: for variable x, put ||x||M,v = v(x); for a term f (t1 , . . . , tn ), put ||f (t1 , . . . , tn )||M,v = f M (||t1 ||M,v , . . . , ||tn ||M,v ). A value (truth degree) ||ϕ||M,v ∈ L of formula ϕ is defined as follows: ||a||M,v = a for each a ∈ K; for formula r(t1 , . . . , tn ), put ||r(t1 , . . . , tn )||M,v = rM (||t1 ||M,v , . . . , ||tn ||M,v ); for formulas ϕ, ψ, and variable x, put ||ϕ o ψ||M,v = ||ϕ||M,v ⊗ ||ψ||M,v ; ||ϕ i ψ||M,v = ||ϕ||M,v → ||ψ||M,v ; ||ϕ c ψ||M,v= ||ϕ||M,v ∧ ||ψ||M,v ; ||ϕ d ψ||M,v = ||ϕ||M,v ∨ ||ψ||M,v ; ||(∀x)ϕ||M,v = w≡x v ||ϕ||M,w with infimum takenover all valuations w which differ from v at most in x; taken over all valuations ||(∃x)ϕ||M,v = w≡x v ||ϕ||M,w with supremum w which differ from v at most in x. Then S = {E ∈ LFml | for some M, v : E(ϕ) = ||ϕ||M,v for each ϕ ∈ Fml } is an L-semantics. Another L-semantics can be obtained as follows. Define a truth degree of a formula ϕ in a structure M by ||ϕ||M = v:X→M ||ϕ||M,v where the infimum ranges over all valuations v in M . Then, S = {E ∈ LFml | for some M : E(ϕ) = ||ϕ||M for each ϕ ∈ Fml }

(1.92)

is an L-semantics. This shows how semantics of predicate fuzzy logic can be seen in the framework of Pavelka’s abstract approach. (3) Probability in finite sets. Let Ω = ∅ be a finite set, ω ∈ Ω be called elementary events. Let L be the real unit interval [0, 1] with its natural order

52


Ω ≤, let Fml be the set of all events over Ω, i.e. Fml = 2 . A probability distribution on Ω is a function p : Ω → [0, 1] satisfying ω∈Ω p(ω) = 1. on Ω by putting Probability distribution induces a probability measure P p Pp (A) = ω∈A p(ω) for each event A ∈ 2Ω . Then

S = {E ∈ LFml | for some p : E(ϕ) = Pp (ϕ)} is an L-semantics. The concept of an L-semantics is rather simple, namely, an L-semantics is but an L-set in Fml . Nevertheless, it serves as a good starting point to further considerations interesting from the logical point of view. In classical logic, a theory is a set T of formulas, i.e. T ⊆ Fml . Theories are of crucial importance in logic. We might have a set Th(M) of all formulas which are true in some structure M as a theory. Or, a theory T can be understood as representing somebody’s knowledge about some domain. For instance, a theory T = {husband (george), . . . , husband (x)imale(x), . . . } contains formulas which assert that (an individual referred to as) george is somebody’s husband and that if somebody is a husband then he is a male. From the point of view of fuzzy approach, however, it is quite appealing to consider a theory as an L-set of formulas. For instance, theory T = {0.8/young(john), 0.9/baby(joe), . . . , 1/baby(x) i young(x)} asserts that john is young in degree (at least) 0.8, that joe is young in degree (at least) 0.9, that the fact that somebody is a baby implies that he/she is young in degree 1, etc. This leads to the following definition. Definition 1.97. A theory over Fml and L (L-theory over Fml ) is any L-set T ∈ LFml . Given an L-semantics S for Fml , we say that E ∈ S is a model of a theory T over Fml and L if T ⊆ E. That is, E is a model of T is for each formula ϕ ∈ Fml we have T (ϕ) ≤ E(ϕ). This says that for each formula ϕ, the truth degree E(ϕ) of ϕ in E is at least as high as the truth degree prescribed to ϕ by theory T . For instance, for the L-semantics of predicate fuzzy logic from Example 1.96 (2), an evaluation E which corresponds to a structure M and a valuation v is a model of a theory T if for each formula ϕ, the truth degree ||ϕ||M,v of ϕ in M under v is greater than or equal to T (ϕ). It is easily seen that this is a natural generalization of the classical concept of a model (just consider L = 2). In Example 1.96 (3), let Ω = {1, . . . , 6} be a set of elementary events which corresponds to tossing a die. If the die is fair, we have p(1) = · · · = p(6) = 16 for the corresponding probability distribution p. However, for a biased die we might have p(1) = 1 3 , p(6) = 12 (6 is more likely to appear than 5). · · · = p(4) = 16 , p(5) = 12 1 1 /{1}, . . . , 12 /{6}} which can be seen Then we might consider a theory T = { 12 as requiring for each possible result i ∈ {1, . . . , 6}, the probability of i is at 1 , i.e. the die cannot be biased too much. For instance, an evaluation E least 12 corresponding to the above distribution p of a biased die, is a model of T . For


53

a fake die containing only even numbers, the corresponding E is not a model of T since for i ∈ {1, 2, 3} we have T ({i}) ≤ 0 = E({i}). Now, we can define the degree of semantic entailment in our abstract setting. Definition 1.98. For an L-semantics S for Fml , a theory T , and a formula ϕ ∈ Fml , the degree ||ϕ||S T to which ϕ semantically follows from T (denoted simply by ||ϕ||T ) is defined by ||ϕ||ST = {||ϕ||E | E ∈ S is a model of T } . Recall that the completeness theorem (for classical logic) says that (for suitable proof systems), formula ϕ follows semantically from a theory T iff ϕ is provable from T . Therefore, an interesting problem arises of whether the degree ||ϕ||T can be obtained by syntactic means, i.e. whether ||ϕ||T is equal or at least can be approximated by a suitably defined “degree of provability” of ϕ from T . However strange the concept of degree of provability may sound at the first encounter, it turns out that it is quite natural. A way to go, showed by Pavelka [74] and inspired by Goguen [45], is the following. We need to prove formulas from L-sets of formulas. The first step is to modify the concept of a deduction rule. An ordinary deduction rule takes formulas as inputs and yields a formula as its output. A deduction rule in our setting takes formulas with their truth-weights from L as inputs and yields a formula with a truth-weight as its output. Then, a truth-weighted proof is a finite sequence ϕ1 , a1 , . . . , ϕn , an of formulas ϕi with their truth-weights ai ∈ L which results from a theory T (L-set of formulas) by a repeated application of deduction rules. The truth-weight an of the last formula ϕn in a proof is considered as a degree of the proof for ϕn . But different proofs can yield different degrees, i.e. we might have a proof ϕ1 , b1 , . . . , ϕm , bm with ϕn = ϕm but an = bm . Therefore, the degree |ϕ|T of provability of ϕ from T is defined as the supremum of degrees of proofs of ϕ over all possible proofs, i.e. |ϕ|T = {a | there exists a proof . . . , ϕ, a from T }. The details are captured by the following definitions. Definition 1.99. An n-ary deduction rule for Fml and L is a pair R = Rsyn , Rsem consisting of a partial function Rsyn : Fml n → Fml (syntatic part) and a function Rsem : Ln → L (semantic part). Analogously as in the ordinary case, a deduction rule R may be visualized by ϕ1 , a1 , . . . , ϕn , an . Rsyn (ϕ1 , . . . , ϕn ), Rsem (a1 , . . . , an ) A rule R enables us to infer that formula Rsyn (ϕ1 , . . . , ϕn ) is valid in degree (at least) Rsem (a1 , . . . , an ) if we know that each ϕi is valid in degree at least ai (i = 1, . . . , n).

54


Example 1.100. (1) Modus ponens for predicate fuzzy logic and for a complete residuated lattice L, see Example 1.96 (2), is a deduction rule ϕ, a , ϕ i ψ, b . ψ, a ⊗ b Returning to the above example, modus ponens enables us to infer young(joe), 0.9 from baby(joe), 0.9 and baby(joe) i young(joe), 1 . (2) Generalization for predicate fuzzy logic is a deduction rule ϕ, a . (∀x)ϕ, a Note that a truth-weight a of the inferred formula is the same as of the input formula. (3) As an example of a deduction rule which is degenerate in the ordinary case, consider ϕ, a , ϕ, b . ϕ, a ∨ b Here, the inferred formula is the same as both the input formulas but we infer the least upper bound a ∨ b of the truth-weights of the input formulas. In addition to deduction rules, a (Hilbert-style) logical system contains a collection of (logical) axioms. In our setting, we may consider an arbitrary L-set A of formulas from Fml , i.e. a theory over Fml and L, as a collection of logical axioms. Definition 1.101. Let R be a set of deduction rules for Fml and L, A (logical axioms) and T (non-logical axioms) be theories over Fml and L, let ϕ ∈ Fml and a ∈ L. An (L-weighted ) proof of ϕ, a from T using A and R is a sequence ϕ1 , a1 , . . . , ϕn , an such that ϕn is ϕ, an = a, and for each i = 1, . . . , n, we have ai = T (ϕi ), or ai = A(ϕi ), or ϕi , ai follows from some ϕj , aj ’s, j < i, by some deduction rule R ∈ R. The number n is called a length of the proof. In such a case, we write T A,R ϕ, a and call ϕ, a provable from Σ using A and R. If T A,R ϕ, a , ϕ is called provable in degree (at least) a from T using A and R. A degree of provability of ϕ , is defined by from T using A and R, denoted by |ϕ|A,R T |ϕ|A,R = {a | T A,R ϕ, a } . T Remark 1.102. (1) The fact that ϕi , ai follows from some ϕj , aj ’s, j < i, by a k-ary deduction rule R ∈ R means that ϕi = Rsyn (ϕi1 , . . . , ϕik ) and ai = Rsem (ai1 , . . . , aik ) for some i1 , . . . , ik ≤ i. (2) We say also “A, R-provable” instead of “provable using A and R”. If A and R are obvious from the context, we omit them, e.g. we write T ϕ, a , etc. instead of T A,R ϕ, a , |ϕ|T instead of |ϕ|A,R T


55

It is usually the case that formulas from Fml are particular strings of symbols of a so-called language. Then, it is sometimes useful to have symbols a of truth degrees a ∈ L as particular symbols of the language. One may then consider the concept of an L-weighted formula. Definition 1.103. An L-weighted formula is a pair ϕ, a where ϕ is a formula and a is a symbol of a truth degree a ∈ L. Remark 1.104. (1) If formulas are strings of symbols of a language, then Lweighted formulas are as well. Therefore, the concept of an L-weighted formula is a concept of syntax. If there is no danger of misunderstanding, we usually do not distinguish between a and a. Therefore, we also write ϕ, a instead of ϕ, a . (2) If L is obvious from the context, we also say a truth-weighted formula or just weighted formula. Terminology is not consistent here. For instance, [71] uses adjective “evaluated” while [49] prefers “weighted”. We use “weighted” since “evaluated” suggest a connection to valuations E and truth degrees ||ϕ||E which is not proper here. (3) There is a correspondence between L-sets (theories) Σ ∈ LFml of formulas and (particular) sets of L-weighted formulas. Namely, for Σ ∈ LFml , one may consider a set TΣ = {ϕ, a | ϕ ∈ Fml , a = Σ(ϕ)} of L-weighted formulas. For a set T of L-weighted formulas such that for each ϕ there is at most one a with ϕ, a ∈ T one may consider an L-set ΣT defined by ΣT (ϕ) = a if ϕ, a ∈ T and ΣT (ϕ) = 0 if there is no a ∈ L with ϕ, a ∈ T . (4) The main reason we introduce the concept of an L-weighted formula is that we might want to have some concepts in a usual fashion from the point of view of syntax and semantics. Namely, a proof is usually defined as a certain sequence of formulas (well-formed strings of symbols) of a language of the corresponding logic. Without the concept of an L-weighted formula, an L-weighted proof is a sequence of pairs ϕ, a where ϕ is a formula and a ∈ L is a truth degree. Truth degrees come from the metalevel (of a corresponding logic) and are not a part of language. That is, without the concept of an Lweighted formula, the concept of an L-weighted proof is not a sequence of wellformed strings of a language. Having the concept of an L-weighted formula as a concept of syntax, L-weighted proofs are sequences of well-formed strings of symbols of language, as usual. In a similar manner, ordinary proofs go from sets of formulas while in the setting of abstract logic, proofs go from L-sets of formulas. However, as we have seen in (3), an L-set of formulas can be seen as an ordinary set of L-weighted formulas. Thus, again, the concept of an L-weighted formula enables us to have a concept of theory in a fashion usual in logic. (5) Note that we could define a truth degree ||ϕ, a ||E of L-weighted formula ϕ, a by ||ϕ, a ||E = a → ||ϕ||E . This way, we have ||ϕ, a ||E = 1 iff a ≤ ||ϕ||E . Furthermore, ||ϕ, 1 ||E = ||ϕ||E , and so, formulas correspond to L-weighted formulas with weight a = 1.

56


A degree |ϕ|T of provability is thus a least upper bound of all a ∈ L such that there exists a proof of ϕ, a from T . However, it may happen that |ϕ|T is not attained by any proof, i.e. there does not exist any proof of ϕ, a for a = |ϕ|T . Therefore, if we find a proof of ϕ, a from T , we have just a ≤ |ϕ|T , i.e. we have only a lower approximation a of the degree |ϕ|T of provability. Note that this is different from the classical notion of a proof (!) since in classical setting, all proofs of a formula are equally important. That is, any proof does the job. Then, we can define the following. Definition 1.105. An abstract logic is a tuple L = Fml , L, S, A, R where Fml is a set of formulas, L is a complete lattice, S is an L-semantics for Fml , A is a theory (of logical axioms) over Fml and L, and R is a set of deduction rules. L is (Pavelka-)sound if for each theory T over Fml and L and each ≤ ||ϕ||ST . L is (Pavelka-)complete if for formula ϕ ∈ Fml we have |ϕ|A,R T = each theory T over Fml and L and each formula ϕ ∈ Fml we have |ϕ|A,R T ||ϕ||ST . Remark 1.106. One can see that the ordinary notions of soundness and completeness are particular cases of those from Definition 1.105 for L being a two-element chain. Another issue of a standard logical agenda which can be introduced in terms of our abstract setting is a pair of mappings Md and Th. Introduce mappings Md : LFml → 2S and Th : 2S → LFml as follows. For an L-set T of formulas Fml define a set Md(T ) of evaluations from S by Md(T ) = {E ∈ S | E is a model of T } .

(1.93)

For a set K of evaluations from S, define an L-set Th(K) of formulas by (Th(K))(ϕ) = {E(ϕ) | E ∈ K} . (1.94) That is, Md(T ) is a set of all models of T while Th(K) is a theory of K. It is Fml easy , ⊆ and S to see that Md and Th form a Galois connection between L 2 , ⊆ . That is, we have T1 ⊆ T2 implies Md(T2 ) ⊆ Md(T1 ) , K1 ⊆ K2 implies Th(K2 ) ⊆ Th(K1 ) ,

(1.95) (1.96)

K ⊆ Md(Th(K)) , T ⊆ Th(Md(T )) ,

(1.97) (1.98)

for any K, K1 , K2 ⊆ S and any T, T1 , T2 ∈ LFml . As a consequence, we also have Md(T ) = Md(Th(Md(T ))) , Th(K) = Th(Md(Th(K))) .

(1.99) (1.100)

1.5 Bibliographical Remarks

57

Remark 1.107. Note that using Md and Th, a degree ||ϕ||ST to which ϕ semantically follows from T can be expressed by ||ϕ||ST = (Th(Md(T )))(ϕ) .

1.5 Bibliographical Remarks Sets and Structures More information related to sets, mappings, relations, and structures can be found in many textbooks, e.g. in [25, 75, 88]. Structures of Truth Degrees Fuzzy sets and fuzzy logic were introduced by Lotfi A. Zadeh [100]. The set of truth degrees proposed by Zadeh is the real unit interval [0, 1] equipped with min as conjunction. The idea to use complete residuated lattices as structures of truth degrees is due to Goguen [44, 45]. Particularly, our justification of adjointness property is basically due to Goguen (cf. also beginning of Chap. 2 in [49]). Residuated lattices have been introduced by Ward and Dilworth [94]. For the point of view of fuzzy logic, residuated lattices have been thorouhly investigated by Höhle, see e.g. [55]. Several useful results concerning residuated lattices can be found e.g. in [10, 47, 49, 71]. The notion of a t-norm goes back to the study of probabilistic metric spaces an particularly to Schweizer and Sklar, see [78] and [79]. A good book on t-norms is [60]. For representation of continuous t-norms by ordinal sums see [67, 63], for definition of ordinal sums of residuated lattices see [48]. The concept of a truth stresser is due to H´ ajek [51], see also [4, 83] for related papers. Fuzzy Sets The first paper on fuzzy sets is Zadeh’s [100]. Fuzzy sets and their applications are well covered e.g. in [35, 46, 47, 61]. The set of truth degrees is usually [0, 1]. The concept of an L-set is due to Goguen [44] (Goguen uses the term L-fuzzy set). Fuzzy sets and fuzzy relations with truth degrees in complete residuated lattices were investigated by Höhle, see e.g. a survey paper [56]. The notion of a fuzzy equivalence is due to Zadeh [99]. Fuzzy equivalences and fuzzy equalities with truth degeees in a complete residuated lattice were studied by H¨ ohle [54, 56, 57]. There are many papers on various aspects of fuzzy equivalence relations, see e.g. [21, 30, 36, 37, 49, 58, 59, 72, 85, 87].

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Pavelka-style Fuzzy Logic The concept of an abstrat fuzzy logic is due to Pavelka [74] (it was inspired by Goguen’s [44, 45]). A thorough investigation of abstract fuzzy logic is provided by [42]. Several particular logical calculi fitting the framework of abstract fuzzy logic (most noticeably: obeying Pavelka-completeness) can be found e.g. in [42, 49, 71, 74].

2 Algebras with Fuzzy Equalities

The present chapter studies algebras with fuzzy equalities. Briefly speaking, an algebra with fuzzy equality is a set equipped with a fuzzy equality and with functions which are compatible with this fuzzy equality. If we interpret fuzzy equality as similarity, compatibility of functions with fuzzy equality says that the functions map similar (tuples of) elements to similar elements. From this point of view, an algebra with fuzzy equality can be seen as a collection of similarity-preserving functions on a set. Functions operating on a set so that close (similar) elements are mapped to close elements have traditionally been the subject of study of calculus and functional analysis. The concept of closeness has been almost exclusively formalized using the notion of a metric. However, the very essence of the problem calls for a logical treatment. Namely, formulated verbally, the condition of mapping similar elements to similar ones reads “if arguments of a function are pairwise similar then the results are similar as well”. From a logical point of view, this condition can be described by a logical formula called a compatibility axiom (or congruence axiom). Equivalence relations satisfying compatibility axiom with some given functions are called congruence relations. In bivalent setting, congruence relations do not constitute appropriate framework for studying similarity-preserving mappings. Namely, ordinary congruence relations are bivalent while similarity is a graded notion. In fuzzy setting, however, one can use fuzzy equivalence/equality relations for modeling of similarity. Nevertheless, the axiom expressing preservation of similarity by functions remains the same as in bivalent setting, namely, the above-mentioned compatibility axiom. While this axiom retains its clear verbal description, its meaning depends on the choice of a conjunction operation (usually a t-norm) and if truth degrees are numbers, it has a numerical significance. As we will see, the meaning of compatibility axiom in fuzzy setting is rich enough to capture interesting aspects of the problem of similarity-preserving functions. The notion of an algebra with fuzzy equality generalizes that of a universal algebra. Namely, with fuzzy equality being the identity relation, the concept of an algebra with fuzzy equality coincides with that of a universal R. Bˇ elohl´ avek and V. Vychodil: Fuzzy Equational Logic, StudFuzz 186, 59–137 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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algebra. This fact delineates the basic way of our investigation. In principle, we study problems traditionally studied in universal algebra. Algebraic properties of algebras with fuzzy equalities are our main concern in this chapter. Our generalized setting leaves many of the results obtained in universal algebra particular cases of more general results. In addition to that, there are several new aspects in our setting which are hidden in the ordinary bivalent setting. Analogously to the ordinary case, algebras with fuzzy equalities are structures of a fragment of predicate fuzzy logic. Several aspects interesting from this point of view will be investigated in subsequent chapters. ∗

∗

∗

Section 2.1 introduces the concept on an algebra with fuzzy equality and presents examples. Basic structural notions and constructions for algebras with fuzzy equalities are introduced and studied in Sect. 2.2. Section 2.3 studies direct and subdirect products of algebras with fuzzy equalities. Section 2.4 deals with terms, term functions, and related issues. Sections 2.5, 2.6, and 2.7 study more advanced constructions of direct unions, direct limits, and reduced products. Section 2.8 deals with classes of algebras with fuzzy equalities and with operators over these classes. Section 2.9 presents selected approaches to study of algebras from the point of view of fuzzy logic with comments.

2.1 Definition and Examples A type is a triplet ≈, F, σ where ≈ ∈ F and σ is a mapping σ : F ∪ {≈} → N0 with σ(≈) = 2. Each f ∈ F is called a function symbol, ≈ is a relation symbol called a symbol for fuzzy equality. Mapping σ assigns an arity σ(f ) to every function symbol f ∈ F . Symbol ≈ stands always for a binary relation symbol. If there is no danger of confusion, a type will be denoted simply by F . Definition 2.1. An L-equality of type ≈, F, σ is a triplet algebra with M = M, ≈M , F M such that M, F M is an algebra of type F, σ and ≈M M M M is an L-equality on M such that each f ∈ F is compatible with ≈ . The M , denoted by ske(M), is called a skeleton of M . algebra M, F The fact that M, F M is an algebra of type F, σ means that F M = {f M : M σ(f ) → M | f ∈ F }, i.e., F M contains a σ(f )-ary function f M on M for each f ∈ F . Recall from Definition 1.80 that the compatibility condition says that (a1 ≈M b1 ) ⊗ · · · ⊗ (an ≈M bn ) ≤ f M (a1 , . . . , an ) ≈M f M (b1 , . . . , bn ) for each n-ary f ∈ F and every a1 , b1 , . . . , an , bn ∈ M . An algebra with L-equality will also be simply called an L-algebra. If L is obvious, we say also an algebra with fuzzy equality. Following common usage, we sometimes identify an L-algebra M with its support set M . Moreover, we write

2.1 Definition and Examples

61

M, ≈M , f1M , . . . , fnM instead of M, ≈M , {f1M , . . . , fnM } and sometimes also omit the superscript M from ≈M , F M , and f M . In the following, we often call the degree (a ≈M b ) the degree of similarity between a and b which due to the intended interpretation of ≈M . The concept of an L-algebra is not artificial. It is obvious that the compatibility axiom expresses a natural constraint on the operations requiring that each f M maps similar arguments to similar results. If one takes, e.g., L = [0, 1], this constraint has a numerical character. Moreover, this constraint is expressed in a simple fragment of first-order fuzzy logic. Namely, f M is compatible with ≈M if and only if a first-order formula saying that “if a1 and b1 are similar and · · · and an and bn are similar then f M (a1 , . . . , an ) and f M (b1 , . . . , bn ) are similar” is true in M. Therefore, unlike ordinary algebras, L-algebras M have two non-trivial parts. First, the “functional part”, which is an ordinary algebra M, F M . Second, the “relational part” which is a set with a fuzzy equality, namely, M, ≈M . Remark 2.2. (1) For L = 2 (two-element Boolean algebra), the only L-equality ≈ on M is the ordinary identity relation, i.e. (a ≈ b ) = 1 for a = b , (a ≈ b ) = 0 for a = b . It is therefore easy to see that 2-algebras coincide with ordinary algebras. This is the first way the notion of an L-algebra generalizes the notion of an (ordinary) algebra. (2) Taking this point of view, one may consider a structure L of truth degrees as a parameter and think of the theory of L-algebras as a parametrized theory. Setting L = 2, we get the theory of ordinary universal algebras. As we will see in the following, several results are valid for each L (each complete residuated lattice). However, there are results which are true only for L’s satisfying special properties and so, additional properties of L are important. (3) Obviously, taking arbitrary L and a crisp L-equality ≈M , L-algebras M M can be identified with ordinary algebras. This is the M = M, ≈ , F second way the notion of an L-algebra generalizes the notion of an algebra. The following are some examples of L-algebras. Example 2.3. Take a four-element set L = {0, a, b, 1} of truth degrees linearly ordered by 0 < a < b < 1. Consider a complete residuated lattice L on L given by ⊗ defined by a ⊗ b = min(a, b). Fig. 2.1 and Fig. 2.2 show an example of an L-algebra M. In particular, M = {0 , u , . . . , z , 1 } is the support of M. A fuzzy equality ≈M on M is defined by the first table of Fig. 2.1. We have 0 ≈M u = 0, 0 ≈M v = a, etc. There are two operations on M, denoted by ∧M and ∨M , which are defined in the second and the third table of Fig. 2.1. M M M One can check that both ∧ and ∨ are compatible with ≈ . Moreover, M M is a lattice with its Hasse diagram depicted in the left part of M, ∧ , ∨ Fig. 2.2. Therefore, M is a lattice with fuzzy equality. M (both the lattice part and the fuzzy equality part) is depicted in the right part of Fig. 2.2. Fuzzy equality ≈M is visualized using gray color. A degree of fuzzy equality corresponds to darkness of gray. White corresponds to 0, light gray to a, dark

62


Fig. 2.1. Lattice with L-equality (definition)

1

1 z

x

w u

z y

v 0

x

w u

y v

0

Fig. 2.2. Lattice with L-equality (diagrams)

gray to b, and black to 1. For instance, 0 ≈M v = a since 0 and v are contained in a light gray box, v ≈M y = b since v and y are contained in a dark gray box, etc. Clearly, black and white boxes can be omitted. Example 2.4. Let U be a set equipped with an L-equality ≈U . Let M = S(U ) be the set of all permutations of U compatible with ≈U , i.e. the set of all bijective mappings for which we have (u ≈U v ) ≤ (π(u ) ≈U π(v )). π onMU M where The triplet M = M, ≈ , ◦ M π ≈ σ = u ,v (u ≈U v ) → (π(u ) ≈U σ(v )) (2.1) and ◦M denotes the composition of permutations, is an L-algebra. Note that π ≈M σ is the degree to which it is true that similar elements are mapped to similar ones. Moreover, the restriction of M to bijective mappings satisfying (u ≈U v ) = (π(u ) ≈U π(v )), is a group with fuzzy equality. To prove this assertion, we first show that π ≈M σ = u (π(u ) ≈U σ(u )) . (2.2) On the one hand, we clearly have


π ≈M

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σ = u ,v (u ≈U v ) → (π(u ) ≈U σ(v )) ≤ ≤ u (u ≈U u ) → (π(u ) ≈U σ(u )) = = u 1 → (π(u ) ≈U σ(u )) = u (π(u ) ≈U σ(u )) .

On the other hand,

u (π(u )

≈U σ(u )) ≤ π ≈M σ

holds true iff for each u , v ∈ U we have U U U u (π(u ) ≈ σ(u )) ≤ (u ≈ v ) → (π(u ) ≈ σ(v )) , i.e. iff (u ≈U v ) ⊗

u (π(u )

≈U σ(u )) ≤ π(u ) ≈U σ(v )

which is true. Indeed, compatibility of σ yields (u ≈U v ) ⊗ u (π(u ) ≈U σ(u )) ≤ (σ(u ) ≈U σ(v )) ⊗ (π(u ) ≈U σ(u )) ≤ π(u ) ≈U σ(v ) . It is now routine to check that ≈M is an L-equality on M : reflexivity and symmetry follow directly from definition, and for transitivity we have on account of (2.2) for each u ∈ U that (π ≈M σ) ⊗ (σ ≈M θ) ≤ (π(u ) ≈U σ(u )) ⊗ (σ(u ) ≈U θ(u )) ≤ π(u ) ≈U θ(u ) , that is, (π ≈M σ) ⊗ (σ ≈M θ) ≤

u (π(u )

≈U θ(u )) = π ≈M θ ,

proving transitivity of ≈M . If (π ≈M σ) = 1 then, using (2.2), (π(u ) ≈U σ(u )) = 1 for each u ∈ U . Since ≈U is an L-equality, we have that π(u ) = σ(u ) for each u ∈ U , whence π = σ. Therefore, ≈M is an L-equality on S(U ). To verify that ◦M is compatible with ≈M take any π, π , , ∈ M . We have (π ≈M π ) ⊗ ( ≈M ) ≤ (π ◦ ≈M π ◦ ) iff for each u ∈ U we have (π ≈M π ) ⊗ ( ≈M ) ≤ ((π(u )) ≈U (π (u ))) which is true: (π ≈M π ) ⊗ ( ≈M ) ≤

≤ (π(u ) ≈U π (u )) ⊗ (π (u )) ≈U (π (u )) ≤ ≤ (π(u )) ≈U (π (u )) ⊗ (π (u )) ≈U (π (u )) ≤

≤ (π(u )) ≈U (π (u )) . To sum up, M = M, ≈M , ◦M is an algebra with fuzzy equality. Example 2.5. Let L be the standard L ukasiewicz algebra on the unit interval. Consider U = {a , . . . , f }, and let ≈U be given by Fig. 2.3. Following Example 2.4, there are the four compatible permutations on U : π1 = idU , π2 = aa bb cc df ee df , π3 = ba ba cc dd ee ff , π4 = ba ba cc df ee df .

64


Fig. 2.3. Example of compatible permutations

The resulting L-algebra M = {π1 , . . . , π4 }, ≈M , ◦M of compatible permutations is depicted in Fig. 2.3. Note that in this particular case, M can be seen an a group with L-equality, because ske(M) is a group (so-called Klein’s group: π1 is the neutral element and each permutation is the inverse for itself). Example 2.6. Let C : LU → LU be an L-closure operator on X, see Remark 1.93 (2). Furthermore, let SC = {A ∈ LX | C(A) = A} be the system of all closed fuzzy setsof C, see [7]. Then SC is a complete lattice with respect to ⊆ where infima coincide with intersections and for suprema we have i∈I Ai = C( i∈I Ai ). It is easy to verify that (2.3) i∈I (Ai ≈ Bi ) ≤ ( i∈I Ai ≈ i∈I Bi ) = ( i∈I Ai ≈ i∈I Bi ) , (2.4) i∈I (Ai ≈ Bi ) ≤ ( i∈I Ai ≈ i∈I Bi ) . Hence, using (1.89), (2.4) it follows that i∈I (Ai ≈ Bi ) ≤ i∈I Ai ≈ i∈I Bi = = S i∈I Ai , i∈I Bi ∧ S i∈I Bi , i∈I Ai ≤ ≤ S C i∈I Ai , C i∈I Bi ∧ S C i∈I Bi , C i∈I Ai = = C i∈I Ai ≈ C i∈I Bi = i∈I Ai ≈ i∈I Bi . For I = {1, 2} we can use (1.28), (2.3) and the previous inequality to get (A1 ≈ B1 ) ⊗ (A2 ≈ B2 ) ≤ (A1 ∧ A2 ) ≈ (B1 ∧ B2 ) , (A1 ≈ B1 ) ⊗ (A2 ≈ B2 ) ≤ (A1 ∨ A2 ) ≈ (B1 ∨ B2 ) . Therefore, SC , ≈, ∧, ∨ is an L-algebra. Particularly, since the identity mapX ping is an L-closure operator C (sending A to A) for which SC = L , we get X that L , ≈, ∩, ∪ is an L-algebra, i.e. a lattice with fuzzy equality. Example 2.7. Let X and Y be a set of objects and a set of attributes, respectively, and I be an L-relation between X and Y . Let I(x , y ) be interpreted as a degree to which the object x has the attribute y . Furthermore, let us define for all L-sets A ∈ LX , B ∈ LY the L-sets A↑ ∈ LY , B ↓ ∈ LX by A↑ (y ) = x ∈X A(x ) → I(x , y ) , B ↓ (x ) = y ∈Y B(y ) → I(x , y ) .


65

Now, put B(X, Y, I) = {A, B ∈ LX × LY | A↑ = B, B ↓ = A} and define for A1 , B1 , A2 , B2 ∈ B(X, Y, I) a binary relation ≤ by A1 , B1 ≤ A2 , B2 iff A1 ⊆ A2 (or, iff B2 ⊆ B1 ; both ways are equivalent). The structure B(X, Y, I), ≤ is called a fuzzy concept lattice induced by X, Y , and I. The elements of A, B of B(X, Y, I) are naturally interpreted as concepts hidden in (the input data represented by) I. Namely, A↑ = B and B ↓ = A says that B is the collection of all attributes shared by all objects from A, and A is the collection of all objects sharing all attributes from B. Note that these conditions represent exactly the definition of a concept as developed in the so-called Port-Royal logic. A and B are called the extent and the intent of the concept A, B , respectively, and represent the collection of all objects and all attributes covered by the particular concept. Furthermore, ≤ models a subconcept-superconcept hierarchy – concept A1 , B1 is a subconcept of A2 , B2 iff each object from A1 belongs to A2 (dually for attributes). Put Ext(X, Y, I) = {A ∈ LX | A, B ∈ B(X, Y, I) for some B ∈ LY }, i.e. Ext(X, Y, I) consists of all extents of the concepts from B(X, Y, I). Now, it can be shown (see [7, 8]) that for a fuzzy concept lattice B(X, Y, I), Ext(X, Y, I) is an L-closure system in X and that for each L-closure system S in X there are some X, Y , and I, such that S = Ext(X, Y, I). This means that B(X, Y, I), ≈, ∧, ∨ , where (A1 , B1 ≈ A2 , B2 ) = (A1 ≈ A2 ) and ∧ and ∨ are the infimum and supremum induced by the corresponding operations in Ext(X, Y, I), is an L-algebra (see Example 2.6). Example 2.8. The following is a particular example of a fuzzy concept lattice. ukasiewicz Consider a complete residuated lattice on L = {0, 12 , 1} given by L structure. Figure 2.4 contains a data table with fuzzy attributes describing a fuzzy relation I between sets X (planets) and Y (fuzzy attributes of planets). The corresponding fuzzy concept lattice contains 38 concepts. These are depicted in Fig. 2.6. The corresponding fuzzy concept lattice B(X, Y, I), ≤ is shown in the left part of Fig. 2.5. Fuzzy equality ≈ on B(X, Y, I) (see Example 2.7) is depicted in the right part of Fig. 2.5. Truth degrees 0, 12 , and 1 are

Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune Pluto

(Me) (Ve) (Ea) (Ma) (Ju) (Sa) (Ur) (Ne) (Pl)

size distance small (s) large (l) far (f) near (n) 1 0 0 1 1 0 0 1 1 0 0 1 1/2 1 0 1 1/2 0 1 1 1/2 0 1 1 1/2 1/2 1 0 1/2 1/2 1 0 1 0 1 0

Fig. 2.4. Data table with fuzzy attributes

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2 Algebras with Fuzzy Equalities 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.

38 28

36 25

27

37 13

33

34

35 26 11

24 21

12 23 8

22 10 32 20

9 19 5

18 17

6 16 3

31 30

4

7

15 2

14

29

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.

1

Fig. 2.5. Fuzzy concept lattice with L-equality

denoted by white, gray, and black boxes. Denote a concept no. i by Ci . Then, for instance, C8 ≈ C2 = 0, C8 ≈ C3 = 12 , etc. Note that [6] shows how the above-discussed fuzzy equality on fuzzy concept lattice B(X, Y, I) can be used to form a factor lattice of B(X, Y, I). Elements of the factor lattice can be seen as similarity-based clusters of concepts from B(X, Y, I). The factor lattice itself thus represents an approximate version of B(X, Y, I). The clusters in our example are apparent from the right part of Fig. 2.5. This is important from the application point of view since B(X, Y, I) can be large and its factor lattice, which is smaller, is more comprehensible for a user. For more information see the last paragraph of Remark 2.13. Example 4.18 on page 183 shows a possible application of algebras with fuzzy equalities in formal specification of humanistic systems. Further examples illustrating notions related to algebras with fuzzy equalities will be shown in the course of the following sections.

2.2 Subalgebras, Congruences, and Morphisms The goal of this section is to investigate subalgebras, morphisms, congruences, and factor algebras, i.e. the very fundamental algebraic constructions. All principal properties of such constructions, being developed in universal algebra, will generalize for every complete residuated lattice L as a structure of truth degrees. Remark 2.9. Before delving into the basic structural notions, an important remark is in order. As mentioned above, each algebra M = M, ≈M , F M can be thought of as having two parts, a skeleton ske(M) (functional part)

2.2 Subalgebras, Congruences, and Morphisms no. Me Ve Ea 1. 0 0 0 2. 0 0 0 3. 0 0 0 4. 0 0 0 5. 1/2 1/2 1/2 6. 1/2 1/2 1/2 7. 0 0 0 8. 1/2 1/2 1/2 9. 1/2 1/2 1/2 10. 1/2 1/2 1/2 11. 1 1 1 12. 1 1 1 13. 1 1 1 14. 0 0 0 15. 0 0 0 16. 0 0 0 17. 0 0 0 18. 0 0 0 19. 1/2 1/2 1/2 20. 1/2 1/2 1/2 21. 1/2 1/2 1/2 22. 0 0 0 23. 1/2 1/2 1/2 24. 1/2 1/2 1/2 25. 1/2 1/2 1/2 26. 1 1 1 27. 1 1 1 28. 1 1 1 29. 0 0 0 30. 0 0 0 31. 0 0 0 32. 1/2 1/2 1/2 33. 1/2 1/2 1/2 34. 0 0 0 35. 1/2 1/2 1/2 36. 1/2 1/2 1/2 37. 1 1 1 38. 1 1 1

extent Ma Ju Sa 0 0 0 0 0 0 1/2 0 0 1/2 0 0 1/2 0 0 1/2 0 0 1/2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 1/2 1/2 0 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 1 1/2 1/2 0 1 1 1/2 1 1 1/2 1 1 1/2 1 1 1/2 1 1 1/2 1 1 1 1 1 1 1 1 1 1 1 1 1 1

Ur 0 1/2 0 1/2 0 1/2 1/2 0 1/2 1/2 0 1/2 1/2 0 1/2 0 1/2 1 0 1/2 1 1 0 1/2 1 0 1/2 1 1/2 1/2 1 1/2 1 1 1/2 1 1/2 1

Ne 0 1/2 0 1/2 0 1/2 1/2 0 1/2 1/2 0 1/2 1/2 0 1/2 0 1/2 1 0 1/2 1 1 0 1/2 1 0 1/2 1 1/2 1/2 1 1/2 1 1 1/2 1 1/2 1

Pl 0 0 0 1/2 0 1/2 1 0 1/2 1 0 1/2 1 0 0 0 1/2 1/2 0 1/2 1/2 1 0 1/2 1 0 1/2 1 0 1/2 1/2 1/2 1/2 1 1/2 1 1/2 1

s 1 1 1 1 1 1 1 1 1 1 1 1 1 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2 0 0 0 0 0 0 0 0 0 0

67

intent l f n 1 1 1 1 1 1/2 1/2 1 1 1/2 1 1/2 1/2 1/2 1 1/2 1/2 1/2 0 1 0 0 1/2 1 0 1/2 1/2 0 1/2 0 0 0 1 0 0 1/2 0 0 0 1 1 1 1 1 1/2 1/2 1 1 1/2 1 1/2 1/2 1 0 1/2 1/2 1 1/2 1/2 1/2 1/2 1/2 0 0 1 0 0 1/2 1 0 1/2 1/2 0 1/2 0 0 0 1 0 0 1/2 0 0 0 1 1 1/2 1/2 1 1/2 1/2 1 0 1/2 1/2 1/2 1/2 1/2 0 0 1 0 0 1/2 1/2 0 1/2 0 0 0 1/2 0 0 0

Fig. 2.6. Extracted concepts

and a set with L-equality M, ≈M (relational part) which are connected via the compatibility condition. Traditional universal algebra deals with the functional part only. However, it is obvious that when developing structural properties of L-algebras, we will face situations where (1) only the functional part is important, (2) only the relational part is important, (3) both the

68


functional and the relational parts are important. For (1), we can obviously use well-known results established in universal algebra. However, since we want our text to be self-contained, we omit the parts of proofs dealing with the functional part only if they are easy to see. M M be an L-algebra of type F . An LDefinition 2.10. Let M = M, ≈ , F N N is called a subalgebra of M if ∅ = N ⊆ M , each algebra N = N, ≈ , F function f N ∈ F N is a restriction of f M to N , and ≈N is a restriction of ≈M to N . A subuniverse of M is any subset N ⊆ M which is closed under all functions of M. That is, N ⊆ M is a subuniverse of M iff for each n-ary f M ∈ F M and every a1 , . . . , an ∈ N we have f M (a1 , . . . , an ) ∈ N . The collection of all subuniverses of M will be denoted by Sub(M). If ≈, F, σ is a type with nullary function symbols (i.e., symbols of constants) and M is an L-algebra of that type then each subuniverse N ∈ Sub(M) is non-empty because for each nullary function symbol c ∈ F , the corresponding constant cM ∈ M belongs to N . On the contrary, if ≈, F, σ does not contain nullary function symbols one can easily see that ∅ ∈ Sub(M). Thus, ∅ ∈ Sub(M) if and only if M is an L-algebra of a type without nullary function symbols. An L-algebra M = M, ≈M , F M is said to be trivial if M = {a }. Since the universe of a trivial L-algebra contains only one element, all mappings {a }n → {a } are trivially compatible with ≈M . In the case of trivial Lalgebras, the L-equality ≈M does not represent any constraint for functions f M and vice versa. An L-algebra which is not trivial is called non-trivial . For convenience, we write N ∈ Sub(M) if N is a subalgebra of M. It is almost immediate that there is a bijective correspondence between subalgebras and non-empty subuniverses of M. Namely, a support set of a subalgebra of M is a non-empty subuniverse and, conversely, a non-empty subuniverse of M equipped with the restrictions of f M ’s and the restriction of ≈M is a subalgebra of M. It follows from the ordinary case that Sub(M) is closed under arbitrary intersections and, therefore, it is a closure system under the subsethood relation ⊆. We denote the corresponding closure operator by [ ]M , i.e. for N ⊆ M , [N ]M is the least subuniverse of M containing N and we have [N ]M = {N | N ∈ Sub(M) and N ⊆ N } . (2.5) If [N ]M is non-empty then the corresponding subalgebra of M is called the subalgebra of M generated by N . If |N | < κ for an infinite cardinal κ then the corresponding subalgebra of M is called κ-generated. If N is finite then the corresponding subalgebra of M is called finitely generated. Hence, a subalgebra is finitely generated if and only if it is ω-generated (ω is the least infinite cardinal). ∗

∗

∗

2.2 Subalgebras, Congruences, and Morphisms

69

In ordinary case, congruences correspond to abstract views on algebraic systems: one can put congruent (i.e. equivalent from a certain abstract point of view) elements together and form a factor algebra which provides a view on the original algebra under which congruent elements are indistinguishable. In fuzzy setting, it is natural to consider graded indistinguishability of elements. This leads to the notion of a congruence on an L-algebra. Definition 2.11. Let M be an L-algebra of type F . An L-relation θ on M is called a congruence on M if (i) θ is an L-equivalence relation on M , (ii) θ is compatible with ≈M , (iii) all functions f M ∈ F M are compatible with θ, see Definition 1.80 and Remark 1.81. Remark 2.12. (1) In what follows, the (ordinary) sets of all L-equivalences and congruences on an L-algebra M are denoted by EqL (M ) and ConL (M). Evidently, ConL (M) ⊆ EqL (M ). (2) It is immediate that congruences on 2-algebras correspond to ordinary congruences on algebras since condition (ii) of Definition 2.11 is trivially satisfied by ordinary congruence relations, generalizing thus the well-known concept of a congruence. (3) Note that due to Lemma 1.82, condition (ii) may be equivalently replaced by requiring ≈M ⊆ θ, i.e. (a ≈M b ) ≤ θ(a , b ) for arbitrary a , b ∈ M . Remark 2.13. Let L be a complete residuated lattice with ⊗ being ∧ (i.e. L is a complete Heyting algebra). If L = [0, 1], this means that ⊗ is the minimum which is probably the most common choice of “fuzzy conjunction”. It is straightforward to verify that a binary L-relation on M is an L-equality if and only if each a-cut a θ, a ∈ L is an ordinary equivalence relation and 1 θ is the identity relation. Furthermore, for an L-algebra M, θ satisfies the compatibility condition w.r.t. the operations of M if and only if each a θ satisfies the compatibility condition w.r.t. operations of the corresponding ordinary algebra M, F M . Indeed, for a1 , b1 ∈ a θ, . . . , an , bn ∈ a θ we have a ≤ θ(ai , bi ) for every i = 1, . . . , n, thus a ≤ θ(a1 , b1 ) ⊗ · · · ⊗ θ(an , bn ) ≤ θ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) by the assumption ⊗ = ∧, i.e. f M (a1 , . . . , an ), f M (b1 , . . . , bn ) ∈ a θ. Con· · ⊗ θ(an , bn ). Then we have ai , bi ∈ a θ for versely, put a = θ(a1 , b1 ) ⊗ ·M every i = 1, . . . , n, and so f (a1 , . . . , an ), f M (b1, . . . , bn ) ∈ a θ. Hence, it follows that a ≤ θ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) . Therefore, for a congruence θ on an L-algebra M, Sθ = {a θ | a ∈ L} is a system of binary relations on M indexed by L which satisfies: (i) a ≤ b implies b θ ⊆ a θ; (ii) for each a , b ∈ M there exists a greatesta ∈ L with a , b ∈ a θ; (iii) for each a ∈ L, a θ is a congruence on M, F M containing a ≈M . One

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may verify that conversely, if S = {θa | a ∈ L} is a system of binary relations on M satisfying (i)–(iii) then putting θS (a , b ) = {a | a , b ∈ θa }, θS is a congruence on M. Furthermore, θ = θSθ and S = SθS . This way of looking at congruences of L-algebras (but only when ⊗ is ∧!) gives the following interpretation: An ordinary congruence on an ordinary algebra may be thought of as representing an abstract view of the algebra – a view under which one does not distinguish congruent elements. A congruence θ on an L-algebra may be thought of as a hierarchic system (the hierarchy a θ is a congruence on M, F M , supplied by L) of abstract views a θ. Since each one may form the ordinary factor algebra M, F M /a θ. Clearly, the smaller a, the bigger a θ, and so the coarser the factorization. This fact has non-trivial applications. In [6], a method of factorization of concept lattices is presented. As shown in Example 2.7, a fuzzy concept lattice B(X, Y, I) can be naturally turned into an L-algebra B(X, Y, I), ≈, ∧, ∨ . A fuzzy concept lattice B(X, Y, I) represents a set of natural clusters (concepts) contained in the input data X, Y, I . If there are too many concepts in B(X, Y, I), the fuzzy concept lattice is hardly graspable by human mind. If ⊗ is ∧, one can form the factor lattice B(X, Y, I)/a ≈ for any a. The factor lattice B(X, Y, I)/a ≈ may be thought of as a simplified version of B(X, Y, I). The choice of a controls the coarseness of the factorization. The point of [6] is that, in fact, one may form the factor lattice even for general ⊗ (not necessarily ∧) in which case a ≈ is a tolerance relation (i.e., reflexive and symmetric) which is compatible with the lattice structure of B(X, Y, I). Since (complete) lattices can be factorized even by compatible tolerance relations, a method for general ⊗ is available, see [6]. It is also worth to note that the factor lattice can be computed directly from the data table, without the need to compute the whole B(X, Y, I). For details we refer to [13]. Example 2.14. Consider a fuzzy concept lattice B(X, Y, I) with fuzzy equality from Example 2.8 (see Fig. 2.5). Figure 2.7 depicts six congruences on B(X, Y, I). Again, truth degrees 0, 12 , and 1 are denoted by white, gray, and black boxes. Notice that the congruence from part (a) is a fuzzy equality while those from (b)–(f) are not. Intuitively, one can see that each congruence induces clusters of concepts from B(X, Y, I). By definition, ConL (M) ⊆ EqL (M ). The following theorem shows that ConL (M) inherits even the complete lattice structure of EqL (M ). Theorem 2.15. ConL (M), ⊆ is a complete sublattice of EqL (M ), ⊆ . Proof. Take an index of congru i ∈ I} set I and a family {θi | θi ∈ ConL (M), ences. Obviously, i∈I θi ∈ EqL (M ) and (a ≈M b ) ≤ i∈I θi (a , b ). For every f M ∈ F M and arbitrary a1 , b1 , . . . , an , bn ∈ M , we have, n n j=1 j=1 θi (aj , bj ) ≤ i∈I θi (aj , bj ) ≤ i∈I ≤ i∈I θi f M (a1 , . . . , an ), f M (b1 , . . . , bn ) ,

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.

(a)

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(c)

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(e)

(d)

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(b)

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Fig. 2.7. Examples of congruences

Fig. 2.7. Examples of congruences

(f)

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thus every f M ∈ F M is compatible with i∈I θi . Hence, ConL (M), ⊆ is closed under arbitrary infima showing that infima in ConL (M), ⊆ coincide with infima in EqL (M ), ⊆ . To complete the proof we show that ConL (M), ⊆ and EqL (M ), ⊆ agree on their suprema. First note that the supremum of θi (i ∈ I) in EqL (M ) equals θ ◦ · · · ◦ θ . Indeed, denote ik i1 ,...,ik ∈I i1 i1 ,...,ik ∈I θi1 ◦ · · · ◦ θik by θ. One can see that each θi , i ∈ I is contained in θ (i.e. θi ⊆ θ), θ is contained in any Lequivalence containing all θi ’s, and that θ is reflexive (due to reflexivity of θi ) and symmetric (due to symmetry of θi and commutativity of ⊗). Moreover, θ(a , b ) ⊗ θ(b , c ) = = i1 ,...,ik ∈I θi1 ◦ · · · ◦ θik (a , b ) ⊗ θj1 ◦ · · · ◦ θjl (b , c ) ≤ j1 ,...,jl ∈I ≤ i1 ,...,ik ∈I θi1 ◦ · · · ◦ θik ◦ θj1 ◦ · · · ◦ θjl (a , c ) ≤ j1 ,...,jl ∈I ≤ i1 ,...,im ∈I θi1 ◦ · · · ◦ θim (a , c ) = θ(a , c ) , thus θ is also transitive showing that θ is the supremum of θi ’s in EqL (M ). Now it suffices to show that θ ∈ ConL (M). By Lemma 1.82, condition (ii) of Definition 2.11 is satisfied trivially. For (i) take any n-ary function f M ∈ F M and arbitrary a1 , b1 , . . . , an , bn ∈ M . We have θ(a1 , b1 ) ⊗ · · · ⊗ θ(an , bn ) = n = j=1 ij1 ,...,ijk ∈I θij1 ◦ · · · ◦ θijkj (aj , bj ) = j n = i11, ...,i1k1 ∈I c11, ...,c1(k1 −1) ∈M j=1 θij1 (aj , cj1 ) ⊗ θij2 (cj1 , cj2 ) ⊗ · · · .. .. . . in1,...,inkn ∈I

.. .. . . cn1,...,cn(kn −1) ∈M

· · · ⊗ θijkj (cj(kj −1) , bj ) .

Now for any of θij1 (aj , cj1 ), θij2 (cj1 , cj2 ), . . . , θijkj (cj(kj −1) , bj ), j ∈ J we have θi11 (a1 , c11 ) ≤ θi11 f M (a1 , . . . , an ), f M (c11 , a2 , . . . , an ) , .. . θin1 (an , cn1 ) ≤ θin1 f M (c11 , c21 . . . , c(n−1)1 , an ), f M (c11 , . . . , cn1 ) , θi12 (c11 , c12 ) ≤ θi12 f M (c11 , . . . , cn1 ), f M (c12 , c21 , . . . , cn1 ) , .. . θinkn (cn(kn −1) , bn ) ≤ θinkn f M (b1 , . . . , b(n−1) , cn(kn −1) ), f M (b1 , . . . , bn ) . Hence, we finish the proof with i11, ...,i1k1 ∈I .. .. . . in1,...,inkn ∈I

≤ =

c11, ...,c1(k1 −1) ∈M .. .. . . cn1,...,cn(kn −1) ∈M

n j=1

θij1 (aj , cj1 ) ⊗ θij2 (cj1 , cj2 ) ⊗ · · · · · · ⊗ θijkj (cj(kj −1) , bj ) ≤

M M l1 ,...,lm ∈I θl1 ◦ · · · ◦ θlm f (a1 , . . . , an ), f (b1 , . . . , bn ) M θ f (a1 , . . . , an ), f M (b1 , . . . , bn ) .

Altogether, θ ∈ ConL (M).

=


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Lemma 2.16. For congruences θ1 , θ2 ∈ ConL (M) the following conditions are equivalent: (i) θ1 ◦ θ2 = θ2 ◦ θ1 , (ii) θ1 ∨ θ2 = θ1 ◦ θ2 , (iii) θ2 ◦ θ1 ⊆ θ1 ◦ θ2 . Proof. “(i) ⇒ (iii)” is trivial. “(iii) ⇒ (ii)”:We only need to show θ1 ∨ θ2 ⊆ θ1 ◦ θ2 which follows easily from θ1 ∨ θ2 = i1 ,...,ik ∈{1,2} θi1 ◦ · · · ◦ θik and from θi ◦ θi ⊆ θi , the latter relationship is a reformulation of transitivity. “(ii) ⇒ (i)”: It is easily seen that θ1 ◦ θ2 = θ1 ∨ θ2 = (θ1 ∨ θ2 )−1 = (θ1 ◦ θ2 )−1 = θ2−1 ◦ θ1−1 = θ2 ◦ θ1 . An important notion is that of a congruence generated by a finite collection of pairs. A natural meaning of this notion is the following: We are given a collection of pairs ai , bi of elements of M and are looking for the least congruence containing all ai , bi ’s. In other words, we are looking for the least abstract view on M under which all ai and bi are indistinguishable. In fuzzy setting, it is natural to provide additionally the degrees of required indistinguishability of ai and bi . In what follows, we present tractable descriptions of congruences generated by prescribed pairs of elements and their required indistinguishability degrees. Definition 2.17. Let M be an L-algebra. For a binary L-relation R in M we denote by θ(R) the least congruence on M containing R. Particularly, for any b1 , c1 , . . . , bk , ck ∈ M and arbitrary truth degrees a1 , . . . , ak ∈ L we denote θ({a1/b1 , c1 ,. . ., ak/bk , ck }) by θ(a1/b1 , c1 , . . . , ak/bk , ck ). It is the least congruence θ ∈ ConL (M) such that θ(b1 , c1 ) ≥ a1 , . . . , θ(bk , ck ) ≥ ak , i.e. θ(a1/b1 , c1 , . . . , ak/bk , ck ) is generated by the finite L-relation R = {a1/b1 , c1 , . . . , ak/bk , ck }. A congruence θ(a/b , c ) ∈ ConL (M) is called a principal congruence on M. Remark 2.18. Note that due toTheorem 2.15, θ(R) exists for every L-relation R ∈ LM ×M . Namely, θ(R) = {θ ∈ ConL (M) | R ⊆ θ}. Example 2.19. Consider the congruences from Example 2.14, see Fig. 2.7. (a) is θ(0.5/C1 , C38 ), (b) is θ(1/C15 , C29 ), (c) is θ(1/C8 , C12 ), (d) is θ(1/C2 , C7 ), (e) is θ(1/C1 , C14 ), and (f) is θ(1/C1 , C11 ). Lemma 2.20. The following are properties of principal congruences: (i) θ(a/b , c ) = θ(a/c , b ), ak a1 ak (ii) θ(a1/ b1 , c1 , . . . , /bk , ck ) =θ( /b1 , c1 ) ∨ · · · ∨ θ( /bk , ck ), (iii) θ = b ,c ∈ M θ(θ(b ,c )/b , c ) = b ,c ∈ M θ(θ(b ,c )/b , c ).

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Proof. (i) follows directly from the definition. (ii): It is easily seen that θ(a1/b1 , c1 , . . . , ak/bk , ck ) is always contained in θ(a1/b1 , c1 ) ∨ · · · ∨ θ(ak/bk , ck ). For the converse inequality observe that θ(ai/bi , ci ) ⊆ θ(a1/b1 , c1 , . . . , ak/bk , ck ) for every i = 1, . . . , k. Thus, property (ii) holds. (iii): We have θ(b ,c )/b , c ) ⊆ θ(b ,c )/b , c ) ⊆ θ ⊆ b ,c ∈M θ( b ,c ∈M θ( ⊆ b ,c ∈M θ(θ(b ,c )/b , c ) proving the claim.

Remark 2.21. The so-called Mal’cev lemma provides a description of principal congruences. Mal’cev lemma can be generalized to fuzzy setting. However, it involves notions concerning terms, term functions and term algebras which are discussed in Sect. 2.4. Therefore, we postpone the generalized Mal’cev lemma and present it on page 94 as Theorem 2.82. We are going to define the concept of a factor L-algebra by a congruence θ. For this purpose, we utilize the concept of a factor set with fuzzy equality by a fuzzy equivalence (Definition 1.91). Definition 2.22. Let θ be a congruence on an L-algebra M of type F. A factor L-algebra of M by θ is an L-algebra M/θ = M/θ, ≈M/θ , F M/θ of type F such that (i) M/θ, ≈M/θ is the factor set with L-equality of M, ≈M by θ, see Definition 1.91 and Remark 1.92; (ii) f M/θ [a1 ]θ , . . . , [an ]θ = f M (a1 , . . . , an ) θ for every n-ary function f M/θ ∈ F M/θ and arbitrary a1 , . . . , an ∈ M . Remark 2.23. (1) Condition (i) says that elements of M/θ are ordinary equivalence classes of 1 θ. By convention, these classes are denoted by [a ]θ instead of [a ]1 θ . That is, M/θ = {[a ]θ | a ∈ M } where [a ]θ = {a | θ(a , a ) = 1}. Furthermore, condition (i) says that ≈M/θ is defined by [a ]θ ≈M/θ [b ]θ = θ(a , b ). (2) A factor L-algebra is well-defined. First, ≈M/θ is a well-defined Lon factor set M/θ. Namely, since θ is a congruence on M = equality M, ≈M , F M , θ is compatible with ≈M , see Remark 2.12 (3). Therefore, ≈M/θ is a well-defined L-equality on M/θ by Lemma 1.90. Second, each f M/θ ∈ F M/θ is a well-defined function on M/θ which is compatible with ≈M/θ . Namely, for each n-ary function f M/θ ∈ F M/θ and a1 , b1 , . . . , an , bn ∈ M , we have [a1 ]θ ≈M/θ [b1 ]θ ⊗ · · · ⊗ [an ]θ ≈M/θ [bn ]θ = θ(a1 , b1 ) ⊗ · · · ⊗ θ(an , bn ) ≤ ≤ θ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) = = f M (a1 , . . . , an ) θ ≈M/θ f M (b1 , . . . , bn ) θ = = f M/θ [a1 ]θ , . . . , [an ]θ ≈M/θ f M/θ [b1 ]θ , . . . , [bn ]θ .


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Applying this inequality for [a1 ]θ ≈M/θ [b1 ]θ = 1, . . . , [an ]θ ≈M/θ [bn ]θ = 1 yields that if [a1 ]θ = [b1 ]θ , . . . , [an ]θ = [bn ]θ , then f M/θ ([a1 ]θ , . . . , [an ]θ ) = f M/θ ([b1 ]θ , . . . , [bn ]θ ). Hence, every function f M/θ ∈ F M/θ is well defined. Furthermore, the above inequality itself says that every f M/θ ∈ F M/θ is compatible with ≈M/θ . Remark 2.24. An important role in the definition of a factor L-algebra is played by the 1-cut 1 θ of a congruence θ on M. It is important to note that in general, a congruence θ on an L-algebra is not determined by 1 θ, not even if θ is an L-equality (in fact, all L-equalities on M have a common 1-cut of the form {a , a | a ∈ M }). Furthermore, one can easily verify that for a con gruence θ ∈ ConL (M), 1 θ is an ordinary congruence relation on M, F M . Therefore, the functional part of M/θ coincides with M, F M /1 θ. Clearly, this issue is completely degenerate in the ordinary case since if L = 2 then θ and 1 θ coincide (modulo the relationship between crisp fuzzy relations and the corresponding ordinary relations). Example 2.25. Fig. 2.8 contains the skeletons of factor lattices with fuzzy equalities which result by factorization of a fuzzy concept lattice with fuzzy equality from Example 2.8 by congruences (a)–(f) from Example 2.14. Consider in more detail case (e) and denote the corresponding congruence by θ. The elements of the factor L-algebra (factor lattice with fuzzy equality) are the following classes of θ:

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 2.8. Examples of factor L-algebras

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Fig. 2.9. Factor lattice with L-equality

[C1 ]θ = {C1 , C14 } ,

[C2 ]θ = {C2 , C15 } ,

[C3 ]θ = {C3 , C16 } , [C5 ]θ = {C5 , C8 , C19 , C23 } , [C11 ]θ = {C11 , C26 } ,

[C4 ]θ = {C4 , C7 , C17 , C18 , C22 } , [C6 ]θ = {C6 , C9 , C10 , C20 , C21 , C24 , C25 } , [C12 ]θ = {C12 , C13 , C27 , C28 } ,

[C29 ]θ = {C29 } , [C32 ]θ = {C32 , C33 , C35 , C36 } ,

[C30 ]θ = {C30 , C31 , C34 } , [C37 ]θ = {C37 , C38 } .

The corresponding factor lattice is shown (once again) in the left part of Fig. 2.9. The fuzzy equality on the factor lattice is shown in the right part of Fig. 2.9. ∗

∗

∗

In the sequel, a morphism of L-algebras is defined as a mapping that preserves both their operations f M and their fuzzy equality ≈M . All properties well-known from ordinary case generalize in full scope. Definition 2.26. Let M and N be L-algebras of type F . A mapping h : M → N is called a morphism (or homomorphism) of M to N if (i) h(f M (a1 , . . . , an )) = f N (h(a1 ), . . . , h(an )) for every n-ary f ∈ F and arbitrary a1 , . . . , an ∈ M ; (ii) h is a ≈-morphism of M, ≈M to N, ≈N (Definition 1.83 ). The fact that h : M → N is a morphism is denoted by h : M → N. Furthermore, • an injective morphism is called a monomorphism, • a morphism such that

a ≈M b = h(a ) ≈N h(b )

for all a , b ∈ M

is called an embedding, • a surjective morphism is called an epimorphism,

(2.6)


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• an epimorphism which is an embedding is called an isomorphism, • a morphism h : M → M is called an endomorphism, • an isomorphism h : M → M is called an automorphism. For an epimorphism h : M → N, the L-algebra N is called an image of M. We say M is isomorphic to N, written M ∼ = N, if there exists an isomorphism h : M → N. Let idM denote the identity mapping on M (we use idM to emphasize that M is a universe of M). Remark 2.27. Note that condition (ii) requires a ≈M b ≤ h(a ) ≈N h(b ) which can be verbally described as “if a and b are similar, then h(a ) and h(b ) are similar”. Namely, a ≈M b ≤ h(a ) ≈N h(b ) is true for all a , b ∈ M iff formula (∀x)(∀y) x ≈ y i h(x) ≈ h(y) is fully true (i.e. has truth degree 1) in an appropriate structure encompassing both M and N. Condition h(f M (a1 , . . . , an )) = f N (h(a1 ), . . . , h(an )) is equivalent to saying that h is an (ordinary) morphism of the skeletons ske(M) and ske(N). Theorem 2.28. Let h : M → N be a morphism. Then we have (i) if M ∈ Sub(M) then h(M ) ∈ Sub(N), (ii) if N ∈ Sub(N) then h−1 (N ) = {a ∈ M | h(a ) ∈ N } ∈ Sub(M), (iii) if h : M → N is an embedding then M ∼ = h(M), where h(M) is a subalgebra of N induced by h(M ). Proof. (i): Take an n-ary f N ∈ F M . For arbitrary b1 , . . . , bn ∈ h(M ) there are c1 , . . . , cn ∈ M such that b1 = h(c1 ), . . . , bn = h( n ). Since M ∈ cM Sub(M), N N we get f (b1 , . . . , b1 ) = f h(c1 ), . . . , h(c1 ) = h f (c1 , . . . , cn ) ∈ h(M ). Furthermore, each f N is trivially compatible with the restriction of ≈N on h(M ). Hence, h(M ) ∈ Sub(N). (ii): Take an n-ary f M ∈ F M . For each a1 , . . . , an ∈ h−1 (N ) we have h f M (a1 , . . . , an ) = f N h(a1 ), . . . , h(an ) ∈ N because h(a1 ), . . . , h(an ) ∈ N . Therefore, f M (a1 , . . . , an ) ∈ h−1 (N ). Evidently, each f M is compatible with the restriction of ≈M on h−1 (N ). (iii): Suppose h : M → N is an embedding. (i) yields that h(M ) ∈ Sub(N). Thus, g : M → h(M), where g(a ) = h(a ) for every a ∈ M is a surjective embedding, i.e. M ∼ = h(M). Theorem 2.29. Suppose g : M → M and h : M → M are morphisms. The composition g ◦ h is a morphism (g ◦ h) : M → M . Proof. Theorem 1.86 yields that h is an ≈-morphism. The rest follows from the ordinary case.

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Theorem 2.30. Let h : M → M , g : M → M be morphisms such that h ◦ g = idM , g ◦ h = idM . Then M ∼ = M . Proof. Evidently, h is bijective since for every a ∈ M we have h(g(a )) = a , and from h(a ) = h(b ) it follows that a = g(h(a )) = g(h(b )) = b . Clearly, h−1 = g since for h(a ) = b we have g(b ) = g(h(a )) = a = h−1 (b ). Hence, we can apply Theorem 1.86 (ii) to see that a ≈M b = h(a ) ≈M h(b ) for all a , b ∈ M . Thus, h is an isomorphism. Theorem 2.31. Let h : M → N be a morphism. Then θh ∈ ConL (M). Proof. From Theorem 1.85 and Theorem 1.87 it follows that a ≈M b ≤ θh (a , b ) for all a , b ∈ M and θh ∈ EqL (M ). Now it is sufficient to show that θh is also compatible with functions f M ∈ F M . For n-ary f M ∈ F M and arbitrary a1 , b1 , . . . , an , bn ∈ M , the following inequality holds: n n N i=1 θh (ai , bi ) = i=1 h(ai ) ≈ h(bi ) ≤ ≤ f N h(a1 ), . . . , h(an ) ≈N f N h(b1 ), . . . , h(bn ) = = h f M (a1 , . . . , an ) ≈N h f M (b1 , . . . , bn ) = = θh f M (a1 , . . . , an ), f M (b1 , . . . , bn ) . Thus, θh ∈ ConL (M).

Definition 2.32. For every L-algebra M and θ ∈ ConL (M) a mapping hθ : M → M/θ, where hθ (a ) = [a ]θ for all a ∈ M , is called a natural mapping. Theorem 2.33. A natural mapping hθ from an L-algebra M to a factor Lalgebra M/θ is an onto morphism (i.e. an epimorphism). Proof. For any a , b ∈ M we have

a ≈M b ≤ θ(a , b ) = [a ]θ ≈M/θ [b ]θ = hθ (a ) ≈M/θ hθ (b ) . Furthermore, for n-ary f M ∈ F M and arbitrary a1 , . . . , an ∈ M we have hθ f M (a1 , . . . , an ) = f M (a1 , . . . , an ) θ = = f M/θ [a1 ]θ , . . . , [an ]θ = f M/θ hθ (a1 ), . . . , hθ (an ) . Surjectivity of hθ is evident.

Remark 2.34. In what follows we call hθ a natural morphism. In the rest of this section we present some of the basic results on relationships between morphisms and congruence relations in fuzzy setting. Note that in a more general setting, some of the results are presented in [10]. Theorem 2.35 (first isomorphism theorem). Let h : M → N be a surjective morphism. Then there is an isomorphism g : M/θh → N such that hθh ◦ g = h.


79

Fig. 2.10. First isomorphism theorem

M

g1 d

g2 N2

N1 g1

g2

M

Fig. 2.11. Diagonal fill-in

Proof. Introduce g by putting g [a ]θh = h(a ) for all elements a ∈ M . Evidently, hθh ◦ g = h. Furthermore, [a ]θh ≈M/θh [b ]θh = θh (a , b ) = h(a ) ≈N h(b ) = g [a ]θh ≈N g [b ]θh . The mapping g is surjective as h is surjective and g [a ]θh = h(a ). Take any n-ary f M/θh ∈ F M/θh and arbitrary [a1 ]θh , . . . , [an ]θh ∈ M/θh . We have g f M/θh [a1 ]θh , . . . , [an ]θh = g f M a1 , . . . , an θ = h = h f M (a1 , . . . , an ) = f N h(a1 ), . . . , h(an ) = = f N g [a1 ]θh , . . . , g [an ]θh . Thus g : M/θh → N is an isomorphism and hθh ◦ g = h, see Fig. 2.10.

An easy extension of Theorem 2.35 says that every morphism h : M → M can be expressed as h = g ◦ g , where g : M → N is an epimorphism and g : N → M is an embedding. The following theorem deals in more detail with this kind of decomposition. Theorem 2.36. Let h : M → M be a morphism. Moreover, let g1 : M → N1 , g2 : M → N2 be surjective morphisms and let g1 : N1 → M , g2 : N2 → M be embeddings such that h = g1 ◦ g1 = g2 ◦ g2 . Then there exists a unique morphism d : N1 → N2 such that g1 = d ◦ g2 , g2 = g1 ◦ d. Proof. Let us introduce a mapping d : N1 → N2 defined for a ∈ N1 by d(a ) = g2 (b ),

where b ∈ M, g1 (b ) = a .

(2.7)

In ordinary case, d is called a diagonal fill-in [97]. First, we prove that d is a well-defined morphism. Due to the surjectivity of g1 , for every a ∈ N1 there

80


is some b ∈ M such that g1 (b ) = a . Clearly, for g1 (b ) = g1 (b ) = a we have g1 (g1 (b )) = g1 (g1 (b )), thus it follows that g2 (g2 (b )) = g2 (g2 (b )) since g1 ◦ g1 = g2 ◦ g2 . But g2 is an embedding, that is g2 (b ) = g2 (b ), i.e. d is a well-defined mapping. For every n-ary f N1 ∈ F N1 and arbitrary elements a1 , . . . , an ∈ N1 there are b1 , . . . , bn ∈ M such that g1 (bi ) = ai for i = 1, . . . , n. Hence, f N2 d(a1 ), . . . , d(an ) = f N2 g2 (b1 ), . . . , g2 (bn ) = = g2 f M (b1 , . . . , bn ) = d f N1 (a1 , . . . , an ) since g1 f M (b1 , . . . , bn ) = f N1 g1 (b1 ), . . . , gn (bn ) = f N1 (a1 , . . . , an ), i.e. d is compatible with f N1 ∈ F N1 . Take a1 , a2 ∈ N1 . Thus, there are b1 , b2 ∈ M such that g1 (b1 ) = a1 , g1 (b2 ) = a2 . It readily follows that

a1 ≈N1 a2 = g1 (b1 ) ≈N1 g1 (b2 ) = g1 (g1 (b1 )) ≈M g1 (g1 (b2 )) =

= g2 (g2 (b1 )) ≈M g2 (g2 (b2 )) = g2 (b1 ) ≈N2 g2 (b2 ) = = d(a1 ) ≈N2 d(a1 ) , i.e. d is an embedding. Furthermore, for a , a ∈ N1 , b , b ∈ M such that a = g1 (b ), a = g1 (b ) we have g1 (a ) = g1 (g1 (b )) = g2 (g2 (b )) = g2 (d(a )) = (d ◦ g2 )(a ) , g2 (b ) = d(a ) = d(g1 (b )) = (g1 ◦ d)(b ) , i.e. d is a morphism satisfying the required conditions. The situation is depicted in Fig. 2.7. It suffices to check the uniqueness of d. So, let d : N1 → N2 be a morphism such that g1 = d ◦ g2 , g2 = g1 ◦ d . Clearly, d ◦ g2 = g1 = d ◦ g2 . Since g2 is an embedding, we have d = d . Definition 2.37. Suppose M is an L-algebra and φ, θ ∈ ConL (M), θ ⊆ φ. Then we let φ/θ denote an L-relation on M/θ defined by (φ/θ) [a ]θ , [b ]θ = φ(a , b ) (2.8) for all a , b ∈ M . Theorem 2.38. Let φ, θ ∈ ConL (M), θ ⊆ φ. Then φ/θ ∈ ConL (M/θ). Proof. Clearly, φ/θ is an L-equivalence. For every a , b ∈ M we have [a ]θ ≈M [b ]θ = θ(a , b ) ≤ φ(a , b ) = (φ/θ) [a ]θ , [b ]θ . Take n-ary f M/θ ∈ F M/θ and [a1 ]θ , [b1 ]θ . . . , [an ]θ , [bn ]θ ∈ M/θ. We have n n i=1 (φ/θ) [ai ]θ , [bi ]θ = i=1 φ(ai , bi ) ≤ M ≤ φ f (a1 , . . . , an ), f M (b1 , . . . , bn ) = = (φ/θ) f M (a1 , . . . , an ) θ , f M (b1 , . . . , bn ) θ = = (φ/θ) f M/θ ([a1 ]θ , . . . , [an ]θ ), f M/θ ([b1 ]θ , . . . , [bn ]θ ) . Altogether, φ/θ ∈ ConL (M/θ).


81

Theorem 2.39 (second isomorphism theorem). Suppose M is an Lalgebra and φ, θ ∈ ConL (M), θ ⊆ φ. Then the mapping h : (M/θ)/(φ/θ) → M/φ defined by h [[a ]θ ]φ/θ = [a ]φ is an isomorphism. Proof. For every [[a ]θ ]φ/θ , [[b ]θ ]φ/θ ∈ (M/θ)/(φ/θ) we have, [[a ]θ ]φ/θ ≈(M/θ)/(φ/θ) [[b ]θ ]φ/θ = (φ/θ) [a ]θ , [b ]θ = = φ(a , b ) = [a ]φ ≈M/φ [b ]φ . Since for all a ∈ M we have h [[a ]θ ]φ/θ = [a ]φ , h is a surjective ≈-morphism. Take any n-ary function f (M/θ)/(φ/θ) ∈ F (M/θ)/(φ/θ) and arbitrary elements [[a1 ]θ ]φ/θ , . . . , [[an ]θ ]φ/θ ∈ (M/θ)/(φ/θ). h f (M/θ)/(φ/θ) [[a1 ]θ ]φ/θ , . . . ,[[an ]θ ]φ/θ = h f M/θ [a1 ]θ , . . . ,[an ]θ φ/θ = = h [f M (a1 , . . . , an )]θ φ/θ = f M (a1 , . . . , an ) φ = = f M/φ [a1 ]φ , . . . , [an ]φ = f M/φ h [[a1 ]θ ]φ/θ , . . . , h [[an ]θ ]φ/θ . Thus h : (M/θ)/(φ/θ) → M/φ is an isomorphism.

Definition 2.40. Suppose M is an L-algebra, N ⊆ M and θ ∈ ConL (M). Let N θ = {a ∈ M | [a ]θ ∩ N = ∅}, put Nθ = N θ M , and let θ|N denote the restriction of θ to N . In other words, N θ is a union of congruence classes being incident with N . Lemma 2.41. Let N be a subalgebra of M, θ ⊆ ConL (M). Then (i) N θ is the universe of Nθ , (ii) θ|N ∈ ConL (N). Proof. (i): Take any n-ary f N ∈ F N and a1 , . . . , an ∈ N θ . By Definition 2.40 there are elements b1 , . . . , bn ∈ N for which we have a1 ∈ [b1 ]θ , . . . , an ∈ [bn ]θ , in other words θ(a1 , b1 ) = · · · = θ(an , bn ) = 1. Hence, it follows that θ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) = 1. Since the result f M (b1 , . . . , bn ) is in N , we have f M (a1 , . . . , an ) ∈ N θ . (ii) is easy to check. Theorem 2.42 (third isomorphism theorem). Let N be a subalgebra of an L-algebra M, θ ⊆ ConL (M). Then N/(θ|N ) ∼ = Nθ /(θ|N θ ). Proof. Take a mapping h : N/(θ|N ) → N θ /(θ|N θ ), where h([a ]θ|N ) = [a ]θ|N θ for all a ∈ M . It is routine to verify that h is the required isomorphism. Lemma 2.43. Let M and N be L-algebras such that M ∈ Sub(N) and let θ ∈ ConL (M), φ ∈ ConL (N), θ ⊆ φ|M . Then a mapping h : M/θ → N/φ defined by h([a ]θ ) = [a ]φ is a morphism.

82


Proof. The claim is a consequence of Theorem 2.39 and Theorem 2.42. Indeed, let h(φ|M )/θ : M/θ → (M/θ)/((φ|M )/θ) be the natural morphism, where (φ|M )/θ ∈ ConL (M/θ) is defined by (2.8). From Theorem 2.39 it follows that there is an isomorphism g : (M/θ)/((φ|M )/θ) → M/(φ|M ) such that we have g(h(φ|M )/θ ([a ]θ )) = [a ]φ|M for every a ∈ M . Due to Theorem 2.42, there is an isomorphism g : M/(φ|M ) → Mφ /(φ|M φ ), where g ([a ]φ|M ) = [a ]φ|M φ . Obviously, Mφ /(φ|M φ ) ∈ Sub(N/φ). Put h = h(φ|M )/θ ◦ g ◦ g . Clearly, we have h([a ]θ ) = g ([a ]φ|M ) = [a ]φ|M φ = [a ]φ for all a ∈ M . Lemma 2.44. Let h : M → N be a morphism and let φ ∈ ConL (M) such that φ ⊆ θh . Then h = hφ ◦ g, where g : M/φ → N is a uniquely determined morphism. Proof. Since φ ⊆ θh from Lemma 2.43 it follows that there is a morphism g : M/φ → M/θh , where g [a ]φ = [a ]θh , a ∈ M . Moreover, Theorem 2.35 yields that there is an embedding g : M/θh → N, where g [a ]θh = h(a ), [a ]θh ∈ M/θh . Thus, we can put g = g ◦ g . Clearly, h(a ) = g [a ]θh = g g [a ]φ = g [a ]φ = g(hφ (a )) = (hφ ◦ g)(a ) for every a ∈ M , i.e. h = hφ ◦ g. It remains to check the uniqueness of g. Since hφ is surjective, hφ ◦ g = h = hφ ◦ k implies g = k. For θ1 , θ2 ∈ ConL (M) with θ1 ⊆ θ2 we put [θ1 , θ2 ] = {φ ∈ ConL (M) | θ1 ⊆ φ ⊆ θ2 } .

(2.9)

It is easily seen that [θ1 , θ2 ] is a complete sublattice of ConL (M). Theorem 2.45. Let an L-algebra M and θ ∈ ConL (M) be given. Then the mapping h : [θ, M ×M ] → ConL (M/θ) defined by h(φ) = φ/θ is a lattice isomorphism. Proof. For every φ1 , φ2 ∈ [θ, M ×M ] we have φ1 ⊆ φ2 iff φ1 /θ ⊆ φ2 /θ iff to prove thath is a bijective mapping. h(φ1 ) ⊆ h(φ2 ). Now it is sufficient Suppose that (φ1 /θ) [a ]θ , [b ]θ = (φ2 /θ) [a ]θ , [b ]θ for every a , b ∈ M . Then also φ1 (a , b ) = φ2 (a , b ) for all a , b ∈ M . Thus, the mapping h is injective. To show surjectivity, first take any ϕ ∈ ConL (M/θ) and define an L-relation φ on M by φ(a , b ) = ϕ [a ]θ , [b ]θ . We will show that φ is an L-equivalence on M for which θ ⊆ φ and h(φ) = ϕ. First, we check conditions (i), (ii) of Definition 2.11. Clearly, φ is an Lequivalence relation and for every a , b ∈ M we have a ≈M b ≤ [a ]θ ≈M/θ [b ]θ ≤ ϕ [a ]θ , [b ]θ = φ(a , b ) , which is equivalent to (ii), see Lemma 1.82. To check condition (i), take any n-ary f M ∈ F M and a1 , b1 , . . . , an , bn ∈ M . Now the following inequality holds,

2.3 Direct and Subdirect Products

n i=1

n

83

i=1 ϕ [ai ]θ , [bi ]θ ≤ M/θ ≤ϕ f [a1 ]θ , . . . , [an ]θ , f M/θ [b1 ]θ , . . . , [bn ]θ = = ϕ f M (a1 , . . . , an ) θ , f M (b1 , . . . , bn ) θ = = φ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) .

φ(ai , bi ) =

Thus, φ ∈ [θ, M ×M ] ⊆ ConL (M). Clearly, h(φ) = φ/θ = ϕ. To sum up, h is a surjective mapping. Hence, h is a lattice isomorphism.

2.3 Direct and Subdirect Products Direct product is a basic construction yielding larger L-algebras from collections of input L-algebras. Subdirect product can be thought of as a derived construction since it is a special subalgebra of a direct product. This section introduces basic properties of both constructions. It is worth to note that some results presented in this section will generalize only for certain subclasses of residuated lattices as the structures of truth degrees. Definition 2.46. Let I be an index product of aQfamily set. A direct {Mi | i ∈ I} of L-algebras Mi = Mi , ≈Mi , F Mi is an L-algebraQ i∈I Mi Q ofQtype F such that M = i∈I Mi and for every n-ary function f i∈I Mi ∈ F i∈I Mi and a1 , . . . , an ∈ M we have f

Q

i∈I Mi

(a1 , . . . , an )(i) = f Mi (a1 (i), . . . , an (i)), Q

for all i ∈ I and the L-equality ≈ Q

(a ≈

i∈I Mi

(2.10)

i∈I Mi

b) =

is defined by Mi b (i) i∈I a (i) ≈

(2.11)

for all a , b ∈ M . Q Remark 2.47. (1) Note that if I = ∅, then | i∈I M i | = 1. Q (2) A direct product as defined above is a well-defined L-algebra. Suppose i∈I Mi is a direct product of a family {Mi | i ∈ I} of L-algebras. L-relation Q Q ≈ i∈I Mi is evidently reflexive and symmetric. Now for every a , b , c ∈ i∈I M i we have Q Q a ≈ i∈I Mi b ⊗ b ≈ i∈I Mi c = Mi = b (i) ⊗ j∈I b (j) ≈Mj c (j) ≤ i∈I a (i) ≈ ≤ i,j∈I a (i) ≈Mi b (i) ⊗ b (j) ≈Mj c (j) ≤ ≤ i∈I a (i) ≈Mi b (i) ⊗ b (i) ≈Mi c (i) ≤ Q ≤ i∈I a (i) ≈Mi c (i) = a ≈ i∈I Mi c . Q

Q

i∈I Mi b = 1 we have a (j) ≈Mj b (j) = Hence, ≈ i∈I Mi is transitive. Q For a ≈ Q M i b = 1 implies a = b . Altogether, ≈ i∈I Mi 1 for every j ∈ I, thus a ≈ i∈I

84


Fig. 2.12. Example of direct product Q

is an L-equality relation. The compatibility ofQfunctions with ≈ i∈I Mi follows Q M M i ∈ F i∈I i and arbitrary from the definition. Q Indeed, for every n-ary f i∈I a1 , b1 , . . . , an , bn ∈ i∈I M i , we have Q Q a1 ≈ i∈I Mi b1 ⊗ · · · ⊗ an ≈ i∈I Mi bn = nj=1 i∈I aj (i) ≈Mi bj (i) ≤ n ≤ i∈I j=1 aj (i) ≈Mi bj (i) ≤ ≤ i∈I f Mi (a1 (i), . . . , an (i)) ≈Mi f Mi (b1 (i), . . . , bn (i)) = Q Q Q = f i∈I Mi (a1 , . . . , an ) ≈ i∈I Mi f i∈I Mi (b1 , . . . , bn ) . Q Q Hence, ≈ i∈I Mi is an L-equality compatible with all functions of i∈I Mi . Example 2.48. Let L be a three-element L ukasiewicz chain. The direct product of two linearly ordered lattices with L-equality is depicted in Fig. 2.12. The L-equalities are depicted the same way as in Example 2.3. Q of a family {Mi | i ∈ I} of Definition 2.49. Let i∈I Mi be a direct product Q Mj , where πj (a ) = L-algebras. For every j ∈ I a mapping πj : i∈I Mi → Q a (j) is called a projection map on the j-th coordinate of i∈I Mi or, shortly, Q a j-th projection of i∈I Mi . Q Theorem 2.50. For a direct product i∈I Mi of a family {Mi | i ∈ I} of Lalgebras, the j-th projection πj is an epimorphism for every j ∈ I. Proof. Q Take any j ∈ I. For arbitrary a ∈ Mj we have πj (b ) = a for every b ∈ i∈I Mi , where b (j) = a . Thus, πj is a surjective mapping. Moreover, Q a ≈ i∈I Mi b = i∈I a (i) ≈Mi b (i) ≤ a (j) ≈Mj b (j) = πj (a ) ≈Mj πj (b ) . Q Q Q For any n-ary f i∈I Mi ∈ F i∈I Mi and a1 , . . . , an ∈ i∈I Mi we have, Q Q πj f i∈I Mi (a1 , . . . , an ) = f i∈I Mi (a1 , . . . , an )(j) = = f Mj a1 (j), . . . , an (j) = f Mj πj (a1 ), . . . , πj (an ) . Hence, πj is an epimorphism.


M

h

85

i∈I Mi

πi

hi Mi

Fig. 2.13. Direct product property

Theorem 2.51. For every family of morphisms Q {hi : M → Mi | i ∈ I} there is a uniquely determined morphism h : M → i∈I Mi such that h ◦ πi = hi for every i ∈ I. Q Proof. Let us have a mapping h : M → i∈I M i defined by h(a )(i) = hi (a ) for every i ∈ I, a ∈ M . We have a ≈M b ≤ hi (a ) ≈Mi hi (b ) since hi is supposed to be a morphism for every i ∈ I. Thus, Q a ≈M b ≤ i∈I hi (a ) ≈Mi hi (b ) = h(a ) ≈ i∈I Mi h(b ) . Furthermore, for n-ary f M ∈ F M , and a1 , . . . , an ∈ M we have h f M (a1 , . . . , an ) (i) = hi f M (a1 , . . . , an ) = f Mi hi (a1 ), . . . , hi (an ) = Q = f Mi h(a1 )(i), . . . , h(an )(i) = f i∈I Mi h(a1 ), . . . , h(an ) (i) for every i ∈ I. Hence h is a morphism. Clearly, h ◦Qπi = hi for every i ∈ I. It suffices to check the uniqueness. Let g : M → i∈I Mi be a morphism satisfying g ◦ πi = hi for all i ∈ I. Hence, g ◦ πi = h ◦ πi for every i ∈ I. Thus, for arbitrary a ∈ M , we have g(a )(i) = h(a )(i) for all i ∈ I which implies g(a ) = h(a ). Hence, h is determined uniquely. Remark 2.52. The unique existence of morphism h described by Theorem 2.51 is Q occasionally referred to as the direct product property of family {πj : i∈I Mi → Mj | j ∈ I}, cf. [97], and is illustrated in in Fig. 2.13. Lemma 2.53. For a direct product M = M1 × M2 , we have (i) θπ1 ∧ θπ2 = ≈M , (ii) θπ1 ∨ θπ2 = M ×M , (iii) θπ1 ◦ θπ2 = θπ2 ◦ θπ1 .

(a , b ) = θπ1 (a , b ) ∧ θπ2 (a , b ). That is ∧ θ Proof. For (i), we have θ π π 1 2 M1 θπ1 ∧ θπ2 (a , bQ) = a (1) ≈ b (1) ∧ a (2) ≈M2 b (2), but this is exactly the M i for I = {1, 2}. Thus, (i) holds. definition of ≈ i∈I (ii): For any a , b ∈ M1 × M2 , we have θπ1 (a , a (1), b (2) ) = 1, and θπ2 (a (1), b (2) , b ) = 1 .

(2.12)

By transitivity, for each congruence θ ∈ ConL (M) containing both θπ1 and θπ2 we have θ(a , b ) = 1, whence θπ1 ∨ θπ2 = M ×M proving (ii). (iii): It is immediate that (θπ1 ◦ θπ2 )(a , b ) = 1 for every a , b ∈ M . Thus, θπ1 ◦ θπ2 = θπ1 ∨ θπ2 from which we get (iii).

86


Definition 2.54. A congruence θ ∈ ConL (M) is called a factor congruence, if there is a congruence θ∗ ∈ ConL (M) such that (i) θ ∧ θ∗ = ≈M , (ii) [a ]θ ∩ [b ]θ∗ = ∅ for all a , b ∈ M . The pair θ, θ∗ is called a pair of factor congruences on M. Remark 2.55. The proof of Lemma 2.53 yields that for every M = M1 × M2 the couple θπ1 , θπ2 is a pair of factor congruences on M. Moreover, condition (ii) of Definition 2.54 implies that for every a , b ∈ M there is some c ∈ M such that θ∗ (a , c ) ⊗ θ(c , b ) = 1. Hence, from (ii) it follows that θ ◦ θ∗ = θ∗ ◦ θ = θ ∨ θ∗ = M ×M , this is easy to check. Theorem 2.56. If θ, θ∗ is a pair of factor congruences on an L-algebra M, then M ∼ = M/θ × M/θ∗ . Proof. Put h(a ) = [a ]θ , [a ]θ∗ for all a ∈ M . For every a , b ∈ M , we have

a ≈M b = (θ ∧ θ∗ )(a , b ) = θ(a , b ) ∧ θ∗ (a , b ) = ∗

= [a ]θ ≈M/θ [b ]θ ∧ [a ]θ∗ ≈M/θ [b ]θ∗ = ∗

∗

= [a ]θ , [a ]θ∗ ≈M/θ×M/θ [b ]θ , [b ]θ∗ = h(a ) ≈M/θ×M/θ h(b ) . Condition (ii) of Definition 2.54 yields that for every a , b ∈ M there is some c ∈ M such that c ∈ [a ]θ ∩ [b ]θ∗ . That is, c ∈ [a ]θ and c ∈ [b ]θ∗ , i.e. [a ]θ = [c ]θ , [b ]θ∗ = [c ]θ∗ . Therefore, we obtain h(c ) = [c ]θ , [c ]θ∗ = [a ]θ , [b ]θ∗ , i.e. h is surjective. Finally, we have to check the compatibility with functions. Take any n-ary f M ∈ F M and arbitrary a1 , . . . , an ∈ M . We have, h f M (a1 , . . . , an ) = f M (a1 , . . . , an ) θ , f M (a1 , . . . , an ) θ∗ = ∗ = f M/θ ([a1 ]θ , . . . , [an ]θ ), f M/θ ([a1 ]θ∗ , . . . , [an ]θ∗ ) = ∗ = f M/θ×M/θ [a1 ]θ , [a1 ]θ∗ , . . . , [an ]θ , [an ]θ∗ = ∗ = f M/θ×M/θ h(a1 ), . . . , h(an ) . Hence, h is an isomorphism, M ∼ = M/θ × M/θ∗ .

Remark 2.57. The previous theorem yields that every L-algebra is isomorphic to a direct product. Namely, since ≈M , M ×M is a pair of factor congruences on M, whence M ∼ = M/≈M × M/(M ×M ). Definition 2.58. An L-algebra M is said to be directly indecomposable if M is not isomorphic to a direct product of two non-trivial L-algebras. Theorem 2.59. An L-algebra is directly indecomposable iff ≈M , M ×M is the only one pair of its factor congruences. Moreover, every finite L-algebra is isomorphic to a direct product of directly indecomposable L-algebras.


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∼ N × N iff there is a Proof. Remark 2.55 and Theorem 2.56 yield that M = ∗ ∼ ∗ pair of factor congruences θ, θ on M, such that N ∼ = M/θ and N = M/θ . M So clearly, M is directly indecomposable iff ≈ , M ×M is the only one pair of factor congruences of M. Trivial L-algebras are directly indecomposable. Any finite non-trivial Lalgebra M either is directly indecomposable or M ∼ = N × N where |N | < n and |N | < n. Thus, by induction one can prove that M is isomorphic to a direct product of directly indecomposable L-algebras. Definition 2.60. If a , b ∈ M and h : M, ≈M → N, ≈N is a mapping, we N say a and b if h(a ) ≈ h(b ) < 1. A family of mappings h separates that M Ni → Ni , ≈ | i ∈ I separates pointsiff for every hi : M, ≈ a , b ∈ M with a ≈M b < 1 there is an index j ∈ I such that hj : M, ≈M → Nj , ≈Nj separates a and b . Lemma 2.61. For a family of morphisms {hi : M → Mi | i ∈ I}, the following conditions are equivalent: (i) the family {hi : M → M Qi | i ∈ I} separates points, (ii) a mapping h : M → i∈I Mi , where h(a )(i) = hi (a ) for every i ∈ I, a ∈ M is an injective morphism, (iii) 1 i∈I θhi = {a , a | a ∈ M }. Proof. “(i) ⇒ (ii)”: Theorem 2.51 yields that h is a morphism. For a = b we have a ≈M b < 1. Thus, if {hi : M → Mi | i ∈ I} separates points, then there is some i0 ∈ I such that hi0 (a ) ≈Mi0 hi0 (b ) < 1, which implies Q is injective. h(a ) ≈ i∈I Mi h(b ) < 1, i.e. h (a , a ) = 1, thus {a , a | a ∈ M } ⊆ “(ii) ⇒ (iii)”: Evidently, θ h i i∈I 1 i∈I θhi . Let us assume that (ii) holds. Then for a = b we obtain Q 1 > h(a ) ≈ i∈I Mi h(b ) = i∈I hi (a ) ≈Mi hi (b ) = = i∈I θhi (a , b ) = i∈I θhi (a , b ) , (iii). that is a , b ∈ 1 i∈I θhi . Hence, (ii) implies ( “(iii) ⇒ (i)”: Let a = b , i.e. θ a , b ) < 1 by (iii). This gives i∈I hi Mi hi (b ) < 1 , i∈I hi (a ) ≈ i.e., there is i0 ∈ I such that hi0 (a ) ≈Mi0 hi0 (b ) < 1, showing that hi0 separates a and b . Therefore, {hi : M → Mi | i ∈ I} separates points. Lemma Q 2.62. Let {hi : M → Mi | i ∈ I} be a family of morphisms and let h : M → i∈I Mi denote a morphism, where h(a )(i) = hi (a ) for every i ∈ I, a ∈ M . The morphism h is an embedding iff M i∈I θhi (a , b ) = a ≈ b . Q Proof. Recall that h(a ) ≈ i∈I Mi h(b ) = i∈I θhi (a , b ) for all a , b ∈ M . The rest is evident.

88


∗

∗

∗

In bivalent case, every algebra can be represented by a subdirect product of subdirectly irreducible algebras. In fuzzy case, this is not true in general. In the subsequent development, we introduce a sufficient condition for subdirect representation. However, unlike the bivalent case, we will also show that there are L-algebras, which are not subdirectly representable. Definition 2.63. Let M be an L-algebra of type F . The L-algebra M is said to be a subdirect product of a family {Mi | i ∈ I} of L-algebras of type F if Q (i) M is a subalgebra of i∈I Mi , (ii) πi (M ) = Mi for every i ∈ I. Q An embedding h : M → i∈I Mi is called subdirect if h(M) is a subdirect product of the family {Mi | i ∈ I}. Lemma 2.64. If θQi ∈ ConL (M) for every i ∈ I and i∈I θi = ≈M , then the mapping g : M Q → i∈I M/θi , where g(a )(i) = [a ]θi is a subdirect embedding. If h : M → i∈I Mi is a subdirect embedding, then there is a family of congruences {θi ∈ ConL (M) | i ∈ I} such that i∈I θi = ≈M and Mi ∼ = M/θi for every i ∈ I. Proof. Since g defined by g(a )(i) = [a ]θi is a morphism, using Lemma 2.62 we can deduce that g is an embedding. Moreover, g(M )(i) = {[a ]θi | a ∈ M } = M/θi for every i ∈ I, thus g isQa subdirect embedding. Furthermore, let h : M → i∈I Mi be a subdirect embedding. Put θi = θhi for every i ∈ I (recall that hi : M → Mi , where hi (a ) = h(a )(i) is a surjective morphism). Now by applying Lemma 2.62 one can conclude that i∈I θi = ≈M . Since every hi is surjective, Theorem 2.35 yields that Mi ∼ = M/θi for every i ∈ I. Definition 2.65. An L-algebra Q M is subdirectly irreducible if for every subdirect embedding h : M → i∈I Mi there is an index j ∈ I such that h ◦ πj : M → Mj is an isomorphism. Theorem 2.66. An L-algebra M is subdirectly irreducible iff M either is trivM ial, congruence in ConL (M) − {≈ }. In the latter case or there is a least M ConL (M) − {≈ } is a principal congruence and ConL (M), ⊆ contains exactly one atom. Proof. “⇒”: Suppose, by contradiction, that M is not a trivial L-algebra and ConL (M) − {≈M } does not have the least element. Then ConL (M) − {≈M } = ≈M . M Put I = Con QL (M)−{≈ }. Using Lemma 2.64, there is a subdirect embedding h : M → θ∈I M/θ, where h(a )(θ) = [a ]θ for every a ∈ M and θ ∈ I. Moreover, for every congruence θ ∈ I we have θ ⊃ ≈M . Thus, there are a , b ∈ M such that a ≈M b < θ(a , b ), i.e.


89

a ≈M b < θ(a , b ) = [a ]θ ≈M/θ [b ]θ = = hθ (a ) ≈M/θ hθ (b ) = (h ◦ πθ )(a ) ≈M/θ (h ◦ πθ )(b ) . Hence, for every θ ∈ I the mapping h ◦ πθ : M → M is not an isomorphism. Consequently, the L-algebra M is not subdirectly irreducible. “⇐”: A trivial L-algebra MQis subdirectly irreducible. Indeed, for every subdirect embedding h : M → i∈I Mi , each L-algebra Mi must be trivial, because πi (h(M )) = πi (h({a })) = Mi holds for every i ∈ I by virtue of the assumption. Hence, h ◦ πi : M → Mi is an isomorphism for all i ∈ I. Altogether, So suppose M is non-trivial and M is subdirectlyM irreducible. let θ = ConL (M) − {≈ } > ≈M , i.e. θ is the least congruence in M M ConL (M) − {≈ Q }. Then there are a , b ∈ M such that θ(a , b ) > a ≈ b . Let h : M → i∈I Mi be a subdirect embedding. Now we have, a ≈M b = i∈I h(a )(i) ≈Mi h(b )(i) < θ(a , b ) . Therefore, it follows that there is an index i ∈ I such that h(a )(i) ≈Mi h(b )(i) θ(a , b ), that is, (h ◦ πi )(a ) ≈Mi (h ◦ πi )(b ) θ(a , b ), so θh◦πi θ. Since θ is the least congruence in ConL (M) − {≈M }, we readily obtain θh◦πi = ≈M . Thus, h ◦ πi : M → Mi is an isomorphism and the L-algebra M is subdirectly irreducible. If θ is the least congruence in ConL (M) − {≈M }, then there are elements a , b ∈ M such that θ(a , b ) > a ≈M b . Obviously, we have θ(θ(a ,b )/a , b ) ⊆ θ. The converse inclusion holds since θ(θ(a ,b )/a , b ) > a ≈M b and θ is the least congruence with θ(a , b ) > a ≈M b . Thus, we have θ(θ(a ,b )/a , b ) = θ, i.e. θ is a principal congruence. Theorem 2.67 (representation theorem). Let M be a non-trivial Lalgebra such that for every distinct a , b ∈ M there exists θa ,b ∈ ConL (M), where θa ,b (a , b ) = a ≈M b , and θa ,b is ∧-irreducible in ConL (M). Then M is isomorphic to a subdirect product of a family of subdirectly irreducible L-algebras. Proof. Let θa ,b be ∧-irreducible in ConL (M). As a result, [θa ,b , M ×M ] − {θa ,b } has the least element. Now from Theorem 2.45 andTheorem 2.66 it follows that M/θa ,b is subdirectly irreducible. We have a ,b ∈I θa ,b = ≈M , where I = {a , b | a , b ∈ M and a = b }. Thus, Q Lemma 2.64 yields that there is a subdirect embedding h : M → a ,b ∈I M/θa ,b , where h(c )(a , b ) = [c ]θa ,b for all a , b , c ∈ M , a = b . Hence, M is isomorphic to a subdirect product of subdirectly irreducible L-algebras. The previous representation theorem is sort of abstract. It only delimits a condition under which an L-algebra M has a subdirect representation. In the following we will focus on the existence of suitable ∧-irreducible elements in ConL (M). First, we prove a technical lemma.

90


Lemma 2.68. Let {θi | i ∈ I} ⊆ ConL (M) be a directed system of congru∈ {θi | i ∈ I}, there is θi , i ∈ I ences, i.e. for any finite number θi1 , . . . , θik such that θij ⊆ θi , for all j = 1, . . . , k. Then i∈I θi = i∈I θi . Proof. Clearly, i∈I θi ⊆ i∈I θi . We have to show “⊇”: θi1 (a , c1 ) ⊗ · · · ⊗ θik (ck−1 , b ) ≤ i1 ,...,ik ∈I i∈I θi (a , b ) = c1 ,...,ck−1 ∈M θi (a , c1 ) ⊗ · · · ⊗ θi (ck−1 , b ) ≤ ≤ i∈I c1 ,...,ck−1 ∈M ≤ i∈I θi (a , b ) = i∈I θi (a , b ) holds for every a , b ∈ M .

Lemma 2.69. Suppose M is a non-trivial L-algebra. Then for every distinct elements a , b ∈ M : (i) there is a maximal θa ,b ∈ ConL (M) such that θa ,b (a , b ) = a ≈M b ; (ii) if a ≈M b is ∧-irreducible in L then θa ,b is ∧-irreducible in ConL (M). Proof. (i): Let Ia ,b = θ ∈ ConL (M) | θ(a , b ) = a ≈M b . It is easy to observe that Ia ,b , ⊆ is a partially ordered set, Ia ,b ⊆ ConL (M). Moreover, M I ⊆ Ia ,b of congruences we have a ,b = ∅. For every chain ≈ ∈ Ia,b implies I M ( ( θ a , b ) = θ a , b ) = a ≈ b due to Lemma 2.68. That is, i i i∈I i∈I θ ∈ I . Hence, every chain I ⊆ I is bounded from above. Now using i a , b a , b i∈I Zorn Lemma it readily follows that there is a maximal element θa ,b ∈ Ia ,b . (ii): Put J = [θa ,b , M ×M ] − {θa ,b } and suppose a ≈M b to be an ∧irreducible element of L. By maximality of θa ,b , θ(a , b ) > θa ,b (a , b ) for each θ ∈ J. Thus, it follows that J (a , b ) = θ∈J θ(a , b ) > a ≈M b . As a consequence, J ∈ J, i.e. θa ,b is ∧-irreducible in ConL (M). If the lattice part of L is a finite chain then each 1 = a ∈ L is ∧-irreducible. Thus, for finite linearly ordered residuated lattices we can use Theorem 2.67 and Lemma 2.69 to obtain the following consequence. Corollary 2.70. If L is a finite chain then every non-trivial L-algebra is isomorphic to a subdirect product of subdirectly irreducible L-algebras. Remark 2.71. (1) In [16] we have presented more general criterion for subdirect representation. This criterion is, however, more technical than the one given by Lemma 2.69. Let M be an L-algebra. If for every distinct b , b ∈ M we have a b ≈M b < (2.13) a > b ≈M b θ( /b , b ) (b , b ), then M is isomorphic to a subdirect product of subdirectly irreducible Lalgebras. Indeed, take a maximal θb ,b ∈ ConL (M) with θb ,b (b , b ) = b ≈M

2.4 Terms, Term L-Algebras

91

b in ConL (M) due to (2.13). Namely, θb ,b ∨ . Clearly, θb ,b is ∧-irreducible a θ( / b , b ) is the least congruence in [θb ,b , M ×M ] − {θb ,b }. a > b ≈M b Now apply Theorem 2.67. It is easily seen that if b ≈M b is ∧-irreducible in L, then (2.13) holds trivially. (2) The subdirect representation does not pass for every L. In other words, for certain structures of truth degrees, there are still L-algebras which cannot be isomorphic to a subdirect product of subdirectly irreducible L-algebras. An example follows.

Example 2.72. Take a complete residuated lattice L on the real unit interval [0, 1]. Let us have an L-algebra M = M, ≈M , ∅ of type F = ∅, where M = {a , b }, and a ≈M b = 0. It is easy to see that every reflexive and symmetric binary L-relation θ on M is a congruence on M since M is a two-element set with a ≈M b = 0, and F M = ∅. Obviously, every congruence θ ∈ ConL (M) is uniquely determined by the truth degree θ(a , b ) ∈ L. Thus, there is a one-to-one correspondence between congruences from ConL (M) and truth degrees from L. Moreover, ConL (M) is isomorphic to the lattice part of L. For θ ∈ ConL (M), Theorem 2.45 yields ConL (M/θ) ∼ = [θ, M ×M ] ∼ = [c, 1], where θ(a , b ) = c. Since [c, 1] − {c} does not have theQleast element, M/θ is subdirectly reducible. As a consequence, if h : M → i∈I Mi is a subdirect embedding, then every Mi ∼ = M/θh◦πi is subdirectly reducible, i.e. M is not isomorphic to a subdirect product of subdirectly irreducible L-algebras.

2.4 Terms, Term L-Algebras The notion of a term is defined as usual. As we will see, results known from the ordinary case generalize naturally to fuzzy case. In addition to that, there are new natural properties in fuzzy case which are degenerate in the ordinary case (cf. Theorem 2.76 and Remark 2.77). Definition 2.73. Let X be a set of variables. Let F be a type such that X ∩ F = ∅. The set T (X) of terms of type F is the smallest set such that (i) X ⊆ T (X), (ii) If f ∈ F , f is n-ary and t1 , . . . , tn ∈ T (X), then f (t1 , . . . , tn ) ∈ T (X). Every t ∈ T (X) is called a term of type F over X. Definition 2.74. Let t be a term of type F over X. For a variable x ∈ X let |t|x denote the number of occurrences of x in term t. That is, if t is a variable, then 1 for t = x , |t|x = 0 otherwise , if t is of the form f (t1 , . . . , tn ), then |t|x = |t1 |x +· · ·+|tn |x . Variable x ∈ X has an occurrence in t ∈ T (X) if |t|x > 0. For t ∈ T (X) we write t(x1 , . . . , xk )

92


instead of t to indicate that the variables occurring in t are among x1 , . . . , xk ∈ X. A term t is n-ary if the number of variables appearing explicitly in t is at most n. For t ∈ T (X) we define var(t) ⊆ X by var(t) = {x ∈ X | |t|x > 0}. Definition 2.75. Given a term t(x1 , . . . , xn ) of type F over X and given an L-algebra M of type F let tM denote a mapping tM: M n → M such that (i) if t is a variable xi then tM (a1 , . . . , an ) = ai , (ii) if t is of the form f (t1 , . . . , tk ) then M tM (a1 , . . . , an ) = f M tM 1 (a1 , . . . , an ), . . . , tk (a1 , . . . , an ) . Mapping tM is called a term function on M corresponding to term t. Theorem 2.76. Let t(x1 , . . . , xn ) be a term of type F over X. Then for every L-algebra M we have the following properties: (i) θ(a1 , b1 )|t|x1 ⊗ · · · ⊗ θ(an , bn )|t|xn ≤ θ tM (a1 , . . . , an ), tM (b1 , . . . , bn ) for every θ ∈ ConL (M) and arbitrary a1 ,b1 , . . . , an , bn ∈ M ; (ii) h tM (a1 , . . . , an ) = tN h(a1 ), . . . , h(an ) for every morphism h : M → N and arbitrary a1 , . . . , an ∈ M . Proof. The first statement can be proved using structural induction. Take any t ∈ T (X) and θ ∈ ConL (M). If t is a variable xi , 1 ≤ i ≤ n, then (i) holds trivially. Suppose that t is of the form f (t1 , . . . , tk ) and (i) holds for every t1 , . . . , tk . Now it follows that θ(a1 , b1 )|t|x1 ⊗ · · · ⊗ θ(an , bn )|t|xn = k = i=1 θ(a1 , b1 )|ti |x1 ⊗ · · · ⊗ θ(an , bn )|ti |xn ≤ k M ≤ i=1 θ tM i (a1 , . . . , an ), ti (b1 , . . . , bn ) ≤ ≤ θ tM (a1 , . . . , an ), tM (b1 , . . . , bn ) .

Hence, (i) holds. (ii) follows from the ordinary case.

Remark 2.77. (1) Estimation (i) of Theorem 2.76 is degenerate and has no interesting meaning in the ordinary case. In fuzzy case, however, (i) provides a non-trivial (possibly numerical, when truth degrees are numbers) estimation of similarity (degree of equivalence) of results of term functions which can be interpreted as describing sensitivity of a term function. (2) Since ≈M is a congruence relation, by applying (i) of Theorem 2.76 to M ≈ we obtain (a1 ≈M b1 )|t|x1 ⊗ · · · ⊗ (an ≈M bn )|t|xn ≤ ≤ tM (a1 , . . . , an ) ≈M tM (b1 , . . . , bn ) .

(2.14) |t|xi

and (3) If ⊗ is ∧, the powers |t|xi can be removed from θ(ai , bi ) (ai ≈M bi )|t|xi . This has the following consequence. In general, the estimation describing sensitivity of a term function depends on the structure of the corresponding term, hence the number of occurrences of variables is important. If ⊗ = ∧, the structure of the term does not play any role.


93

Definition 2.78. Given a type F and a set of variables X, if T (X) = ∅ then the term L-algebra of type F over X, is an L-algebra T(X) = T (X), ≈T(X) , F T(X) of type F with functions defined by f T(X) (t1 , . . . , tn ) = f (t1 , . . . , tn ) for any n-ary f defined by

T(X)

∈F

T(X)

and t1 , . . . , tn ∈ T (X), and L-equality ≈T(X)

T(X)

t≈

(2.15)

t =

for t = t , otherwise .

1 0

(2.16)

for all t, t ∈ T (X). The set X is called a set of free generators of T(X). Theorem 2.79. Let T(X) be a term L-algebra of type F in variables X. Every mapping h : X → M , where M = M, ≈M , F M is an L-algebra of type F , can be extended to a morphism h : T(X) → M, where h(x) = h (x) for all x ∈ X. Proof. The homomorphic extension h : T(X) → M can be defined induc tively by h (x) = h(x) for every variable x ∈ X and h (f (t1 , . . . , tn )) = M f h (t1 ), . . . , h (tn ) for every n-ary f ∈ F and terms t1 , . . . , tn ∈ T (X). Since ≈T(X) is a crisp identity relation, h is evidently an ≈-morphism, the rest is evident. Hence, h is a homomorphic extension of h. Example 2.80. Homomorphic extension h described by Theorem 2.79 can be used to define important structural notions which involve terms. In the following we show examples. (1) Let M be an L-algebra, a1 , . . . , an ∈ M . For X = {x1 , . . . , xn }, put h(xi ) = ai (i = 1, . . . , n). One can see that h : T (X) → M is a mapping such that h (t) = tM (a1 , . . . , an ) for any term t(x1 , . . . , xn ). Hence, h (t) is the value of term function tM applied to a1 , . . . , an . (2) For each term L-algebra T(X) we can consider an L-algebra M of the same type such that M = N0 , ≈M is the crisp L-equality on M , and for each n-ary f M ∈ F M we have f M (a1 , . . . , an ) = a1 + · · · + an (a1 , . . . , an ∈ M ). Now introduce h : X → M by putting h(x) = 1, h(y) = 0 if x = y. Then h (t) = |t|x for each t ∈ T (X). (3) Analogously as in (2), for T(X) consider M with M = 2X , ≈M being the crisp L-equality on M , f M (a1 , . . . , an ) = a1 ∪ · · · ∪ an (a1 , . . . , an ∈ M ). For the homomorphic extension h of mapping h : X → M where h(x) = {x} (x ∈ X), we have h (t) = var(t). ∗

∗

∗

In the rest of this section, we illustrate the role of term functions by providing a description of two notions introduced above. In the first case, the theorem we prove results by a straightforward generalization of the corresponding theorem from the ordinary case. In the second case, however, the theorem and its proof in fuzzy setting is technically considerably more complicated compared to the ordinary case.

94


Theorem 2.81. Let M be an L-algebra and N ⊆ M . Then for any set X of variables satisfying |N | ≤ |X| we have (2.17) [N ]M = tM (a1 , . . . , ak ) | t ∈ T (X), a1 , . . . , ak ∈ N . Proof. Denote the right hand side of (2.17) by N . Take a variable x ∈ T (X). For each a ∈ N we have xM (a ) = a ∈ N , i.e. N ⊆ N . Furthermore, we show that N is closed under all functions f M ∈ F M . Indeed, take an n-ary M f M ∈ F M and tM 1 (a11 , . . . , a1k1 ), . . . , tn (an1 , . . . , ankn ) ∈ N , where t1 , . . . , tn are terms t1 (x11 , . . . , x1k1 ), . . . , tn (xn1 , . . . , xnkn ). We may safely assume that aij = akl iff xij = xkl . Now we clearly have that M M f M (tM 1 (a11 , . . . , a1k1 ), . . . , tn (an1 , . . . , ankn )) = t (a1 , . . . , am )

for some a1 , . . . , am ∈ N and t = f (t1 , . . . , tn ) ∈ T (X). Thus, N is a subuniverse of M which contains N . Since [N ]M is the least subuniverse of M containing N , we have [N ]M ⊆ N . Conversely, each a ∈ N can be obtained by applying finitely many functions on finitely many elements from N , i.e., N ⊆ [N ]M . Hence, [N ]M = N , proving (2.17). The following theorem generalizes the well-known Mal’cev lemma describing principal congruences. Note that although on verbal level (and for the truth degree a = 1) the theorem reads the same way as in the ordinary case, the proof is technically much more involved in fuzzy setting. Theorem 2.82. Suppose M is an L-algebra of type F and let c , d ∈ M . For every set of variables X = {x, y1 , . . . , yk }, terms p1 , . . . , pn ∈ T (X) and arbitrary elements s1 , t1 , . . . , sn , tn ∈ M such that {sj , tj } = {c , d }, 1 ≤ j ≤ n and b , b , e1 , . . . , ek ∈ M let Π (b , b , p1 , . . . , pn , s , t , e ) denote n−1 M M M M b ≈M pM pi+1 (si+1 , e ) ⊗ pM 1 (s1 , e ) ⊗ n (tn , e ) ≈ b , i=1 pi (ti , e ) ≈ where e is an abbreviation for e1 , . . . , ek , s is an abbreviation for s1 , . . . , sn and t is an abbreviation for t1 , . . . , tn . Then for θ(a/c , d ) ∈ ConL (M) and every b , b ∈ M we have n a i=1 |pi |x ⊗ Π (b , b , p1 , . . . , pn , s , t , e ) . (2.18) θ(a/c , d )(b , b ) = X={x,y1 ,...,yk }; e1 ,...,ek ∈M p1 ,...,pn ∈ T (X); {sj ,tj } ={c ,d }, 1 ≤ j ≤ n

Proof. First, let θ∗ be an L-relation on M defined by the right side of (2.18). We have to check that θ(a/c , d ) = θ∗ . In what follows we use the fact that if for every aithere is b j with ai ≤ bj and for every bi there is aj such that bi ≤ aj then i∈I ai = j∈J bj . “θ(a/c , d ) ⊆ θ∗ ”: It is sufficient to prove that θ∗ (c , d ) ≥ a and θ ∈ ConL (M). The first condition is obvious. Indeed, take a term p(x) = x, elements s1 = c , t1 = d , and observe that θ∗ (c , d ) ≥ a|p|x ⊗ c ≈M pM (c ) ⊗ pM (d ) ≈M d = a1 ⊗ 1 ⊗ 1 = a . ∗


95

Thus, θ∗ (c , d ) ≥ a. Moreover, take b , b ∈ M and a binary term p(x, y) = y. It follows that θ∗ (b , b ) ≥ a|p|x ⊗ b ≈M pM (c , b ) ⊗ pM (d , b ) ≈M b = = a0 ⊗ (b ≈M b ) ⊗ 1 = b ≈M b , i.e. b ≈M b ≤ θ∗ (b , b ). Clearly, we can put b = b , thus the foregoing inequality yields 1 = b ≈M b ≤ θ∗ (b , b ). Hence, θ∗ is reflexive. Furthermore, for every set of variables X = {x, y1 , . . . , yk }, arbitrary terms p1 , . . . , pn ∈ T (X) and elements s1 , t1 , . . . , sn , tn ∈ M such that {sj , tj } = {c , d }, 1 ≤ j ≤ n and b , b , e1 , . . . , ek ∈ M it is easy to see that n−1 M M M M b ≈M pM pi+1 (si+1 , e ) ⊗ pM 1 (s1 , e ) ⊗ n (tn , e ) ≈ b = i=1 pi (ti , e ) ≈ n−2 M M M = b ≈M pM pn−i−1 (tn−i−1 , e ) ⊗ n (tn , e ) ⊗ i=0 pn−i (sn−i , e ) ≈ M ⊗ pM 1 (s1 , e ) ≈ b .

Hence, the following equality holds, n

a

i=1

|pi |x

⊗ Π (b , b , p1 , . . . , pn , s , t , e ) = n

=a

i=1

|pi |x

⊗ Π (b , b , pn , . . . , p1 , t , s , e ) ,

where s is an abbreviation for sn . . . , s1 and t is an abbreviation for tn , . . . , t1 . This equality ensures that θ∗ (b , b ) = θ∗ (b , b ), i.e. θ∗ is symmetric. Consider b , b , b ∈ M , two sets of variables X1 = {x1 , y11 , . . . , y1k1 }, X2 = {x2 , y21 , . . . , y2k2 }, terms p11 , . . . , p1n1 ∈ T (X1 ), p21 , . . . , p2n2 ∈ T (X2 ) and elements s11 , t11 , . . . , s1n1 , t1n1 ∈ M , s21 , t21 , . . . , s2n2 , t2n2 ∈ M abbreviated by s1 , t1 , s2 , t2 such that {s1j , t1j } = {c , d }, {s2l , t2l } = {c , d }, 1 ≤ j ≤ n1 , 1 ≤ l ≤ n2 . Moreover, let us have e11 , . . . , e1k1 ∈ M , e21 , . . . , e2k2 ∈ M abbreviated by e1 and e2 . Without loss of generality, we can assume that x1 = x2 and (X1 − {x1 }) ∩ (X2 − {x2 }) = ∅ (if this is not so, one can rename variables in X1 and X2 accordingly). In other words, all terms p11 , . . . , p1n1 , p21 , . . . , p2n2 can be thought of as terms in variables X1 ∪ X2 . Using transitivity of ≈M we have n1 −1 M p1i (t1i , e1 ) ≈M pM b ≈M pM 11 (s11 , e1 ) ⊗ 1(i+1) (s1(i+1) , e1 ) ⊗ i=1 M ⊗ pM b ⊗ b ≈M pM 1n1 (t1n1 , e1 ) ≈ 21 (s21 , e2 ) ⊗ ⊗

n2 −1 i=1

≤ b ≈M

M M M pM p2(i+1) (s2(i+1) , e2 ) ⊗ pM b ≤ 2i (t2i , e2 ) ≈ 2n2 (t2n2 , e2 ) ≈ n1 −1 M p1i (t1i , e1 ) ≈M pM pM 11 (s11 , e1 ) ⊗ 1(i+1) (s1(i+1) , e1 ) ⊗ i=1

M M ⊗ pM p21 (s21 , e2 ) ⊗ 1n1 (t1n1 , e1 ) ≈ n2 −1 M M M ⊗ i=1 p2i (t2i , e2 ) ≈M pM 2(i+1) (s2(i+1) , e2 ) ⊗ p2n2 (t2n2 , e2 ) ≈ b .

Hence, it follows that

96


n1

a

i=1

|p1i |x1

⊗ Π (b , b , p11 , . . . , p1n1 , s1 , t1 , e1 ) ⊗ n2

⊗a

n1

i=1

|p2i |x2

n2

⊗ Π (b , b , p21 , . . . , p2n2 , s2 , t2 , e2 ) ≤

≤ a i=1 |p1i |x1 ⊗ a i=1 |p2i |x2 ⊗ Π (b , b , p11 , . . . , p1n1 , p21 , . . . , p2n2 , s , t , e ) = n n2 1 = a i=1 |p1i |x1 + i=1 |p2i |x2 ⊗ Π (b , b , p11 , . . . , p1n1 , p21 , . . . , p2n2 , s , t , e ) , where e is an abbreviation for elements e11 , . . . , e1k1 , e21 , . . . , e2k2 , s denotes s11 , . . . , s1n1 , s21 , . . . , s2n2 , analogously for t . Taking into account the definition of θ∗ , we can conclude that θ∗ (b , b ) ⊗ θ∗ (b , b ) ≤ θ∗ (b , b ), i.e. θ∗ is transitive. Now we will check compatibility with operations of M. Take any m-ary f M ∈ F M and elements a1 , b1 , . . . , am , bm ∈ M . For every 1 ≤ i ≤ m let us have a set Xi = {xi , yi1 , . . . , yiki } of variables, terms pi1 , . . . , pini ∈ T (Xi ), arbitrary elements si1 , ti1 , . . . , sini , tini ∈ M such that {sij , tij } = {c , d } (1 ≤ j ≤ ni ) denoted by si , ti and elements ei1 , . . . , eiki ∈ M denoted by ei . Again, we can assume that Xi , Xj are pairwise disjoint for every 1 ≤ i, j ≤ m, i = j, i.e. for every 1 ≤ i ≤ m, 1 ≤ l ≤ ni we have pil ∈ T (X ), where X = X1 ∪ · · · ∪ Xm . Put n = max(n1 , . . . , nm ). We are going to show that we can enlarge every sequence pi1 , . . . , pini of terms by defining new terms and suitable elements to obtain a sequence pi1 , . . . , pini , pi(ni +1) , . . . , pin which satisfies certain conditions. After that, all sequences have the same length, so it will be possible to apply the compatibility of f M with ≈M simultaneously. First of all, put X = X ∪{z1 , . . . , zm }, where z1 , . . . , zm ∈ X . All terms pij can be thought as terms of the form pij (x1 , y11 , . . . , y1k1 , x2 , y21 , . . . , ymkm , z1 , . . . , zm ). For every sequence pi1 , . . . , pini such that ni < n, let pij = zi for all ni < j ≤ n. We can conclude that ni

a

j=1

|pij |xi

⊗ Π (ai , bi , pi1 , . . . , pini , si , ti , ei ) = n

=a

j=1

|pij |xi

⊗ Π (ai , bi , pi1 , . . . , pin , si , ti , e ) ,

where e denotes e11 , . . . , emkm , b1 , . . . , bm , furthermore, si denotes si1 , . . . , sin , and ti denotes elements ti1 , . . . , tin such that {sij , tij } = {c , d }, ni < j ≤ n. Indeed, since thenew terms pij = zi do not have any occurrence of xi , we have ni n a j=1 |pij |xi = a j=1 |pij |xi and the values of pM ij ’s do not depend on newly defined elements sij , tij for j > ni . Thus, M M M pM 1 ⊗ 1 ⊗ ··· ⊗ 1 = ini (tini , ei ) ≈ bi = pini (tini , ei ) ≈ bi ⊗ (n−ni )-times

M M = pM pi(ni +1) (si(ni +1) , e ) ⊗ ini (tini , e ) ≈ M M M ⊗ pM pi(ni +2) (si(ni +2) , e ) ⊗ · · · ⊗ pM in (tin , e ) ≈ bi . i(ni +1) (ti(ni +1) , e ) ≈ Hence, it is possible to state


m

n−1

97

pM j(i+1) (sj(i+1) , e ) ⊗

M M aj ≈M pM j1 (sj1 , e ) ⊗ i=1 pji (tji , e ) ≈ M ⊗ pM bj ≤ jn (tjn , e ) ≈ M ≤ f M (a1 , . . . , am ) ≈M f M pM 11 (s11 , e ), . . . , pm1 (sm1 , e ) ⊗ M n−1 M ⊗ i=1 f M pM 1i (t1i , e ), . . . , pmi (tmi , e ) ≈ M ≈M f M pM ⊗ 1(i+1) (s1(i+1) , e ), . . . , pm(i+1) (sm(i+1) , e ) M M M M M ⊗ f p1n (t1n , e ), . . . , pmn (tmn , e ) ≈ f (b1 , . . . , bm ) . M Moreover, the value of every f M pM 1i (s1i , e ), . . . , pmi (smi , e ) can be expressed as the resulting value of the term f p1i (x1 , y), p2i (x2i , y) . . . , pmi (xmi , y) in variables (X ∪ {xji | j ≥ 2}) − {x2 , . . . , xm } for elements eMtogether with elements {tji , sji | j ≥ 2}. In the case of f M pM 1i (t1i , e ), . . . , pmi (tmi , e ) , we can proceed analogously. Thus, we can claim, m ni |pij |x j=1 i ⊗ Π (ai , bi , pi1 , . . . , pini , si , ti , ei ) = i=1 a m n m = a i=1 j=1 |pij |xi ⊗ i=1 Π (ai , bi , pi1 , . . . , pin , si , ti , e ) ≤ n m ≤ a j=1 |p1j |x1 ⊗ i=1 Π (ai , bi , pi1 , . . . , pin , si , ti , e ) ≤ j=1

n

≤a

j=1

|p1j |x1

⊗

⊗ Π(f (a1 , . . . , am ), f M (b1 , . . . , bm ) , f (p11 , . . . , pm1 ), . . . , f (p1n , . . . , pmn ), s1 , t1 , e ) . n Clearly, f (p11 , . . . , pm1 ), . . . , f (p1n , . . . , pmn ) are terms and j=1 |p1j |x1 is the number of occurrences of the variable x1 in these terms. This observation yields the compatibility of f M with θ∗ . Together with the foregoing results, θ∗ ∈ ConL (M) and θ∗ (c , d ) ≥ a. Consequently, θ(a/c , d ) ⊆ θ∗ . M

“θ(a/c , d ) ⊇ θ∗ ”: Take a set X = {x, y1 , . . . , yk } of variables, terms p1 , . . . , pn ∈ T (X) and arbitrary elements s1 , t1 , . . . , sn , tn ∈ M abbreviated by s , t such that {sj , tj } = {c , d }, 1 ≤ j ≤ n and b , b , e1 , . . . , ek ∈ M , where e is an abbreviation for e1 , . . . , ek . It is easy to see that n

a

i=1

|pi |x

= a|p1 |x ⊗ · · · ⊗ a|pn |x ≤

≤ θ(a/c , d )(c , d )|p1 |x ⊗ · · · ⊗ θ(a/c , d )(c , d )|pn |x . Theorem 2.76 yields θ(a/c , d )(c , d )|p|x ≤ θ(a/c , d ) pM (c, e ), pM (d , e ) for every term p(x, y) and any b , b , e1 , . . . , ek ∈ M . Thus, we have

98

2 Algebras with Fuzzy Equalities n

n

⊗ Π (b , b , p1 , . . . , pn , s , t , e ) = a i=1 |pi |x ⊗ b ≈M pM 1 (s1 , e ) ⊗ n−1 M M M ⊗ i=1 pi (ti , e ) ≈M pM b ≤ i+1 (si+1 , e ) ⊗ pn (tn , e ) ≈ n |pi |x M a i=1 ≤a ⊗ θ( /c , d ) b , p1 (s1 , e ) ⊗ n−1 M ⊗ i=1 θ(a/c , d ) pM ⊗ i (ti , e ), pi+1 (si+1 , e ) M ⊗ θ(a/c , d ) pn (tn , e ), b ≤ |p1 |x a ≤ θ(a/c , d ) b , pM ⊗ 1 (s1 , e ) ⊗ θ( /c , d )(c , d ) M n−1 ⊗ i=1 θ(a/c , d ) pi (ti , e ), pM i+1 (si+1 , e ) ⊗ |pi+1 |x a a ⊗ θ( /c , d )(c , d ) ⊗ θ( /c , d ) pM ≤ n (tn , e ), b M M M ≤ θ(a/c , d ) b , p1 (s1 , e ) ⊗ θ(a/c , d ) p1 (s1 , e ), p1 (t1 , e ) ⊗ n−1 M ⊗ i=1 θ(a/c , d ) pM i (ti , e ), pi+1 (si+1 , e ) ⊗ M . ⊗ θ(a/c , d ) pM ⊗ θ(a/c , d ) pM i+1 (si+1 , e ), pi+1 (ti+1 , e ) n (tn , e ), b

a

i=1

|pi |x

Hence, we can apply the transitivity of θ(a/c , d ) repeatedly to obtain n

a

i=1

|pi |x

⊗ Π (b , b , p1 , . . . , pn , s , t , e ) ≤ θ(a/c , d )(b , b ) .

Since X = {x, y1 , . . . , yk }, p1 , . . . , pn ∈ T (X) and b , b , s , s1 , t1 , . . . , sn , tn ∈ M , e1 , . . . , ek ∈ M have been chosen arbitrarily, we have proved the desired inequality. Altogether, θ∗ = θ(a/c , d ).

2.5 Direct Unions In general, there is no obvious way to naturally equip the union i∈I Mi of universes of L-algebras Mi with functions so that it became an L-algebra. In some cases, however, it is possible to define functions on i∈I Mi such that M equipped with such functions and suitable L-equality turns into an i i∈I L-algebra. For instance, when {Mi | i ∈ I} is a directed family of L-algebras, we can define a direct union of {Mi | i ∈ I} which is a well-defined L-algebra. Definition 2.83. A partially ordered index set I, ≤ is called directed, if I = ∅ and for every i, j ∈ I there is k ∈ I such that i, j ≤ k. A family {Mi | i ∈ I} of L-algebras of type F , where I, ≤ is a directed index set and Mi ∈ Sub(Mj ) whenever i ≤ j is called a directed family of L-algebras. Let κ be an infinite cardinal. A family {Mi | i ∈ I} = ∅ of L-algebras of type F is called a κ-directed family, if for every J ⊆ I, |J| < κ there exists an index i ∈ I such that Mj ∈ Sub(Mi ) for all j ∈ J. Lemma 2.84. Let κ be an infinite cardinal. Then (i) every κ-directed family {Mi | i ∈ I} is a directed family for some partial order ≤ on I; (ii) every directed family is an ω-directed family.

2.5 Direct Unions

99

Proof. (i): Let {Mi | i ∈ I} be a κ-directed family of L-algebras. It is easily seen that we can define a partial order ≤ on I by i≤j

iff Mi ∈ Sub(Mj ) .

(2.19)

Moreover, for J = {i, j} ⊆ I we have |J| = 2 < κ, i.e. there is some k ∈ I such that Mi , Mj ∈ Sub(Mk ), that is i, j ≤ k. Hence, I, ≤ is a directed index set and {Mi | i ∈ I} is a directed family of L-algebras. (ii): Let {Mi | i ∈ I} be a directed family of L-algebras. Let J ⊆ I, |J| < ω, i.e. J = {i1 , . . . , in }. Using the induction on elements of J, it follows that there is an index i ∈ I such that Mi1 , . . . , Min ∈ Sub(Mi ). Thus, {Mi | i ∈ I} is a ω-directed family of L-algebras. Definition 2.85. For a directed family of L-algebras of type F {Mi | i ∈ I} M i i∈I i∈I Mi such that M , ≈ , F we define an L-algebra i∈I Mi = i i∈I for every n-ary f ∈ F , and arbitrary a1 , . . . , an ∈ i∈I Mi we put f

i∈I

Mi

(a1 , . . . , an ) = f Mj (a1 , . . . , an ) ,

(2.20)

where a1 ∈ Mi1 , . . . , an ∈ Min , j ∈ I, and i1 , . . . , in ≤ j. For every elements a , b ∈ i∈I Mi such that a ∈ Mi , b ∈ Mj we define a ≈ i∈I Mi b by

(a ≈

b ) = (a ≈Mk b ) , (2.21) where k ∈ I, i, j ≤ k. The L-algebra i∈I Mi is called a direct union of a directed family {Mi | i ∈ I} of L-algebras. Given a κ-directed family {Mi | i ∈ I}, the direct union of a directed family {Mi | i ∈ I} with a partial order ≤ on I defined κ by (2.19) is called the κ-direct union of {Mi | i ∈ I} and is denoted by i∈I Mi . Remark 2.86. (1) The direct union i∈I Mi of a directed family {Mi | i ∈ I} is a well-defined L-algebra. First observe, that for finitely many i1 , . . . , in ∈ I there is always an index j ∈ I such that i1 , . . . , in ≤ j (this follows from the definition of directed index set). Moreover,{Mi | i ∈ I} is a directed family exists. so we have Mi1, . . . , Min ∈ Sub(Mj ), i.e. f i∈I Mi (a1 , . . . , an ) always The fact that i∈I Mi equipped with functions f i∈I Mi ∈ F i∈I Mi is an algebra follows from the ordinary case. i∈I Mi is well-defined. Indeed, it suffices to check that The L-relation ≈ a ≈ i∈I Mi b is independent on the choice of indices i, j, k ∈ I used in (2.21). Let a ∈ Mi , a ∈ Mi , b ∈ Mj , b ∈ Mj . Take k ≥ i, j and k ≥ i , j . Since I, ≤ is directed, there is an index l ∈ I such that l ≥ k, k . As a consequence, Mk , Mk ∈ Sub(Ml ). Thus, it follows that

i∈I

Mi

(a ≈Mk b ) = (a ≈Ml b ) = (a ≈Mk b ) .

Therefore, a ≈ i∈I Mi b is independent on the choice of i, j, k ∈ I. have a ∈ Mj for some j ∈ I. Since Furthermore, for a ∈ i∈I Mi we a ≈Mj a = 1, it follows that a ≈ i∈I Mi a = 1 (reflexivity). Symmetry follows from the symmetry of all ≈Mi ’s. For every a , b , c ∈ M there is an index j ∈ I such that a , b , c ∈ Mj , thus

100


(a ≈

i∈I

Mi

b ) ⊗ (b ≈

i∈I

Mi

c ) = (a ≈Mj b ) ⊗ (b ≈Mj c ) ≤

≤ (a ≈Mj c ) = (a ≈

i∈I

Mi

c) ,

is transitive. Moreover, if a ≈ i∈I b = 1 then for j ∈ I such i.e. ≈ i∈I Mj ( a ≈ b ) = 1. Thus, ( a ≈ i∈I Mi b ) = 1 implies that a , b ∈ Mj we have M a = b . Altogether, ≈ i∈I i is an L-equality. i∈I Mi ∈ F i∈I Mi is compatible with It suffices to verify, that every f M ≈ i∈I i . For every n-ary f ∈ F and arbitrary a1 , b1 , . . . , an , bn ∈ i∈I Mi there is an index j ∈ I such that a1 , b1 , . . . , an , bn ∈ Mj . Thus, Mi

(a1 ≈

i∈I

Mi

b1 ) ⊗ · · · ⊗ (an ≈ = (a1 ≈

Mj

i∈I

Mi

Mi

bn ) =

b1 ) ⊗ · · · ⊗ (an ≈Mj bn ) ≤

≤ f Mj (a1 , . . . , an ) ≈Mj f Mj (b1 , . . . , bn ) =

=f

i∈I

Mi

(a1 , . . . , an ) ≈

i∈I

Mi

f

i∈I

Mi

(b1 , . . . , bn ) .

Hence, i∈I Mi is a well-defined L-algebra. (2) Clearly,every L-algebra M is isomorphic to a trivial direct union. Namely, M ∼ = i∈I Mi , where I = {1} and M1 is M. (3) If {Mi | i ∈ I} is a directed family of L-algebras, where I is finite, then evidently I has the greatest element, let us denote it by k. Clearly, for every i ∈ I we have Mi ∈ Sub(Mk ), that is i∈I Mi ∼ = Mk . The following theorem presents a representation of L-algebras as κ-direct unions of κ-directed families of subalgebras generated by less than κ generators. In particular, the theorem says that every L-algebra can be composed from its finitely generated subalgebras. Recall that by a κ-generated L-algebra we mean M such that M = [M ]M for some M ⊆ M with |M | < κ. Theorem 2.87. Let κ be any infinite cardinal. Every L-algebra is isomorphic to a κ-direct union of a κ-directed family of its κ-generated subalgebras. Proof. Let M be an L-algebra. Put IM = {M ⊆ M | κ > |M |}. For each M ∈ IM we identify [M ]M with the subalgebra of M with universe [M ]M . Obviously, each [M ]M is κ-generated. Furthermore, for J ⊆ IM2 such that < κ = κ. As a |J| < κ, we have |M | < κ for every M ∈ J. Thus, M ∈J M consequence, M ∈J M ∈ IM . It is evident that [M ]M ∈ Sub M ∈J M M for every M ∈ J. So, {[M ]M | M ∈ IM } is a κ-directed family of κ-generated subalgebras of M. Now observe that a ∈ M is contained in[{a }]M and where {a } ∈ IM . It is thus possible to define a mapping h : M → {[M ]M | M ∈ IM } by h(a ) = a (a ∈ M ). For all a , b ∈ M we have

a ≈M b = a ≈[{a ,b }]M b = h(a ) ≈

{[M ]M | M ∈IM }

h(b ) ,

i.e. h is an ≈-morphism. For any n-ary f ∈ F and a1 , . . . , an ∈ M we have,

2.5 Direct Unions

101

h f M (a1 , . . . , an ) = h f [{a1 ,...,an }]M (a1 , . . . , an ) = = f [{a1 ,...,an }]M (a1 , . . . , an ) = f {[M ]M | M ∈IM } h(a1 ), . . . , h(an ) . Since for each finite M ⊆ M ,[M ]M is a subalgebra of M, h is a surjective embedding which yields M ∼ = {[M ]M | M ∈ IM }. Theorem 2.87 and Lemma 2.84 give the following Corollary 2.88. Every L-algebra is isomorphic to the direct union of its finitely generated subalgebras. The following asserion says that any direct union can be thought of as a derived construction since it is isomorphic to a factorization of certain subalgebra of a direct product. Lemma 2.89. Let {Mi | i ∈ I} be a directed family of L-algebras of type F . Then i∈I Mi ∼ = N/θ, where Q (i) N ∈ Sub( M ), where Qi∈I i such that a (k) = a (i) for all i ≥ k , N = a ∈ i∈I M i | there is k ∈ I (ii) θ ∈ ConL (N) such that θ(a , b ) = i∈I k≥i a (k) ≈Mk b (k). Proof. For the sake of brevity, let a ∈ N(j) denote the fact that a (j) = a (i) for all i ≥ j. Clearly, if a ∈ N(j) , then a ∈ N(l)Qfor every l ≥ j. First, we will check that N is a non-empty subuniverse of i∈I Mi . Evidently, N = ∅. Furthermore, for any n-ary f ∈ F and arbitrary elements a1 , . . . , an ∈ N there is an index j ∈ I, such that a1 ∈ N(j) , . . . , an ∈ N(j) . Since Mj ∈ Sub(Mi ) for all i ≥ j, it follows that Q f i∈I Mi (a1 , . . . , an )(j) = f Mj a1 (j), . . . , an (j) = Q = f Mi a1 (i), . . . , an (i) = f i∈I Mi (a1 , . . . , an )(i) Q Q for every i ≥ j, i.e. f i∈I Mi (a1 , . . . , an ) ∈ N(j) . Hence, N ∈ Sub( i∈I Mi ). In the the rest of the proof, we will take advantage of the following claim: for every elements a ∈ N(j) , b ∈ N(j ) we have θ(a , b ) = a (m) ≈Mm b (m)

for every m ≥ j, j .

(2.22)

“≤” of (2.22): Take m ≥ j, j , where a ∈ N(j) , b ∈ N(j ) . For arbitrary index i ∈ I there is some m ∈ I such that i, m ≤ m . Moreover, from a (m ) = a (m), b (m ) = b (m), Mm ∈ Sub(Mm ) it follows that Mk b (k) ≤ a (m ) ≈Mm b (m ) = a (m) ≈Mm b (m) . k≥i a (k) ≈ Hence, we have θ(a , b ) = i∈I k≥i a (k) ≈Mk b (k) ≤ a (m) ≈Mm b (m). “≥” of (2.22): Let us have a ∈ N(j) , b ∈ N(j ) and let m ≥ j, j . We can take i = m. Since Mm ∈ Sub(Mk ) for all k ≥ m, we readily obtain a (m) ≈Mm b (m) = k≥m a (k) ≈Mk b (k). That is, a (m) ≈Mm b (m) ≤ i∈I k≥i a (k) ≈Mk b (k) = θ(a , b ) .

102


Altogether, claim (2.22) holds true. Now it is easy to check that θ ∈ ConL (N). Reflexivity and symmetry of θ are obvious. For every a , b , c ∈ N , we have θ(a , b ) = a (m) ≈Mm b (m), θ(b , c ) = b (m ) ≈Mm c (m ) for certain m, m ∈ I due to (2.22). Evidently, a , b ∈ N(m) , b , c ∈ N(m ) . Thus, for m ≥ m, m , we have a , b , c ∈ N(m ) . Using this observation, it follows that θ(a , b ) ⊗ θ(b , c ) = a (m) ≈Mm b (m) ⊗ b (m ) ≈Mm c (m ) = = a (m ) ≈Mm b (m ) ⊗ b (m ) ≈Mm c (m ) ≤ ≤ a (m ) ≈Mm c (m ) = θ(a , c ) . Hence, θ is transitive. In an analogous way it is possible to prove that every n-ary f N ∈ F N is compatible with θ. For a1 , b1 ∈ N(m1 ) , . . . , an , bn ∈ N(mn ) , and m ∈ I such that m ≥ m1 , . . . , mn we have n M mi bi (mi ) = ni=1 ai (m) ≈Mm bi (m) ≤ i=1 ai (mi ) ≈ ≤ f Mm a1 (m), . . . , an (m) ≈Mm f Mm b1 (m), . . . , bn (m) = = f N (a1 , . . . , an )(m) ≈Mm f N (b1 , . . . , bn )(m) = = θ f N (a1 , . . . , an ), f N (b1 , . . . , bn ) . That is, we have θ ∈ ConL (N). Let us introduce mappings hi : Mi → N/θ (i ∈ I) defined by hi (a ) = [a ]θ , where a ∈ N such that for every j ≥ i we have a (j) = a . Every hi is a well-defined mapping since for a , a ∈ N , where a (j) = a (j) = a for all j ≥ i we have θ(a , a ) = 1, i.e. [a ]θ = [a ]θ . Furthermore, for any n-ary f Mi ∈ F Mi and arbitrary a1 , . . . , an ∈ Mi it is obvious that f N/θ hi (a1 ), . . . , hi (an ) = f N/θ [a1 ]θ , . . . , [an ]θ = = f N (a1 , . . . , an ) θ = hi f Mi (a1 , . . . , an ) . Hence, every hi : Mi → N/θ is compatible with operations. Moreover, hi is a restriction of hjon Mi whenever i ≤ j. Now it is possible to define a mapping h : i∈I Mi → N/θ by h(a ) = hi (a ) for all a ∈ Mi , N/θ h(a1 ), . . . , h(an ) = h f Mi (a1 , . . . , an ) . Furthermore, i ∈ I. Clearly, f for a , b ∈ i∈I Mi , a ∈ Mi , b ∈ Mj we can take k ≥ i, j, thus

a≈

i∈I

Mi

b = a ≈Mk b = a (k) ≈Mk b (k) ,

where a , b ∈ N such that for every l ≥ k we have a (l) = a , b (l) = b , i.e. a , b ∈ N(k) . Moreover, we can apply (2.22) to obtain

a (k) ≈Mk b (k) = θ(a , b ) = [a ]θ ≈N/θ [b ]θ = = hk (a ) ≈N/θ hk (b ) = hi (a ) ≈N/θ hj (b ) = h(a ) ≈N/θ h(b ) . Consequently, h is an embedding. Furthermore, for any [b ]θ ∈ N/θ there is an index i ∈ I such that b (j) = a for some a ∈ Miand every j ≥ i. Thus, h(a ) = hi (a ) = [b ]θ , i.e. h is surjective. Altogether, i∈I Mi ∼ = N/θ.

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103

2.6 Direct Limits This section focuses on the construction of a direct limit. In ordinary case, every algebra is isomorphic to a direct limit of finitely presented algebras. In the subsequent development we will present an analogous characterization for L-algebras. Furthermore, direct limits are important from the point of view of quasivariety theory, see Sect. 4.8. We begin our development with the definition of a (weak ) direct family. This notion is similar to that one from ordinary case, but we have to postulate an additional condition (which holds true automatically in ordinary case) to be able to generalize some required properties of direct limits. Definition 2.90. A weak direct family of L-algebras of type F consists of: (i) a directed index set I, ≤ , (ii) a family {Mi | i ∈ I} of pairwise disjoint L-algebras of type F , (iii) a family {hij : Mi → Mj | i ≤ j} of morphisms, where hii = idMi

for every i ∈ I ,

(2.23)

hik = hij ◦ hjk

for all i, j, k ∈ I, where i ≤ j ≤ k .

(2.24)

A weak direct family is called a direct family if for every a ∈ Mi , b ∈ Mj there exists k ∈ I, i, j ≤ k such that for each l ∈ I, k ≤ l we have hik (a ) ≈Mk hjk (b ) = hil (a ) ≈Ml hjl (b ) .

(2.25)

Remark 2.91. (1) A (weak) direct family of L-algebras is usually denoted simply by {Mi | i ∈ I}. If there is no danger of confusion, we will not mention the morphisms hij : Mi → Mj explicitly. (2) In general, there are weak direct families which do not satisfy (2.25). Take L with L = [0, 1] as a structure of truth degrees and a family {Mi | i ∈ [0, 1)} of L-algebras, where Mi = {ai , bi }, ≈Mi , ∅ , and ai ≈Mi bi = i. Furthermore, morphisms hij : Mi → Mj (i ≤ j) defined by hij (ai ) = aj , hij (bi ) = bj evidently satisfy (2.23) and (2.24). Therefore, [0, 1), ≤ together with {M i | i ∈ [0, 1)} and hij ’s is a weak direct family. On the other hand, for ai , bj ∈ m∈[0,1) Mm , and every k ≥ i, j there is l > k, i.e. hik (ai ) ≈Mk hjk (bj ) = ak ≈Mk bk = k < < l = al ≈Ml bl = hil (ai ) ≈Ml hjl (bj ) , showing that {Mi | i ∈ [0, 1)} is not a direct family. Lemma 2.92. Let {Mi | i ∈ I} be a weak direct family of L-algebras. For every a ∈ Mi , b ∈ Mj and arbitrary k, l ∈ I such that i, j ≤ k ≤ l we have hik (a ) ≈Mk hjk (b ) ≤ hil (a ) ≈Ml hjl (b ), Mk hjk (b ) = m≥l him (a ) ≈Mm hjm (b ) . k≥i,j hik (a ) ≈

(2.26) (2.27)

104


Proof. Equation (2.26): Using (2.24) we have hik (a ) ≈Mk hjk (b ) ≤ hkl (hik (a )) ≈Ml hkl (hjk (b )) = = (hik ◦ hkl )(a ) ≈Ml (hik ◦ hkl )(b ) = hil (a ) ≈Ml hjl (b ) . Equation (2.27): Take an index k0 ≥ i, j. For m0 ∈ I such that m0 ≥ k0 , l we can use (2.26) to get hik0 (a ) ≈Mk0 hjk0 (b ) ≤ him0 (a ) ≈Mm0 hjm0 (b ) ≤ ≤ m≥l him (a ) ≈Mm hjm (b ) , by which follows the “≤”-part of (2.27). The converse inequality is trivial.

Remark 2.93. If L is a Noetherian residuated lattice then every weak direct family is a direct family. Indeed, for any a ∈ Mi and b ∈ Mj there are indices k1 , . . . , kn ≥ i, j such that n Mk hjk (b ) = m=1 hikm (a ) ≈Mkm hjkm (b ) k≥i,j hik (a ) ≈ Thus, we can take k ∈ I with k ≥ k1 , . . . , kn . Now (2.26) gives hikm (a ) ≈Mkm hjkm (b ) ≤ hik (a ) ≈Mk hjk (b ) for each m = 1, . . . , n. Therefore, hik (a ) ≈Mk hjk (b ) is the greatest one of all hik (a ) ≈Mk hjk (b ) for k ≥ i, j. Since for each l ≥ k we have hik (a ) ≈Mk hjk (b ) ≤ hil (a ) ≈Ml hjl (b ) by (2.26), it follows that in fact hik (a ) ≈Mk hjk (b ) = hil (a ) ≈Ml hjl (b ) for all l ≥ k proving (2.25). Definition 2.94. For every weak direct family of L-algebras {Mi | i ∈ I} let θ∞ denote a binary L-relation on i∈I Mi defined by (2.28) θ∞ (a , b ) = k≥i,j hik (a ) ≈Mk hjk (b ) for all a ∈ Mi , b ∈ Mj . Remark 2.95. If {Mi | i ∈ I} is a direct family, then (2.28) can be equivalently expressed without using the general suprema. Indeed, taking into account (2.25) and (2.27), for any a ∈ Mi , b ∈ Mj there is k0 ≥ i, j such that θ∞ (a , b ) = k≥i,j hik (a ) ≈Mk hjk (b ) = m≥k0 him (a ) ≈Mm hjm (b ) = = hik0 (a ) ≈Mk0 hjk0 (b ) . Lemma 2.96. Let F be a type of L-algebras and let {Mi | i ∈ I} be a weak direct family of L-algebras of type F . The following are properties of θ∞ : (i) θ∞ (a , b ) = θ∞ hil (a ), hjl (b) for all a ∈ Mi , b ∈ Mj , l ≥ i, j; on i∈I Mi ; (ii) θ∞ is an L-equivalence (iii) θ∞ a , hik (a ) = 1 for every a ∈ Mi , k ≥ i;

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105

(iv) for every n-ary f ∈ F , a1 ∈ Mi1 , b1 ∈ Mj1 , . . . , an ∈ Min , bn ∈ Mjn , and k ≥ i1 , j1 , . . . , in , jn we have θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤ ≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) . Proof. (i): Clearly, using (2.27) we have θ∞ (a , b ) = k≥i,j hik (a ) ≈Mk hjk (b ) = m≥l him (a ) ≈Mm hjm (b ) = = m≥l hlm (hil (a )) ≈Mm hlm (hjl (b )) = θ∞ (hil (a ), hjl (b )) . (ii): We have θ∞ (a , a ) = k≥i hik (a ) ≈Mk hik (a ) = 1 for every a ∈ Mi , i.e. θ∞ is reflexive. Symmetry is obvious. Hence, it suffices to check transitivity. Let a ∈ Mi , b ∈ Mj , c ∈ Mk , and let l ≥ i, j, k. Furthermore, using (2.26) and (i) together with monotony of ⊗ it follows that θ∞ (a , b ) ⊗ θ∞ (b , c ) = θ∞ (hil a ), hjl (b ) ⊗ θ∞ (hjl b ), hkl (c ) = = m≥l him (a ) ≈Mm hjm (b ) ⊗ n≥l hjn (b ) ≈Mn hkn (c ) = = m,n≥l him (a ) ≈Mm hjm (b ) ⊗ hjn (b ) ≈Mn hkn (c ) = = n≥l hin (a ) ≈Mn hjn (b ) ⊗ hjn (b ) ≈Mn hkn (c ) ≤ ≤ n≥l hin (a ) ≈Mn hkn (c ) = θ∞ (a , c ) . Hence, θ∞ is an L-equivalence. (iii): Let us have a ∈ Mi , k ≥ i. Take l ∈ I such that l ≥ k, i. From reflexivity of θ∞ together with (i) it follows that θ∞ (a , hik (a )) = θ∞ hil (a ), hkl (hik (a )) = θ∞ (hil (a ), hil (a )) = 1 . (iv): For an n-ary f ∈ F , arbitrary am ∈ Mim , bm ∈ Mjm , m = 1, . . . , n, and k ≥ i1 , j1 , . . . , in , jn , we can use the compatibility of f Ml with ≈Ml assumed for every l ∈ I to get θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) = n = m=1 km ≥k him km (am ) ≈Mkm hjm km (bm ) = n = k1 ,...,km ≥k m=1 him km (am ) ≈Mkm hjm km (bm ) = n = l≥k m=1 him l (am ) ≈Ml hjm l (bm ) ≤ ≤ l≥k f Ml hi1 l (a1 ), . . . , hin l (an ) ≈Ml f Ml hj1 l (b1 ), . . . , hjn l (bn ) = = l≥k f Ml hkl (hi1 k (a1 )), . . . , hkl (hin k (an )) ≈Ml f Ml hkl (hj1 k (b1 )), . . . , hkl (hjn k (bn )) = = l≥k hkl f Mk (hi1 k (a1 ), . . . , hin k (an )) ≈Ml hkl f Mk (hj1 k (b1 ), . . . , hjn k (bn )) = = θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) , which is the desired inequality.

106


Condition (iv) of Lemma 2.96 is similar to that of compatibility, but in this case, (iv) expresses a compatibility with respect to homomorphic images. Now we define suitable operations on the factorization of i∈I Mi by θ∞ . Definition 2.97. Let {Mi | i ∈ I} be a (weak of L-algebras family ) direct lim Mi /θ M , ≈ , F lim Mi , of type F . Then the L-algebra lim Mi = i ∞ i∈I where (i) of i∈I Mi by θ∞ , i /θ∞ is a factorization i∈I M (ii) f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ ∞ for every n-ary f ∈ F and arbitrary [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i∈I Mi /θ∞ such that a1 ∈ Mi1 , . . . , an ∈ Min , and k ∈ I, k ≥ i1 , .. . , in , (iii) [a ]θ∞ ≈lim Mi [b ]θ∞ = θ∞ (a , b ) for all [a ]θ∞ , [b ]θ∞ ∈ i∈I Mi /θ∞ is called a direct limit of a (weak ) direct family {Mi | i ∈ I}. Remark 2.98. A direct limit lim Mi of a weak direct family {Mi | i ∈ I} is a well-defined L-algebra. Obviously, ≈lim Mi is an L-equality, see Definition 1.91 and Remark 1.92. It remains to show that each f lim Mi is well defined and compatible with ≈lim Mi . First, we show that f Mk (hi1 k (a1 ), . . . , hin k (an )) θ ∞ given by (iii) does not depend on the chosen k ∈ I. Thus, take k ∈ I with k ≥ i1 , . . . , in , and arbitrary l ≥ k, k . Lemma 2.96 gives θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Ml (hi1 l (a1 ), . . . , hin l (an )) = = θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), hkl f Mk (hi1 k (a1 ), . . . , hin k (an )) = 1 . That is, f Mk (hi1 k (a1 ), . . . , hin k (an )) θ = f Ml (hi1 l (a1 ), . . . , hin l (an )) θ ∞ ∞ and analogously for k . Hence, M f k (hi1 k (a1 ), . . . , hin k (an )) θ∞ = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ∞ . Moreover, f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ does not depend on a1 , . . . , an chosen from classes [a1 ]θ∞ , . . . , [an ]θ∞ , because for bm ∈ [am ]θ∞ , bm ∈ Mjm (m = 1, . . . , n), and k ≥ i1 , j1 , . . . , in , jn we have 1 = θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤ ≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) = = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ∞ ≈lim Mi f Mk (hj1 k (b1 ), . . . , hjn k (bn )) θ∞ . Hence, f Mk (hi1 k (a1 ), . . . , hin k (an )) θ = f Mk (hj1 k (b1 ), . . . , hjn k (bn )) θ , ∞ ∞ i.e. f lim Mi is well defined. lim Mi ∈ F lim Mi and arbitrary It remains to check the compatibility. Takef [a1 ]θ∞ , [b1 ]θ∞ , . . . , [an ]θ∞ , [bn ]θ∞ ∈ i∈I Mi /θ∞ , where am ∈ Mim , bm ∈ Mjm (m = 1, . . . , n). For k ∈ I such that k ≥ i1 , j1 , . . . , in , jn , Lemma 2.96 together with the definition of ≈lim Mi yield

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107

[a1 ]θ∞ ≈lim Mi [b1 ]θ∞ ⊗ · · · ⊗ [an ]θ∞ ≈lim Mi [bn ]θ∞ = = θ∞ (a1 , b1 ) ⊗ · · · ⊗ θ∞ (an , bn ) ≤ ≤ θ∞ f Mk (hi1 k (a1 ), . . . , hin k (an )), f Mk (hj1 k (b1 ), . . . , hjn k (bn )) = = f Mk (hi1 k (a1 ), . . . , hin k (an )) θ ≈lim Mi ∞ M f k (hj1 k (b1 ), . . . , hjn k (bn )) θ = ∞ lim Mi lim Mi lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ ≈ [b1 ]θ∞ , . . . , [bn ]θ∞ . =f f Hence, lim Mi is a well-defined L-algebra. Lemma 2.99. Let lim Mi be a weak direct limit i | i ∈ I}. Then for of {M /θ∞ we have M every n-ary f lim Mi , and [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i i∈I f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = f Mk (a1 , . . . , an ) θ , (2.29) ∞

where

a1 , . . . , an

∈ Mk , and

am

∈ [am ]θ∞ for each m = 1, . . . , n.

) = 1 for each m = 1, . . . , n, we can take an index Proof. Since θ∞ (am , am l ∈ I such that l ≥ k, i1 , . . . , in and apply Definition 2.97 and Lemma 2.96: M f k (a1 , . . . , an ) θ = M ∞ = hkl f k (a1 , . . . , an ) θ = f Ml (hkl (a1 ), . . . , hkl (an )) θ = ∞ ∞ = f Ml (hi1 l (a1 ), . . . , hin l (an )) θ = f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ , ∞

proving (2.29).

Definition 2.100. Let {Mi | i ∈ I} be a weak direct family of L-algebras of type F . A family {hi : Mi → lim Mi | i ∈ I} of morphisms hi : Mi → lim Mi , where hi (a ) = [a ]θ∞ for every i ∈ I, a ∈ Mi is called a limit cone of the weak direct family {Mi | i ∈ I}. Let N be an L-algebra of type F . A family {gi : Mi → N | i ∈ I} of morphisms is said to satisfy a direct limit property (DLP) with respect to a weak direct family {Mi | i ∈ I} if gi = hij ◦ gj for all i ≤ j and for every family {gi : Mi → N | i ∈ I} of morphisms with gi = hij ◦ gj for all i ≤ j, there exists a unique morphism g : N → N such that gi = gi ◦ g for every i ∈ I, see Fig. 2.14.

Mi

hij

Mj

gi

gi gi

gj N

hij

Mi

N

g

Fig. 2.14. Direct limit property

Mj gj

N

108


Remark 2.101. Every hi : Mi → lim Mi of a limit cone of {Mi | i ∈ I} is indeed a morphism. Clearly, for all i ∈ I, a , b ∈ Mi we have a ≈Mi b ≤ k≥i hik (a ) ≈Mk hik (b ) = θ∞ (a , b ) = = [a ]θ∞ ≈lim Mi [b ]θ∞ = hi (a ) ≈lim Mi hi (b ) . Furthermore, for an n-ary f ∈ F , and a1 , . . . , an ∈ Mi : hi f Mi (a1 , . . . , an ) = f Mi (a1 , . . . , an ) θ = ∞ Mi lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = = f (hii (a1 ), . . . , hii (an )) θ = f ∞ = f lim Mi hi (a1 ), . . . , hi (an ) , i.e. hi is a morphism. Moreover, Lemma 2.96 yields θ∞ (hij (a ), a ) = 1, that is hi (a ) = [a ]θ∞ = [hij (a )]θ∞ = hj (hij (a )), i.e. hi = hij ◦ hj . Remark 2.102. If I, ≤ is a finite directed index set, then I has the greatest element. In consequence, every weak direct family {Mi | i ∈ I} is a direct family since for every a ∈ Mi , b ∈ Mj (2.25) is satisfied trivially for k being the greatest element of I. Consequently, θ∞ (a , b ) = hik (a ) ≈Mk hjk (b ). Moreover, for hk : Mk → lim Mi we have

a ≈Mk b = hkk (a ) ≈Mk hkk (b ) = θ∞ (a , b ) = hk (a ) ≈lim Mi hk (b ) , and for every [c ]θ∞ ∈ i∈I Mi /θ∞ , c ∈ Mi we have hk (hik (c )) = [hik (c )]θ∞ = [c ]θ∞ . Hence, hk is an isomorphism (hk is compatible with operations since it is a part of the limit cone), Mk ∼ = lim Mi . In other words, the direct limit is trivial for finite I, ≤ . Theorem 2.103. Let {Mi | i ∈ I} be a weak direct family of L-algebras. Then the limit cone {hi : Mi → lim Mi | i ∈ I} of {Mi | i ∈ I} satisfies the direct limit property with respect to {Mi | i ∈ I}. Proof. Let us have a family {gi : Mi → N | i ∈ I} of morphisms such that gi = hij ◦ gj for all i ≤ j. We have to check the existence and uniqueness of a morphism h : lim Mi → N, where gi = hi ◦ h for every i ∈ I. a ∈ Existence of h: Every i∈I Mi belongs to some Mi , hence we can /θ∞ → N by M define a mapping h : i i∈I where a ∈ Mi . h [a ]θ∞ ) = gi (a , (2.30) For every a ∈ Mi , b ∈ Mj we have

[a ]θ∞ ≈lim Mi [b ]θ∞ = θ∞ (a , b ) = k≥i,j hik (a ) ≈Mk hjk (b ) ≤ ≤ k≥i,j gk (hik (a )) ≈N gk (hjk (b )) = = k≥i,j gi (a ) ≈N gj (b ) = gi (a ) ≈N gj (b )

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109

Thus, [a ]θ∞ = [b ]θ∞ implies gi (a ) = gj (b ). Hence, h defined by (2.30) is a well-defined ≈-morphism. Now for every n-ary f ∈ F , and [a1 ]θ∞ , . . . , [an ]θ∞ ∈ i∈I Mi /θ∞ there are indices i1 , . . . , in ≤ l such that a1 ∈ Mi1 , . . . , an ∈ Min . Hence, using Lemma 2.96 it follows that h f lim Mi ([a1 ]θ∞ , . . . , [an ]θ∞ ) = h f Ml hi1 l (a1 ), . . . , hin l (an ) θ∞ = = gl f Ml (hi1 l (a1 ), . . . , hin l (an )) = f N gl (hi1 l (a1 )), . . . , gl (hin l (an )) = = f N gi1 (a1 ), . . . , gin (an ) = f N h([a1 ]θ∞ ), . . . , h([an ]θ∞ ) . That is, h : lim Mi → N is the required morphism satisfying gi (a ) = h([a ]θ∞ ) = h(hi (a )) = (hi ◦ h)(a ) for all a ∈ Mi , i ∈ I. Uniqueness of h: Let h : lim Mi → N be a morphism satisfying gi = hi ◦h . Every a ∈ i∈I Mi belongs to Mi for a suitable i ∈ I, that is h [a ]θ∞ = h (hi (a )) = gi (a ) = h(hi (a )) = h [a ]θ∞ .

Therefore, h is determined uniquely.

Theorem 2.104. Let {Mi | i ∈ I} be a weak direct family of L-algebras of type F and let N be an L-algebra of type F . There is a family {gi : Mi → N | i ∈ I} of morphisms satisfying DLP w.r.t. {Mi | i ∈ I} iff N ∼ = lim Mi . Proof. “⇒”: For N being lim Mi , the limit cone {hi : Mi → lim Mi | i ∈ I} satisfies direct limit property w.r.t. {Mi | i ∈ I}. Hence, it is sufficient to prove that for any families {gi : Mi → N | i ∈ I}, {gi : Mi → N | i ∈ I} satisfying DLP w.r.t. {Mi | i ∈ I} we have N ∼ = N . Thus, if both {gi : Mi → N | i ∈ I} and {gi : Mi → N | i ∈ I} satisfy DLP w.r.t. {Mi | i ∈ I}, there are uniquely determined morphisms g : N → N , g : N → N, where gi = gi ◦ g , gi = gi ◦ g, see Fig. 2.15. Consequently gi = gi ◦ g = (gi ◦ g) ◦ g = gi ◦ (g ◦ g ) , thus g ◦ g = idN , similarly g ◦ g = idN . So, we can apply Theorem 2.30 to claim that g, g are mutually inverse isomorphisms between N and N . Thus, for N being lim Mi it follows N ∼ = lim Mi . “⇐”: Obviously, for N ∼ = lim Mi we can take morphisms gi : Mi → N such that gi = hi ◦ h, where h : lim Mi → N is an isomorphism. It is routine to check that {gi : Mi → N} satisfies DLP w.r.t. {Mi | i ∈ I}. Mi gi

gi N

Mi

g

N

gi

gi N

g

Fig. 2.15. Morphisms between N and N

N

110


∼ T(X)/θ(R), where Definition 2.105. Every L-algebra M such that M = R ∈ LT (X)×T (X) is said to be presented by X, R . If both X and R are finite, then M is said to be finitely presented by X, R . Remark 2.106. Note that every L-algebra of type F is isomorphic to T(X)/θ, where T(X) is a term L-algebra of type F , X is a suitable set of variables, and θ ∈ ConL (T(X)). Indeed, we can take |X| ≥ |M |, and a surjective mapping h : X → M . Due to Theorem 2.79, there exists a surjective morphism h : T(X) → M. Finally, from Theorem 2.35 it follows that M ∼ = T(X)/θh . . From this point of view, every Consequently, M ∼ T(X)/θ(R) for R = θ = h L-algebra is presented by some X, R . Let us stress that such X and R are not finite in general. Theorem 2.107. Every L-algebra is isomorphic to a direct limit of a direct family of finitely presented L-algebras. Proof. Let M be an L-algebra. Due to Remark 2.106, we can identify M with T(X)/θ(R) for some set of variables X and R ∈ LT (X)×T (X) . Recall that for every Y ⊆ X we can consider a restriction θ(R)|T(Y ) of θ(R) on T(Y ). Now let us assume an index set I = Y, S | Y ⊆ X, Y is finite, S is a finite restriction of θ(R)|T(Y ) . We can define a partial order ≤ on I by Yi , Si ≤ Yj , Sj

iff Yi ⊆ Yj and Si ⊆ Sj .

It is easily seen that I, ≤ is directed. For the sake of brevity, we will denote indices of the form Yi , Si , Yj , Sj , . . . simply by i, j, . . . We introduce morphisms hij : T(Yi )/θ(Si ) → T(Yj )/θ(Sj ) (i ≤ j) de fined by hij [t]θ(Si ) = [t]θ(Sj ) , see Lemma 2.43. Clearly, every hij satisfies (2.23) and (2.24). Thus, I, ≤ , {T(Yi )/θ(Si ) | i ∈ I} together with hij ’s is a weak direct family. Moreover, it is even a direct family. Indeed, take elements [ti ]θ(Si ) ∈ T(Yi )/θ(Si ), [tj ]θ(Sj ) ∈ T(Yj )/θ(Sj ). There is k ≥ i, j such that Yk = Yi ∪ Yj and Sk (ti , tj ) = θ(R)(ti , tj ). Clearly, for every l ≥ k we have hik [ti ]θ(Si ) ≈T(Yk )/θ(Sk ) hjk [tj ]θ(Sj ) = [ti ]θ(Sk ) ≈T(Yk )/θ(Sk ) [tj ]θ(Sk ) = = θ(Sk )(ti , tj ) = θ(R)(ti , tj ) = θ(Sl )(ti , tj ) = [ti ]θ(Sl ) ≈T(Yl )/θ(Sl ) [tj ]θ(Sl ) = = hil [ti ]θ(Si ) ≈T(Yl )/θ(Sl ) hjl [tj ]θ(Sj ) , showing that I, ≤ , {T(Yi )/θ(Si ) | i ∈ I} together with hij ’s is a direct family. Now we show that there is a family {hi : T(Yi )/θ(Si ) → T(X)/θ(R) | i ∈ I} of morphisms satisfying DLP w.r.t. {T(Yi )/θ(Si ) | i ∈ I}. Then T(X)/θ(R) ∼ = ) on account of Theorem 2.104. lim T(Yi )/θ(S i Put hi [t]θ(Si ) = [t]θ(R) for all t ∈ T (Yi ). Due to Lemma 2.43, every hi is a morphism. Evidently, hi = hij ◦ hj for all i, j ∈ I with i ≤ j. Take a family

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{gi : T(Yi )/θ(Si ) → N | i ∈ I} of morphisms with gi = hij ◦ gj (i ≤ j), and define h : T(X)/θ(R) → N by (2.31) h [t]θ(R) = gi [t]θ(Si ) , where t ∈ T (X), and i ∈ I such that var(t) ⊆ Yi . Let us note that the value of h [t]θ(R) does not depend on the chosen index i ∈ I since for every i, j ∈ I such that var(t) ⊆ Yi , Yj we can take l ∈ I, i, j ≤ l. Thus, gi [t]θ(Si ) ≈N gj [t]θ(Sj ) = gl hil [t]θ(Si ) ≈N gl hjl [t]θ(Sj ) = = gl [t]θ(Sl ) ≈N gl [t]θ(Sl ) = 1 , i.e. gi [t]θ(Si ) = gj [t]θ(Sj ) . Moreover, h is a well-defined ≈-morphism. Indeed, take ti , tj ∈ T (X) such that var(ti ) ⊆ Yi , and var(tj ) ⊆ Yj . For k ∈ I, k ≥ i, j such that θ(R)(ti , tj ) = θ(Sk )(ti , tj ), it follows that [ti ]θ(R) ≈T(X)/θ(R) [tj ]θ(R) = θ(R)(ti , tj ) = θ(Sk )(ti , tj ) = = [ti ]θ(Sk ) ≈T(Yk )/θ(Sk ) [tj ]θ(Sk ) ≤ gk [ti ]θ(Sk ) ≈N gk [tj ]θ(Sk ) = = gk hik [ti ]θ(Si ) ≈N gk hjk [tj ]θ(Sj ) = = gi [ti ]θ(Si ) ≈N gj [tj ]θ(Sj ) . Hence, [ti ]θ(R) = [tj ]θ(R) implies gi [ti ]θ(Si ) = gj [tj ]θ(Sj ) . Altogehter, h is a well-defined ≈-morphism. Furthermore, take any n-ary f ∈ F and arbitrary elements [t1 ]θ(R) , . . . , [tn ]θ(R) . We can consider an index k ∈ I for which var(ti ) ⊆ Yk for each i = 1, . . . , n. Thus, h f T(X)/θ(R) ([t1 ]θ(R) , . . . , [tn ]θ(R) ) = h [f (t1 , . . . , tn )]θ(R) = = gk [f (t1 , . . . , tn )]θ(Sk ) = gk f T(Yk )/θ(Sk ) ([t1 ]θ(Sk ) , . . . , [tn ]θ(Sk ) ) = = f N gk [t1 ]θ(Sk ) , . . . , gk [t1 ]θ(Sk ) = f N h [t1 ]θ(R) , . . . , h [tn ]θ(R) . Hence, h is a morphism. In addition to that, gi [t]θ(Si ) = h [t]θ(R) = h hi [t]θ(Si ) , for every t ∈ T (Yi ), i.e. gi = hi ◦ h. Finally, we will check the uniqueness of h. Let h : T(X)/θ(R) → N be a morphism satisfying gi = hi ◦ h for all i ∈ I. It is immediate, that for t ∈ T (X), var(t) ⊆ Yi we have h [t]θ(R) = h hi [t]θ(Si ) = gi [t]θ(Si ) = h hi [t]θ(Si ) = h [t]θ(R) . Altogether, h : T(X)/θ(R) → N is a uniquely determined morphism such that gi = hi ◦ h for every i ≤ j. Consequently, M ∼ = lim T(Yi )/θ(Si ). Theorem 2.108. Let {Mi | ∈ I} be a direct family of L-algebras. For a morphism h : N → lim Mi from a finitely presented L-algebra N to lim Mi , there exists k ∈ I and a morphism g : N → Mk such that h = g ◦ hk .

112


Fig. 2.16. Morphism g from T(X)/θ(R) to some Mk

Proof. Since N is supposed to be finitely presented, we can identify N with some T(X)/θ(R), where X, R are finite. Thus, let us assume a morphism h : T(X)/θ(R) → lim Mi is given. It is obvious that T(X)/θ(R) is generated by [x]θ(R) | x ∈ X . For every variable x ∈ X there is an index ix ∈ I such that h [x]θ(R) ∈ hix (Mix ). Since there are only finitely many variables in X, we can choose an index j ∈ I such that j ≥ ix for each x ∈ X. Clearly, h [x]θ(R) ∈ hix (Mix ) = hj (hix j (Mix )) ⊆ hj (Mj ) for each x ∈ X. Therefore, h(hθ(R) (X)) ⊆ hj (Mj ), where hθ(R) is the natural morphism sending elements of T (X) to T (X)/θ(R). Following this observa tion, for each x ∈ X there is ax ∈ Mj such that h [x]θ(R) = hj (ax ) ∈ hj (Mj ). Hence, we introduce a mapping → Mj by putting v(x) = ax (x ∈ X). v: X By definition, hj (v(x)) = h [x]θ(R) for each x ∈ X. Since for v we have hj (v (t)) = h [t]θ(R) for all t ∈ T (X), it follows that v ◦ hj = hθ(R) ◦ h. The situation is depicted in Fig. 2.16. Recall that R is finite, i.e. Supp(R) = {t1 , t1 , . . . , tm , tm }. Since {Mi | ∈ I} is a direct family of L-algebras, it follows from (2.25) that for each i = 1, . . . , m there is an index ki ∈ I such that θ∞ (v (ti ), v (ti )) = hjki (v (ti )) ≈Mki hjki (v (ti )), see Remark 2.95. Hence, for an index k ∈ I such that k ≥ k1 , . . . , km we have θ(R)(ti , ti ) = [ti ]θ(R) ≈T(X)/θ(R) [ti ]θ(R) ≤ h [ti ]θ(R) ≈lim Mi h [ti ]θ(R) = = hj (v (ti )) ≈lim Mi hj (v (ti )) = θ∞ (v (ti ), v (ti )) = = hjki (v (ti )) ≈Mki hjki (v (ti )) ≤ hjk (v (ti )) ≈Mk hjk (v (ti )) . Thus, R(ti , ti ) ≤ hjk (v (ti )) ≈Mk hjk (v (ti )) = θv ◦hjk (ti , ti ) for each i = 1, . . . , m. Since θv ◦hjk ∈ ConL (Mk ), and θ(R) is generated by R, we readily obtain θ(R) ⊆ θv ◦hjk . Finally, put g [t]θ(R) = (v ◦ hjk )(t) = hjk (v (t)). For [t]θ(R) , [t ]θ(R) ∈ T (X)/θ(R) it follows that [t]θ(R) ≈T(X)/θ(R) [t ]θ(R) = θ(R)(t, t ) ≤ θv ◦hjk (t, t ) = = hjk (v (t)) ≈Mk hjk (v (t )) = g [t]θ(R) ≈Mk g [t ]θ(R) ,

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Fig. 2.17. Direct limit of a weak direct family

i.e. g is a well-defined ≈-morphism. For any n-ary f ∈ F and arbitrary elements [t1 ]θ(R) , . . . , [tn ]θ(R) ∈ T (X)/θ(R) we have g f T(X)/θ(R) [t1 ]θ(R) , . . . , [tn ]θ(R) = g [f (t1 , . . . , tn )]θ(R) = = hjk v f (t1 , . . . , tn ) = hjk f Mj v (t1 ), . . . , v (tn ) = = f Mk hjk (v (t1 )), . . . , hjk (v (tn )) = f Mk g([t1 ]θ(R) ), . . . , g([tn ]θ(R) ) . Hence, g : T(X)/θ(R) → Mk is a morphism. In addition to that, h [t]θ(R) = hj (v (t)) = hk (hjk (v (t))) = hk g [t]θ(R) = (g ◦ hk ) [t]θ(R) holds for every [t]θ(R) ∈ T (X)/θ(R), i.e. h = g ◦ hk .

Remark 2.109. Let us stress that the existence of a morphism described by Theorem 2.108 is limited to direct families {Mi | ∈ I}. In bivalent case, every weak direct family is a direct family, i.e. Theorem 2.108 then coincides with the well-known image factorization theorem for ordinary algebras. It follows from Remark 2.93 that for Noetherian residuated lattices, Theorem 2.108 is true for all weak direct families. However, the following example illustrates that postulating (2.25) is necessary for general residuated lattices. Example 2.110.Take L = [0, 1]. Let us have a family {Mi | i ∈ N} of Li . algebras Mi = Mi , ≈Mi , ∅ such that Mi = {ai , bi }, and ai ≈Mi bi = 2i+1 1 2 3 M1 M2 M3 That is, a1 ≈ b1 = 3 , a2 ≈ b2 = 5 , a3 ≈ b3 = 7 , . . . , see Fig. 2.17. Clearly, N, ≤ is a directed index set, the universe sets Mi (i ∈ N) are pairwise disjoint, and {hij : Mi → Mj | i ≤ j}, where hij (ai ) = aj , and hij (bi ) = bj is a family of morphisms satisfying (2.23) and (2.24). Altogether, N, ≤ with {Mi | i ∈ N}, and {hij : Mi → Mj | i ≤ j} is a weak direct family. On the other hand, it is not a direct family, because for i, k ∈ N with i ≤ k we have hik (ai ) ≈Mk hik (bi ) < hi,k+1 (ai ) ≈Mk+1 hi,k+1 (bi ). Moreover, we have θ∞ (ai , bi ) = k≥i hik (ai ) ≈Mk hik (bi ) = 12 , i.e. i∈N Mi /θ∞ = [ai ]θ∞ , [bi ]θ∞ , see Fig. 2.17. Since lim Mi is of the empty type (F lim Mi = ∅), it readily follows that T (X) = X. Thus, for X = {x, y}, and R ∈ LT (X)×T (X) , where R(x, y) = R(y, x) = 12 , and R(x, x) = R(y, y) = 1, we have θ(R) = R. Therefore, T(X)/θ(R) is finitely presented.

114


Let h : T(X)/θ(R) → lim Mi be defined by h [x]θ(R) = [aj ]θ∞ and h [y]θ(R) = [bj ]θ∞ . Clearly, h is an ≈-morphism and thus a morphism. Suppose h = g ◦ hk , where g : T (X)/θ(R) → Mk and hk : Mk → lim Mi is a morphism of the limit cone of lim Mi . In this case, h = g ◦ hk yields g [x]θ(R) = ak and g [y]θ(R) = bk . Thus, g cannot be an ≈-morphism since

1 2

k 2k+1 .

An important thing to stress is that our generalized direct limit has a special property which is degenerated in ordinary case. If L is infinite and {Mi | i ∈ I} is a weak direct family which is not a direct family, there can be elements a ∈ Mi , b ∈ Mj the homomorphic images of which are distinct in every Mk for k ≥ i, j. However, it can happen that θ∞ (a , b ) = 1, i.e. [a ]θ∞ = [b ]θ∞ due to the general suprema used in (2.28). Such a situation is apparently ill at least from the standpoint of compatibility with ordinary direct limits. Indeed, the skeleton ske(lim Mi ) (i.e. an ordinary algebra being the functional part of lim Mi ) is then not isomorphic to the ordinary direct limit of skeletons ske(Mi ). On the other hand, such a situation cannot occur for direct families of L-algebras. Theorem 2.111. Let {Mi | i ∈ I} be a direct family of L-algebras. Then ske(lim Mi ) ∼ = lim{ske(Mi ) | i ∈ I} .

(2.32)

Proof. The claim is almost evident. It remains to show that for a ∈ Mi , b ∈ θ∞ (a , b ) = 1 iff there exists k ≥ i, j such that hik (a ) = hjk (b ) Mj we have since then i∈I Mi /θ∞ coincides with its crisp counterpart. So, assume that θ∞ (a , b ) = 1. Since {Mi | i ∈ I} is a direct family, there is some index k ≥ i, j such that hik (a ) ≈Mk hjk (b ) = 1, that is, hik (a ) = hjk (b ). The converse implication holds trivially. Altogether, ske(lim Mi ) ∼ = lim ske(Mi ) since the corresponding functions on ske(lim Mi ) and lim ske(Mi ) are defined the same way. Example 2.112. Consider L = [0, 1]. We can take a weak direct family from Remark 2.91 (2) on page 103. It is evident that lim Mi is a trivial L-algebra but there is not any j ∈ I such that hij (ai ) = hij (bi ). On the other hand, lim ske(Mi ) is a two-element (ordinary) algebra. In consequence, Theorem 2.111 is not true for general weak direct families of L-algebras. ∗

∗

∗

In the rest of this section, we show basic relationships between the notions of a direct union, direct limit, and some of the notions introduced earlier. In the literature, ordinary direct limit is often defined as a factorization of a special subalgebra of a direct product. This generalization does not require the algebras of a direct family to be pairwise disjoint. Since this approach uses

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only the notions we have already generalized for L-algebras (subalgebras, direct products and congruences), we can introduce an alternative generalization of a direct limit. In what follows, we will show that there is a natural relationship between both of these generalizations (the former approach and the one introduced below). For every directed index set I, ≤ and a family {hij : Mi → Mj | i ≤ j} of morphisms satisfying (2.23), (2.24) we can define a set M by Q M = a ∈ i∈I M i | there is i ∈ I such that for j, k ∈ I, i ≤ j ≤ k (2.33) we have hjk (a (j)) = a (k) . Q In other words, M represents a subset of i∈I M i every element of which respects morphisms {hij : Mi → Mj | i ≤ j}. Namely, for every a ∈ M there is an index i ∈ I such that for j ∈ I, i ≤ j we have hij (a (i)) = a (j). Furthermore, we define a binary L-relation θ on M by θ (a , b ) = i∈I k≥i a (k) ≈Mk b (k) (2.34) for every a , b ∈ M . Theorem 2.113. Let I, ≤ be a directed index set, {hij : Mi → Mj | i ≤ j} be a family of morphisms satisfying conditions (2.23), (2.24). Then ∅ = M ∈ Q Sub( i∈I Mi ), and θ ∈ ConL (M ), where M is the subalgebra given by M . Q Proof. We will show that M is a non-empty subuniverse of i∈I M i . EviQ Q dently, M = ∅. For an n-ary f i∈I Mi ∈ F i∈I Mi and elements a1 , . . . , an ∈ M there exists j ∈ I such that hjk (ai (j)) = ai (k) for each i = 1, . . . , n and k ≥ j. So, it follows that Q hjk f i∈I Mi (a1 , . . . , an )(j) = hjk f Mj (a1 (j), . . . , an (j)) = = f Mk hjk (a1 (j)), . . . , hjk (an (j)) = f Mk a1 (k), . . . , an (k) = =f

Q

i∈I Mi

(a1 , . . . , an )(k) .

Q Hence, f i∈I Mi (a1 , . . . , an ) ∈ M . That is, ∅ = M ∈ Sub( i∈I M i ). Therefore, we can consider the subalgebra M whose universe set is M . Introduce binary L-relation θ defined on M by (2.34). Reflexivity and symmetry of θ follows directly from reflexivity and symmetry of each ≈Mi . Now, for a , b , c ∈ M we have Mk b (k) ⊗ j∈I l≥j b (l) ≈Ml c (l) ≤ i∈I k≥i a (k) ≈ ≤ i,j∈I k≥i, l≥j a (k) ≈Mk b (k) ⊗ b (l) ≈Ml c (l) ≤ ≤ i,j∈I k≥i,j a (k) ≈Mk b (k) ⊗ b (k) ≈Mk c (k) ≤ ≤ i,j∈I k≥i,j a (k) ≈Mk c (k) ≤ m∈I m ≥m a (m ) ≈Mm c (m ) , Q

the last inequality follows from the fact that for every i, j ∈ I we can choose an index m ∈ I such that m ≥ i, j, in which case k≥i,j a (k) ≈Mk c (k) ≤

116


a (m ) ≈Mm c (m ) ≤ m∈I m ≥m a (m ) ≈Mm c (m ). Hence, θ is transitive. Compatibility with operations can be checked analogously. Briefly, for an n-ary f M ∈ F M and a1 , b1 , . . . , an , bn ∈ M we have m ≥m

θ (a1 , b1 ) ⊗ · · · ⊗ θ (an , bn ) = n = m=1 im ∈I km ≥im am (km ) ≈Mkm bm (km ) ≤ n ≤ i1 ,...,in ∈I k≥i1 ,...,in m=1 am (k) ≈Mk bm (k) ≤ n ≤ i∈I j≥i m=1 am (j) ≈Mj bm (j) ≤ ≤ i∈I j≥i f Mj (a1 (j), . . . , an (j)) ≈Mj f Mj (b1 (j), . . . , bn (j)) = = i∈I j≥i f M (a1 , . . . , an )(j) ≈Mj f M (b1 , . . . , bn )(j) = = θ f M (a1 , . . . , an ), f M (b1 , . . . , bn ) .

Clearly, ≈M ⊆ θ . Altogether, θ ∈ ConL (M ).

Lemma 2.114. Let {Mi | i ∈ I} be a weak direct family of L-algebras. Then θ∞ (a , b ) = θ (a , b ), where a ∈ Mi , b ∈ Mj , a , b ∈ M , and hik (a ) = a (k), hjl (b ) = b (l) for every k, l ∈ I, i ≤ k, j ≤ l. Proof. Let us note that we have Ml b (l) = k≥i,j l≥k a (l) ≈Ml b (l) . k∈I l≥k a (l) ≈

(2.35)

Indeed, the “≥”-part of (2.35) is trivial. Conversely, for every k ∈ I there is an index m ∈ I such that i, j, k ≤ m. Thus, Ml b (l) ≤ l≥m a (l) ≈Ml b (l) l≥k a (l) ≈ implies the “≤”-part of (2.35). Now, using (2.26) and (2.35) we obtain θ (a , b ) = k∈I l≥k a (l) ≈Ml b (l) = k≥i,j l≥k a (l) ≈Ml b (l) = = k≥i,j l≥k hil (a ) ≈Ml hjl (b ) = k≥i,j hik (a ) ≈Mk hjk (b ) = θ∞ (a , b ) ,

which is the desired equality.

Theorem 2.115. For every weak direct family {Mi | i ∈ I} of L-algebras we have lim Mi ∼ = M /θ . Proof. For a ∈ Mi put h [a ]θ∞ = [a ]θ , where a ∈ M satisfies a (k) = hik (a ) for each k ≥ i. It is easily seen that such an element a ∈ M always exists. Moreover, Lemma 2.114 yields [a ]θ∞ ≈lim Mi [b ]θ∞ = θ∞ (a , b ) = θ (a , b ) = [a ]θ ≈M

/θ

[b ]θ ,

where a ∈ Mi , b ∈ Mj and a , b ∈ M such that hik (a ) = a (k), hjl (b ) =

b (l) for every k, l ∈ I, i ≤ k, j ≤ l. Clearly, h : i∈I Mi /θ∞ → M is a well-defined ≈-morphism. Furthermore, h is surjective since for every element c ∈ M there is an index i ∈ I such that for all k ≥ i we have hik (c (i)) = c (k), that is h [c (i)]θ∞ = [c ]θ .

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Take an n-ary f ∈ F and arbitrary elements a1 , . . . , an ∈ i∈I Mi /θ∞ . h ( a ) = a (k) for each m = We can take a1 , . . . , an ∈ M , where im k m m 1, . . . , n, and k ≥ im . Since f M a1 , . . . , an ∈ M , there is j ∈ I such that i1 , . . . , in ≤ j and hjk f M a1 , . . . , an (j) = f M a1 , . . . , an (k) for all k ≥ j. Now, taking into account previous observations, we have h f lim Mi [a1 ]θ∞ , . . . , [an ]θ∞ = h f Mj (hi1 j (a1 ), . . . , hin j (an )) θ = ∞ Mj = h f (a1 (j), . . . , an (j)) θ = h f M (a1 , . . . , an )(j) θ = ∞ ∞ M M /θ = f [a1 ]θ , . . . , [an ]θ = a1 , . . . , an θ = f = f M /θ h [a1 ]θ∞ , . . . , h [an ]θ∞ .

Altogether h : lim Mi → M is an isomorphism, i.e. lim Mi ∼ = M .

We established that every direct limit is isomorphic to M /θQ , i.e. it can be thought of as a special factorization of a suitable subalgebra of i∈I Mi . On the other hand, the following theorem shows that every M /θ is isomorphic to a direct limit of some weak direct family of L-algebras.

Theorem 2.116. Let I, ≤ be directed index set. Let {hij : Mi → Mj | i ≤ j} be a family of morphisms satisfying (2.23) and (2.24). Then there is a weak direct family {gij : Ni → Nj | i ≤ j} such that lim Ni ∼ = M /θ . Proof. Suppose we have a family of morphisms {hij : Mi → Mj | i ≤ j} satisfying conditions (2.23), (2.24). Since Mi ’s are not assumed to be pairwisedisjoint, we will introduce a new family {Ni | i ∈ I} of L-algebras, where Ni = {a , i | a ∈ Mi } , a , i ≈Ni b , i = a ≈Mi b , f Ni a1 , i , . . . , an , i = f Mi (a1 , . . . , an ), i , for every n-ary f ∈ F , a , b , a1 , . . . , an ∈ Mi , and i ∈ I. Evidently, Ni ∼ = Mi and I, ≤ together with Ni ’s and {gij : Ni → Nj | i ≤ j}, where gij (a , i ) = hij (a ), i is a weak direct family of L-algebras.

, where θN ∈ ConL (N ) Moreover, Theorem 2.115 yields lim Ni ∼ = N /θN

∼

. is defined by (2.34). Hence, it is sufficient to check that M /θ = N /θN

∼

Evidently, M = N since every a ∈ M is in one-to-one correspondence with a ∈ N , where a (i) = a (i), i for all i ∈ I. Clearly, h [a ]θ = [a ]θ N

defines an isomorphism h : M /θ → N /θN . Thus, lim Ni ∼ = M /θ . Let us note that due to Lemma 2.89, every direct union i∈I Mi is isomorphic to N/θ, where N can be thought of as M for a special family of morphisms. In consequence, it is possible to treat a direct union as a special direct limit. Theorem 2.117. Let {Mi | i ∈ I} be a directed family of L-algebras. Then ∼ i∈I Mi = lim Ni for some direct family {Ni | i ∈ I} of L-algebras.

118


Proof. For indices i, j ∈ I satisfying i ≤ j we put hij (a ) = a (a ∈ Mi ). satisfying (2.23), Clearly, {hij : Mi → Mj | i ≤ j}, is a family of morphisms ∼ M (2.24), and (2.25). Due to Lemma 2.89, we have i = N/θ. Hence, i∈I

∼ M /θ since N (see Lemma 2.89) is defined the same way as M M = i i∈I

for hij ’s with hij (a ) = a . Trivially, θ = θ . Now we can use Theorem 2.116 to claim i∈I Mi ∼ = M /θ ∼ = lim Ni , where {Ni | i ∈ I} is a weak direct family of L-algebras defined as in the proof of Theorem 2.116. A moment’s reflection shows that {Ni | i ∈ I} is a direct family since for a , i ∈ Ni , b , j ∈ Nj and arbitrary k ≥ i, j we have gik a , i ≈Nk gjk b , j = hik (a ), k ≈Nk hjk (b ), k =

= hik (a ) ≈Mk hjk (b ) = a ≈Mk b = a ≈ i.e. {Ni | i ∈ I} is a direct family, completing the proof.

i∈I

Mi

b,

The following assertion shows that direct limits can be constructed by means of direct unions and factorizations. Theorem 2.118. Every direct limit of a weak direct family of L-algebras is isomorphic to a direct union of a directed family of L-algebras. Proof. Let us have a weak direct family of L-algebras which consists of a directed index set I, ≤ , a family of L-algebras {Mi | i ∈ I}, and a family of morphisms {hij : Mi → Mj | i ≤ j}. Recall that for every hi : Mi → lim Mi and the corresponding congruence θhi ∈ ConL (Mi ) we have θhi (a , b ) = hi (a ) ≈lim Mi hi (b ) = θ∞ (a , b ) = k≥i hik (a ) ≈Mk hik (b ) , for all i ∈ I, a , b ∈ Mi . Hence, θhi is a restriction of θ∞ on Mi , i.e. θhi = θ∞ |Mi . For a family {Mi /θhi | i ≤ j} of factor L-algebras we can define a family of mappings gij : Mi /θhi → Mj /θhj such that gij ([a ]θhi ) = [hij (a )]θhj for all i ≤ j. Thus, for every i, j ∈ I, i ≤ j and arbitrary a , b ∈ Mi we have [a ]θhi ≈Mi /θhi [b ]θhi = θhi (a , b ) = θ∞ (a , b ) = θ∞ hij (a ), hij (b ) = = θhj hij (a ), hij (b ) = [hij (a )]θhj ≈Mj /θhj [hij (b )]θhj = = gij [a ]θhi ≈Mj /θhj gij [b ]θhi . Therefore, every gij is a well-defined ≈-morphism. Moreover, for every n-ary f Mi /θhi and arbitrary elements [a1 ]θhi , . . . , [an ]θhi ∈ Mi /θhi it follows that gij f Mi /θhi [a1 ]θhi , . . . , [an ]θhi = gij f Mi (a1 , . . . , an ) θhi = = hij f Mi (a1 , . . . , an ) θhj = f Mj hij (a1 ), . . . , hij (an ) θhj = = f Mj /θhj [hij (a1 )]θhj , . . . , [hij (an )]θhj = = f Mj /θhj gij [a1 ]θhi , . . . , gij [an ]θhi , i.e., gij (i ≤ j) are embeddings. We check that gij ’s satisfy conditions (2.23) and (2.24). Equation (2.23) holds trivially and for i ≤ j ≤ k, [a ]θhi ∈ Mi /θhi we have

2.6 Direct Limits

119

gik [a ]θhi = [hik (a )]θhk = [hjk (hij (a ))]θhk = = gjk [hij (a )]θhj = gjk gij [a ]θhi = (gij ◦ gjk ) [a ]θhi , showing (2.24). Therefore, I, ≤ , {Mi /θhi | i ∈ I}, and morphisms gij (i ≤ j) form a weak direct family. Now, we can consider a direct limit lim Mi /θhi . In what follows, we will prove that lim Mi ∼ = lim Mi /θhi , and also that the direct limit lim Mi /θhi is isomorphic to a direct union of a suitable directed family of L-algebras. First, we will check lim Mi ∼ = lim Mi /θhi . Note that every member gi of the limit cone {gi : Mi /θhi → lim Mi /θhi | i ∈ I} associated with lim Mi /θhi is an embedding. Indeed, due to Remark 2.101, every gi is a morphism satisfying gi = gij ◦ gj . Moreover, for every k ≥ i, a , b ∈ Mi we have [a ]θhi ≈Mi /θhi [b ]θhi = θhi (a , b ) = θ∞ (a , b ) = = θ∞ hik (a ), hik (b ) = θhk hik (a ), hik (b ) . Taking into account this observation, we readily obtain [a ]θhi ≈Mi /θhi [b ]θhi = k≥i θhk hik (a ), hik (b ) = = k≥i [hik (a )]θhk ≈Mk /θhk [hik (b )]θhk = = k≥i gik [a ]θhi ≈Mk /θhk gik [b ]θhi = = gi [a ]θhi ≈lim Mi /θhi gi [b ]θhi for every gi : Mi /θhi → lim Mi /θhi and [a ]θhi , [b ]θhi ∈ Mi /θhi . Let hi be the natural morphism hi : Mi → Mi /θhi (i ∈ I). Clearly, for the composed morphism hi ◦ gi : Mi → lim Mi /θhi we have gi (hi (a )) = gi [a ]θhi = gj gij [a ]θhi = gj [hij (a )]θhj = gj (hj (hij (a ))) i.e. hi ◦ gi = hij ◦ (hj ◦ gj ) for every i ≤ j. Since {hi : Mi → lim Mi } satisfies DLP w.r.t. weak direct family {Mi | i ∈ I}, there is a uniquely determined morphism g : lim Mi → lim Mi /θhi such that hi ◦ gi = hi ◦ g for every i ∈ I, see Fig. 2.18 (left). Moreover, due to Theorem 2.35 a family {ki : Mi /θhi → lim Mi | i ∈ I}, where ki [a ]θhi = [a ]θ∞ = hi (a ) is a family of morphisms for which ki [a ]θhi = hi (a ) = hj (hij (a )) = kj [hij (a )]θhj = kj gij [a ]θhi ,

Fig. 2.18. Morphisms between lim Mi and lim Mi /θhi

120


i.e. ki = gij ◦ kj holds for every i ≤ j. Since {gi : lim Mi → lim Mi /θhi | i ∈ I} satisfies DLP w.r.t. weak direct family {Mi /θhi | i ∈ I}, there is a morphism g : lim Mi /θhi → lim Mi such that ki = gi ◦ g for every i ∈ I, see Fig. 2.18 (right). Now, from hi = hi ◦ ki = hi ◦ gi ◦ g = hi ◦ g ◦ g , hi

◦ gi = hi ◦ g = hi ◦ ki ◦ g = hi ◦ gi ◦ g ◦ g ,

it readily follows that g ◦ g = idlim Mi , g ◦ g = idlim Mi /θhi . Thus, we can apply Theorem 2.30 to get lim Mi ∼ = lim Mi /θhi . Finally, it suffices to check the following claim: (2.36) lim Mi /θhi ∼ gi Mi /θhi | i ∈ I , = but this is almost evident. Clearly, gi Mi /θhi ∈ Sub(lim Mi /θhi ) by Theo M ∈ = g ◦ g , i.e. g /θ rem 2.28. Furthermore, for i ≤ j we have g i ij j i i h i Sub(gj Mj /θhj ). That is, gi Mi /θhi | i ∈ I is a directed family of L gi Mi /θhi | i ∈ I to be algebras. Hence, we consider h : lim Mi /θhi → the identity automorphism on lim M /θ . Since every i h i element of lim Mi /θhi gi Mi /θhi | i ∈ I , we have can be expressed by some gi [a ]θhi ∈ gi [a ]θhi ≈lim Mi /θhi gj [b ]θhj = = gk gik [a ]θhi ≈lim Mi /θhi gk gjk [b ]θhj = = gk gik [a ]θhi ≈gk (Mk /θhk ) gk gjk [b ]θhj = = gk gik [a ]θhi ≈ {gi (Mi /θhi ) | i∈I} gk gjk [b ]θhj = = gi [a ]θhi ≈ {gi (Mi /θhi ) | i∈I} gj [b ]θhj for every gi [a ]θhi , gj [b ]θhj , and k ≥ i, j. Analogously, for any n-ary f ∈ F and arbitrary gim [am ]θhi , im ≤ k, m = 1, . . . , n it follows that m f lim Mi /θhi . . . , gim [am ]θhi , . . . = m ,... = = f lim Mi /θhi . . . , gk gim k [am ]θhi m = f gk (Mk /θhk ) . . . , gk gim k [am ]θhi ,... = m = f {gi (Mi /θhi ) | i∈I} . . . , gk gim k [am ]θhi ,... = m = f {gi (Mi /θhi ) | i∈I} . . . , gim [am ]θhi , . . . . m Altogether, lim Mi ∼ gi Mi /θhi | i ∈ I . = lim Mi /θhi ∼ =

2.7 Reduced Products In this section we propose a generalization of reduced products and show its connection with previously introduced constructions. In addition to that, in

2.7 Reduced Products

121

Sect. 4.8 we show its non-trivial applications. Note that the generalization presented below is probably not the only one possible. A survey on reduced products and ultraproducts in fuzzy setting will be available in [19]. We define a reduced product of L-algebras by means of previously defined constructions similarly Q as in ordinary case. The key issue is how to define a congruence relation on i∈I Mi with respect to a filter F over I (see Sect. 1.1). Recall that in ordinary case we put a , b ∈ θF iff {i ∈ I | a (i) = b (i)} ∈ F Q for every a , b ∈ i∈I M i . Thus, on the verbal level: “a , b ∈ θF iff the set of indices on which a equals to b is large (i.e. belongs to a filter F ).” In what follows, we will proceed in two steps. First, we try to generalize the notion of “being equal on indices from X ∈ F ”. Then, using such a graded equality with respect to some index set, we define an L-relation representing for every Q a , b ∈ i∈I M i a degree to which a equals to b over a large set of indices. In the sequel, we use an ordinary filter. That is, we do not fuzzify the notion of a filter itself. We denote a filter by F , and the elements of F will be denoted X, Y, Z, . . . (there is no danger of confusion with the symbol of a type of an L-algebra and with sets of variables, because we use a fixed type and we do not use variables in the rest of this section). of the same type Definition 2.119. Let {Mi | i ∈ I} be a family of L-algebras Q and let F be a filter over I. Then for every a , b ∈ i∈I M i and X ∈ F we define a truth degree [[a ≈ b ]]X ∈ L as follows, [[a ≈ b ]]X = i∈X a (i) ≈Mi b (i). (2.37) Remark 2.120. (1) On the verbal level, [[a ≈ b ]]X expresses a truth degree to which it is true that a is equal to b over all indices taken from X. (2) An easy but important observation is that the truth degree [[a ≈ b ]]X depends only on the truth degrees a (i) ≈Mi b (i) for i ∈ X. Lemma 2.121. Let {Mi | i ∈ I} be a family of L-algebras of the same type and let F be a filter over I. Then (i) for every X, Y ∈ F , such that X ⊆ Y we have [[a ≈ b ]]Y ≤ [[a ≈ b ]]X , Q a ≈ i∈I Mi b ≤ [[a ≈ (ii) b ]]X for every X ∈ F , (iii) X∈F [[a ≈ b ]]X = X1 ,...,Xn ∈F [[a ≈ b ]]X1 ∩···∩Xn . Proof. (i) follows directly by Definition Q2.119. (ii): Since X ⊆ I ∈ F , (i) yields a ≈ i∈I Mi b = [[a ≈ b ]]I ≤ [[a ≈ b ]]X . (iii): The “≤”-part follows easily since for each X ∈ F , X = X ∩ · · · ∩ X. Conversely, if X1 , . . . , Xn ∈ F then X1 ∩ · · · ∩ Xn ∈ F because F is closed under finite intersections. Hence, the “≥”-part is also evident. Q Now we use [[a ≈ b ]]X to define a suitable L-relation on i∈I M i .

122


Definition 2.122. Let {Mi | i ∈ I} be a family of L-algebras of the Q same type and let F be a filter over I. We define a binary L-relation θF on i∈I M i by θF (a , b ) = X∈F [[a ≈ b ]]X (2.38) Q for all a , b ∈ i∈I M i . Remark 2.123. Since X ∈ F are thought of as large subsets, θF (a , b ) can be understood as a degree to which “there is a large X such that a equals b over all indices from X”. Theorem 2.124. Let {Mi | i ∈ I} be a familyQof L-algebras of the same type and let F be a filter over I. Then θF ∈ ConL ( i∈I Mi ). Q

Proof. From Lemma 2.121 it readily follows that ≈ i∈I Mi ⊆ θF . Moreover, reflexivity and symmetry of θF follow directly from reflexivity and symmetry of every ≈Mi , respectively. Q Q Thus, it suffices to check transitivity and compatibility with functions of i∈I Mi . Using Lemma 2.121, for a , b , c ∈ i∈I M i we have θF (a , b ) ⊗ θF (b , c ) = X∈F [[a ≈ b ]]X ⊗ Y ∈F [[b ≈ c ]]Y = = X,Y ∈F [[a ≈ b ]]X ⊗ [[b ≈ c ]]Y = = X,Y ∈F i∈X a (i) ≈Mi b (i) ⊗ j∈Y b (j) ≈Mj c (j) ≤ ≤ X,Y ∈F i,j∈X∩Y a (i) ≈Mi b (i) ⊗ b (j) ≈Mj c (j) ≤ ≤ X,Y ∈F i∈X∩Y a (i) ≈Mi b (i) ⊗ b (i) ≈Mi c (i) ≤ ≤ X,Y ∈F i∈X∩Y a (i) ≈Mi c (i) = X,Y ∈F [[a ≈ c ]]X∩Y = = X∈F [[a ≈ c ]]X = θF (a , c ) . Thus, θF is transitive. For compatibility withQfunctions take an n-ary f and arbitrary elements a1 , b1 , . . . , an , bn ∈ i∈I M i . We have

Q

i∈I Mi

θF (a1 , b1 ) ⊗ · · · ⊗ θF (an , bn ) = n = X1 ,...,Xn ∈F i=1 ji ∈Xi a (ji ) ≈Mji b (ji ) ≤ Q Q ≤ X1 ,...,Xn ∈F f i∈I Mi (a1 , . . . , an ) ≈ f i∈I Mi (b1 , . . . , bn ) X1 ∩···∩Xn = Q Q = X∈F f i∈I Mi (a1 , . . . , an ) ≈ f i∈I Mi (b1 , . . . , bn ) X = Q Q = θF f i∈I Mi (a1 , . . . , an ), f i∈I Mi (b1 , . . . , bn ) . Q Altogether, θF is a congruence on i∈I Mi . Finally, we introduce reduced product of L-algebras. Definition 2.125. Let {Mi | i ∈ I}be of L-algebras of Q the same type Q a family /θ M denoted by and let F be a filter over I. Then i F i∈I F Mi is called a reduced product of {Mi | i ∈ I} modulo F .


123

Q

Remark 2.126. Clearly, θF and the corresponding F Mi are determined by {Mi | i ∈ I} and a filter F over I. In borderline cases, θF behaves the same way as in ordinary case. Indeed, when F is an improper filter (i.e. ∅ ∈ F ), we have Q Q θF (a , b ) = 1 for all a , b ∈ i∈I M i . Thus, F Mi is a trivial (one-element) } ∈ F for some L-algebra. If F is a trivial filter (i.e. F is a proper filter and {i0Q i0 ∈ I) it follows that θF = ≈Mi0 for Qcertain i0 ∈ I. Thus, F Mi ∼ = Mi0 . Q Q M . Finally, if F = {I} then clearly θF = ≈ i∈I Mi , i.e. F Mi ∼ = i∈I i ∗

∗

∗

In the ordinary case, reduced products are isomorphic to particular direct limits and conversely, direct limits are isomorphic to particular subalgebras of reduced products. In the subsequent development, we present analogous characterizations. Let {Mi | i ∈ I} be a family of L-algebras of the same type and Q let F be a . For filter over I. For every X ∈ F we can consider a direct product i∈X M Q i Q Q MX i∈X Mi , M , ≈ = ≈ brevity, let MX denote i∈X Mi . That is, MX = i∈X i Q and for an n-ary function symbol f let f MX denote f i∈X Mi . It readily follows that a ≈MX b = i∈X a (i) ≈Mi b (i) = [[a ≈ b ]]X . (2.39) In addition to that, filter F can be partially ordered using the ordinary set inclusion. Moreover, F, ⊇ can be thought of as a (downward) directed index set. Indeed, for every X, Y ∈ F it follows that X, Y ⊇ X ∩ Y ∈ F , i.e. for every X, Y ∈ F there is Z ∈ F such that X, Y ⊇ Z. Clearly, MX , MY are disjoint for all X, Y ∈ F , X = Y . For every X, Y ∈ F , X ⊇ Y , we can consider a morphism hXY : MX → MY defined by hXY (a )(i) = a (i)

(2.40)

for every a ∈ MX and i ∈ Y . Due to Theorem 2.51, hXY is the uniquely determined morphism induced by family {πi : MX → Mi | i ∈ Y } of projections. It is easily seen that {hXY : MX → MY | X ⊇ Y, X, Y ∈ F } satisfies conditions (2.23) and (2.24). As a consequence, {MX | X ∈ F } is a weak direct family of L-algebras. Theorem 2.127. Let {Mi | i ∈ I}Qbe a family of L-algebras of the same type and let F be a filter over I. Then F Mi ∼ = lim MX . Q Proof. We present a family {hX : MX → F Mi } of Q morphisms satisfying DLP with respect to {MX | X ∈ F }. Then we get F Mi ∼ = lim MX as a consequence of Theorem 2.104. Q Q Recall that I ∈ F , and MI stands for i∈I Mi .QThus, hIX : i∈I Mi → MX (X ∈ F ) are a surjective morphisms. For a , b ∈ i∈I M i we have θhIX (a , b ) = hIX (a ) ≈MX hIX (b ) = i∈X hIX (a )(i) ≈Mi hIX (b )(i) = = i∈X a (i) ≈Mi b (i) = [[a ≈ b ]]X ≤ θF (a , b ) .

124

2 Algebras with Fuzzy Equalities i∈I Mi

hθ F

hθF

hIX

hIX

i∈I Mi

F Mi

gI

gX

hX F Mi

MX

g

MX

N

Q Fig. 2.19. hX : MX → F Mi satisfies DLP w.r.t. {MX | X ∈ F }

Q Q Therefore, θhIX ⊆ θF for every X ∈ F . Let hθF : i∈I Mi → F Mi denote the natural morphism. Hence, from Lemma 2.44 it followsQ that for every X ∈ F there is a uniquely determined morphism hX : MX → F Mi satisfying hθF = hIX ◦ hX , see Fig. 2.19 (left). Moreover, we can apply (2.24) to obtain hθF = hIY ◦ hY = hIX ◦ hXY ◦ hY , i.e. hIX ◦ hXY ◦ hY = hIX ◦ hX . Thus, the surjectivity of hIX yields hX = hXY ◦ hY . Let us have a family {gX : MX → N | X ∈ F } of morphisms satisfying gX = hXY ◦ gY for every X, Y ∈ F , X Q ⊇ Y . It remains to show that there is a uniquely determined morphism g : F Mi → N such that gX = hX ◦ g for Q every X ∈ F . First, for a , b ∈ i∈I M i it follows that θF (a , b ) = X∈F [[a ≈ b ]]X = X∈F i∈X a (i) ≈Mi b (i) = = X∈F i∈X hIX (a )(i) ≈Mi hIX (b )(i) = X∈F hIX (a ) ≈MX hIX (b ) ≤ ≤ X∈F gX (hIX (a )) ≈N gX (hIX (b )) = X∈F gI (a ) ≈N gI (b ) = = gI (a ) ≈N gI (b ) = θgI (a , b ) . Hence, Q θF ⊆ θgI . Due to Lemma 2.44 there is a uniquely determined morphism g : F Mi → N, where gI = hθF ◦ g, see Fig. 2.19 (right). Finally, gI = hθF ◦g = hIX ◦hX ◦g and gI = hIX ◦gX by the assumption. AsQa consequence, gX = hX ◦g due to the surjectivity of hIX . Thus, {hQ X : MX → F Mi } satisfies DLP with respect to {MX | X ∈ F }. Altogether, F Mi ∼ = lim MX . Theorem 2.128. Let {Mi | i ∈ I} be a weak direct family. ThenQ there is a filter F over I such that lim Mi is isomorphic to a subalgebra of F Mi . Proof. Introduce F by putting X∈F

iff

there is i ∈ I such that {k ∈ I | k ≥ i} ⊆ X ⊆ I.

(2.41)

Obviously, if X ∈ F and X ⊆ Y ⊆ I then Y ∈ F . If X, Y ∈ F then there are indices i, j ∈ I such that {k ∈ I | k ≥ i} ⊆ X and {k ∈ I | k ≥ j} ⊆ Y , i.e. {k ∈ I | k ≥ i, j} ⊆ X ∩ Y . Since I, ≤ is a directed index set there is l ≥ i, j. Thus, {k ∈ I | k ≥ l} ⊆ X ∩ Y , showing X ∩ Y ∈ F . So, F defined by (2.41) is a filter over I. Consider M and θ defined by (2.33) and (2.34), respectively. We claim that θ equals θF |M (recall that θF |M denotes the restriction of θF on M ). It suffices to show that for any a , b ∈ M we have


i∈I

k≥i

a (k) ≈Mk b (k) =

X∈F

k∈X

125

a (k) ≈Mk b (k) .

“≤”: For each i ∈ I we have {k ∈ I | k ≥ i} ∈ F , i.e. the “≤”-part is obvious. “≥”: Take X ∈ F . By definition of F , there is i ∈ I with {k ∈ I | k ≥ i} ⊆ X. Thus, k≥i a (k) ≈Mk b (k) ≥ k∈X a (k) ≈Mk b (k), showing “≥”. Using the claim together with Theorem 2.115 and Theorem 2.42, we get lim Mi ∼ = M /θ ∼ = M /(θF |M ) ∼ = (M )θF/(θF |(M )θF ) . Q Since (M )θF/(θF |(M )θF ) is Q a subalgebra of F Mi we get that lim Mi is isomorphic to a subalgebra of F Mi . As we have seen, for {Mi | i ∈ I} and a filter F over I, {MX | X ∈ F } is a weak direct family of L-algebras. Suppose {MX | X ∈ F } is a direct family of L-algebras and let us look whether the essential property (2.25) has some natural translation in terms of the properties of F . Definition 2.129. Let {Mi | i ∈ I} be a family of L-algebras of the same type. A Q filter F over I is safe with respect to {Mi | i ∈ I} if for every a , b ∈ i∈I M i there is X ∈ F such that θF (a , b ) = [[a ≈ b ]]X . If F is safe with respect to any family of L-algebras of the same type then FQis called safe. If F is safe with respect to a family {Mi | i ∈ I} then F Mi is called a safe reduced product of {Mi | i ∈ I} modulo F . Remark 2.130. Safeness of a filter F with respect to a given family of Lalgebras is a non-trivial property. (1) If F = {I} then F is safe. Also every trivial and improper filterQis safe. (2) If θF (a , b ) is compact element of L (see Sect. 1.1) for Q all a , b ∈ i∈I M i then F is safe w.r.t. {Mi | i ∈ I}. Indeed, for any a , b ∈ i∈I M i there are n X1 , . . . , Xn ∈ F such that θF (a , b ) = i=1 [[a ≈ b ]]Xi . Since X1 ∩ · · · ∩ Xn ∈ F , it follows that θF (a , b ) ≤ [[a ≈ b ]]X1 ∩···∩Xn . The converse inequality holds trivially. In particular, if L is a Noetherian residuated lattice then every filter is safe. (3) Take L = [0, 1] as the structure of truth degrees. Let us have an index set N. We can consider a family {Mi | i ∈ N} of L-algebras of empty type, where Mi = {a , b } and a ≈Mi b = 1 − 1i for every i ∈ N. Thus, a ≈M1 b = 0, a ≈M2 bQ= 12 , a ≈M3 b = 23 , etc. Let F be the Fréchet filter over N. Take a , b ∈ i∈N Mi , where a (i) = a and b (i) = b for all i ∈ N. Clearly, we have θF (a , b ) = 1 but [[a ≈ b ]]X < 1 for every X ∈ F . Hence, F is not safe w.r.t. {Mi | i ∈ N}. Lemma 2.131. Let {Mi | i ∈ I} be a family of L-algebras of the same type and let F be a filter over I. Then for a ∈ MX , b ∈ MY , and any Z ∈ F such that X, Y ⊇ Z we have where a , b ∈

Q

hXZ (a ) ≈MZ hYZ (b ) = [[a ≈ b ]]Z , i∈I M i

satisfy hIX (a ) = a and hIY (b ) = b .

(2.42)

126


Proof. Clearly, we have

hXZ (a ) ≈MZ hYZ (b ) = i∈Z hXZ (a )(i) ≈Mi hYZ (b )(i) = = i∈Z hXZ (hIX (a ))(i) ≈Mi hYZ (hIY (b ))(i) = = i∈Z hIZ (a )(i) ≈Mi hIZ (b )(i) = i∈Z a (i) ≈Mi b (i) = [[a ≈ b ]]Z ,

which is the desired equality.

Theorem 2.132. A filter F is safe w.r.t. a family {Mi | i ∈ I} of L-algebras of the same type if and only if {MX | X ∈ F } is a direct family of L-algebras. Proof. “⇒”: Let F be safe w.r.t. {Mi | i ∈ I}. Take a ∈ MX , b ∈ MY . We have to show that there is Z ∈ F such that X, Y ⊇ Z and hXZ (a ) ≈MZ hYZ (b ) = hXZ (a ) ≈MZ hYZ (b )

Q is satisfied for every Z ∈ F , where Z ⊇ Z . Let us have a , b ∈ i∈I M i such that hIX (a ) = a and hIY (b ) = b . Since F is safe, we have θF (a , b ) = [[a ≈ b ]]Z0 for certain Z0 ∈ F . Put Z = Z0 ∩ X ∩ Y . Clearly, for every Z ∈ F such that Z ⊇ Z , it follows that θF (a , b ) = [[a ≈ b ]]Z0 = [[a ≈ b ]]Z ≥ [[a ≈ b ]]Z . Moreover, Lemma 2.121 (i) yields [[a ≈ b ]]Z ≤ [[a ≈ b ]]Z . Altogether, using Lemma 2.131, we obtain hXZ (a ) ≈MZ hYZ (b ) = [[a ≈ b ]]Z = [[a ≈ b ]]Z = hXZ (a ) ≈MZ hYZ (b ) . Hence, {MX | X ∈ F } is a direct family of L-algebras. ∈ F } be a direct family of L-algebras. From (2.25) it “⇐”: Let {MX | X Q follows that for a , b ∈ i∈I M i there is some Z ∈ F such that [[a ≈ b ]]Z = hIZ (a ) ≈MZ hIZ (b ) = hIZ (a ) ≈MZ hIZ (b ) = [[a ≈ b ]]Z holds for every Z ∈ F , Z ⊇ Z . Thus, we have θF (a , b ) = X∈F [[a ≈ b ]]X ≤ ≤ X∈F [[a ≈ b ]]X∩Z = X∈F [[a ≈ b ]]Z = [[a ≈ b ]]Z . The converse inequality follows from the definition of θF . That is, filter F is safe with respect to a family {Mi | i ∈ I} of L-algebras. Remark 2.133. The construction of a safe reduced product is compatible with its ordinary counterpart in the sense of preserving the skeletons: Q Q ske(Mi ) . ske( Mi ) ∼ = ske(lim MX ) ∼ = lim{ske(MX ) | X ∈ F } ∼ = F

F

This is an immediate consequence of Theorem 2.111 and Theorem 2.127 and Theorem 2.132.QOn the contrary, we can use Remark 2.130 (3) to observe that Q ske( F Mi ) ∼ = F ske(Mi ) does not hold for general reduced products. Reduced products will be used in Sect. 4.8 where we establish a connection between safe reduced products and closure properties of certain implicationally definable classes of L-algebras.

2.8 Class Operators

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2.8 Class Operators In this section we present basic notions related to classes of L-algebras. Definition 2.134. For a class K of L-algebras of the same type we define the following operators: H(K) = {h(M) | M ∈ K, h is a morphism} , I(K) = {M | M is isomorphic to some N ∈ K} , S(K) = {M | M is a subalgebra of some N ∈ K} , P(K) = {M | M is a direct product of a family P ⊆ K} , PS (K) = {M | M is a subdirect product of a family P ⊆ K} , U(K) = {M | M is a direct union of a directed family U ⊆ K} , L(K) = {M | M is a direct limit of a direct family L ⊆ K} , PR (K) = {M | M is a safe reduced product of a family P ⊆ K} . That is, H(K) is the class of all homomorphic images of L-algebras from K. I(K) is the class of all L-algebras isomorphic to some N ∈ K. S(K) is the class of all subalgebras of L-algebras from K. P(K) and PS (K) denote the classes of all direct and subdirect products of families of L-algebras from K, respectively. U(K) denotes the class of all direct unions of directed families of L-algebras from K. L(K) denotes the class of all direct limits of direct families of Lalgebras from K. Finally, PR (K) denotes the class of all safe reduced products of families of L-algebras from K. Remark 2.135. (1) Class operators may be composed. That is, we may have HS, HHPHS, and so on. (2) If O1 , O2 are two operators on classes of L-algebras of the same type we put O1 ≤ O2 iff O1 (K) ⊆ O2 (K) for every class K of L-algebras of type F . It is easy to see that ≤ is a partial order on operators on classes of L-algebras of the same type. Definition 2.136. A class K of L-algebras of type F is said to be closed under an operator O if O(K) ⊆ K. An operator O is said to be idempotent iff O = OO. Definition 2.137. A class K of L-algebras closed under the operator I is called an abstract class of L-algebras. We adopt the following convention for class operators on abstract classes of L-algebras. Let O be an operator on classes of algebras. If K is an abstract class of L-algebras, then O(K) denotes an abstract class IO(K). Roughly speaking, we add to O(K) all isomorphic copies of L-algebras from O(K). The notion of an abstract class will be used mainly in Sect. 4.5 to avoid technical complications. Let us note that for abstract classes we have several convenient consequences, e.g. P is an idempotent operator on abstract classes of L-algebras while this is not so for general classes of L-algebras.

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Definition 2.138. Let K be a class of L-algebras of the same type. Let M be an L-algebra generated by M ⊆ M , i.e. M = [M ]M . If for each N ∈ K and every ≈-morphism h : M , ≈M → N, ≈N there exists a homomorphic extension h : M → N of h (that is h is an morphism and h (a ) = h(a ) for every a ∈ M ), we say that M has a universal mapping property (UMP) for K over M . In this case M is said to be a set of free generators of M over K. Theorem 2.139. Suppose that M has UMP M . Let N ∈ K. for KNover M → N, ≈ there exists a unique Then for every ≈-morphism h : M , ≈ morphism h : M → N extending h. Proof. Suppose there are two homomorphic extensions, h , h : M → N. Since M is generated by M , Theorem 2.81 yields that for every a ∈ M there is a term t ∈ T (X), |X| = |M | such that tM (a1 , . . . , ak ) = a for some a1 , . . . , ak ∈ M . Thus, h (a ) = h tM (a1 , . . . , ak ) = tN h (a1 ), . . . , h (ak ) = = tN h (a1 ), . . . , h (ak ) = h tM (a1 , . . . , ak ) = h (a ) .

Hence, h admits a unique homomorphic extension h .

Theorem 2.140. Suppose that M has UMP for K over M and let N, N ∈ K. If g : M , ≈M → N, ≈N is an ≈-morphism and h : N → N is a morphism then (g ◦ h) = g ◦ h. Proof. For each a ∈ M we have (g ◦h)(a ) = h(g (a )) = h(g(a )) = (g ◦h)(a ) because g is an extension of g. Thus, g ◦ h is a morphism which extends g ◦ h. Since g ◦ h is an ≈-morphism, Theorem 2.139 gives that (g ◦ h) = g ◦ h. Theorem 2.141. For a type F , a set X of variables, and a class K of Lalgebras of type F , if T(X) exists, then it has a universal mapping property for K over X. Proof. Consequence of Theorem 2.79.

2.9 Related Approaches Algebras with fuzzy equalities represent but one particular approach to the notion of an algebra from the point of view of fuzzy logic. This approach is a modest one – at the beginning, the only basic concept of ordinary algebras which becomes fuzzy is the equality on the universe. Doing so, our approach to algebras fits the framework of predicate fuzzy logic. Namely, algebras with fuzzy equalities are just the structures of predicate fuzzy logic with equality symbol as the only relation symbol. Needless to say, there can be many ways to approach algebraic concepts from the point of view of fuzzy logic. The aim of this section is to present some of them which appeared in the literature. If possible, we discuss relationships to algebras with fuzzy equalities.

2.9 Related Approaches

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Metric Algebras Recall from Sect. 1.3 that a generalized metric on a set M is a mapping : M × M → [0, ∞] satisfying conditions listed on page 42. For the sake of brevity, we call a generalized metric simply a metric throughout this section. In [96] (see also [95]), Weaver studies so-called metric algebras. A metric algebra is basically an algebra equipped with a metric on its support such that operations of the algebra are in a particular sense compatible with the metric. The metric can be seen as a constraint for the operations. Metric algebras thus represent an approach to the idea of extending algebras by adding constraints. Since the very idea of a metric is to describe closeness of elements of the universe, metric algebras can be seen as providing an alternative to algebras with fuzzy equalities. An obvious question is that of the relationship between metric algebras and algebras with fuzzy equalities. The goal of this section is to look at it. We start with basic notions related to metric algebras. A metric algebra of type F is a triplet M = M, M , F M , where (i) M, F M is a classical algebra of type F , (ii) M is a metric on M . Recall from the beginning of this section that we allow M (a , b ) = ∞. The notion of a metric algebra itself does not include any constraint on functions. In order to introduce a constraint, Weaver uses implications between so-called atomic inequalities and a notion of an equicontinuous satisfaction. Given a metric algebra M = M, M , F M and a mapping v : X → M , we utilize the usual notion of a value !t!M,v of a term t ∈ T (X) in M under a valuation v: for variable x ∈ X, put !x!M,v = v(x); for a term f (t1 , . . . , tn ) ∈ T (X), put !f (t1 , . . . , tn )!M,v = f M (!t1 !M,v , . . . , !tn !M,v ), cf. Remark 1.96 (2). An atomic inequality is an expression of the form (t, t ) α, where t, t ∈ T (X) and α ∈ [0, ∞]. (t, t ) α is δ-true in M under v if M !t!M,v , !t !M,v ≤ α + δ. (t, t ) α is true in M under v if it is δ-true in M under v for δ = 0. An implication (between atomic inequalities) is an expression of the form (s1 , s1 ) α1 c · · · c (sn , sn ) αn i (t, t ) β .

(2.43)

Let K be a class of metric algebras. K satisfies (2.43) if for each M ∈ K and each valuation v we have: if (si , si ) αi is true in M under v for each i = 1, . . . , n, then (t, t ) β is true in M under v. K satisfies (2.43) equicontinuously, if for each ε > 0 there is δ > 0 such that for each M ∈ K and each valuation v we have: if (si , si ) αi is δ-true in M under v for each i = 1, . . . , n, then (t, t ) β is ε-true in M under v. K has equicontinuous functions if for any n-ary f ∈ F , K satisfies (x1 , y1 ) 0 c · · · c (xn , yn ) 0 i i (f (x1 , . . . , xn ), f (y1 , . . . , yn )) 0

(2.44)

equicontinuously. M has equicontinuous functions if K = {M} has equicontinuous functions.

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Remark 2.142. (1) If K satisfies (2.43) equicontinuously then K satisfies (2.43). In more detail, if each (si , si ) αi is true (i.e., 0-true) in M under v then it is also δ-true. Therefore, if (si , si ) αi (i = 1, . . . , n) are true in M under v then (t, t ) β is ε-true for each ε > 0 which immediately gives that (t, t ) β is true (i.e., 0-true). (2) Condition (2.44) can be seen as the classical compatibility axiom, see Remark 1.81 (2), which is formulated in terms of metric algebras, i.e., as an implication between atomic inequalities instead of the classical identities. (3) In certain cases, equicontinuity of functions of a metric algebra M is trivial. For example, if {M (a , b ) | a , b ∈ M and a = b } has a lower bound γ > 0 then M has equicontinuous functions since for each ε > 0 one can put δ = γ2 . M then satisfies (2.44) equicontinuously, because (xi , yi ) 0 is δ-true ( γ2 -true) in M under v iff v(xi ) = v(yi ). Note that such a situation occurs if M is finite, or more generally, if {M (a , b ) | a , b ∈ M and a = b } is finite. (4) M has equicontinuous functions iff functions of M are uniformly continuous. (5) Let us note that the existence of δ > 0 for each ε > 0 in the definition of equicontinuous satisfaction is required for a whole class K. That is, the fact that each M ∈ K has equicontinuous functions does not imply in general that K has equicontinuous functions. We are now going to look at the relationship between algebras with fuzzy equalities and metric algebras with equicontinuous functions. As mentioned above, algebras with fuzzy equalities and metric algebras can be seen as two different approaches to functions on a set which are compatible with closeness/similarity of elements of this set. In addition to this, as described in Theorem 1.78, there is a close relationship between metrics and fuzzy equalities. We are going to show that L-algebras with L given by a continuous Archimedean t-norm can be turned into metric algebras with equicontinuous functions. Lemma 2.143. Let L be a residuated lattice given by a continuous Archimedean t-norm ⊗, ≈M be an L-equivalence on M , f M : M n → M be a function compatible with ≈M . Then ≈M f M (a1 , . . . , an ), f M (b1 , . . . , bn ) ≤ ≈M (a1 , b1 ) + · · · + ≈M (an , bn ) for each a1 , b1 , . . . , an , bn ∈ M . Proof. Let g be the additive generator of ⊗. First, we claim that for each a1 , . . . , an ∈ L we have g(a1 ⊗ · · · ⊗ an ) ≤ g(a1 ) + · · · + g(an ) .

(2.45)

For n = 1, the claim is trivial. By induction, suppose g(a2 ⊗ · · · ⊗ an ) ≤ n g(a2 ) + · · · + g(an ). By Theorem 1.43, we have g( i=1 ai ) = g g (−1) (g(a1 ) +


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n n n g(i=2 ai )) . Now, if g(a1 )+g( i=2 ai ) ≤ g(0), we have g( i=1 ai ) = g(a1 )+ n n n g(a ) by induction assumption. If g(a ) + g( g( i=2 ai ) ≤ i i=1 i=2 ai ) n 1n n g(0), we have g( i=1 ai ) = g(0) < g(a1 ) + g( i=2 ai ) ≤ i=1 g(ai ), showing that (2.45) holds. Thus, applying this claim and the fact that g is decreasing, we get ≈M f M (a1 , . . . , an ), f M (b1 , . . . , bn ) = = g f M (a1 , . . . , an ) ≈M f M (b1 , . . . , bn ) ≤ ≤ g(a1 ≈M b1 ⊗ · · · ⊗ an ≈M bn ) ≤ g(a1 ≈M b1 ) + · · · + g(an ≈M bn ) = = ≈M (a1 , b1 ) + · · · + ≈M (an , bn ) , which is the desired inequality.

Theorem 2.144. Let lattice given by a continuous Archi L be a residuated medean t-norm ⊗, M, ≈M , F M be an L-algebra. Then M, ≈M , F M is a metric algebra with equicontinuous functions. M Proof. Theorem 1.78 yields that ≈M is a metric since ≈ is an L-equality. Thus, it remains to check that for each n-ary f ∈ F , M, ≈M , F M satisfies (2.44) equicontinuously. If f is a nullary function symbol, the claim is trivial. So suppose n > 0. Given ε > 0, put δ = nε . Suppose ≈M (ai , bi ) ≤ δ for each i = 1, . . . , n. Using Lemma 2.143 we have ≈M f M (a1 , . . . , an ), f M (b1 , . . . , bn ) ≤ ≤ ≈M (a1 , b1 ) + · · · + ≈M (an , bn ) ≤ nδ = ε . Hence, M, ≈M , F M satisfies (2.44) equicontinuously.

Remark 2.142 (4) yields the following corollary. Corollary 2.145. Let lattice given by a continuous Archi L be a residuated medean t-norm ⊗, M, ≈M , F M be an L-algebra. Then each f M ∈ F M is a uniformly continuous function with respect to ≈M . Note that the nontrivial point of the above assertions is that if the functions f M of the L-algebra are compatible with ≈M , then f M are uniformly continuous w.r.t. ≈M . The converse transformation is not possible in general. Namely, one cannot use the pseudo-inverse g (−1) of g to transform a metric algebra with equicontinuous functions into an L-algebra. An example follows. Example 2.146. For a complete residuated lattice L given by an arbitrary Archimedean t-norm, one can find a finite metric algebra M = continuous M, M , F M with a single unary function f M ∈ F M such that f M is not compatible with ≈M . For instance, put M = {a , b , c , d }, and let f M and ≈M be defined by the following tables.

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a b c d

fM c d c d

≈M a b c d

a 1 x 0 0

b x 1 0 0

c 0 0 1 y

d 0 0 y 1

Suppose 1 > x > y. It is easily seen that ≈M is an L-equality. Furthermore, ≈M equals ≈≈M by Theorem 1.78. Since M is finite, M, ≈M , F M is a metric algebra with equicontinuous functions, see Remark 2.142 (3). On the other hand, f M is not compatible with ≈≈M , because ≈≈M coincides with ≈M and a ≈M b = x y = f (a ) ≈M f (b ). Therefore, M, ≈M , F M cannot be turned into an L-algebra by using (1.82). Therefore, although the idea behind equicontinuity of operations in metric algebras is similar to that of compatibility with a given L-equality, metric algebras deal with a different type of restrictions on operations. On the one hand, L-algebras can be thought of as more general than metric algebras because L need not be linearly ordered. On the other hand, if we restrict ourselves only to L’s given by continuous Archimedean t-norms, compatibility with a given L-equality is more restrictive than the equicontinuity of functions as we have seen in Theorem 2.144 and Example 2.146. For further information on metric algebras we refer to [95, 96] and [91]. Fuzzy Congruences of Murali and Samhan In this section, we take a brief look at the approach developed in [68] and [77] and its relationship to algebras with fuzzy equalities. The authors develop the conceptof a congruence in fuzzy setting. Their approach goes as follows. Let M, F M be an ordinary algebra. Consider the real unit interval [0, 1] as a set of truth degrees. A binary [0, 1]-relation (i.e. a fuzzy relation with truth degrees from [0, 1]) θ in M is called a fuzzy congruence on M, F M if θ is a fuzzy equivalence with min as conjunction operation (i.e., transitivity says min(θ(a , b ), θ(b , c )) ≤ θ(a , c )) which satisfies min(θ(a1 , b1 ), . . . , θ(an , bn )) ≤ θ(f M (a1 , . . . , an ), f M (b1 , . . . , bn )) (2.46) for each n-ary f M ∈ F M and every a1 , b1 , . . . , an , bn ∈ M . [77], definition of a factor algebra of the author presents the following In M M by a fuzzy congruence θ on M, F . A fuzzy M, F factor algebra of an ordinaryalgebra M, F M by a fuzzy congruence θ on M, F M is an ordinary algebra Mθ , FθM such that (i) Mθ = {a θ | a ∈ M }, where a θ is a fuzzy set in M defined by (a θ)(b ) = θ(a , b ) for all a , b ∈ M . (ii) f Mθ (a1 θ, . . . , an θ) = f M (a1 , . . . , an ) θ for any n-ary f Mθ ∈ F Mθ and a1 θ, . . . , an θ ∈ Mθ .


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The following are basic remarks from the point of view of algebras with fuzzy equalities. Remark 2.147. (1) On the one hand, M, F M is an ordinary algebra, i.e. it is equipped with crisp equality relation. On the other hand, congruences on M, F M are considered as fuzzy relations. Proceeding this way, one does not have some (usual) relationships between congruences and morphisms. For instance, in the ordinary case we have that all congruences of an algebra can be obtained as kernels of morphisms, i.e. are of the form θh = {a , b | h(a ) = h(b )} where h is a morphism of the algebra into some other algebra. For algebras with fuzzy equalities, this fact remains valid. Namely, for a congruence θ on an L-algebra M we have θ = θhθ (see Definition 1.83, Definition 2.32, and Theorem 2.33). For the approach of [68, 77] this is no longer true since each kernel θh is a crisp relation. (2) From the point of view of the choice of a structure of truth degrees, [68, 77] deal with a set L = [0, 1] of truth degrees equipped with ⊗ = min. That is, they deal with a particular complete residuated lattice L given by a M coincides t-norm ⊗ = min on [0, 1]. Recall that an ordinary algebra M, F with the L-algebra M, ≈M , F M with ≈M being a crisp equality. Furthermore, notice that with ⊗ = min, condition (2.46) expressing compatibility of a fuzzy congruence coincides with our condition of compatibility, see Definition 1.80 and Remark 1.81 (1). From this point of view, the concept of a fuzzy congruence on an ordinary algebra as considered in [68, 77] is a special case of the concept of a congruence on an algebra with fuzzy equality. Note also that in addition to the fact that [0, 1] equipped with min is only a particular example of a complete residuated lattice, one may lose insight into some important algebraic concepts. For instance, recall from Corollary 2.70 and Example 2.72 that the subdirect representation theorem for L-algebras is true only for certain classes of complete residuated lattices. However, it is not true for L = [0, 1]. As another example, the left side of compatibility condition (2.46) depends only on the least θ(ai , bi ). This is a very particular property which is not available (and may even be considered strange) in the general case of ⊗, see also Remark 2.13. (3) The concept of an algebra is one of the fundamental concepts in mathematics. When generalizing to fuzzy setting, one should be as general as possible. Namely, any theory of algebras developed in fuzzy setting can be seen as useful only when one finds a rich class of examples for which the general algebraic concepts and results will bring new insights and results. We showed several such examples existing in the literature in Sect. 2.1. Considering only [0, 1] with min is restrictive from this point of view since it rules out both existing examples as well as examples which can appear in future work in fuzzy logic. Consider now the concept of a fuzzy factor algebra. The following two remarks summarize basic observations and show an advantage of factorization as presented in the framework of L-algebras.

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Remark 2.148. Except for the fact that [77] consider ordinary algebras (without fuzzy equalities), the main distinction between the concept of a fuzzy factor algebra and that of a factor L-algebra (Definition 2.22) is the universes. While the elements of the universe Mθ of a fuzzy factor algebra are fuzzy sets a θ in M defined by (a θ)(b ) = θ(a , b ), the elements of the universe M/θ of a factor L-algebra M/θ are crisp sets [a ]θ in M (i.e. ordinary subsets of M ) defined by [a ]θ = {b | a , b ∈ 1 θ}. Recall from Sect. 1.3 (page 46 and onwards) that of the two possibilities of what the elements of a factor set of a set with fuzzy equality should be, whether a θ or [a ]θ , we decided to take [a ]θ since it is more simple than a θ (namely, [a ]θ = 1 a θ) and since both ways lead to isomorphic sets with fuzzy equality (Lemma 1.90). Now, if we would take a θ and would define a factor structure Mθ of an L-algebra M = M, ≈M , F M by a congruence θ on M in the spirit of [77], Mθ would be isomorphic to M/θ. Therefore, the approach of [77], modified to the setting of algebras with fuzzy equalities, yields a factor structure which is isomorphic to but more complex than the factor L-algebra of Definition 2.22. In more detail, for a congruence θ on an L-algebra M,let a fuzzy factor L-algebra be M with Mθ and FθM defined as defined as an L-algebra Mθ = Mθ , ≈M θ , Fθ in (i) and (ii) of the definition of a fuzzy factor algebra on page 132 and with ≈M θ defined by (iii) (a θ ≈M θ b θ) = θ(a , b ). Then, as an immediate consequence of Lemma 1.90 we get: Theorem 2.149. Under the above notation, Mθ is isomorphic to M/θ with a mapping h : Mθ → M/θ defined by h(a θ) = [a ]θ being an isomorphism. Remark 2.150. Consider L-algebra M = M, ≈M , F M . Then ≈M is a an M in the sense of [77] (without reservations if fuzzy congruence on M, F L = [0, 1] and ⊗ = min, and with appropriate modification of the concepts of L). Therefore, we can consider a fuzzy factor al [77] for general gebra M≈M , F≈MM (facrotization of M, F M by ≈M in the sense of [77]). Now, M≈M , F≈MM encompasses the same information as M, ≈M , F M since the L-equality ≈M can be reconstructed from M≈M , and vice versa. Namely, M≈M = {a ≈M | a ∈ M } and a ≈M b = (a ≈M )(b ). Now, for instance, morphisms of L-algebras are required to preserve L-equalities, i.e. to be ≈morphisms (Definition 2.26), while morphisms of ordinary algebras (which are considered in [77])are, of course, not. As a consequence, there are morphisms from M≈M , F≈MM which are not morphisms from M, ≈M , F M since they are not ≈-morphisms. This shows that due to “loss of similarity information” the concept of an L-algebra with its ramifications cannot be appropriately captured by framework provided by [68, 77]. We close our visit on fuzzy congruences by a remark on factor congruences. Remark 2.151. In [77], instead of Definition 2.54 (ii), the author requires θ ◦ θ∗ = M ×M (denote this condition by (ii’)). As we know from Remark 2.55,


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(ii) implies (ii’). In the ordinary case, both (ii) and (ii’) are equivalent. This, ∗ however, the case might not be in general since we may have θ ◦ θ (a , b ) = 1, ∗ i.e. c ∈M θ(a , c )⊗θ (b , c ) = 1, although there is no c ∈ M with θ(a , c ) = 1 and θ∗ (b , c ) = 1, i.e. c ∈ [a ]θ and c ∈ [b ]θ∗ . Therefore, (ii) is stronger than (ii’) in general. This fact reflects itself in Theorem 4.2 of [77] which is a generalization of the well-known theorem on factor congruences. Contrary to our approach, Theorem 4.2 of [77] needs an additional assumption. Fuzzy Subalgebras Let us briefly comment on so-called fuzzy (sub)algebras. Fuzzy subalgebras started with Rosenfeld’s fuzzy groups [76]. Since then, many papers developing this approach for general algebras but also for many special algebras appeared, see e.g. [20, 29, 34, 66, 76, 81, 101] (this selection is probably not representative). Recall that a fuzzy subalgebra in an ordinary algebra M = M, F M is an L-set A in M such that for each n-ary operation f M ∈ F M we have A(a1 ) ⊗ · · · ⊗ A(an ) ≤ A(f M (a1 , . . . , an )) for every a1 , . . . , an ∈ M . This condition is satisfied iff a first-order formula saying that “if a1 belongs to A and · · · and an belongs to A then f M (a1 , . . . , an ) belongs to A”. Therefore, from the point of view of predicate fuzzy logic, this condition generalizes a condition required for an ordinary subset A of M to be a subalgebra (to be precise: a subuniverse). Note that that there is no fuzzy equality on M involved in the concept of a fuzzy subalgebra. On the other hand, a subalgebra of an algebra with fuzzy equality is again an algebra with fuzzy equality (i.e. with a crisp set as a universe), see Definition 2.10. In a sense, the distinction between the theory of fuzzy subalgebras and the theory of algebras with fuzzy equalities lies in what becomes fuzzy: in fuzzy subalgebras, it is the universe set, while in algebras with fuzzy equalities, it is the equality on the universe. While most parts of universal algebra can be developed in the setting of algebras with fuzzy equalities, we are not aware of a development of these parts in the setting of fuzzy subalgebras. Nevertheless, fuzzy subalgebras and algebras with fuzzy equalities are complementary and it is thus an open problem to look at the combination of both of these approaches, i.e. to consider fuzzy subalgebras A in an algebra M with fuzzy equality ≈M , possibly such that A is compatible with ≈M , i.e. A(a ) ⊗ (a ≈M b ) ≤ A(b ). Recall also Head’s paper [53] which is one of few existing attempts which can lead stop the boom of papers (very often without any motivations and providing only easy-to-get results) on fuzzy subalgebras. Fuzzy Functions Another approach to universal algebras from the point of view of fuzzy sets might concern the functions f M of an algebra. Namely, several approaches

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have been developed to give a reasonable meaning to the concept “fuzzy function”, see e.g. [35]. In a series of papers, see e.g. [33], Demirci considers sets equipped with fuzzy equivalence/equality relations and compatible fuzzy functions. Fuzzy functions used by Demirci are fuzzy relations that need to satisfy certain properties which make them behave like functions. Ordinary functions can be seen as particular fuzzy functions. Our approach is thus a particular approach of Demirci’s. In his papers, Demirci considers only binary fuzzy functions. Moreover, he does not consider general structural algebraic notions which we have developed for algebras with fuzzy equalities. A development of universal algebraic results in the setting of Demirci is an open problem. Note, however, that with fuzzy functions, things get technically more complicated. For instance, it is not immediately obvious how terms and terms functions should look like in the setting of fuzzy functions. More generally, structures with fuzzy functions should possibly be studied in a framework of a suitable predicate fuzzy logic. This would provide a unifying framework and make it possible to use general results of predicate fuzzy logic. Also, this would perhaps suggest a way to develop logical calculi for reasoning about fuzzy identities in Demirci’s setting analogous the calculi developed in Chap. 3 and Chap. 4. Note also that in the early development of predicate fuzzy logic by Nov´ ak, function symbols were interpreted in fuzzy structures by so-called fuzzy functions of type 2, see [70]. Given a fuzzy set A in a set M , a function f : M n → M is said to be an n-ary fuzzy function of type 2 in A if A(a1 ) ⊗ · · · ⊗ A(an ) ≤ A(f (a1 , . . . , an )) for every a1 , . . . , an ∈ M which is is just the condition used in the definition of a fuzzy subalgebra. Later on, fuzzy functions of type 2 were replaced by ordinary functions in the semantics of predicate fuzzy logic.

2.10 Bibliographical Remarks The concept of a general/universal algebra goes back to the end of nineteenth century. However, the birth of contemporary universal algebra is being attributed to Birkhoff and his [22, 23, 24]. A good book on universal algebra is [27]. Universal algebra as a research field deals with the general concept of an algebra, i.e. a set with operations. It is concerned with topics related to structural properties of algebras. It disregards problems which result as consequences of special nature of particular algebras. Results of general algebra are useful if one comes up with a new kind of algebra. Then, many properties are immediately available as instances of results of universal algebra. Except for that, universal algebras are semantical structures of a fragment of predicate logic with an equality symbol as the only relational symbol. This fragment is the framework of so-called equational reasoning which is studied in theoretical computer science, see e.g. [1, 62]. The concept of an algebra with fuzzy equality appeared in [9]. It results by putting together the concept of a set with equality (Definition 1.79) and that

2.10 Bibliographical Remarks

137

of a universal algebra and by connecting them via compatibility condition (Definition 1.80). Bibliographical comments on these notions were presented in Sect. 1.5. The present chapter contains results most of which are contained in [16] and [92]. Some of the results on algebras with fuzzy equalities can be obtained from more general results on fuzzy relational structures presented in [10]. However, since we do not want to introduce concepts on the level of generality as they appear in [10] (e.g., a degree to which a fuzzy relation is a congruence), we do not refer to [10]. Note that reduced products and their special case of ultraproducts play a crucial role in universal algebra and model theory. A thorough study on this topic can be found in [32]. There is also a chapter [27] on reduced products and their applications in universal algebra. In [31], the authors present a generalization of an ultraproduct for structures equipped with relations whose truth degrees form a compact Hausdorff space. In fuzzy setting, however, there is only a small effort in studying reduced products and ultraproducts. An exception is Gottwald’s [47] where the author presents an approach to ultraproducts in fuzzy setting which is based on [31]. Reference [98] deals with ultraproducts in first-order Pavelka-style logic (but with limiting assumptions). A further study of reduced products and ultraproducts in fuzzy setting is an important open problem. The present chapter deals with basic results of universal algebra in fuzzy setting. Needless to say, it might be interesting to develop other parts of universal algebra and, in particular, to look at phenomena which are hidden in crisp setting. This seems to be an open problem.

3 Fuzzy Equational Logic

Equational logic deals with identities (equations) like x+y ≈ y+x, x·(y+z) ≈ x · y + x · z, dec(inc(x)) ≈ x, etc. Identities are simple formulas which can be interpreted in algebras. Thus, given an identity and an algebra, either the identity is true or false in the algebra. For instance, x ◦ y ≈ y ◦ x is true in the algebra Z of all integers if ◦ is interpreted by addition of integers, but it is false in the algebra of all square real matrices (say, of dimension 5 × 5) if ◦ is interpreted by matrix multiplication. The two most important aspects dealt with in equational logic are reasoning over identities and definability (specification of requirements) using sets of identities. Reasoning: Deriving new identities from known ones follows some intuitive rules. A trivial example is “derive x ≈ x” (more precisely, “derive x ≈ x from an arbitrary set of known identities”, i.e. no specific identities are needed to be known), “derive x + z ≈ x + z”, or generally, “derive t ≈ t” where t is an arbitrary term. Another trivial example is “derive x + y ≈ y + x from y + x ≈ x + y”. Although a bit complex, a rule “derive x · (y + z) ≈ x · z + x · y from x · (y + z) ≈ x · y + x · z and x + y ≈ y + x” is also being used without reservations. In his seminal paper, Birkhoff [23] showed that there are five simple rules which are sufficient for deriving new identities from known ones. More precisely, an identity t ≈ s can be derived from a set Σ of identities using these rules if an only if t ≈ s is true in every algebra satisfying each identity from Σ. This result is the well-known Birkhoff’s completeness theorem of equational logic. Definability: Interesting classes of algebras are often “definable by identities.” This is to say, there is a set Σ of identities such that an (interesting) class K of algebras contains just algebras which satisfy each identity from Σ. For instance, the class of all commutative semigroups is definable by Σ = {x ◦ y ≈ y ◦ x, x ◦ (y ◦ z) ≈ (x ◦ y) ◦ z}. Birkhoff [23] showed that a class K of algebras is definable by identities if and only if K is closed under formation of homomorphic images, subalgebras, and direct products of algebras from K. This result is the well-known Birkhoff’s variety theorem.

R. Bˇ elohl´ avek and V. Vychodil: Fuzzy Equational Logic, StudFuzz 186, 139–170 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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The aim of this chapter is to develop equational logic in fuzzy setting. We focus mainly on the two above-mentioned Birkhoff’s results, i.e. completeness and variety theorem. A logical calculus which we develop will be called a fuzzy equational logic. The basic features of fuzzy equational logic are the following. Like in case of ordinary equational logic, formulas of fuzzy equational logic are identities. However, identities will be interpreted in algebras with fuzzy equalities. That is, algebras with fuzzy equalities are semantical structures of fuzzy equational logic. From this point of view, Chap. 2 can be seen as developing modeltheoretical constructions of fuzzy equational logic. Furthermore, we develop fuzzy equational logic in Pavelka style, see Sect. 1.4. In particular, we deal with fuzzy sets of identities, a degree of provability, and a degree of semantic entailment. ∗ ∗ ∗ The rest of this chapter is organized as follows. Section 3.1 introduces basic concepts of syntax and semantics. In Sect. 3.2 we prove (Pavelka-) completeness of fuzzy equational logic for arbitrary complete residuated lattice L. Section 3.3 develops varieties and free algebras with fuzzy equalities. In Sect. 3.4, we prove Birkhoff’s variety theorem in our setting. Section 3.5 we deal with two topics in varieties in fuzzy setting which are degenerate in the ordinary setting.

3.1 Syntax and Semantics We start with basic concepts of syntax and semantics of fuzzy equational logic. We define the concepts of a language, term, formula, structure, value of a term, and truth degree of a formula. Suppose ≈, F, σ is a type (page 60) and L is a complete residuated lattice. A language of fuzzy equational logic of type ≈, F, σ for L consists of a binary relation symbol ≈ called a symbol of (fuzzy) equality, function symbols f ∈ F with their arities σ(f ) ∈ N0 , symbols a of truth degrees (a ∈ L), (at least denumerable) set X of variables with X ∩ F = ∅, and auxiliary symbols (parentheses, etc.). If ≈, F, σ and L are clear from the context, a language of type ≈, F, σ for L is called shortly a language. Remark 3.1. (1) Until otherwise mentioned, we will use a fixed type denoted simply by F . T(X) always refers to a term L-algebra of type F where X is (at least) denumerable set of variables with its universe T (X) denoting the set of all terms of type F over X. (2) For the sake of convenience and since there is no danger of misunderstanding, we identify each a with a ∈ L, i.e. we identify each symbol of truth degree with the truth degree itself. Terms (of type F over X) of fuzzy equational logic are those defined in Definition 2.73. Formulas of fuzzy equational logic are identities.

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Definition 3.2. Let F be a type. An identity of type F over X is an expression t ≈ t , where t, t ∈ T (X). Remark 3.3. (1) We can see that a language of fuzzy equational logic is a part of a language of predicate fuzzy logic as introduced in Example 1.96 (2). In more detail, if we put R = {≈} and K = L in Example 1.96 (2), a language of fuzzy equational logic of type ≈, F, σ results from a language of predicate fuzzy logic of Example 1.96 (2) by removing of symbols of connectives and quantifiers. (2) Also terms of fuzzy equational logic coincide with terms of predicate fuzzy logic of Example 1.96 (2). Formulas of fuzzy equational logic (identities) are just the atomic formulas of predicate fuzzy logic with R = {≈}. (3) Up to now, we did not make use of symbols a of truth degrees in developing syntactic notions. Symbols a will be used when introducing the notions of a proof and provability. Note that we do not consider a as formulas of fuzzy equational logic, cf. Example 1.96 (2). We are now going to introduce basic concepts of semantics of fuzzy equational logic. Structures of fuzzy equational logic are algebras with fuzzy equalities. As usual, given an algebra M with fuzzy equality, a valuation (of variables from X) in M is a mapping v : X → M assigning to each variable x ∈ X some element v(x) ∈ M . Then, given an L-algebra M and a valuation v : X → M , a value !t!M,v of a term t, and a truth degree !t ≈ t !M,v of an identity t ≈ t are defined in a straightforward way. The details follow. Definition 3.4. Let M be an L-algebra of type F , v : X → M be a valuation, t be a term. A value tM,v of t in M under v is defined as follows: (i) if t is a variable x, then !t!M,v = v(x), (ii) if t is of the form f (t1 , . . . , tn ), then !t!M,v = f M !t1 !M,v , . . . , !tn !M,v . Remark 3.5. (1) !t!M,v is thus defined in a usual way. (2) It follows directly from Theorem 2.79 that for each term t(x1 , . . . , xn ) we have (i) !t!M,v = tM (a1 , . . . , an ) for any valuation v : X → M satisfying v(x1 ) = a1 , . . . , v(xn ) = an , (ii) !t!M,v = v (t) where v is the homomorphic extension of v. By (i), !t!M,v can be seen as a value of a term function tM induced by t. By (ii), !t!M,v can be seen as a value assigned to a term t by the (unique) homomorphic extension v of v. Definition 3.6. Let M be an L-algebra of type F , v : X → M be a valuation, t ≈ t be an identity. A truth degree t ≈ t M,v of t ≈ t in M under v is defined by !t ≈ t !M,v = !t!M,v ≈M !t !M,v .

(3.1)

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A truth degree t ≈ t M of t ≈ t in M is defined by !t ≈ t !M = v:X→M !t ≈ t !M,v .

(3.2)

Remark 3.7. (1) A truth degree !t ≈ t !M,v is defined in the usual sense of predicate fuzzy logic, see Example 1.96 (2). !t ≈ t !M,v can be seen as follows. First, we evaluate terms t and t and get elements !t!M,v and !t !M,v of M . Then, we take a fuzzy equality ≈M of M and let !t ≈ t !M,v be just the degree to which !t!M,v and !t !M,v are ≈M -equal. (2) A truth degree !t ≈ t !M can be seen as a degree to which t ≈ t is true in M under all valuations v : X → M . ∗

∗

∗

As to the concepts developed so far, we have seen that fuzzy equational logic can be considered as a certain fragment of predicate fuzzy logic as introduced in Example 1.96 (2). Our aim is to develop further concepts of fuzzy equational logic in Pavelka style. To this end, we proceed by presenting fuzzy equational logic in terms of abstract logic as defined in Definition 1.105. This gives us for free the concepts of a theory, model, degree of semantic entailment, proof, degree of provability, soundness, and completeness in fuzzy equational logic (see Sect. 1.4). Nevertheless, we recall all of these concepts for the particular setting of fuzzy equational logic. Recall from Definition 1.105 (page 56) that an abstract logic is a tuple L = Fml , L, S, A, R where Fml is a set of formulas, L is a structure of truth degrees, S is an L-semantics for Fml , A is a theory of logical axioms, and R is a set of deduction rules. In the following, we present fuzzy equational logic as a particular abstract logic. That is, we define its components Fml , L, S, A, and R. For a set Fml of formulas, we take the set of all identities, i.e. Fml = {t ≈ t | t, t ∈ T (X)}. For L we take an arbitrary complete residuated lattice. Recall that according to Definition 1.105, L is required to be a complete lattice. We assume that L is, moreover, a complete residuated lattice. That is, L is a complete lattice with an additional structure on it. For S we take an L-semantics S for Fml defined by S = {E ∈ LFml | for some L-algebra M : E(t ≈ t ) = !t ≈ t !M for each t ≈ t ∈ Fml },

(3.3)

cf. (1.92). For A, we take an empty L-set of formulas, i.e. A(t ≈ t ) = 0 for each identity t ≈ t . For R we take a set of deduction rules (ERef)–(ESub) for Fml and L introduced below. Before presenting the rules, introduce the following concepts. A subterm of a term t is any substring of t which is a term itself. For a variable


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x ∈ X and a term r ∈ T (X), the pair (x/r) is called a substitution of r for x. Given a substitution (x/r) and a term t ∈ T (X), let t(x/r) denote the term resulting from t by substitution of r for every occurrence of x in t. Now, R consists of the following five deduction rules for Fml and L: (ERef) :

(ERep) :

t ≈ t, 1

,

(ESym) :

t ≈ t , a , t ≈ t, a

(ETra) :

t ≈ t , a , t ≈ t , b , t ≈ t , a ⊗ b

t ≈ t , a t ≈ t , a , (ESub) : , s ≈ s , a t(x/r) ≈ t (x/r), a

where t, t , t , r, s ∈ T (X), a, b ∈ L are arbitrary, x ∈ X, term s has an occurrence of t as a subterm and s is a term resulting from s by substitution of one occurrence of t by t . These rules can be seen as allowing for the following inference steps (in parentheses, we attach names of the rules which will be used for the sake of brevity): (from the empty set of assumptions) infer t ≈ t, 1 (reflexivity); from t ≈ t , a infer t ≈ t, a (symmetry); from t ≈ t , a and t ≈ t , b infer t ≈ t , a ⊗ b (transitivity); from t ≈ t , a infer s ≈ s , a where s is a term containing t as a subterm and s results from s by replacing one occurrence of t by t (replacement); (ESub) from t ≈ t , a infer t(x/r) ≈ t (x/r), a (substitution).

(ERef) (ESym) (ETra) (ERep)

Remark 3.8. Rules (ERef)–(ESub) generalize the corresponding rules from the ordinary case, see [23, 27]. Namely, the corresponding rules of ordinary equational logic are: infer t ≈ t (reflexivity); from t ≈ t infer t ≈ t (symmetry); from t ≈ t and t ≈ t infer t ≈ t (transitivity); from t ≈ t infer s ≈ s where s is a term containing t as a subterm and s results from s by replacing one occurrence of t by t (replacement, instead of “one occurrence” we can have “some occurrences” which is equivalent in the ordinary case); from t ≈ t infer t(x/r) ≈ t (x/r) (substitution). We have thus defined a particular abstract logic. This gives us automatically the concepts defined in Sect. 1.4. However, in order for these concepts to fit better into our desired framework, we need to extend them somewhat. In particular, we want to deal with L-algebras instead of evaluations and with classes of L-algebras instead of the corresponding sets of evaluations. The following remark clarifies the basic relationships. Remark 3.9. Due to (3.3), there is the following relationship between Lalgebras and evaluations from S. Each L-algebra M induces an evaluation EM ∈ S by EM (t ≈ t ) = !t ≈ t !M for each identity t ≈ t . For an evaluation E ∈ S, however, there might be whole class of L-algebras M such that E = EM .

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Using mappings Md and Th defined by (1.93) and (1.94), we can introduce mappings Mod and Id by Mod(Σ) = {M | EM ∈ Md(Σ)}

and Id(K) = Th({EM | M ∈ K})

where Σ ∈ L is a theory and K is a class of L-algebras. Now, we can say that an L-algebra M is a model of a theory Σ if M ∈ Mod(Σ) (i.e., if EM is a model of Σ, see Definition 1.97). It is easy to see that M is a model of Σ ∈ LFml iff Σ(t ≈ t ) ≤ !t ≈ t !M for each identity t ≈ t ∈ Fml . Then, Mod(Σ) is a class of L-algebras and Id(K) is an L-set of identities over X for which we have Fml

Mod(Σ) = {M | M is a model of Σ} , (Id(K))(t ≈ t ) = {!t ≈ t !M | M ∈ K}

(3.4) (3.5)

for each identity t ≈ t , cf. (1.93) and (1.94). Occasionally, if we want to make the set X of variables expicit, we write IdX (K) instead of Id(K). Also, we write Id(M) instead of Id({M}). Then, like Md and Th, Mod and Id form a Galois connection and satisfy thus (1.95)–(1.100) with sets K’s of evaluations replaced by classes K’s of L-algebras. As in Remark 1.107, we can define a degree !t ≈ t !Σ to which t ≈ t semantically follows from Σ ∈ LFml by !t ≈ t !Σ = (Id(Mod(Σ)))(t ≈ t ). Putting fuzzy equational logic into a setting of abstract logic gives us thus (with a small modification concerning evaluations vs. L-algebras) automatically semantical concepts in Pavelka style. Notice that all the concepts of fuzzy equational logic are uniquely determined by the choice of a type ≈, F, σ and a complete residuated lattice L. Therefore, we might speak of a fuzzy equational logic given by ≈, F, σ and L. The following definition gives a summary of semantical concepts of a fuzzy equational logic given by ≈, F, σ and L. Definition 3.10. A theory is an L-set Σ of identities, i.e. Σ ∈ LFml . An L-algebra M is a model of a theory Σ if M ∈ Mod(Σ). A degree t ≈ t Σ to which t ≈ t semantically follows from a theory Σ is defined by !t ≈ t !Σ = (Id(Mod(Σ)))(t ≈ t ) . For a class K of L-algebras, a truth degree t ≈ t K of t ≈ t in K is defined by !t ≈ t !K = (Id(K))(t ≈ t ) . Remark 3.11. (1) Let us return to some of the concepts of Definition 3.10 and rewrite the definitions in a way which might be more convenient for the reader. We have M is a model of Σ iff for each t ≈ t : Σ(t ≈ t ) ≤ !t ≈ t !M , !t ≈ t !K = M ∈ K !t ≈ t !M = M ∈ K v:X→M !t ≈ t !M,v , !t ≈ t !Σ = !t ≈ t !Mod(Σ) = M ∈ Mod(Σ) !t ≈ t !M = = M ∈ Mod(Σ) v:X→M !t ≈ t !M,v .


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(2) The above concepts are straightforward generalizations of ordinary logical concepts. Namely, M is a model of Σ if each identity is true in M at least in a degree to which it belongs to Σ. Furthermore, !t ≈ t !K can be seen as a degree to which t ≈ t is true in all L-algebras from K. Finally, !t ≈ t !Σ can be seen as a degree to which t ≈ t is true in each L-algebra which is a model of Σ. Definition 3.12. For a theory Σ ∈ LFml , Mod(Σ) is called an equational class of Σ. For a class K of L-algebras, Id(K) is called an equational theory of K. A class K of L-algebras is called an equational class if K = Mod(Σ) for some theory Σ. An L-set Σ of identities is called an equational theory if Σ = Id(K) for some class K of L-algebras. Remark 3.13. Equational class of a theory Σ thus consists of all L-algebras which are models of Σ. A theory of a class K of L-algebras contains identities, each in a degree to which it is true in K. A class of L-algebras is equational if it is definable by an L-set of identities. We now turn to the concepts of a proof and provability. Again, we utilize the notions of abstract logic, see Definition 1.101. In our setting, the concept of an L-weighted formula (Definition 1.103) gives the following. Definition 3.14. An L-weighted identity is a pair t ≈ t , a where t ≈ t is an identity and a is a symbol of a truth degree a ∈ L. Remark 3.15. Remark 1.104 applies here. In particular, an L-weighted identity is a string of symbols of a language and is thus a part of syntax of fuzzy equational logic; we also write t ≈ t , a instead of t ≈ t , a ; we also say a truth-weighted identity or just weighted identity. We can now define concepts related to provability of a fuzzy equational logic given by ≈, F, σ and L. Definition 3.16. Let R be a set of deduction rules for Fml and L, Σ ∈ LFml be a theory (L-set of identities), let t ≈ t ∈ Fml be an identity and a ∈ L. An (L-weighted ) proof of t ≈ t , a from Σ using R is a sequence t1 ≈ t1 , a1 , . . . , tn ≈ tn , an such that tn ≈ tn is t ≈ t , an = a, and for each i = 1,. . . , n, we have ai = Σ(ti ≈ ti ) or ti ≈ ti , ai follows from some tj ≈ tj , aj ’s, j < i, by some deduction rule R ∈ R. The number n is called a length of the proof. In such a case, we write Σ R t ≈ t , a and call t ≈ t , a provable from Σ using R. If Σ R t ≈ t , a , t ≈ t is called provable in degree (at least) a from Σ using R. A degree of provability of t ≈ t from Σ using R, denoted by |t ≈ t |R Σ , is defined by |t ≈ t |R {a | Σ R t ≈ t , a } . Σ = Remark 3.17. The conventions introduced in Remark 1.102 apply. In particular, R will denote a fixed set consisting of deduction rules (ERef)–(ESub) and will therefore be omitted.

146


∗

∗

∗

Note that it is the Pavelka style what makes fuzzy equational logic interesting. For instance, if we had considered only provability from crisp sets of identities, fuzzy equational logic would, in a sense, collapse to ordinary equational logic. This would also be the case of semantic entailment for crisp sets of identities. First, consider provability from crisp L-sets Σ of identites. That is, for any t ≈ t we either have Σ(t ≈ t ) = 1 or Σ(t ≈ t ) = 0. An easy inspection of deduction rules (ERef)–(ESub) shows that if t ≈ t , a is a member of a proof from crisp Σ then a ∈ {0, 1}. Hence, |t ≈ t |Σ ∈ {0, 1} for every t, t ∈ T (X). Moreover, one can easily see that when applied to weighted identities ti ≈ ti , 1 ’s (i.e., with weights equal to 1), our rules (ERef)–(ESub) yield a weighted identity t ≈ t , 1 if and only if t ≈ t can be obtained from ti ≈ ti ’s by the corresponding rules of ordinary equational logic (see Remark 3.8). Therefore, with crisp Σ, fuzzy equational logic in fact collapses to ordinary equational logic. Once again, even if we allow for degrees of provability in Pavelka style, degrees of provability from crisp theories can only be 0 or 1 and they coincide with “non-provable” or “provable” in ordinary equational logic. Note that this is not the case of general predicate fuzzy logic (one might have a degree of provability other than 0 and 1 even if one proves from a crisp theory). Second, consider semantic entailment for a crisp L-set Σ. Observe that an identity t ≈ t is fully true in an L-algebra M (i.e., !t ≈ t !M = 1) iff for any valuation v we have 1 = !t ≈ t !M,v = !t!M,v ≈M !t !M,v which is true iff for any valuation v, !t!M,v equals !t !M,v . That is, the fact that t ≈ t is fully true in M depends only on the functional part M, F M of M (does not depend on the L-equality ≈M ). Therefore, as one can easily see, semantic entailment from crisp L-sets of identities coincides with semantic entailment of ordinary equational logic. As a consequence, for a crisp Σ, a degree !t ≈ t !Σ to which t ≈ t semantically follows from Σ can only be 0 or 1 and it coincides with “does not follow from Σ” and “follows from Σ” in ordinary equational logic. That is, for crisp Σ’s, semantic entailment would collapse to the ordinary case as well.

3.2 Completeness of Fuzzy Equational Logic In this section we work out a proof of (Pavelka)-completeness of fuzzy equational logic. That is, we show that for any complete residuated lattice, an L-set Σ of identities, and an identity t ≈ t the degree !t ≈ t !Σ to which t ≈ t semantically follows from Σ equals the degree of provability of t ≈ t from Σ. First, we describe semantic entailment using appropriate algebraic means. In particular, we define so-called fully invariant congruences, show that equational theories can be seen as fully invariant congruences on T(X), and use

3.2 Completeness of Fuzzy Equational Logic

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fully invariant congruences to describe semantic entailment. We start by an auxiliary lemma. Lemma 3.18. Suppose that T(X) of type F exists and let M be an L-algebra of type F . For every valuation v : X → M on M and for every endomorphism h of T(X), there is a valuation w : X → M such that !h(t)!M,v = !t!M,w for every term t ∈ T (X). Proof. For v : X → T (X) and endomorphism h : T(X) → T(X), we can put w(x) = !h(x)!M,v for every variable x ∈ X. By structural induction we can easily prove that !h(t)!M,v = !t!M,w . Indeed, for x ∈ X the assertion is trivial. For t of the form f (t1 , . . . , tn ) we have !h(t)!M,v = !f (h(t1 ), . . . , h(tn ))!M,v = f M !h(t1 )!M,v , . . . , !h(tn )!M,v = = f M !t1 !M,w , . . . , !tn !M,w = !f (t1 , . . . , tn )!M,w = !t!M,w . Thus, !h(t)!M,v = !t!M,w for every t ∈ T (X).

Definition 3.19. We call a congruence θ on M fully invariant if for any endomorphism h : M → M we have θ(a , b ) ≤ θ(h(a ), h(b )) (a , b ∈ M ). Note that theories Σ (i.e., L-sets of identities) can be identified with binary L-relations in T (X). Namely, given a theory Σ ∈ LFml , we may consider a corresponding binary L-relation θΣ in T (X) defined for each p, q ∈ T (X) × T (X) by θΣ (p, q) = Σ(p ≈ q). Conversely, given a binary L-relation θ in T (X), we may consider a theory Σθ ∈ LFml defined for each identity p ≈ q ∈ Fml by Σθ (p ≈ q) = θ(p, q). Then we clearly have Σ = ΣθΣ and θ = θΣθ . For convenience, we make use of this possibility and identify L-sets of identities with binary L-relations in T (X) in the subsequent text. Theorem 3.20. IdX (K) is a fully invariant congruence on T(X). Proof. For brevity, we abbreviate “ M∈K v:X→M ” by “ M,v ”. Reflexivity and symmetry of IdX (K) are obvious. In more detail, we have (IdX (K))(t, t) = M,v !t ≈ t!M,v = M,v !t!M,v ≈M !t!M,v = M,v 1 = 1 for each t ∈ T (X), i.e. IdX (K) is reflexive. Furthermore, (IdX (K))(t, t ) = M,v !t ≈ t !M,v = M,v !t!M,v ≈M !t !M,v = = M,v !t !M,v ≈M !t!M,v = M,v !t ≈ t!M,v = = (IdX (K))(t , t) , i.e. IdX (K) is symmetric. Transitivity of IdX (K) follows by (IdX (K))(t, t ) ⊗ (IdX (K))(t , t ) = = M,v !t ≈ t !M,v ⊗ M,v !t ≈ t !M,v ≤ ≤ M,v !t ≈ t !M,v ⊗ !t ≈ t !M,v = = M,v !t!M,v ≈M !t !M,v ⊗ !t !M,v ≈M !t !M,v ≤ ≤ M,v !t!M,v ≈M !t !M,v = M,v !t ≈ t !M,v = (IdX (K))(t, t ) .

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Since ≈T(X) is the identity relation, ≈T(X) ⊆ IdX (K) is trivial, showing compatibility of IdX (K) with ≈T(X) . For any n-ary function symbol f we have ( IdX (K))(t1 , t1 ) ⊗ · · · ⊗ (IdX (K))(tn , tn ) = = M,v !t1 ≈ t1 !M,v ⊗ · · · ⊗ M,v !tn ≈ tn !M,v ≤ ≤ M,v !t1 ≈ t1 !M,v ⊗ · · · ⊗ !tn ≈ tn !M,v = = M,v !t1 !M,v ≈M !t1 !M,v ⊗ · · · ⊗ !tn !M,v ≈M !tn !M,v ≤ ≤ M,v f M !t1 !M,v , . . . , !tn !M,v ≈M f M !t1 !M,v , . . . , !tn !M,v = = M,v !f (t1 , . . . , tn )!M,v ≈M !f (t1 , . . . , tn )!M,v = = M,v !f (t1 , . . . , tn ) ≈ f (t1 , . . . , tn )!M,v = = (IdX (K))(f (t1 , . . . , tn ), f (t1 , . . . , tn )) . Thus, IdX (K) ∈ ConL (T(X)). Finally, we check that IdX (K) is fully invariant. Let h : T(X) → T(X) be an endomorphism. Take M ∈ K. For any valuation v : X → M , Lemma 3.18 yields that there is w : X → M such that !h(t)!M,v = !t!M,w for every t ∈ T (X). Therefore, (IdX (K))(t, t ) ≤ !t ≈ t !M,w = !t!M,w ≈M !t !M,w = = !h(t)!M,v ≈M !h(t )!M,v = !h(t) ≈ h(t )!M,v , yielding (IdX (K))(t, t ) ≤ M,v !h(t) ≈ h(t )!M,v = (IdX (K))(h(t), h(t )). To sum up, IdX (K) is a fully invariant congruence of T(X). We need the following lemma. Lemma 3.21. Suppose that T(X) of type F exists and let θ ∈ ConL (T(X)). Let v : X → T (X)/θ be a valuation. Then there is an endomorphism h on T(X) such that [h(t)]θ = !t!T(X)/θ,v . Proof. Let v : X → T (X)/θ be a valuation. Take any hX : X → T (X) such that hX (x) ∈ v(x) for all x ∈ X. Let h be the homomorphic extension of hX . For a natural morphism hθ : T(X) → T(X)/θ we have (hX ◦ hθ )(x) = [hX (x)]θ = v(x) for all x ∈ X. Thus, Theorem 2.139 and Theorem 2.140 give (hX ◦ hθ ) = v . Using Theorem 2.141 we get h ◦ hθ = (hX ◦ hθ ) = v . That is, for any t ∈ T (X) we have [h(t)]θ = hθ (h(t)) = (h ◦ hθ )(t) = v (t) = !t!T(X)/θ,v for every t ∈ T (X) which is the desired equality.

Now we show the converse assertion to Theorem 3.20 Theorem 3.22. Let θ be a fully invariant congruence on T(X). Then θ is an equational theory such that IdX ({T(X)/θ}) = θ.


149

Proof. We prove the claim by showing !t ≈ t !T(X)/θ = θ(t, t ). “≤”: For a valuation v : X → T (X)/θ such that v(x) = [x]θ (x ∈ X) we have !t!T(X)/θ,v = [t]θ for all t ∈ T (X) (follows from Lemma 3.21 for h such that h(t) = t for each term t). Therefore, !t ≈ t !T(X)/θ ≤ !t ≈ t !T(X)/θ,v = !t!T(X)/θ,v ≈T(X)/θ !t !T(X)/θ,v = = [t]θ ≈T(X)/θ [t ]θ = θ(t, t ) . “≥”: Take any v : X → T (X)/θ. Lemma 3.21 yields that there is an endomorphism h on T(X), such that [h(t)]θ = !t!T(X)/θ,v for every t ∈ T (X). It follows that !t ≈ t !T(X)/θ,v = !t!T(X)/θ,v ≈T(X)/θ !t !T(X)/θ,v = = [h(t)]θ ≈T(X)/θ [h(t )]θ = θ(h(t), h(t )) ≥ θ(t, t ) for all t, t ∈ T (X). Since v is chosen arbitrarily, !t ≈ t !T(X)/θ ≥ θ(t, t ).

Putting Theorem 3.20 and Theorem 3.22 together, we get the following corollary. Corollary 3.23. Let Σ be an L-set of identities. Then Σ is a fully invariant congruence on T(X) iff Σ is an equational theory. Lemma 3.24. The system of all fully invariant congruences on T(X) is a closure system. Proof. Clearly, T (X)×T (X) is a fully invariant congruence on T(X). It is now sufficient to show that for any non-empty family {θi ∈ ConL (T(X)) | i ∈ I} of fully invariant congruences, θ = {θi | i ∈ I} is a fully invariant congruence on T(X). According to Theorem 2.15, it suffices to show that θ(t, t ) ≤ θ(h(t), h(t )) for every endomorphism h on T(X) and t, t ∈ T (X). However, this is true since θ(t, t ) = i∈I θi (t, t ) ≤ i∈I θi (h(t), h(t )) = θ(h(t), h(t )) . Thus, θ is a fully invariant congruence on T(X).

For an L-set Σ of identities denote by Σ the least fully invariant congruence on T(X) such that Σ ⊆ Σ (its existence follows from Lemma 3.24). Theorem 3.25. For any L-set Σ of identities, and every terms t, t ∈ T (X) we have !t ≈ t !Σ = Σ (t, t ). Proof. “≤”: Theorem 3.22 yields T(X)/Σ ∈ Mod(Σ ). Since Σ ⊆ Σ , it follows that Mod(Σ ) ⊆ Mod(Σ). Thus, T(X)/Σ is a model of Σ. Using Theorem 3.22 we get !t ≈ t !Σ = M∈Mod(Σ) !t ≈ t !M ≤ !t ≈ t !T(X)/Σ = = (IdX (T(X)/Σ ))(t, t ) = Σ (t, t ) .

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“≥”: Take any M ∈ Mod(Σ), i.e. !t ≈ t !M ≥ Σ(t ≈ t ) for all t, t ∈ T (X). Thus, Σ ⊆ IdX (M). Theorem 3.20 gives that IdX (M) is a fully invariant congruence containing Σ. Since Σ is the least fully invariant congruence containing Σ, we have Σ ⊆ IdX (M). Therefore, Σ (t, t ) ≤ !t ≈ t !M for every M ∈ Mod(Σ) and arbitrary t, t ∈ T (X). This further yields !t ≈ t !Σ = M∈Mod(Σ) !t ≈ t !M ≥ Σ (t, t )

which is the desired inequality.

Remark 3.26. Since !t ≈ t !Σ = IdX (Mod(Σ))(t ≈ t ), Theorem 3.25 says that IdX (Mod(Σ)) = Σ . Definition 3.27. Let Σ be an L-set of identities. We define a deductive closure Σ of Σ by letting Σ be the least L-set of identities such that Σ ⊆ Σ

(3.6)

Σ (t, t) = 1 ,

(3.7)

Σ (t, t ) ≤ Σ (t , t) ,

(3.8)

Σ (t, t ) ⊗ Σ (t , t ) ≤ Σ (t, t ) ,

(3.9)

Σ (t, t ) ≤ Σ (s, s ) ,

(3.10)

Σ (t, t ) ≤ Σ (t(x/r), t (x/r)) ,

(3.11)

where t, t , t , r, s ∈ T (X), x ∈ X, term s has an occurrence of t as a subterm and s is a term resulting from s by substitution of one occurrence of t by t . Remark 3.28. Note that the existence of Σ follows from the fact that the set of all L-sets of identities which contain Σ and satisfy (3.6)–(3.11) is non-empty (it contains the full L-set) and is closed with respect to arbitrary intersections (easy to check). Now, the next theorem shows that the deductive closure Σ is just the least fully invariant congruence Σ on T(X). Theorem 3.29. Let Σ be an L-set of identities. Then Σ = Σ . Proof. “⊆”: It suffices to check that Σ is a fully invariant congruence containing Σ. By (3.6)–(3.9), Σ is an L-equivalence on T (X) which contains Σ. To show that Σ is a congruence on T(X), we repeatedly use (3.10). For any t1 , s1 , . . . , tn , sn ∈ T (X) and any n-ary f ∈ F , (3.10) yields Σ (ti , si ) ≤ Σ (f (s1 , . . . , si−1 , ti , ti+1 , . . . , tn ), f (s1 , . . . , si−1 , si , ti+1 , . . . , tn )) for each i = 1, . . . , n. Recall that in every i-th step, we have applied (3.10) correctly, because f (s1 , . . . , si−1 , ti , ti+1 , . . . , tn ) contains ti as a subterm and f (s1 , . . . , si−1 , si , ti+1 , . . . , tn ) results from it by replacing ti by si . Hence, (3.9) yields


151

Σ (t1 , s1 ) ⊗ · · · ⊗ Σ (tn , sn ) ≤ n ≤ i=1 Σ f (s1 , . . . , si−1 , ti , . . . , tn ), f (s1 , . . . , si , ti+1 , . . . , tn ) ≤ ≤ Σ f (t1 , . . . , tn ), f (s1 , . . . , sn ) , proving compatibility. Thus, Σ is a congruence. We show that Σ is fully invariant. Take any endomorphism h on T(X), and terms t, t ∈ T (X). Suppose var(t) ∪ var(t ) ⊆ {x1 , . . . , xn }. Since X is denumerable, we can consider paiwise distrinct variables y1 , . . . , yn so that no yi occurs in xj or h(xj ) (j = 1, . . . , n). Using (3.11), we get Σ (t(x1 , . . . , xn ), t (x1 , . . . , xn )) ≤ Σ (t(y1 , . . . , yn ), t (y1 , . . . , yn )) ≤ ≤ Σ (t(h(x1 ), . . . , yn ), t (h(x1 ), . . . , yn )) ≤ · · · ≤ ≤ Σ (t(h(x1 ), . . . , h(xn )), t (h(x1 ), . . . , h(xn ))) = = Σ (h(t(x1 , . . . , xn )), h(t (x1 , . . . , xn ))) , completing the proof. “⊇”: We check that Σ satisfies conditions (3.6)–(3.11). Equation (3.6) follows from definition of Σ . Equations (3.7)–(3.9) are true because Σ is an L-equivalence. Equation (3.10) follows from the compatibility of Σ . Indeed, take t, t , s, s ∈ T (X) such that s has an occurrence of t as a subterm and s is a term resulting from s by substitution of t by t . If s = f (t1 , . . . , tk−1 , t, tk+1 , . . . , tn ) and s = f (t1 , . . . , tk−1 , t , tk+1 , . . . , tn ), compatibility of Σ with f ∈ F and Σ (ti , ti ) = 1 yield k−1 n Σ (t, t ) = i = 1 Σ (ti , ti ) ⊗ Σ (t, t ) ⊗ j = k+1 Σ (tj , tj ) ≤ ≤ Σ f (t1 , . . . , tk−1 , t, tk+1 , . . . , tn ), f (t1 , . . . , tk−1 , t , tk+1 , . . . , tn ) = = Σ (s, s ) . This argument can be used to show Σ (t, t ) ≤ Σ (s, s ) even in general case (one can proceed by structural induction over the rank of s). (3.11): For any x ∈ X and r ∈ T (X) consider a mapping g : X → T (X) defined by g(x) = r and g(y) = y if y = x. The homomorphic extension g of g is an endomorphism on T(X) satisfying g (t) = t(x/r) for each t ∈ T (X). Therefore, Σ (t, t ) ≤ Σ (g (t), g (t )) = Σ (t(x/r), t (x/r)) because Σ is fully invariant.

The next theorem shows that the deductive closure Σ coincides with provability from Σ. Theorem 3.30. Let Σ be an L-set of identities. Then for every t, t ∈ T (X) we have |t ≈ t |Σ = Σ (t, t ).

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Proof. “≤”: Clearly, it suffices to check that if ti ≈ ti , ai is a member of some L-weighted proof, then ai ≤ Σ (ti , ti ). This is obvious if ai = Σ(ti ≈ ti ). Otherwise (i.e., ti ≈ ti , ai is obtained by some inference rule), proceed by induction over i and suppose that the assertion is true for j < i. The following inference rules could have been used: (ERef): ti = ti and ai = 1 = Σ (ti , ti ) by (3.7). (ESym): ti = tj , ti = tj , and ai = aj for some j < i. Using (3.8), ai = aj ≤ Σ (tj , tj ) ≤ Σ (tj , tj ) = Σ (ti , ti ) . (ETra): Suppose ti ≈ ti , ai was obtained by (ETra) from tj ≈ tj , aj and tk ≈ tk , ak (in this order) for some j, k < i That is, we have ti = tj , tj = tk , tk = ti , and ai = aj ⊗ ak . Then, (3.9) yields ai = aj ⊗ ak ≤ Σ (tj , tj ) ⊗ Σ (tk , tk ) = Σ (ti , tk ) ⊗ Σ (tk , ti ) ≤ Σ (ti , ti ) . tj ,

(ERep): ti is obtained from ti by replacement of one occurrence of tj by and ai = aj for some j < i, whence by (3.10) we get ai = aj ≤ Σ (tj , tj ) ≤ Σ (ti , ti ) .

(ESub): ti = tj (x/r), ti = tj (x/r), and ai = aj for some r ∈ T (X) and j < i. Thus, (3.11) gives ai = aj ≤ Σ (tj , tj ) ≤ Σ (tj (x/r), tj (x/r)) = Σ (ti , ti ) . “≥”: It suffices to prove that an L-set D of identities defined by D(t, t ) = |t ≈ t |Σ contains Σ and satisfies (3.6)–(3.11). The required inequality then follows from the fact that Σ is the least L-set with these properties. (3.6): Since t ≈ t , Σ(t ≈ t ) is a proof from Σ, we have Σ ⊆ D. (3.7) follows from the fact that t ≈ t, 1 is a proof, i.e. D(t, t) = 1. (3.8) follows from the fact that each proof δ1 , . . . , δk , t ≈ t , a from Σ can be extended to a sequence δ1 , . . . , δk , t ≈ t , a , t ≈ t, a which is a proof from Σ as well. Hence, D(t, t ) = |t ≈ t |Σ ≤ |t ≈ t|Σ = D(t , t). (3.9): Let δi1 , . . . , δini , t ≈ t , ai (i ∈ I) and δj1 , . . . , δjnj , t ≈ t , bj (j ∈ J) be all proofs of t , ai and . . . , t ≈ t , bj , respectively. the form . . . , t ≈ Thus, D(t, t ) = i∈I ai and D(t , t ) = j∈J bj . For each i ∈ I and j ∈ J we can concatenate the corresponding proofs and use (ETra) we get a sequence δi1 , . . . , δini , 1: t ≈ t , ai , δj1 , . . . , δjnj ,

2: t ≈ t , bj , 3: t ≈ t , ai ⊗ bj .

proof of t ≈ t , ai proof of t ≈ t , bj by (ETra) on 1, 2

which is a proof of t ≈ t , ai ⊗ bj from Σ. Hence, D(t, t ) ⊗ D(t , t ) = i∈I ai ⊗ j∈J bj = = i∈I j∈J (ai ⊗ bj ) ≤ |t ≈ t |Σ = D(t, t ) , which proves (3.9).


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Equations (3.10) and (3.11) follow from the fact that if δ1 , . . . , δk , t ≈ t , a is a proof from Σ and r ≈ s, a is obtained from t ≈ t , a by (ERep) or (ESub), then δ1 , . . . , δk , t ≈ t , a , r ≈ s, a is a proof from Σ as well. We thus have (Pavelka-)completeness for fuzzy equational logic. Theorem 3.31 (completeness). Let L be any complete residuated lattice, X be denumerable set of variables. Let Σ be an L-set of identities over X. Then |t ≈ t |Σ = !t ≈ t !Σ for every t, t ∈ T (X). Proof. Consequence of Theorem 3.25, Theorem 3.29, and Theorem 3.30.

Remark 3.32. A degree of provability may be strictly greater than the weight of any proof. Indeed, let F = {◦}, ◦ be binary, denote by xn the n-th power of ukasiewicz x with respect to ◦, i.e. x3 = (x◦x)◦x, etc. Let L be the standard L algebra on [0, 1]. Define Σ by Σ(x ◦ x ≈ x) = 1, Σ(xn ≈ y n ) = 1 − n1 , and Σ(t ≈ t ) = 0 otherwise. Clearly, 1: x ◦ x ≈ x, 1 , 2: y ◦ y ≈ y, 1 , 3: xn ≈ y n , 1 − 4: x ≈ x n

n−1

1 n

,

, 1 ,

5: xn−1 ≈ xn , 1 , 6: xn−1 ≈ y n , 1 −

1 n

7: y ≈ y , 1 , 8: xn−1 ≈ y n−1 , 1 − n

,

n−1

.. . x ≈ y, 1 − n1 ,

1 n

,

follows from Σ by (ESub) on 1 follows from Σ by (ERep) on 1 by (ESym) on 4 by (ETra) on 5 and 3 by (ERep) on 2 by (ETra) on 6 and 7

is an L-weighted proof of x ≈ y from Σ for any n. Therefore, |x ≈ y|Σ = 1. On the other hand, there is no proof of x ≈ y from Σ the weight of which is 1: by contradiction, let δ1 , . . . , δk be a proof of x ≈ y, 1 from Σ. Thus, δ1 , . . . , δk also is a proof of x ≈ y, 1 from some finite subset Σ ⊆ Σ because δ1 , . . . , δk contains only finitely many weighted identities of the form xn ≈ y n , 1 − n1 . Consider now an L-algebra M = M, ≈M , ◦M such that M = {a , b }, a ◦M a = a , b ◦M b = b , a ◦M b = b ◦M a = a , and ≈M is an L-equality on 1 for m = 1 + max {n ∈ N | Σ (xn ◦ y n ) = 0}. M satisfying a ≈M b = 1 − m Obviously, M ∈ Mod(Σ ). Since δ1 , . . . , δk is a proof of x ≈ y, 1 from Σ , we get !x ≈ y!M = 1 due to the soundness of fuzzy equational logic. On the 1 < 1, a contradiction to !x ≈ y!M = 1. other hand, a ≈M b = 1 − m Remark 3.33. For readers familiar with Pavelka-style fuzzy logic: As observed by Pavelka [74], we cannot have graded style completeness for arbitrary complete residuated lattice even in the case of propositional logic (the less so for the first-order case [49, 71]). However, since (ERef)–(ESub) can be used in Pavelka-style first-order fuzzy logic as derived rules (more precisely: derived

154


rules in first-order fuzzy logic with the usual inference rules where the relation symbol ≈ is confined in an obvious sense by axioms of reflexivity, symmetry, transitivity, and compatibility), our result implies that the equational fragment (i.e. restriction to formulas of the form of identities) of first-order fuzzy logic is completely axiomatizable (in Pavelka style) using any complete residuated lattice as the structure of truth degrees.

3.3 Varieties, Free L-Algebras In this section, we show basic properties of classes of L-algebras closed under formation of homomorphic images, subalgebras, and direct products. Following the ordinary case, we call them varieties. Definition 3.34. A class K of L-algebras of type F is called a variety if it is closed under homomorphic images, subalgebras and direct products, i.e., if H(K) ⊆ K, S(K) ⊆ K and P(K) ⊆ K. If K is a class of L-algebras of the same type let V(K) denote the least variety containing K. We say that V(K) is a variety generated by K. Remark 3.35. (1) Each variety is non-empty since it contains a trivial Lalgebra (i.e., an L-algebra with a one-element universe). Namely, a trivial L-algebra is a direct product of an empty family of L-algebras. (2) Note that V(K) exists since for a given type F , the class of all varieties is non-empty (it contains the largest variety – the class of all L-algebras of type F ) and closed under intersections (easy to check). In the following, we describe the operator V using a single operator consisting of H, S, and P. Theorem 3.36. We have SH ≤ HS, PS ≤ SP, PH ≤ HP. Furthermore, H, S, and IP are idempotent. Proof. Suppose M ∈ SH(K). Then for some M ∈ K and some epimorphism h : M → N, M is a subalgebra of N. Theorem 2.28 yields that the inverse image h−1 (M) is a subalgebra of M . Moreover, h(h−1 (M)) = M. Hence, it follows that M ∈ HS(K). Q suitable subalgebras Mi of Mi ∈ K If M ∈ PS(K), Q then M = i∈I Mi for Q (i ∈ I). Since i∈I Mi is a subalgebra of i∈I Mi , we have M ∈ SP(K). Let M ∈ PH(K). Then thereQare L-algebras Mi andQ epimorphisms Q hi : . A mapping h : i∈I Mi → i∈I Mi Mi → Mi (i ∈ I) such that M = i∈I MiQ defined by Q h(b )(i) = hi (b (i)) for all b ∈ i∈I Mi is an epimorphism. Indeed, for a , b ∈ i∈I Mi we have, Q a ≈ i∈I Mi b = i∈I a (i) ≈Mi b (i) ≤ i∈I hi (a (i)) ≈Mi hi (b (i)) = Q = i∈I h(a )(i) ≈Mi h(b )(i) = h(a ) ≈ i∈I Mi h(b ) .

3.3 Varieties, Free L-Algebras Q

Q

155

Q

For any n-ary f i∈I Mi ∈ F i∈I Mi , a1 , . . . , an ∈ i∈I Mi , we have Q Q h f i∈I Mi (a1 , . . . , an ) (i) = hi f i∈I Mi (a1 , . . . , an )(i) = = hi f Mi a1 (i), . . . , an (i) = f Mi hi (a1 (i)), . . . , hi (an (i)) = Q = f Mi h(a1 )(i), . . . , h(an )(i) = f i∈I Mi h(a1 ), . . . , h(an ) (i) Q Q for each i ∈ I, i.e. h f i∈I Mi (a1 , . . . , an ) = f i∈I Mi h(a1 ), . . . , h(aQ n) . ∼ The idempotency of H, S is evident. Let M ∈ IPIP(K), i.e. M = i∈I Mi Q ∼ and Mi = j∈Ji Nij for all iQ∈ I, where every Nij belongs to K. Thus, there Q are isomorphisms h : M → i∈I Mi and hi : Mi → Q j∈Ji Nij (i ∈ I). Put K = {i, j | i ∈ I, j ∈ Ji }. Now, a mapping g : M → i,j ∈K Nij defined by g(a )(i, j ) = hi h(a )(i) (j) is an isomorphism. For a , b ∈ M we have Q a ≈M b = h(a ) ≈ i∈I Mi h(b ) = i∈I h(a )(i) ≈Mi h(b )(i) = Q = i∈I hi (h(a )(i)) ≈ j∈Ji Nij hi (h(b )(i)) = = i∈I j∈Ji hi (h(a )(i))(j) ≈Nij hi (h(b )(i))(j) = = i∈I j∈Ji g(a )(i, j ) ≈Nij g(b )(i, j ) = Q = i,j ∈K g(a )(i, j ) ≈Nij g(b )(i, j ) = g(a ) ≈ i,j∈K Nij g(b ) . The rest follows from the ordinary case. Hence, IP is idempotent.

The following theorem says that the variety operator V is just the composition of H, S, and P. Theorem 3.37. For the class operators V and HSP we have V = HSP. Proof. The proof is fully analogous to that one in the ordinary case. For every class K of L-algebras of the same type we have HSP(K) ⊆ HSP(V(K)). Since H(V(K)) ⊆ V(K), S(V(K)) ⊆ V(K) and P(V(K)) ⊆ V(K) we obtain HSP(K) ⊆ HSP(V(K)) ⊆ HS(V(K)) ⊆ H(V(K)) ⊆ V(K) . Hence, HSP(K) is contained in V(K). Now it suffices to show that HSP(K) is a variety, i.e. HSP(K) is closed under H, S and P. Using Theorem 3.36 we have H(HSP) = HHSP = HSP and S(HSP) = SHSP ≤ HSSP = HSP, thus HSP(K) is closed under homomorphic images and subalgebras. Furthermore, P(HSP) = PHSP ≤ HPSP ≤ HSPP. As P ≤ IP, I ≤ H and IP is idempotent, it readily follows that HSPP ≤ HSIPIP = HSIP ≤ HSHP ≤ HHSP = HSP , thus HSP(K) is closed under direct product. Altogether HSP(K) is a variety and HSP = V. We now introduce so-called free L-algebras and show their properties and their role. In particular, we show that each variety is fully determined by its free L-algebras. Namely, a variety is just the class of all homomorphic images of its free L-algebras (Corollary 3.42 and Remark 3.43) and, moreover, it is generated by a single free L-algebra (Theorem 3.44).

156


Definition 3.38. Let K be a class of L-algebras of the same type, X be a set of variables. Put ΦK (X) = {φ ∈ ConL (T(X)) | T(X)/φ ∈ IS(K)} ,

(3.12)

i.e. ΦK (X) is the set of all congruences φ on T(X) such that the factor Lalgebra T(X)/φ is isomorphic to some subalgebra of some M ∈ K. Let (3.13) θK (X) = ΦK (X) denote the intersection of all congruences from ΦK (X). Then FK (X) = T(X)/θK (X) , (3.14) where X = [x]θK (X) | x ∈ X is the set of generators of T(X)/θK (X), is called a K-free L-algebra over X. For x ∈ X we write x instead of [x]θK (X) .

Theorem 3.39. Let K be a class of L-algebras of type F , X be a set of variables such that T(X) exists. Then (i) any mapping g : X → M where M ∈ K can be extended to a morphism from FK (X) to M; (ii) FK (X) has the universal mapping property for K over X. Proof. (i): Let M ∈ K and take a mapping g : X, ≈FK (X) → M, ≈M . Furthermore, let n : T(X) → FK (X) denote the natural morphism, i.e. n(t) = [t]θK (X) for all t ∈ T (X). Suppose nX is a restriction of n on X, i.e. nX is a mapping nX : X → X. Thus, a universal mapping property of T(X) implies that there is a morphism k : T(X) → M extending nX ◦g. Moreover, it follows from Theorem 2.35 that T(X)/θk is isomorphic to a subalgebra of M ∈ K, thus θk ∈ ΦK (X), θK (X) ⊆ θk . It is immediate that g is an ≈-morphism, since the following inequality holds for every x, y ∈ X: x ≈FK (X) y = θK (X)(x, y) ≤ ≤ θk (x, y) = g(nX (x)) ≈M g(nX (y)) = g(x) ≈M g(y) . Now using Theorem 2.39 we obtain FK (X)/(θk /θK (X)) ∼ = T(X)/θk . Hence, there are mappings h, i and j, h

i

j

FK (X) −−→ FK (X)/(θk /θK (X)) −−→ T(X)/θk −−→ M , where h : FK (X) → FK (X)/(θk /θK (X)) is a natural morphism, i is an isomorphism between FK (X)/(θk /θK (X)) and T(X)/θk . Finally, using Theorem 2.35, j : T(X)/θk → M is defined by j([t]θk ) = k(t) for every t ∈ T (X). Thus, (h ◦ i ◦ j)(x) = j i h(x) = j i [x]θk /θK (X) = j [x]θk = = k(x) = g nX (x) = g [x]θK (X) = g(x) . Altogether h ◦ i ◦ j is a homomorphic extension of g. (ii) is a consequence of (i).

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157

Theorem 3.40. Suppose T(X) exists. Then FK (X) ∈ ISP(K). Thus if K is closed under I, S, P, in particular if K is a variety, then FK (X) ∈ K. Proof. Theorem 2.45 and Lemma 2.64 yield that forQFK (X) = T(X)/θK (X) there is a subdirect embedding h : T(X)/θK (X) → θ∈ΦK (X) T(X)/θ. Using PS (K) ⊆ SP(K) and Theorem 3.36 we obtain FK (X) ∈ IPS ({T(X)/θ | θ ∈ ΦK (X)}) = = IPS ({T(X)/θ | θ ∈ ConL (T(X)) and T(X)/θ ∈ IS(K)}) ⊆ ⊆ IPS IS(K) ⊆ ISPIS(K) = ISPS(K) ⊆ ISSP(K) = ISP(K) completing the proof.

Theorem 3.41. Let K denote a class of L-algebras of the same type, M ∈ K. Then for some sufficiently large set of variables X we have M ∈ H(FK (X)). Proof. Take a set of variables X such that |M | ≤ |X|. Every surjective mapping h : X → M has a surjective homomorphic extension h : FK (X) → M due to Theorem 3.39 (i). Hence, we have M ∈ H(FK (X)). Thus, we have the following consequence. Corollary 3.42. K ⊆ H FK (X) | X is a set of variables .

Remark 3.43. If K is a variety then every FK (X) belongs to K. Thus, the previous corollary yields K = H FK (X) | X is a set of variables , i.e. every variety is determined by its K-free L-algebras. The following theorem shows that every variety is closed under the formation of direct unions. This fact suffices to prove that every variety K is determined by the only one K-free L-algebra FK (X) over a denumerable set X of variables. Theorem 3.44. Every variety K of L-algebras is closed under direct unions. Moreover, K = HSP({FK (X)}) for X being a denumerable set of variables. Proof. Let {M i ∈ K | i ∈ I} be a directed family of L-algebras.QLemma 2.89 yields that i∈I Mi ∼ = N/θ for a suitable L-algebra N ∈ Sub( i∈I Mi ) and θ ∈ ConL (N). Since K = HSP(K), we have N ∈ SP(K) = K and N/θ ∈ H({N}) ⊆ K, i.e. i∈I Mi ∈ I(K) = K. Thus, every variety is closed under the formation of direct unions. Let us have M ∈ K. Every finitely generated L-algebra [M ]M , is an image of FK (X), where X is a denumerable set of variables. Indeed, let h : X → [M ]M be any mapping such that h(X) = M . This mapping can be extended to a morphism h : FK (X) → [M ]M . Take an element a ∈ [M ]M . Since [M ]M is finitely generated, we can express a = tM (a1 , . . . , an ), where M = {a1 , . . . , an }, t ∈ T (X). Thus for xi1 , . . . , xin ∈ X such that h (xi1 ) = a1 , . . . , h (xin ) = an we have

158


h [t(xi1 , . . . , xin )]θK (X) = h tFK (X) (xi1 , . . . , xin ) = = tM h (xi1 ), . . . , h (xin ) = a . Hence, h : FK (X) → [M ]M is a surjective morphism. Consequently, [M ]M ∈ H({FK (X)}) ⊆ HSP({FK (X)}). Using Corollary 2.88 it follows that M∼ [M ] ∈ HSP({FK (X)}) | M is a finite subset of M = M

for every M ∈ K. Since HSP({FK (X)}) is a variety, i.e. it is closed under direct unions, we have M ∈ HSP({FK (X)}), i.e. K ⊆ HSP({FK (X)}). The converse inclusion follows from Theorem 3.40. Theorem 3.45. Let K be a variety of L-algebras. (i) K is closed under direct limits of weak direct families of L-algebras from K; (ii) K is closed under reduced products of L-algebras from K. Proof. (i): Let {Mi ∈ K | i ∈ I} be a weak direct family of L-algebras. It is easily seen that Theorem 2.118 yields lim Mi ∈ IUH({Mi | i ∈ I}) ⊆ IUH(K) ⊆ K due to Theorem 3.44. Q Q (ii): Obviously, F Mi = i∈I Mi /θF ∈ HP(K) ⊆ K for every family {Mi ∈ K | i ∈ I}.

3.4 Equational Classes of L-Algebras In this section, we prove Birkhoff’s variety theorem [23] in the setting of fuzzy equational logic. The following lemma characterizes interpretations of terms in homomorphic images, subalgebras, and direct products. Lemma 3.46. Suppose that T(X) of type F exists and let M, Mi (i ∈ I) be L-algebras of type F . (i) For h : M → M , and a valuation v : X → M we have a morphism h !t!M,v = !t!M ,v◦h for all t ∈ T (X); (ii) for every valuation v : X → N , where N ∈ Sub(M) we have !t!M,v = !t!N,v for all t ∈ T (X); Q (iii) for every valuation v : X → i∈I M i and every j ∈ I we have !t!Q Mi ,v (j) = !t!Mj ,v◦πj for all t ∈ T (X). i∈I

Proof. (i): Theorem 2.140 and Remark 3.5 (2) yield h !t!M,v = h(v (t)) = (v ◦ h)(t) = (v ◦ h) (t) = !t!M ,v◦h for any term t ∈ T (X). (ii): Take an embedding → M defined by h(a ) = a for all a ∈ N . h: N By (i) we get !t!N,v = h !t!N,v = !t!M,v◦h Q = !t!M,v for any t ∈ T (X). (iii): Since every j-th projection πj : i∈I Mi → Mj is a morphism, we can apply (i) to get !t!Q Mi ,v (j) = πj !t!Q Mi ,v = !t!Mj ,v◦πj . i∈I

i∈I

3.4 Equational Classes of L-Algebras

159

Lemma 3.46 (i) can be used for h being the natural morphism hθ : M → M/θ. Then we obtain the following consequence. Corollary 3.47. Suppose that T(X) of type F exists and M is an L-algebra of type F . Then for every θ ∈ ConL (M) we have !t!M,v θ = !t!M/θ,w , where w(x) = [v(x)]θ for all variables x ∈ X. Lemma 3.48. Let M be an L-algebra, v : X → M be a valuation. Then for any terms t, t ∈ T (X) we have !t ≈ t !M,v = !t!M,v ≈M !t !M,v = v (t) ≈M v (t ) = θv (t, t ) . Proof. Follows by Definition 1.83 and Remark 3.5 (2).

Lemma 3.49. Let M be an L-algebra. Then for any t, t ∈ T (X) we have (i) (ii) (iii) (iv)

!t ≈ t !M = !t ≈ t !N for every N ∼ = M; !t ≈ t !M ≤ !t ≈ t !h(M) for every epimorphism h; !t ≈ t !M ≤ !t ≈ t !N for every N ∈ Sub(M); !t ≈ t !{Mi |i∈I} ≤ !t ≈ t !Q Mi for every family {Mi | i ∈ I}. i∈I

Proof. We postpone the proof of (i) after proving (ii). (ii): Let h : M → N be an epimorphism, i.e. h(M) = N. Take any valuation v : X → N . For v we can choose w : X → M such that h(w(x)) = v(x) for each variable x ∈ X. Thus, w ◦ h = v. Applying Lemma 3.46 (i), we get !t ≈ t !M ≤ !t ≈ t !M,w = !t!M,w ≈M !t !M,w ≤ ≤ h !t!M,w ≈N h !t !M,w = !t!N,w◦h ≈N !t !N,w◦h = = !t!N,v ≈N !t !N,v = !t ≈ t !N,v = !t ≈ t !h(M),v . Therefore, !t ≈ t !M ≤ !t ≈ t !h(M),v for any v : X → h(M ) from which the claim follows immediately. (i): Consider M ∼ = N. Thus, there is an isomorphism h : M → N. Since both h and its inverse h−1 are epimorphisms, we can apply (ii) twice to get !t ≈ t !M ≤ !t ≈ t !h(M) = !t ≈ t !N and

!t ≈ t !N ≤ !t ≈ t !h−1 (N) = !t ≈ t !M .

Putting it together, !t ≈ t !M = !t ≈ t !N . (iii): Following the idea of (ii), for any valuation v : X → N where N ∈ Sub(M), Lemma 3.46 (ii) gives !t ≈ t !M ≤ !t ≈ t !M,v = !t!M,v ≈M !t !M,v = = !t!N,v ≈N !t !N,v = !t ≈ t !N,v . Hence, !t ≈ t !M ≤ !t ≈ t !N . (iv): Consider aQ family {Mi | i ∈ I} of L-algebras of the same type. For any valuation v : X → i∈I M i , we can consider a family {vi : X → Mi | i ∈ I} of

160


valuations such that vi (x) = πi (v(x)) for each i ∈ I and any variable x ∈ X. That is, vi = v ◦ πi (i ∈ I). Now we can apply Lemma 3.46 (iii) to get !t ≈ t !{Mi |i∈I} = i∈I !t ≈ t !Mi ≤ i∈I !t ≈ t !Mi ,vi = = i∈I !t!Mi ,vi ≈Mi !t !Mi ,vi = = i∈I !t!Mi ,v◦πi ≈Mi !t !Mi ,v◦πi = = i∈I !t!Q Mi ,v (i) ≈Mi !t !Q Mi ,v (i) = i∈I

=

!t!Q

i∈I Mi ,v

Q

≈

i∈I Mi

i∈I

!t !Q

i∈I Mi ,v

= !t ≈ t !Q

Since v was arbitrary, we get !t ≈ t !{Mi |i∈I} ≤ !t ≈ t !Q

i∈I Mi

i∈I Mi ,v

.

.

The following lemma shows that the equational theory of a class K of Lalgebras coincides with those of classes of isomorphic copies, homomorphic images, subalgebras, and direct products of K. Lemma 3.50. For a class K of L-algebras we have IdX (K) = IdX (I(K)) = IdX (H(K)) = IdX (S(K)) = IdX (P(K)) . Proof. First we show IdX (K) = IdX (I(K)). Since K ⊆ I(K) we have IdX (K) ⊇ IdX (I(K)). Conversely, for any N ∈ I(K) there is M ∈ K such that N ∼ = M. For such M, Lemma 3.49 (i) gives !t ≈ t !M = !t ≈ t !N for all t, t ∈ T (X). This yields !t ≈ t !K ≤ !t ≈ t !N for any N ∈ I(K), i.e. IdX (K) ⊆ IdX (I(K)). Next, since for O = H, O = S, or O = IP we have K ⊆ O(K), it follows that IdX (K) ⊇ IdX (H(K)), IdX (K) ⊇ IdX (S(K)), and IdX (K) ⊇ IdX (IP(K)) which yields IdX (K) ⊇ IdX (P(K)) since IdX (K) = IdX (I(K)). We thus need to establish the converse inclusions, i.e. to verify (IdX (K))(t ≈ t ) ≤ (IdX (O(K)))(t ≈ t ) for all t, t ∈ T (X). However, this is a consequence of Lemma 3.49 (ii)–(iv). Indeed, we check the inequality for each operator separately. “H(K)”: For any M ∈ H(K) there is an epimorphism h : M → M where M ∈ K. Lemma 3.49 (ii) thus gives !t ≈ t !K ≤ !t ≈ t !M ≤ !t ≈ t !M for all t, t ∈ T (X). This further yields !t ≈ t !K ≤ M∈H(K) !t ≈ t !M = !t ≈ t !H(K) for all t, t ∈ T (X). Therefore, IdX (K) ⊆ IdX (H(K)). “S(K)”: Briefly, for any M ∈ S(K) there is M ∈ K with M ∈ Sub(M ). Therefore, Lemma 3.49 (iii) gives !t ≈ t !K ≤ !t ≈ t !M ≤ !t ≈ t !M yielding IdX (K) ⊆ IdX (S(K)). Q “P(K)”: Take i∈I Mi ∈ P(K), i.e. {Mi | i ∈ I} ⊆ K. By Lemma 3.49 (iv), !t ≈ t !K ≤ !t ≈ t !{Mi |i∈I} ≤ !t ≈ t !Q from which it follows that IdX (K) ⊆ IdX (P(K)). Theorem 3.51. Every equational class is a variety.

i∈I Mi

3.4 Equational Classes of L-Algebras

161

Proof. Let K = Mod(Σ). We show that K is closed under H, S, and P. Let O denote any of H, S, or P. Take any M ∈ O(K). Lemma 3.50 yields Σ ⊆ IdX (Mod(Σ)) = IdX (K) = IdX (O(K)) ⊆ IdX (M) , i.e. for any t ≈ t we have Σ(t ≈ t ) ≤ !t ≈ t !M whence M ∈ Mod(Σ). This shows that K is closed under O. Lemma 3.52. For a class K of L-algebras and t, t ∈ T (X) we have !t ≈ t !K = !t ≈ t !FK (X) = [t]θK (X) ≈FK (X) [t ]θK (X) = (θK (X))(t, t ) . Proof. We prove the assertion by showing !t ≈ t !K ≤ !t ≈ t !FK (X) ≤ ≤ [t]θK (X) ≈FK (X) [t ]θK (X) ≤ (θK (X))(t, t ) ≤ !t ≈ t !K . “!t ≈ t !K ≤ !t ≈ t !FK (X) ”: Theorem 3.40 yields FK (X) ∈ ISP(K). Thus, using Lemma 3.50 we get !t ≈ t !K = (IdX (K))(t ≈ t ) = (IdX (ISP(K)))(t ≈ t ) ≤ !t ≈ t !FK (X) . “!t ≈ t !FK (X) ≤ ([t]θK (X) ≈FK (X) [t ]θK (X) )”: Take w : X → T (X)/θK (X) sending x to [x]θK (X) . We have !t ≈ t !FK (X) = v:X→T (X)/θK (X) !t!FK (X),v ≈FK (X) !t !FK (X),v ≤ ≤ !t!FK (X),w ≈FK (X) !t !FK (X),w = [t]θK (X) ≈FK (X) [t ]θK (X) . “([t]θK (X) ≈FK (X) [t ]θK (X) ) ≤ (θK (X))(t, t )” is true by definition since FK (X) is T(X)/θK (X). “(θK (X))(t, t ) ≤ !t ≈ t !K ”: We check that for any L-algebra M ∈ K and any valuation v : X → M , (θK (X))(t, t ) ≤ !t ≈ t !M,v . Due to Lemma 3.48, it suffices to prove θK (X) ⊆ θv . Clearly, θv ∈ ConL (T(X)). Furthermore, Theorem 2.35 gives that T(X)/θv is isomorphic to a subalgebra of M which gives T(X)/θv ∈ IS(K). Thus, by Definition 3.38, θv ∈ ΦK (X), i.e. θK (X) ⊆ θv . The required inequality now readily follows. Theorem 3.53. If K is a variety of L-algebras and X a denumerable set of variables, then K = Mod(IdX (K)). Proof. Denote K = Mod(IdX (K)). K is a variety by Theorem 3.51. Obviously, K ⊆ K . It suffices to check the converse inclusion. Clearly, IdX (K ) ⊆ IdX (K) because K ⊆ K . Conversely, IdX (K) ⊆ IdX (K ) is true iff for each M ∈ K we have (IdX (K))(t ≈ t ) ≤ !t ≈ t !M which is true by definition of K . We thus have IdX (K ) = IdX (K). By Lemma 3.52, (θK (X))(t, t ) = !t ≈ t !K = !t ≈ t !K = (θK (X))(t, t ) for all t, t ∈ T (X), i.e. FK (X) coincides with FK (X). Hence, Theorem 3.44 yields K = HSP(FK (X)) = HSP(FK (X)) = K .

162


Theorem 3.54. Let K be a class of L-algebras of the same type, X be a denumerable set of variables. Then the following are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii)

K is an equational class, K is closed under H, S, and P, K is closed under HSP, K = HSP(K), K = HSP(K ) for some class K of L-algebras, K = Mod(IdX (K)), K = Mod(IdX (K )) for some class K of L-algebras.

Proof. “(i) ⇒ (ii)”: By Theorem 3.51. “(ii) ⇒ (iii)”: By Theorem 3.37. “(iii) ⇒ (iv)”: (iv) is a restatement of (iii). “(iv) ⇒ (v)”: Trivial. “(v) ⇒ (vi)”: Apply Theorem 3.37 and Theorem 3.53. “(vi) ⇒ (vii)”: Trivial. “(vii) ⇒ (i)”: By definition.

In the rest of this section, we show several examples of equational classes. Example 3.55. Consider a type F = {·, −1 , 1}, where · is a binary function symbol, −1 is a unary function symbol, and 1 is symbol of a constant (nullary function symbol). For an arbitrary L and a, b, c ∈ L, consider weighted identities G1: x · (y · z) ≈ (x · y) · z, a , G2: x · 1 ≈ x, b , 1 · x ≈ x, b , G3: x · x−1 ≈ 1, c , x−1 · x ≈ 1, c , and the corresponding theory ΣG (i.e., ΣG (x · (y · z) ≈ (x · y) · z) = a, etc., see Remark 3.15). A class K of all L-algebras satisfying G1–G3 is a variety. It is easy to see that K contains, among other L-algebras, all groups endowed with compatible L-equalities. In addition to that, K is determined by truth degrees a, b, c ∈ L. Namely, for a = b = c = 0, K is the class of all L-algebras. On the other hand, if a = b = c = 1, then K is the class of all groups with L-equalities. Example 3.56. Let L be the standard product algebra on [0, 1]. Consider an L-equality depicted in the left table of Fig. 3.1. The operations described in the middle and the right pair of tables of Fig. 3.1 are compatible with the L-equality. Denote by M1 and M2 the L-algebras corresponding to the operations of the middle and left part with 1M1 = b and 1M2 = a . Both M1 and M2 satisfy identities of commutativity and associativity in degree 1, i.e. y) · z!M we have !x · y ≈ y · x!Mi = 1 and !x · (y · z) ≈ (x · −1 i = 1 for i = 1, 2. Furthermore, we have !x · 1 ≈ x!M1 = 1 and x · x ≈ 1M = 3/4 (M1 has 1 inverse elements w.r.t. b = 1M1 in degree 3/4), while !x · 1 ≈ x!M2 = 3/4

3.5 Properties of Varieties

163

Fig. 3.1. L-equality and operations of Example 3.56

Fig. 3.2. “Almost semilattice” of Example 3.57

(a = 1M2 is a neutral element of M2 in degree 3/4) and x · x−1 ≈ 1M = 1. 2 Thus, for instance M1 is a model of ΣG with a = 1, b = 1, c = 0.5, but is not a model of ΣG for a = 1, b = 1, c = 0.8. Example 3.57. Let L be the standard L ukasiewicz algebra on [0, 1]. Consider a type which consists of a single binary function symbol ◦. We define an Lequality ≈M and a function ◦M on the universe set M = {a , b , c , d , e , f } by tables in Fig. 3.2. One can check that ◦M is compatible with ≈M , i.e. M = M, ≈M , ◦M is an L-algebra. Note that ◦M on ske(M) is idempotent but it is neither associative nor commutative. On the other hand, we have !x ◦ y ≈ y ◦ x!M = !x ◦ (y ◦ z) ≈ (x ◦ y) ◦ z!M = 7/8. This can be read: “L-algebra M is commutative in degree 7/8 and associative in degree 7/8”. As a consequence, M is a member of some variety of L-algebras given by a theory to which the identity x ◦ x ≈ x of idempotency belongs in degree 1, and both the identities of associativity and commutativity belong in degree 7/8.

3.5 Properties of Varieties When approaching an area (a well-developed theory, for instance) from a point of view of fuzzy logic, one can (should, perhaps) ask the following two “methodological questions”: First, how to proceed in order to obtain an appropriate generalization of the area such that the ordinary things (definitions, theorems, algorithms, methods) become special cases of their “fuzzy counterparts”? Second, what new phenomena come up in fuzzy setting (“new” since they might be completely hidden, and thus not interesting, in the ordinary case)? Concerning equational logic, the preceding sections of this chapter showed a feasible answer to the first question. In this section, we give

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two examples which may be supplied as (examples of) answers to the second question. Crisply Generated Varieties The following class operator comes up naturally in a fuzzy setting. Definition 3.58. For a class K of L-algebras of the same type, a new class F(K) is defined by F(K) = {M | M is an L-algebra such that for some N ∈ K : ske(M) = ske(N)} . Operator F is called fuzzification. We have M ∈ F(K) iff there is N ∈ K such that the functional parts (i.e. ordinary algebras M, F M and N, F N ) of M and N coincide. Remark 3.59. For L = 2 we have F(K) = K. Therefore, F is trivial and thus not interesting in the ordinary case. We show, that in general, F can be used to characterize equational classes given by crisp L-sets of identities. Lemma 3.60. Let Σ be a crisp L-set of identities. Then for any L-algebras M, N with ske(M) = ske(N) we have M ∈ Mod(Σ) iff N ∈ Mod(Σ). Proof. Take M and N such that ske(M) = ske(N), i.e. the functional parts of M and N coincide. Observe that for any valuation v : X → M and t ∈ T (X) we have !t!M,v = !t!ske(M),v = !t!ske(N),v = !t!N,v . Thus, !t ≈ t !M,v = 1 iff !t!M,v ≈M !t !M,v = 1 iff !t!M,v = !t !M,v iff !t!N,v = !t !N,v iff !t!N,v ≈N !t !N,v = 1 iff !t ≈ t !N,v = 1. Therefore, M ∈ Mod(Σ) iff for each t ≈ t such that Σ(t ≈ t ) = 1 we have !t ≈ t !M,v = 1 for any valuation v : X → M iff for each t ≈ t such that Σ(t ≈ t ) = 1 we have !t ≈ t !N,v = 1 for any valuation v : X → N iff N ∈ Mod(Σ), proving the claim. The following assertion shows that varieties closed under the fuzzification operator are exactly the equational classes of crisp L-sets of identities. Theorem 3.61. Let K be a variety of L-algebras. Then K = F(K) iff there is a crisp L-set Σ of identities such that K = Mod(Σ). Proof. “⇒”: Let K be a variety such that K = F(K). Consider the subclass K ⊆ K such that K = {M ∈ K | ≈M is crisp}. Put Σ = IdX (K ). Evidently, Σ is a crisp L-set of identities. We show K = Mod(Σ) by proving both inclusions.


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“⊆”: Let N ∈ K. It remains to show that N is a model of Σ. Consider M such that ≈M is crisp and ske(M) = ske(N). Since K is closed under F, we have M ∈ K. Therefore, M ∈ K , i.e. M ∈ Mod(Σ). Lemma 3.60 yields N ∈ Mod(Σ). “⊇”: Clearly, K = Mod(IdX (K)) ⊇ Mod(IdX (K )) = Mod(Σ). “⇐”: Let Σ be a crisp L-set of identities such that K = Mod(Σ). Take an L-algebra N ∈ K. For any L-algebra M such that ske(M) = ske(N) we have M ∈ K by Lemma 3.60. Therefore, F(K) ⊆ K. The converse inclusion is trivial. Remark 3.62. One can further show that if K is a variety closed under F then there exists a variety Kc of classical algebras such that K can be reconstructed from Kc by means of fuzzification. In fact, K = {M ∈ K | ≈M is crisp} is a class of L-algebras with crisp equalities which is closed under isomorphic images, subalgebras and direct products. Moreover, if h(M) is an image of M ∈ K which has crisp L-equality then h(M) ∈ K . Thus, one can verify that Kc = {ske(M) | M ∈ K} is a (classical) variety of ordinary algebras. In addition to that, K = {M | M is L-algebra such that ske(M) ∈ Kc }. See also [15, 52]. Graded Permutability of Varieties Permutability of congruences of ordinary algebras is one of the most studied issues in universal algebra. Namely, permutability simplifies several properties and conditions of algebras. Note that an ordinary algebra M is called permutable (has permuting congruences) if θ ◦ φ = φ ◦ θ for any two congruences θ, φ of M. A class K of ordinary algebras is called permutable if each of its members is a permutable algebra. For instance, groups, rings, and Boolean algebras are examples of permutable algebras. Permutability of a variety of algebras was characterized using a simple term condition by Mal’cev [64] in 1954. Mal’cev’s ingenious result, left without any interest for almost two decades, started a long series of results known as Mal’cev-type characterization of properties of algebras and classes of algebras. Mal’cev showed that a variety is permutable if and only if there is a ternary term p(x, y, z) such that each algebra of the variety satisfies both p(x, y, y) ≈ x and p(x, x, y) ≈ y. In fuzzy setting, permutability of congruences can be introduced as a graded property. That is, we might speak of a degree to which two congruences are permutable. We are going to show that even in this case, a Mal’cev-type characterization is possible. In fuzzy set theory, there are several sound definitions of a composition of fuzzy relations, all of which coincide in the ordinary case, see e.g. [10, 47]. For our purposes, we present another generalization of one of the most common compositions of L-relations. For binary L-relations R1 , R2 on U , we define binary L-relations R1 ◦n R2 , R1 ◦ω R2 on U by

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n c ∈U (R1 (a , c ) ⊗ R2 (c , b )) , (R1 ◦ω R2 )(a , b ) = c ∈U n∈N (R1 (a , c ) ⊗ R2 (c , b ))n ,

(R1 ◦n R2 )(a , b ) =

(3.15) (3.16)

where n ∈ N and a , b ∈ U . Observe that for n = 1 we obtain the usual notion of composition of L-relations. Let us stress that in general we can have (R1 ◦ R2 )(a , b ) = 1, although there is no c ∈ U such that R1 (a , c ) = 1 and R2 (c , b ) = 1. On the other hand, such a situation cannot happen if 1 is an ∨-irreducible element of L. Now we present the notion of a graded permutability. Recall that in the ordinary case, congruences θ, φ are called permutable iff θ ◦ φ = φ ◦ θ, which is equivalent to θ ◦ φ ⊆ φ ◦ θ. We are going to generalize classical permutability using the subsethood degree “S” which generalizes the classical inclusion “⊆”, see (1.66). For θ, φ ∈ ConL (M), we define a degree Per(θ, φ) to which θ, φ are permutable by Per(θ, φ) = S(θ ◦ω φ, φ ◦ θ) . For an L-algebra M we put Per(M) =

θ,φ∈ConL (M)

Per(θ, φ) .

For a variety K of L-algebras we define a degree Per(K) by Per(K) = M∈K Per(M) .

(3.17)

Thus, Per(K) represents a degree to which a variety K is permutable. Observe that for L = 2, Per(K) = 1 iff K is permutable in the classical sense. The relationship between the classical permutability and the degree given by Per(K) will be commented later on. Now we turn our attention to a term characterization of Per(K). Recall that according to [64], congruences of any algebra of a variety K are permutable iff there exists a ternary term p(x, y, z) such that identities p(x, y, y) ≈ x and p(x, x, y) ≈ y are true in K. In fuzzy case, however, we deal with a degree of permutability Per(K) ∈ L and with truth degrees !· · ·!K ∈ L of identities. We start by three auxiliary results. Lemma 3.63. Let K be a variety of L-algebras and let X = {x1 , . . . , xn } be a set of variables. Then for every M ∈ K and θ ∈ ConL (M) we have θK (X) p(x1 , . . . , xn ), q(x1 , . . . , xn ) ≤ ≤ θ pM (a1 , . . . , an ), q M (a1 , . . . , an ) , (3.18) where p, q ∈ T (X) and a1 , . . . , an ∈ M . Proof. Let v : X → M be a valuation, where v(xi ) = ai for i = 1, . . . , n. Clearly, v induces a valuation w : X → M/θ such that w(xi ) = [v(xi )]θ = [ai ]θ for all i = 1, . . . , n. Thus, θK (X) p(x1 , . . . , xn ), q(x1 , . . . , xn ) = !p ≈ q!K ≤ !p ≈ q!M/θ,w = = !p!M/θ,w ≈M/θ !q!M/θ,w = !p!M,v θ ≈M/θ !q!M,v θ = = θ !p!M,v , !q!M,v = θ pM (a1 , . . . , an ), q M (a1 , . . . , an ) , which is the desired inequality.


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Lemma 3.64. Let K be a variety of L-algebras, X = {x1 , . . . , xm , y1 , . . . , yn } and Y = {x1 , . . . , xm , y} be sets of variables. Then for every binary L-relation R on T (X)/θK (X) such that Supp(R) = {yi , yj | 1 ≤ i, j ≤ n} we have θ(R) pFK (X) (x1 , . . . , xm , y1 , . . . , yn ), q FK (X) (x1 , . . . , xm , y1 , . . . , yn ) ≤ ≤ !p(x1 , . . . , xm , y, . . . , y) ≈ q(x1 , . . . , xm , y, . . . , y)!K

(3.19)

for arbitrary terms p, q ∈ T (X). Proof. Take a mapping h : X → T (Y )/θK (Y ), where h(xi ) = xi (i = 1, . . . , m), and h(yj ) = y (j = 1, . . . , n). We can assume the variable y to be among y1 , . . . , yn . For every z1 , z2 ∈ X let z1 , z2 ∈ Y with h(z1 ) = z1 and h(z2 ) = z2 . By Theorem 3.39, h admits the uniquely determined homomorphic extension h : FK (X) → FK (Y ). Moreover, θh yi , yj = h (yi ) ≈FK (Y ) h (yj ) = y ≈FK (Y ) y = 1 for 1 ≤ i, j ≤ n. Therefore, R ⊆ θh . As a consequence, θ(R) ⊆ θh . Thus, θ(R) pFK (X) (x1 , . . . , xm , y1 , . . . , yn ), q FK (X) (x1 , . . . , xm , y1 , . . . , yn ) ≤ ≤ h pFK (X) (x1 , . . . , xm , y1 , . . . ) ≈FK (Y ) h q FK (X) (x1 , . . . , xm , y1 , . . . ) = = pFK (Y ) (x1 , . . . , xm , y, . . . , y) ≈FK (Y ) q FK (Y ) (x1 , . . . , xm , y, . . . , y) = = θK (Y ) p(x1 , . . . , xm , y, . . . , y), q(x1 , . . . , xm , y, . . . , y) = = !p(x1 , . . . , xm , y, . . . , y) ≈ q(x1 , . . . , xm , y, . . . , y)!K is true for arbitrary terms p, q ∈ T (X).

Lemma 3.65. Let K be a variety of L-algebras and let M ∈ K. Then we have φ a , pM (a , b , b ) ⊗ θ pM (c , c , b ), b ≤ n ≤ n∈N θ(a , c ) ⊗ φ(c , b ) → (3.20) → φ a , pM (a , c , b ) ⊗ θ pM (a , c , b ), b for every term p(x, y, z) ∈ T (X), elements a , b , c ∈ M , and θ, φ ∈ ConL (M). n Proof. For brevity, put d = n∈N θ(a , c ) ⊗ φ(c , b ) . Since every term has only finitely many occurrences of a variable, it follows that d = n∈N θ(a , c )n ⊗ φ(c , b )n ≤ θ(a , c )|p|x ⊗ φ(c , b )|p|y . Moreover, φ a , pM (a , b , b ) ⊗ θ pM (c , c , b ), b ≤ d → θ(a , c )|p|x ⊗ φ(c , b )|p|y ⊗ φ a , pM (a , b , b ) ⊗ θ pM (c , c , b ), b ≤ d → θ pM (a , c , b ), pM (c , c , b ) ⊗ θ pM (c , c , b ), b ⊗ ⊗ φ a , pM (a , b , b ) ⊗ φ pM (a , b , b ), pM (a , c , b ) ≤ d → φ a , pM (a , c , b ) ⊗ θ pM (a , c , b ), b . Hence, the inequality holds.

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The following assertion presents a connection between graded permutability of a variety and graded validity of Mal’cev identities in a variety. A degree of permutability of a variety K is shown to be equal to a supremum of truth degrees of identities connected in a conjunctive manner which ranges over all ternary terms. Theorem 3.66 (characterization of graded permutability). Let K be a variety of L-algebras and let X = {x, y, z}. Then (3.21) Per(K) = p∈T (X) !p(x, y, y) ≈ x!K ⊗ !p(x, x, y) ≈ y!K . Proof. “≤”: Obviously, Per(K) ≤ Per(M) for every M ∈ K. Take principal congruences θ(1/x, y ), θ(1/y, z ) ∈ ConL (FK (X)). Now using (3.19), we get Per(K) ≤ Per(FK (X)) ≤ Per(θ(1/x, y ), θ(1/y, z )) = = S(θ(1/x, y ) ◦ω θ(1/y, z ), θ(1/y, z ) ◦ θ(1/x, y )) ≤ ≤ θ(1/x, y ) ◦ω θ(1/y, z ) (x, z) → θ(1/y, z ) ◦ θ(1/x, y ) (x, z) ≤ n ≤ n∈N θ(1/x, y )(x, y) ⊗ θ(1/y, z )(y, z) → → p∈T (X) θ(1/y, z ) x, pFK (X) (x, y, z) ⊗ ⊗ θ(1/x, y ) pFK (X) (x, y, z), z = = p∈T (X) θ(1/y, z ) x, pFK (X) (x, y, z) ⊗ ⊗ θ(1/x, y ) pFK (X) (x, y, z), z ≤ ≤ p∈T (X) !p(x, y, y) ≈ x!K ⊗ !p(x, x, y) ≈ y!K . “≥”: Take arbitrary M ∈ K, a , b , c ∈ M , and θ, φ ∈ ConL (M). Properties of free L-algebras together with (3.18) and (3.20) yield p∈T (X) !p(x, y, y) ≈ x!K ⊗ !p(x, x, y) ≈ y!K = = p∈T (X) θK (X)(x, p(x, y, y)) ⊗ θK (X)(p(x, x, y), y) ≤ ≤ p∈T (X) φ a , pM (a , b , b ) ⊗ θ pM (c , c , b ), b ≤ n ≤ p∈T (X) n∈N θ(a , c ) ⊗ φ(c , b ) → ≤ → φ a , pM (a , c , b ) ⊗ θ pM (a , c , b ), b

n θ(a , c ) ⊗ φ(c , b ) → → p∈T (X) φ a , pM (a , c , b ) ⊗ θ pM (a , c , b ), b ≤ n ≤ n∈N θ(a , c ) ⊗ φ(c , b ) → (φ ◦ θ)(a , b ) .

≤

n∈N

Thus, we have p∈T (X) !p(x, y, y) ≈ x!K ⊗ !p(x, x, y) ≈ y!K ≤ ≤ M∈K θ,φ∈ConL (M) a ,b ∈M (θ ◦ω φ)(a , b ) → (φ ◦ θ)(a , b ) = = Per(K) ,


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proving the claim.

Remark 3.67. For L being the two-element Boolean algebra, Theorem 3.66 gives exactly the above-mentioned Mal’cev characterization of congruence permutability for varieties of algebras [64]. Indeed, we have that θ ◦ φ = φ ◦ θ iff θ ◦ φ = θ ◦ω φ ⊆ φ ◦ θ iff S(θ ◦ω φ, φ ◦ θ) = 1. Hence, K is congruence permutable iff Per(K) = 1, i.e. iff there is a ternary term p(x, y, z) such that !p(x, y, y) ≈ x!K = 1 and !p(x, x, y) ≈ y!K = 1 on account of (3.21). Suppose L = L, ∧, ∨, ⊗, →, 0, 1 is a complete residuated lattice, where ⊗ is ∧ (L is a complete Heyting algebra). Then, since ⊗ is idempotent, the definition of Per(θ, φ) simplifies to Per(θ, φ) = S(θ ◦ φ, φ ◦ θ), which is a generalization of θ ◦ φ ⊆ φ ◦ θ in fuzzy setting. Hence, the degree of equality θ ◦ φ ≈ φ ◦ θ is equal to Per(θ, φ) ∧ Per(φ, θ). Per(K) can then be interpreted as a lower estimation of a degree to which θ ◦ φ and φ ◦ θ are equal for all θ, φ ∈ M, M ∈ K. Clearly, if Per(K) = 1, then θ ◦ φ = φ ◦ θ for all congruences θ, φ ∈ ConL (M), where M ∈ K. Hence, for ⊗ = ∧ the meaning of Per(K) corresponds well with the classical permutability. The situation for non-idempotent ⊗ is not so straightforward. Note that the ◦ω -composition has been defined to avoid problems with sensitivity of term functions, see Theorem 2.76 and Remark 2.77. The interpretation of a truth degree (R1 ◦ωR2 )(a , b ) is interesting for L being a BL-algebra (prelinear and divisible residuated lattice, see [10, 49]) on the unit interval [0, 1] with ∧ and ∨ being the minimum and the maximum, respectively. In such a case, ⊗ is a continuous t-norm (Theorem 1.40) and for each a ∈ L, n∈N an is an idempotent (namely, the greatest idempotent which is less or equal to a ∈ L, see Lemma 1.52). Thus, the idempotents of L still play an important role, because (R1 ◦ω R2 )(a , b ) is a supremum of idempotents. For ⊗ being a continuous Archimedean t-norm (0, 1 are its only idempotents), (3.16) simplifies as follows: 1 if there is c ∈ U such that R1 (a , c ) = R2 (c , b ) = 1 , ω (R1 ◦ R2 )(a , b ) = 0 otherwise . That is, R1 ◦ω R2 corresponds with the bivalent relation 1 R1 ◦ 1 R2 . As a consequence, Per(K) = 1 iff for every M ∈ K and θ, φ ∈ ConL (M) we have 1 θ ◦ 1 φ ⊆ 1 (φ ◦ θ). Example 3.68. Consider Example 3.55 again. We show how fuzzy equational logic can be used to estimate the permutability degree of a given variety. For a ternary term x · (y −1 · z), we can use deduction rules of fuzzy equational logic to estimate the provability degree of identities x · (y −1 · y) ≈ x and x · (x−1 · y) ≈ y from G1–G3. We have 1. 2. 3. 4.

y −1 · y ≈ 1, c x · (y −1 · y) ≈ x · 1, c x · 1 ≈ x, b x · (y −1 · y) ≈ x, c ⊗ b

G3, substitution by replacement on 1. G2 by transitivity on 2., 3.

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and 1. 2. 3. 4. 5. 6.

x · x−1 ≈ 1, c (x · x−1 ) · y ≈ 1 · y, c 1 · y ≈ y, b (x · x−1 ) · y ≈ y, c ⊗ b x · (x−1 · y) ≈ (x · x−1 ) · y, a x · (x−1 · y) ≈ y, a ⊗ c ⊗ b

G3 by replacement on 1. G2, substitution by transitivity on 2., 3. G1, substitution by transitivity on 5., 4.

As a consequence of completeness of fuzzy equational logic (Theorem 3.31), we have x · (y −1 · y) ≈ xK ≥ b ⊗ c and x · (x−1 · y) ≈ y K ≥ a ⊗ b ⊗ c. Hence, K is congruence permutable in degree at least a ⊗ b2 ⊗ c2 . Now, let L be the standard product algebra on [0, 1]. Then, for the varieties given by G1–G3 with a = 1, b = 1, c = 0.75 a a = 1, b = 0.75, c = 1, the above lower estimation a ⊗ b2 ⊗ c2 says that both of the varieties are permutable in a degree at least 0.5625. One can proceed in a similar way to obtain further graded properties of classes of L-algebras (modularity, distributivity, regularity of congruences, etc.) and to establish graded Mal’cev-like characterizations. An interesting question is whether it is possible to characterize a large scale of graded properties of varieties at once by generalizing the well-known result of Taylor [84], see also related paper [69]. This seems to be an interesting open problem.

3.6 Bibliographical Remarks Equational logic is perhaps the most known part of universal algebra which is known outside of universal algebra. Namely, equational logic serves as a basis for methods of formal specification developed in theoretical computer science [1, 62]. A very good textbook (focused on computer scientists) on ordinary equational logic is [97]. Ordinary equational logic and ordinary variety theorem were developed by Birkhoff [23]. Since Birkhoff’s seminal contribution, both equational logic and the theory of varieties of universal algebras were investigated in hundreds of papers. A representative selection can be found in journal Algebra Universalis, published since 1971. From this point of view, the present chapter deals only with the basic notions and results of equational logic and theory of varieties in fuzzy setting. Developing further results in this direction is an open problem which might bring new insight to both fuzzy logic (especially to its model theory) and universal algebra. This chapter is based mainly on [9, 11, 16, 89]. Further results on fuzzy equational logic can be found in [93].

4 Fuzzy Horn Logic

Implications between identities have been widely studied in universal algebra. There are numerous results on implicationally defined classes of algebras and provability of implications from theories in form of implications. A survey on implications in the context of universal algebra can be found in monograph [97]. In this chapter, we suggest a generalization of the concept of an implication between identities in fuzzy setting and show that the results on (classical) Horn logic have their analogies in fuzzy setting. As in Chap. 3, we develop our logical calculus in Pavelka style. ∗

∗

∗

In Sect. 4.1, we introduce the concept of an implication and basic concepts of syntax and semantics. Section 4.2 deals with semantic entailment. In Sect. 4.3, we elaborate a general completeness theorem and discuss some of its special cases. In Sect. 4.4 we revisit fuzzy equation logic from the point of view of fuzzy Horn logic. Sections 4.5, 4.6, 4.7, and 4.8 deal with definability of classes of algebras with fuzzy equalities by means of implications between identities.

4.1 Syntax and Semantics We start by recalling the concept of an implication (between identities) as it is usually used in universal algebra. Given terms s1 , s1 , . . . , sn , sn , t, t of type F over variables X, an implication is a formula s1 ≈ s1 c · · · c sn ≈ sn i t ≈ t , s1

(4.1) sn ,

with the intended meaning “if s1 equals and · · · and sn equals then t equals t ”. Such a formula is, in fact, an (ordinary) formula of predicate logic with a language which contains a binary relation symbol ≈, function symbols f ∈ F , and variables x ∈ X. The identities on the left-hand side of (4.1) are called premises, t ≈ t is called a consequent. In the literature on R. Bˇ elohl´ avek and V. Vychodil: Fuzzy Equational Logic, StudFuzz 186, 171–266 (2005) c Springer-Verlag Berlin Heidelberg 2005 www.springerlink.com

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4 Fuzzy Horn Logic

universal algebra, formulas of this form are alternatively called quasi-identities or universal positive Horn clauses. It is sometimes desirable to deal with these formulas as with sentences. That is, instead of (4.1), one considers a formula (∀x1 ) · · · (∀xk )(s1 ≈ s1 c · · · c sn ≈ sn i t ≈ t ) ,

(4.2)

s1 , s1 , . . . , sn , sn , t, t

such that all variables occurring in terms are among variables x1 , . . . , xk . An important generalization of (4.1) is obtained by allowing infinite premises. Namely, given an index set I and terms si , si (i ∈ I), t, t , one can consider a generalized formula ^ si ≈ si i t ≈ t , (4.3) i∈I V where “ ” denotes “generalized conjunction”. The intuitive meaning of (4.3) is “if si equals si for any i ∈ I, then t equals t ”. Obviously, (4.3) is not a first-order formula if I is an infinite set – in such a case (4.3) is an “infinite expression”. On the other hand, if I is finite, (4.1) and (4.3) can be identified. In particular, if I = ∅, both (4.1) and (4.3) simplify to t ≈ t .

(4.4)

Therefore, identities can be thought of as implications without premises. This corresponds well with the intuitive meaning of identities and implications between identities. Formulas (4.1) are long expressions. To save space, one usually introduces the following set-theoretical convention for writing premises. Instead of (4.1) or (4.3), one writes P i (t ≈ t )

(4.5)

where P is a binary relation on T (X) such that s, s ∈ P iff identity s ≈ s is one of the premises of (4.3). Thus, P is a binary relation on terms, called a set of premises, which uniquely describes the conjunction of premises up to their order. Realizing that the order of identities is not important because of associativity and commutativity of conjunction, P fully describes premises of an implication. For instance, implication (4.1) with finite premises can be written as P i (t ≈ t ) where P = {s1 , s1 , . . . , sn , sn }. This notation is handy especially if we deal with implications on a general level – we suppose that P is “some” set of premises, but we do not specify it in more detail. Therefore, in general, we deal with implications whose sets P of premises are possibly infinite. Results concerning implications between identities have shown that it is important which sets P of premises we consider permissible. For instance, model classes of implications with possibly infinite number of premises are closed under isomorphic images, subalgebras, and direct products. Simpler premises of implications naturally lead to further closure properties of model classes. Notice that two extreme cases of implications between identities are those of identities (P is empty) and implications with unrestricted premises (P is arbitrary). Other types of implications result by imposing additional constraints on sets of premises (e.g., sets of premises are


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required to be finite). The following summary gives an overview of types of implications which can be found in the literature [80, 97]. (i) Implications: P is arbitrary, model classes (sur-reflective classes) are abstract classes closed under subalgebras, and direct products. (ii) Finitary implications: P may be infinite, but X is a finite set of variables. In other words, finitary implications contain only finitely many variables. Model classes (semivarieties) are abstract classes closed under formations of subalgebras, direct products, and direct unions. (iii) Horn clauses: P is finite, model classes (quasivarieties) are abstract classes closed under formations of subalgebras, direct products, and direct limits. Alternatively, quasivarieties can be characterized as abstract classes closed under subalgebras and reduced products or as abstract classes closed under subalgebras, direct product, and ultraproducts. (iv) Equation implications: P either is a singleton, i.e. P = {s, s } for some s, s ∈ T (X), or P = ∅. Equation implications are particular Horn clauses. (v) Identities: P = ∅, model classes (varieties) are closed under formations of homomorphic images, subalgebras, and direct products. ∗

∗

∗

We are now going to introduce a logical calculus for dealing with implications between identities in a fuzzy setting. We call our logic a fuzzy Horn logic. We start by introducing basic concepts of syntax and semantics. Fuzzy Horn logic can be seen as an extension of fuzzy equational logic developed in Chap. 3. Compared to it, fuzzy Horn logic has a richer language and has more general formulas called implications. Identities, i.e. formulas of fuzzy equational logic, can be seen as implications with empty premises. As in case of fuzzy equational logic, we develop fuzzy Horn logic in Pavelka style. We start by a language. Suppose ≈, F, σ is a type (page 60) and L is a complete residuated lattice. A language of fuzzy Horn logic of type ≈, F, σ for L consists of a binary relation symbol ≈ called a symbol of (fuzzy) equality, function symbols f ∈ F with their arities σ(f ) ∈ N0 , symbols a of truth degrees (a ∈ L), (at least denumerable) set X of variables with X V ∩ F = ∅, symbols of logical connectives i (implication), c (conjunction), (generalized conjunction), and auxiliary symbols (parentheses, etc.). If ≈, F, σ and L are clear from the context, a language of type ≈, F, σ for L is called shortly a language. Conventions of Remark 3.1 apply also here. Terms (of type F over X) of fuzzy Horn logic are the usual ones introduced in Definition 2.73. Recall that T (X) denotes the set of all terms of (a given) type F over X. Formulas of fuzzy Horn logic are called (P-)implications and are defined in what follows. ∗ ∗ ∗ Before going to the definition of implications and their interpretation, consider the following comments.

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4 Fuzzy Horn Logic

First, when evaluating an implication ϕ i ψ in a standard way (call it the first way), one takes truth degrees !ϕ! and !ψ!, and a truth function → of implication and defines the truth degree !ϕ i ψ! of ϕ i ψ to be !ϕ! → !ψ!. There is, however, a second way to look at evaluating implications in bivalent case. Namely, one takes ϕ and tests whether it is true (!ϕ! equals 1). If not, !ϕ i ψ! equals 1; if yes, !ϕ i ψ! equals !ψ!. Note that in bivalent case, both of the ways yield the same truth degree of ϕ i ψ. While the first way can be directly adopted for fuzzy setting (one just allows !ϕ! and !ψ! to take also intermediate truth degrees and takes a suitable “fuzzy implication” →), the second way is not so straightforward. A general way to go is the following: Pick a threshold a ∈ L such that ϕ is considered sufficiently true iff its truth degree is at least a. In evaluating ϕ i ψ, one takes !ϕ! and compares it to a. If !ϕ! does not exceed the threshold (i.e. a ≤ !ϕ!), set !ϕ i ψ! to 1; otherwise (i.e. a ≤ !ϕ!) set !ϕ i ψ! to !ψ!. In bivalent case, there is only one nontrivial choice of a, namely, a = 1 which yields the usual interpretation of implications. In fuzzy setting, however, the choice of a is not unique. Second, interestingly enough, both of the above ways of interpreting implications are particular cases of a more general approach via truth stressers: let represent a unary logical connective “very true” (i.e. ϕ reads “ϕ is very true”) which is interpreted by a truth stresser ∗ defined on L, see Sect. 1.2. Consider a formula (a i ϕ) i ψ with a a truth constant interpreted by a ∈ L. If ∗ is the identity on L and a = 1, the truth degree of (a i ϕ) i ψ is just the truth degree defined in the first way. If ∗ is the globalization (i.e. a∗ = 1 for a = 1 and a∗ = 0 otherwise), the truth degree of (a i ϕ) i ψ is just the truth degree defined in the second way with a being the threshold. Third, as we are interested in implications between identities with possibly several identities in the premise connected in a conjunctive manner and want to allow each identity to have its own threshold, we deal with formulas ^ (ai i ϕi ) i ψ (4.6) i∈I V is a where ϕi ’s and ψ are identities, ai ’s are the thresholds of ϕi ’s, and “generalized conjunction” interpreted by infimum in L. For convenience and simplicity, we write only ^ ϕi , ai i ψ (4.7) i∈I

instead of (4.6). That is, we omit since its placement is fixed, we write ϕi , ai instead of ai i ϕi , and do not distinguish between constants ai for truth degrees and truth degrees ai themselves. Note that the second convention follows the identification of weighted formulas ϕ, a introduced in Sect. 1.4 with ordinary formulas a i ϕ, see e.g. [49, Sect. 3.3]. Doing so, we deal with formulas which can be thought of as implications P iψ

(4.8)

where P is a fuzzy set of identities ϕi with P (ϕi ) = ai . For our purpose, this is a convenient and sufficiently general way.


∗

∗

175

∗

It will sometimes be desirable to have only particular implications which are “interesting” from some point of view. For instance, we might be interested in implications P i ψ where P is a finite and/or crisp L-set, etc. Therefore, we should always consider implications whose sets of premises belong to a given set P of all “interesting sets of premises”. Another way of looking at P is that P is a constraint which we supply if we need to restrict ourselves only to particular formulas – we allow only those P i ψ where P ∈ P. What properties of P should we require? P should be non-empty in order to have any interesting formulas at all. Another natural requirement on P is the following: if P ∈ P, i.e. P is an interesting set of premises, then each particularization of P should belong to P (should also be interesting). By a particularization of P we mean any set of premises which is more specific than P . This leads to a requirement for P to be closed under substitutions because by substituting terms for variables in formulas we get, in fact, more specific formulas from the original ones. The following definition formalizes our requirements on P. Definition 4.1. For P ∈ LT (X)×T (X) , and endomorphism h : T(X) → T(X) we define an endomorphic image h(P ) ∈ LT (X)×T (X) by (4.9) h(P )(t, t ) = h(s) = t P (s, s ) h(s ) = t

for any t, t ∈ T (X). For P ∈ LT (X)×T (X) we define a set var(P ) ⊆ X by (4.10) var(P ) = {var(t) ∪ var(t ) | P (t, t ) > 0} . A non-empty subset P ⊆ LT (X)×T (X) is called a proper family of premises of type F (in variables X) if for every P ∈ P and any endomorphism h on T(X) we have h(P ) ∈ P. Every P ∈ P is then called an L-set of premises of type F (in variables X). Remark 4.2. (1) Given P ∈ P, the degree P (s, s ) ∈ L can be read “the degree to which identity s ≈ s belongs to P ”. var(P ) can be seen as a set of variables occurring in identities which belong to P in some nonzero degree. (2) Returning to the motivation for our need to define P, the closedness under endomorphic images corresponds with our requirement of closedness under particularizations because each endomorphism h can be seen as a “substitution”: each variable x ∈ X is replaced by term h(x). (3) If P ∈ LT (X)×T (X) is a crisp L-relation then the notion of an endomorphic image as defined above coincides with the classical one. That is, h(P ) = {h(t), h(t ) | t, t ∈ P } . As a trivial consequence, h(∅) = ∅. Example 4.3. The following are examples of proper families of premises. (1) P = LT (X)×T (X) is a proper family of premises (obvious). This family does not represent any constraints on premises. Implications with such

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4 Fuzzy Horn Logic

premises will represent the most general type of formulas used in our investigation. (2) Let P = P ∈ LT (X)×T (X) | var(P ) is finite . Then P is a proper family of premises. Indeed, every endomorphism h on T(X) is determined by its restriction on the set of generators X. Since var(P ) is finite, we can put Y = var(P ) ∪ Z, where Z is defined by Z = {var(h(x)) | x ∈ var(P )}. Now we can define g : Y → T (Y ) by letting g(y) = h(y) for each y ∈ Y . Thus, g (t) = h(t) for all t ∈ T (Y ). For t, t ∈ T (X) such that t, t ∈ T (Y ) we can see that h(s) = t, h(s ) = t implies s, s ∈ T (var(P )), thus P (s, s ) = 0, i.e. h(P )(t, t ) = 0. That is, var(h(P )) = var(g (P )). Moreover, Y is finite since both sets var(P ) and Z are finite. Thus, var(h(P )) ⊆ Y is finite as well. P of this form will be called a family of all finitary premises. Note that if X is (X) . finite, then P = LT (X)×T (3) A family P = P ∈ LT (X)×T (X) | P is finite is a special subfamily of that of (2). It is easy to observe that endomorphic image of every finite Lrelation is finite. Trivially, var(P ) is finite. P of this form is called a proper family of all finite premises. (4) Families defined in (1)–(3) have their “crisp variants”. Clearly, if P is crisp then h(P ) is crisp. Hence, the following families P = {P ∈ LT (X)×T (X) | P is crisp}, P = {P ∈ LT (X)×T (X) | var(P ) is finite and P is crisp}, P = {P ∈ LT (X)×T (X) | P is finite and crisp}, are proper families of premises. Such families will be used to define implications with crisp premises. (5) For each a ∈ L, let Pa denote a subset of LT (X)×T (X) such that Pa = {P | for any t, t ∈ T (X) : P (t, t ) > 0 implies P (t, t ) ≥ a} . It is easy to see that Pa is a proper family of premises since for every P ∈ P, h(P )(t, t ) either is zero or h(P )(t, t ) ≥ a. In fact, we have P0 = LT (X)×T (X) , and P1 denotes the proper family of all crisp premises. (6) P = {∅} is a proper family of premises trivially. As we will see later on, implications with empty premises can be identified with identities. Now we can define the general notion of a P-implication. We define also (L-)weighted implications, see Definition 1.103 and Remark 1.104, and also Definition 3.14 and Remark 3.15. Definition 4.4. Let L be a complete residuated lattice, X be a set of variables, F be a type, P be a proper family of premises of type F in variables X. A P -implication is an expression of the form ^ s ≈ s , P (s, s ) i (t ≈ t ) , (4.11) P (s,s )>0

where P ∈ P and t, t ∈ T (X). For a P-implication ϕ and a truth degree a ∈ L, a couple ϕ, a is called an L-weighted P-implication.


177

Remark 4.5. (1) P-implications are in general “infinite expressions”. Following the previous discussion, we consider (4.11) a shorthand for ^ P (s, s ) i (s ≈ s ) i (t ≈ t ) , (4.12) P (s,s )>0

a formula in language containing a unary connective V , symbols a for truth degrees a ∈ L, and a “generalized conjunction” . Equation (4.12) says “if it is very true that all identities s ≈ s from P are true then t ≈ t is true”, which is the intended meaning of (4.11). (2) As in Remark 1.104 (3), we can identify L-sets of P-implications with ordinary sets of weighted P-implications. Recall that for an L-set Σ of Pimplications we put TΣ = {ϕ, a | Σ(ϕ) = a}. (3) For simplicity, P-implication (4.11) will be denoted by P i (t ≈ t ). If P is finite, P i (t ≈ t ) will be occasionally denoted by t1 ≈ t1 , a1 c t2 ≈ t2 , a2 c · · · c tn ≈ tn , an i t ≈ t , where P (ti , ti ) = ai for i = 1, . . . , n, and Supp(P ) ⊆ {t1 , t1 , . . . , tn , tn }. In the sequel, we will be interested in general proper families of premises and families which result from given families by restrictions. For instance, we are going to study Pavelka-style completeness of fuzzy Horn logic which uses formulas with finitely many premises. That is, given a family P, we will be interested in those P-implications where P ∈ P is a finite L-relation. For any proper family P of premises we thus define the following subsets of P: PFin = {P ∈ P | P is finite} , Pω = {P ∈ P | var(P ) is finite} , Pκ = {P ∈ P | κ > | var(P )|} ,

(4.13) (4.14) (4.15)

where κ is any infinite cardinal. Remark 4.6. (1) PFin is a subset of P which consists of P ∈ P such that P is finite (i.e., Supp(P ) is a finite set). Pω is a subset of P which consists of P ∈ P such that the set of all variables occurring in identities which belong to P in some nonzero degree is finite. Finally, Pκ is a subset of P which consists of P ∈ P such that the set of variables occurring in P ∈ P is strictly lesser than κ. Notice that Pω is a particular case of Pκ for κ = ω. (2) Observe that PFin and Pκ are closed under endomorphic images. Indeed, since each P ∈ PFin is finite, we have that h(P ) is finite, i.e. h(P ) ∈ PFin . In case of Pκ , we have | var(h(P ))| ≤ | var(h(var(P )))| < κ for any P ∈ Pκ (notice that var(h(x)) is finite) from which the claim immediately follows. (3) PFin and Pκ can be empty. Thus, PFin and Pκ need not be proper families of premises. For instance, consider a type F = {c, d, f }, where c, d are constants, and f is a unary function symbol. Put t0 = c, t0 = d, ti = f (ti−1 ), ti = f (ti−1 ) for each i ∈ N. Define P ∈ LT (X)×T (X) so that P (ti , ti ) = 1 (i ∈ N0 ), and P (· · ·) = 0 otherwise. Obviously, P = {P } is a proper family of premises because for any endomorphism h on T(X), h(c) = c and h(d) = d

178

4 Fuzzy Horn Logic

which further gives h(ti ) = ti and h(ti ) = ti (i ∈ N0 ). Clearly, PFin = ∅, i.e. PFin is not a proper family of premises. According to Remark 4.6, PFin and Pκ need not be proper families of premises because it can happen that PFin = ∅ and Pκ = ∅. On the other hand, in the sequel we will be interested only in cases where PFin and Pκ are nonempty in order to have any PFin -implications (Pκ -implications). Thus, from this moment on, we assume the following convention: if we consider PFin or Pκ , we automatically assume that PFin and Pκ are proper families of premises, i.e. we assume that P is a proper family of premises such that PFin or Pκ are non-empty. The following definition introduces a terminology that will be used in the subsequent development. Definition 4.7. PFin defined by (4.13) is called a Horn restriction of P. Pω defined by (4.14) is called a finitary restriction of P. PFin -implications will be called P-Horn clauses. Pω -implications will be called P-finitary implications. Remark 4.8. (1) Note that PFin ⊆ Pω ⊆ Pκ ⊆ P for any infinite κ. Hence, each P-Horn clause is a P-finitary implication, and each P-finitary implication is a P-implication. (2) If P ⊆ PFin , we have PFin = Pω = Pκ = P for any infinite κ. (3) If P is clear from the context, a P-implication (P-Horn clause) can be called a finitary implication (Horn clause). ∗

∗

∗

We are now going to introduce basic concepts of semantics of fuzzy Horn logic. Structures of truth degrees of fuzzy Horn logic will be complete residuated lattices L with truth stressers ∗ . Structures for interpretation of terms and formulas of fuzzy Horn logic are, as in case of fuzzy equational logic, algebras with fuzzy equalities. The concepts of a valuation (i.e. a mapping v : X → M ) and value !t!M,v of term t in an L-algebra M under valuation v are defined as usual (see Definition 3.4). Recall (Definition 3.6) that given terms t, t ∈ T (X), a truth degree !t ≈ t !M,v of an identity t ≈ t in M under a valuation v : X → M is defined by !t ≈ t !M,v = !t!M,v ≈M !t !M,v . A truth degree !P i (t ≈ t )!M,v of implication (4.11) is defined in a straightforward way, taking into account that (4.11) is a shorthand for (4.12) and that connective is interpreted by a truth-stresser ∗ on L. Definition 4.9. Let P i (t ≈ t ) be a P-implication and let ∗ be a truth stresser for a complete residuated lattice L. For an L-algebra M and a valuation v : X → M , we define a truth degree P i (t ≈ t )M,v of P i (t ≈ t ) in M under v with respect to ∗ by !P i (t ≈ t )!M,v = !P !M,v → !t ≈ t !M,v ,

(4.16)


where !P !M,v =

s,s ∈T (X)

P (s, s ) → !s ≈ s !M,v

∗

.

179

(4.17)

A truth degree P i (t ≈ t )M of P i (t ≈ t ) in M with respect to ∗ is defined by (4.18) !P i (t ≈ t )!M = v:X→M !P i (t ≈ t )!M,v . For a weighted P-implication P i (t ≈ t ), a we define a truth degree P i (t ≈ t ), aM,v of P i (t ≈ t ), a in M under v w.r.t. ∗ by a → !P i (t ≈ t )!M,v .

(4.19)

Therefore, we consider ∗ as a parameter controlling the interpretation of (4.11). For the boundary truth stressers of Example 1.14, we have the following. First, for ∗ being the identity we get !P i (t ≈ t )!M,v = → !t ≈ t !M,v . = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v being globalization we get   !t ≈ t !M,v !P i (t ≈ t )!M,v =  1

Second, for

(4.20)

∗

if P (s, s ) ≤ !s ≈ s !M,v for all s, s ∈ T (X) , otherwise .

(4.21)

Denoting by a and b the truth degree of P i (t ≈ t ) as defined by (4.20) and (4.21), respectively, it is easily seen that a ≤ b. Equations (4.20) and (4.21) are thus the boundary cases of (4.16). As we will see, both types of interpretations are reasonable (cf. also discussion in the beginning of this section). Note that in the bivalent case, both (4.20) and (4.21) coincide. Remark 4.10. (1) We will use only one structure of truth degrees and one truth stresser at a time, so there is no danger of confusion if the degree is denoted ∗ simply by !P i (t ≈ t )!M,v . Sometimes, we will use !P i (t ≈ t )!L M,v to point out L with ∗ explicitly. (2) It is easily seen that (4.19) is equal to !P !M,v → (a → !t ≈ t !M,v ). This corresponds well to the intuitive meaning of a weighted implication and also justifies a possible notation t1 ≈ t1 , a1 c t2 ≈ t2 , a2 c · · · c tn ≈ tn , an i t ≈ t , a for a weighted Horn clause t1 ≈ t1 , a1 c t2 ≈ t2 , a2 c · · · c tn ≈ tn , an i t ≈ t , a . (3) Evidently, !P i (t ≈ t )!M,v ≥ b iff !P i (t ≈ t ), b !M,v = 1. (4) For P = ∅, we have ∗ ∗ = =1. !P !M,v = s,s ∈T (X) 0 → !s ≈ s !M,v s,s ∈T (X) 1 Therefore, for any truth stresser

∗

and arbitrary t, t ∈ T (X),

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4 Fuzzy Horn Logic

!∅ i (t ≈ t )!M,v = !∅!M,v → !t ≈ t !M,v = !t ≈ t !M,v . Thus, from the point of view of semantics, identities can be identified with P-implications with empty premises, and the truth degree !∅ i (t ≈ t )!M,v does not depend on ∗ . ∗

∗

∗

Fuzzy Horn logic as developed so far can be seen as a certain fragment of predicate fuzzy logic as introduced in Example 1.96 (2), but with infinite conjunction. As in Chap. 3, we now present fuzzy Horn logic in a setting of abstract logic. That is, we present a particular abstract logic L = Fml , L, S, A, R , see Definition 1.105, which gives us then automatically further concepts we need (theory, semantic entailment, provability, etc.), see Sect. 1.4. For a set Fml of formulas we take the set of all P-implications, where P is a given proper family of premises (of type F in variables X), i.e. Fml = {P i (t ≈ t ) | P ∈ P, t, t ∈ T (X)}. For L we take an arbitrary complete residuated lattice equipped with a truth stresser ∗ . That is, as required by Definition 1.105, L is a complete lattice, but we require an additional structure on it. For S we take an L-semantics S for Fml defined by S = {E ∈ LFml | for some L-algebra M : E(ϕ) = !ϕ!M

(4.22)

for each P-implication ϕ ∈ Fml } , cf. (1.92). For A, we take an empty L-set of formulas, i.e. A(ϕ) = 0 for each Pimplication ϕ. For R we take a set of deduction rules defined later on. Up to R which we specify later, we have defined a particular abstract logic. This gives us automatically all the notions of abstract logic defined in Sect. 1.4. However, as in Chap. 3, in order for these concepts to fit better into our desired framework, we extend them. We want to deal with L-algebras instead of evaluations and with classes of L-algebras instead of the corresponding sets of evaluations. Using mappings Md and Th defined by (1.93) and (1.94), we can introduce mappings Mod and Impl by Mod(Σ) = {M | EM ∈ Md(Σ)}

and

Impl(K) = Th({EM | M ∈ K})

where Σ ∈ LFml is a theory (L-set of P-implications) and K is a class of L-algebras (of type F ). Again, we can then say that an L-algebra M is a model of a theory Σ if M ∈ Mod(Σ). It is easy to see that M is a model of Σ ∈ LFml iff Σ(ϕ) ≤ !ϕ!M for each P-implication ϕ ∈ Fml . Then, Mod(Σ) is a class of L-algebras and Impl(K) is an L-set of P-implications for which we have


Mod(Σ) = {M | M is a model of Σ} , (Impl(K))(ϕ) = {!ϕ!M | M ∈ K}

181

(4.23) (4.24)

for each P-implication ϕ. Remark 4.11. (1) Occasionally, if we want to make L∗ and P explicit, we use ∗ ∗ also ModL (Σ), and also ImplP(K) and ImplL P (K). (2) We also write Implκ(K), Horn(K) instead of ImplPκ(K), ImplPFin(K), respectively. If K = {M}, we write Impl(M) instead of Impl({M}). Since PFin ⊆ Pω ⊆ P, Horn(K) is a restriction of ImplP(K) on P-Horn clauses; Implω(K) is a restriction of ImplP(K) on P-finitary implications. (3) ImplP(K) can be seen as an L-set of P-implications such that a P-implication P i (t ≈ t ) belongs to ImplP(K) in the degree to which P i (t ≈ t ) is true in K. (4) If P and Q are proper families of premises such that Q ⊆ P, we sometimes look at ImplQ(K) as it were an L-set of P-implications, where (ImplP(K))(P i (t ≈ t )) if P ∈ Q , (ImplQ(K))(P i (t ≈ t )) = 0 otherwise , for any P-implication P i (t ≈ t ). For instance, Horn(K) can be seen as an L-set of P-implications such that (Horn(K))(P i (t ≈ t )) = 0 if P is infinite. (5) For a proper family P = {∅} and for any class K of L-algebras of a given ∗ type we get (ImplL P (K))(∅ i (t ≈ t )) = (Id(K))(t ≈ t ) for any t, t ∈ T (X). This is an immediate consequence of Remark 4.10 (4). Like Md and Th, Mod and Impl form a Galois connection and satisfy thus (1.95)–(1.100) with sets K’s of evaluations replaced by classes K’s of L-algebras. As in Remark 1.107, we can define a degree !P i (t ≈ t )!Σ to which a P-implication P i (t ≈ t ) semantically follows from an L-set Σ of P-implications by !P i (t ≈ t )!Σ = (Impl(Mod(Σ)))(P i (t ≈ t )). Notice that all the concepts of fuzzy Horn logic are uniquely determined by the choice of a type ≈, F, σ , a complete residuated lattice L with truth stresser ∗ , and a proper family P of premises. The following definition summarizes semantical concepts of fuzzy Horn logic given by a particular choice of ≈, F, σ , L, ∗ , and P. Definition 4.12. A theory is an L-set Σ of P-implications. An L-algebra M is a model of a theory Σ if M ∈ Mod(Σ). A degree P i (t ≈ t )Σ to which P i (t ≈ t ) semantically follows from a theory Σ is defined by !P i (t ≈ t )!Σ = (Impl(Mod(Σ)))(P i (t ≈ t )). For a class K of L-algebras, a truth degree P i (t ≈ t )K of P i (t ≈ t ) in K is defined by !P i (t ≈ t )!K = (Impl(K))(P i (t ≈ t )).

182

4 Fuzzy Horn Logic

Remark 4.13. (1) For reader’s convenience, let us rewrite the concepts of Definition 4.12: M is a model of Σ iff for each P i (t ≈ t ) : Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!M , !P i (t ≈ t )!K = M ∈ K !P i (t ≈ t )!M = = M ∈ K v:X→M !P i (t ≈ t )!M,v , !P i (t ≈ t )!Σ = !P i (t ≈ t )!Mod(Σ) = = M ∈ Mod(Σ) !P i (t ≈ t )!M = = M ∈ Mod(Σ) v:X→M !P i (t ≈ t )!M,v . (2) If we need to make L and instead of !P i (t ≈ t )!Σ .

∗

∗

explicit, we also use !P i (t ≈ t )!L Σ

Definition 4.14. For a theory Σ ∈ LFml , Mod(Σ) is called an P -implicational class of Σ. A class K of L-algebras is called a P -implicational class if K = Mod(Σ) for some theory Σ. K is called a P -finitary implicaP-Horn class) if K = Mod(Σ) for some L-set Σ of P-finitary tional class (P implications (P-Horn clauses). For a class K of L-algebras, Impl(K) is called an L∗ -implicational P theory of K. An L-set Σ of P-implications is called an L∗ -implicational P -theory if Σ = Impl(K) for some class K of L-algebras. Remark 4.15. (1) Since PFin ⊆ Pω ⊆ P, every P-Horn class is a P-finitary class and every P-finitary class is a P-implicational class. (2) Remark 4.10 (4) yields that for P = {∅} and an L-set Σ of P-implications, Mod(Σ) is the equational class of Σ (see Definition 3.12) where Σ is the L-set of identities such that Σ (t ≈ t ) = Σ(∅ i (t ≈ t )) for any t, t ∈ T (X). Utilizing the notions of abstract logic, we can now define concepts related to provability of a fuzzy Horn logic given by ≈, F, σ , L, ∗ , and P. However, since provability for general systems P of proper premises would require infinitary deduction rules, we restrict ourselves to P-Horn caluses. Definition 4.16. Let R be a set of deduction rules for P-Horn clauses, Γ ∈ LFml be a theory (L-set of P-Horn clauses), let P i (t ≈ t ) ∈ Fml be a PHorn clause and a ∈ L. An (L-weighted ) proof of P i (t ≈ t ), a from Γ using R is a sequence ϕ1 , a1 , . . . , ϕn , an of L-weighted P-Horn clauses ϕi , ai such that ϕn is P i (t ≈ t ), an = a, and for each i = 1, . . . , n, we have ai = Γ (ϕi ) or ϕi , ai follows from some ϕj , aj ’s, j < i, by some deduction rule R ∈ R. The number n is called a length of the proof. In such a case, we write Γ R P i (t ≈ t ), a and call P i (t ≈ t ), a provable from Γ using R. If Γ R P i (t ≈ t ), a , P i (t ≈ t ) is called provable in


183

degree (at least) a from Γ using R. A degree of provability of P i (t ≈ t ) from Γ using R, denoted by |P i (t ≈ t )|R Γ , is defined by |P i (t ≈ t )|R {a | Γ R P i (t ≈ t ), a }. Γ = Remark 4.17. The conventions introduced in Remark 1.102 apply. ∗

∗

∗

Example 4.18. (1) Consider a language with a single binary function symbol ◦ and take P = LT (X)×T (X) . A P-Horn clause x ◦ y ≈ x ◦ z, a i (y ≈ z) can be seen as a type of a graded cancellation rule, saying “if x ◦ y equals x ◦ z in degree (at least) a ∈ L, then y equals z”. Note that in the ordinary case, cancellation cannot be expressed by identities (classes of cancellative algebras are not closed under the formation of homomorphic images). (2) The apparatus of weighted implications can be seen as a tool in formal specification in presence of vagueness. In particular, applications of weighted implications lie mainly in the field of so-called humanistic systems, where the description of a system behavior is influenced by human judgment or perceptions, and is therefore inherently vague. In the following, we present a way to describe approximate knowledge about a simple function-based system. We deal with the problem of human perception of colors and the related problem of color mixture. Needles to say, the problems in question are hardly graspable by bivalent logic (crisp structures) since the notions of “color similarity” and “color indistinguishability” that naturally appear in the problem domain are vague. The color perception itself is a complex neuro-chemical process with a psychological feedback. Denote the set of all colors by M . Equip M with an L-equality relation ≈M the meaning of which is to represent similarity of colors from M . Note that ≈M is a nontrivial L-relation for which all properties of an L-equality seem to be justified. The (additive) mixture of colors can be thought of as an operation on M . Thus, suppose we have a language F = {f }, where f is a binary function symbol and a term f (t, t ) of type F represents a color resulting by the mixture of colors represented by terms t, t . It is a well-known fact [39] that assuming sufficiently high light intensities, if x is indistinguishable from x , and y is indistinguishable from y , then f (x, y) is indistinguishable from f (x , y ). This rule immediately translates into a compatibility condition for f M . M M of type F seems to be a To sum up, an L-algebra M = M, ≈ , f suitable semantical structure enabling us to study color mixture. Weighted implications can be used to define additional constraints on our perception of color mixture. For instance, the weighted P-Horn clause x ≈ x , a c f (x, y) ≈ f (x , y ), b i (y ≈ y ), c

(4.25)

can be read as: “if colors x, x are similar in degree a and if mixtures f (x, y), f (x , y ) are similar in degree b, then y, y are similar (at least) in degree c”.

184

4 Fuzzy Horn Logic

∗

∗

∗

When describing the abstract logic for fuzzy Horn logic, we left a set R of deduction rules unspecified. Deduction rules of fuzzy Horn logic will be described now. Note in advance that our deduction rules do not conform to the notion of a deduction rule as introduced in Sect. 1.4. Recall that according to Pavelka, a deduction rule is a pair R = Rsyn , Rsem , where Rsyn : Fml n → Fml is a partial mapping on the set of formulas and Rsem : Ln → L is a mapping on the set of truth degrees. Weighted formula ϕ, a inferred by R is of the form ϕ = Rsyn (ϕ1 , . . . , ϕn ) and a = Rsem (a1 , . . . , an ), meaning that one infers validity of ϕ in degree (at least) a ∈ L given formulas ϕi valid in degree (at least) ai (i = 1, . . . , n). Contrary to that, some of our rules compute a ∈ L in the inferred weighted formula ϕ, a not only from ai ’s but also from truth degrees (represented by constants) which are present in ϕi ’s. This is, however, only for the sake of convenience. Namely, as we will see in Remark 4.23, all of our deduction rules are in fact derived rules in a suitably extended Pavelka-style first-order fuzzy logic with ordinary deduction rules of the form R = Rsyn , Rsem . Therefore, our deduction rules are partial mappings n

R : (Fml × L) → Fml × L .

(4.26)

A set R of deduction rules will also be called a deductive system. Remark 4.19. (1) Instead of R(ϕ1 , a1 , . . . , ϕn , an ) = ϕ, a , we again use the common notation ϕ1 , a1 , . . . , ϕn , an . (4.27) ϕ, a Described verbally, the deduction rule (4.27) should be read as “From ϕ1 in degree a1 , and · · · and ϕn in degree an infer ϕ in degree a”. (2) An axiom can be thought of as nullary deduction rule, i.e. a mapping A : {∅} → Fml × L. Hence, an axiom is a weighted formula from Fml ×L. In accordance with Remark 4.5, we can denote an axiom P i (t ≈ t ), a also by P i t ≈ t , a . All axioms introduced below are considered as nullary deduction rules, i.e. unlike the abstract approach described in Sect. 1.4, we do not introduce an additional set A of (logical) axioms. In order to introduce our basic set R of deduction rules for fuzzy Horn logic, we define an application of a substitution to L-sets of premises as follows. Definition 4.20. For P ∈ P, let P (x/r) ∈ LT (X)×T (X) denote a binary Lrelation defined by P (x/r) (t, t ) = s(x/r) = t P (s, s ) (4.28) s (x/r) = t

for all terms t, t ∈ T (X).


185

Remark 4.21. Evidently, every substitution (x/r) can be expressed as an endomorphism h on T(X), which is a homomorphic extension of a mapping g : X → T (X), where g(x) = r, and g(y) = y for y ∈ X, y = x. Thus, we have t(x/r) = h(t) for each term t ∈ T (X), i.e. P (x/r) = h(P ). This fact can be easily proved by structural induction. Thus, for a proper family of premises P and any (x/r), we have P (x/r) ∈ P for every P ∈ P. As a consequence, if P i (· · ·) is a P-Horn clause then so is P (x/r) i (· · ·). In what follows, we use a system R of deduction rules (Ref)–(Mon) introduced below. Therefore, if there is no danger of confusion, we omit “R” from R “provable from Γ using R”, “|P i (t ≈ t )|Γ ”, etc. and write simply “prov able from Γ ”, “|P i (t ≈ t )|Γ ”, etc. Until otherwise mentioned, R always stands for the set of deduction rules which consists of (Ref)–(Mon). The first group of rules are the rules of congruence: (Ref) :

(Sym) :

P i (t ≈ t), 1

,

P i (t ≈ t ), a , P i (t ≈ t), a

(Tra) :

P i (t ≈ t ), a , P i (t ≈ t ), b , P i (t ≈ t ), a ⊗ b

(Rep) :

P i (t ≈ t ), a , P i (s ≈ s ), a

where P ∈ PFin , a, b ∈ L, t, t , t , s, s ∈ T (X), and s contains t as a subterm and s results from s by substitution of one occurrence of t in s by t . The second group are the rules of extensivity, substitution, and monotony: (Ext) :

P i (t ≈ t ), P (t, t )

(Sub) :

(Mon) :

,

P i (t ≈ t ), a , P (x/r) i t(x/r) ≈ t (x/r) , a

{Q i (ti ≈ ti ), ai ; i = 1, . . . , n} , P i (t ≈ t ), b n , ∗ Q i (t ≈ t ), b ⊗ i=1 (P (ti , ti ) → ai )

where P, Q ∈ PFin such that Supp(P ) = {t1 , t1 , . . . ,tn , tn }, t, t , r ∈ T (X), x ∈ X, and a1 , . . . , an , a, b ∈ L. Remark 4.22. (1) Deduction rules (Ref)–(Rep) are generalizations of the rules (ERef)–(ERep) presented in Sect. 3.1. Rules (Ref) and (Ext) are nullary, i.e. they can be thought of as axioms. The rule of extensivity (Ext) expresses the relationship between sets of weighted premises and provability degrees.

186

4 Fuzzy Horn Logic

In other words, using (Ext), every P i (t ≈ t ) is provable in degree at least P (t, t ). The rule of substitution is also similar to (ESub) but for technical reasons we use P (x/r) defined by (4.28). (2) On the verbal level, the rule of monotony (Mon) can be read: “if P implies t ≈ t and each premise ti ≈ ti from P is implied by Q, then also t ≈ t is implied by Q”. A finer reading of (Mon) is: “Q implies t ≈ t (at least) in degree to which P implies t ≈ t and to which it is very true that each ti ≈ ti is implied by Q at least in degree n to which ti ≈ ti ∗belongs to P ”. (3) The truth degree b ⊗ i=1 (P (ti , ti ) → ai ) as used in (Mon) is computed using truth degrees of input formulas together with P (ti , ti )’s. In other words, the degree P (ti , ti ) to which the identity ti ≈ ti belongs to premises of P i (t ≈ t ) has an influence on the resulting truth degree. Consider the following example. Take a residuated lattice L = {0, a, 1}, ∨, ∧, ⊗, →, 0, 1 , where 0 < a < 1, and a ⊗ a = 0, the rest is determined uniquely. Furthermore, we equip L by globalization which is Horn truth stresser on L. Now, (Mon) gives r ≈ r , 1 i s ≈ s , a , s ≈ s , a i t ≈ t , 1 . r ≈ r , 1 i t ≈ t , 1 On the other hand, increasing the “threshold” (given by the degree a in the formula s ≈ s , a i t ≈ t , 1 ) from a to 1, we get r ≈ r , 1 i s ≈ s , a , s ≈ s , 1 i t ≈ t , 1 . r ≈ r , 1 i t ≈ t , 0 (4) One way to have deduction rules in Pavelka-style (separate syntactic and semantic parts) is to split the (Mon) rule into separate rules (MonP ) for every set of premises P ∈ P. In this case, it would be possible to distinguish the syntactic and semantic part of a rule by two independent mappings as usual. For the above example, we would then use two different deduction rules (MonP1 ), (MonP2 ). Remark 4.23. We are going to show that (Ref)–(Mon) are derived rules in a natural Pavelka-style first-order fuzzy logic. To that purpose, we assume that some (weighted) formulas are provable in the Pavelka-style logic we work with (we mention them in the course of our demonstration). In particular, we assume that we have formulas guaranteeing reflexivity, symmetry, transitivity, and compatibility of ≈, and formulas guaranteeing the required properties of logical connectives as axioms. As we will work with truth constants (for every a ∈ L we consider a truth constant a), we need to assume appropriate “bookkeeping axioms” for the constants [49]. Namely, for we assume a e a∗ . We use the following deduction rules: modus ponens (MP; from ϕ, a and ϕ i ψ, b infer ψ, a ⊗ b ), logical constant introduction [71] (from ϕ, a infer a i ϕ, 1 ), and truth confirmation [49, 51] (from ϕ, 1 infer ϕ, 1 ). For convenience, we write ϕ instead of ϕ, 1 . Let Supp(P ) = {t1 , t1 , . . . , tn , tn } and put P (ti , ti ) = pi (i = 1, . . . , n).


187

(Ext): Let P i (t ≈ t ). We show p i (P i (t ≈ t )) for p = P (t, t ). Observe that for P (t, t ) = 0, the claim is trivial. Thus, suppose there is some j ∈ {1, . . . , n} with t = tj and t = tj . Therefore, p = P (tj , tj ) = pj . We have Vn Vn i=1 (pi i (ti ≈ ti )) i i=1 (pi i (ti ≈ ti )),

[instance of ϕ i ϕ,]

Vn

i (ti ≈ ti )) i (pj i (tj ≈ tj )),

i=1 (pi

[using (ϕ c ψ) i ϕ,]

Vn

i (ti ≈ ti )) i (pj i (tj ≈ tj )),

i=1 (pi

[by (ϕ i ψ) i ((ψ i χ) i (ϕ i χ)), MP,]

Vn pj i i=1 (pi i (ti ≈ ti )) i (tj ≈ tj ) , [by (ϕ i (ψ i χ)) i (ψ i (ϕ i χ)), MP,]

showing that p i (P i (t ≈ t )). (Mon): We show that if i (t ≈ t )) and ai i (Q i (ti ≈ ti )) b iN(P n for each i = 1, . . . , n then b o i=1 (pi i ai ) i (Q i (t ≈ t )). First,

Q i (ai i (ti ≈ ti )),

[by (ϕ i (ψ i χ)) i (ψ i (ϕ i χ)), MP, i = 1, . . . , n,]

Qi

Vn

i=1 (ai

i (ti ≈ ti )),

[using (ϕ i ψ) i ((ϕ i χ) i (ϕ i (ψ c χ))), MP,]

Vn Q i i=1 (ai i (ti ≈ ti )) , [truth confirmation,]

Q i

Vn

i=1 (ai

i (ti ≈ ti )),

[by (ϕ i ψ) i ( ϕ i ψ), MP,]

Q i Q,

[instance of ϕ i ϕ (Q is of the form ϕ),]

Qi

Vn

i=1 (ai

i (ti ≈ ti )),

[by (ϕ i ψ) i ((ψ i χ) i (ϕ i χ)), MP.]

Now (ψ i χ) i ((ϕ o ψ) i (ϕ o χ)) and ( ϕ o ψ) i (ϕ o ψ) give Nn Nn Vn i=1 (pi i ai ) o Q i i=1 (pi i ai ) o i=1 (ai i (ti ≈ ti )) , Nn Vn ≈ t )) i i=1 (pi i ai ) o i=1 (ai i(t Vn Ni n i i i=1 (pi i ai ) o i=1 (ai i (ti ≈ ti )) . By (ϕ o (ψ c χ)) i ((ϕ o ψ) c (ϕ o χ)), (ϕ o ψ) i ϕ, ((ϕ i ψ) o (ψ i χ)) i (ϕ i χ), and using truth confirmation with isotony of o and (ϕ i ψ) i ( ϕ i ψ): Nn Vn )) i i=1 (pi i ai ) o i=1 (ai i (ti ≈ tiV n i i=1 ((pi i ai ) o (ai i (ti ≈ ti ))) , V n Vn i=1 ((pi i ai ) o (ai i (ti ≈ ti ))) i i=1 (pi i (ti ≈ ti )) . Using transitivity applied on previous formulas, isotony, and MP:

188

4 Fuzzy Horn Logic

Vn (pi i ai ) o Q i i=1 (pi i (ti ≈ ti )) , V n Nn b o i=1 (pi i ai ) o Q i b o i=1 (pi i (ti ≈ ti )) .

Nn

i=1

Taking into account b i (P i (t ≈ t )), we have Vn b o i=1 (pi i (ti ≈ ti )) i (t ≈ t ),

[by (ϕ i (ψ i χ)) i ((ϕ o ψ) i χ), MP,]

Nn b o i=1 (pi i ai ) o Q i (t ≈ t ), [using transitivity, MP,]

Nn b o i=1 (pi i ai ) i (Q i (t ≈ t )), [by ((ϕ o ψ) i χ) i (ϕ i (ψ i χ)), MP,]

which is the desired weighted formula. One can proceed similarly for the rest of the deduction rules. Note that the rule of truth confirmation and axioms ϕ i ϕ , (ϕ i ψ) i ( ϕ i ψ) , ϕ i ϕ , ( ϕ o ψ) i (ϕ o ψ) being used in the previous demonstration naturally correspond to properties of implicational truth stressers, see (1.11), (1.12), (1.13), (1.17), and (1.57). ∗

∗

∗

In the remaining sections we focus on two problems. The first one is completeness of fuzzy Horn logic. The second one is definability of classes of Lalgebras by implicational theories and characterization of definable classes by their closedness under suitable class operators.

4.2 Semantic Entailment In this section we are going to characterize a degree to which a P-implication semantically follows from a given L-set of P-implications. Recall that in fuzzy equational logic, we identified L-sets of identities (over variables X) with binary L-relations on T (X), see Sect. 3.2. Thus, if Σ was an L-set of identities over X, we sometimes denoted Σ(t ≈ t ) by Σ(t, t ) and worked with Σ as with an L-relation and vice versa. This allowed us to identify semantically closed L-sets of identities with particular binary L-relations (fully invariant congruences). In this section, we give an analogous characterization for L-sets of P-implications. The subsequent results are, in fact, generalizations of those of fuzzy equational logic. We will comment on this later on. Let us stress a complication which arises when dealing with general Pimplications: an L-set Σ of P-implications cannot be in general represented by a single binary L-relation on T (X). For instance, we can have P, Q ∈ P,

4.2 Semantic Entailment

189

P = Q, such that Σ(P i (t ≈ t )) = a = b = Σ(Q i (t ≈ t )). So, representing Σ by an L-relation to which t, t belongs in degree Σ(P i (t ≈ t )) does not work any more for P-implications. We resolve the problem of representing Σ as follows. Considering any P-implication P i (t ≈ t ), the L-set P of premises can be seen as an index. Thus, for each P ∈ P we can introduce a separate binary L-relation SP on T (X). Now we can put SP (t, t ) = Σ(P i (t ≈ t )) for any P ∈ P and t, t ∈ T (X). Clearly, a system of all such SP ’s contains full information about the original L-set Σ of P-implications. Returning to the previous example, we have SP (t, t ) = a and SQ (t, t ) = b. Furthermore, given a system {SP | P ∈ P} of binary L-relations on T (X), there is a unique L-set Σ of P-implications with Σ(P i (t ≈ t )) = SP (t, t ) for any P i (t ≈ t ). This particular representation of L-sets of P-implications is introduced in the following definition. Definition 4.24. Let P be a proper family of premises. A system S = SP ∈ LT (X)×T (X) | P ∈ P

(4.29)

is called a P -indexed system of L-relations. For a P-indexed system S of L-relations we define an L-set ΣS of P-implications by ΣS (P i (t ≈ t )) = SP (t, t ),

for every P ∈ P and t, t ∈ T (X).

(4.30)

For every L-set Σ of P-implications we define a P-indexed system SΣ of Lrelations as follows SΣ = SP ∈ LT (X)×T (X) | SP (t, t ) = Σ(P i (t ≈ t )) (4.31) for every P ∈ P and t, t ∈ T (X) . For convenience, if Σ is an L-set of P-implications, then SP ∈ SΣ will be alternatively denoted by ΣP . Remark 4.25. (1) It is immediate that ΣSΣ = Σ, SΣS = S. That is, there is an obvious bijective correspondence between P-indexed systems S of L-relations and L-sets Σ of P-implications, and we can go from S to the corresponding Σ and vice versa. (2) For P-indexed systems S, S of L-implications, we put S ≤ S iff for every P ∈ P we have SP ⊆ SP , i.e. iff ΣS ⊆ ΣS . Consequently, S = S iff SP = SP for every P ∈ P iff ΣS = ΣS . (3) An L-set Σ of identities can be thought of as an L-set of P-implications for P = {∅}. Thus, the corresponding {∅}-indexed system SΣ consists of a single L-relation denoted by Σ∅ . (4) SP denotes an element of a system S. Notice that we use S to denote the subsethood degree, see (1.66). There is no danger of confusion here since SP is a fuzzy relation on terms while S is a fuzzy relation on fuzzy sets. Moreover, elements of S are always used with subscripts (SP , Si , etc.). The following definition introduces particular P-indexed systems of Lrelations that will be important in the subsequent development.

190

4 Fuzzy Horn Logic

Definition 4.26. Suppose L∗ is a complete residuated lattice with implicational truth stresser ∗ . Let Σ be an L-set of P-implications. If the corresponding SΣ satisfies P ⊆ ΣP , ΣP (t, t ) ≤ Σh(P ) (h(t), h(t )) , ∗

S(P, ΣQ ) ≤ S(ΣP , ΣQ ) ,

(4.32) (4.33) (4.34)

for every P, Q ∈ P, t, t ∈ T (X), and every endomorphism h on T(X), then SΣ is called an L∗ -implicational P -indexed system of L-relations. Moreover, if ΣP is a congruence for every P ∈ P, then SΣ is called an L∗ implicational P -indexed system of congruences. Remark 4.27. (1) Since P is supposed to be proper, h(P ) ∈ P for every P ∈ P as required by Definition 4.1. That is, (4.33) is defined correctly. Also note that for Σ, ΣP (t ≈ t ) = Σ(P i (t ≈ t )). (2) Condition (4.34) has equivalent formulations. For instance, (4.34) holds for every P, Q ∈ P iff ∗ ≤ ΣQ (t, t ) (4.35) ΣP (t, t ) ⊗ s,s ∈T (X) P (s, s ) → ΣQ (s, s ) for all t, t ∈ T (X). This equivalent formulation will be used later on. for P = ∅, we can (3) Suppose Σ∅ is a congruence. Then if (4.33) holds claim Σ∅ (t, t ) ≤ Σh(∅) h(t), h(t ) = Σ∅ h(t), h(t ) . Hence, Σ∅ is a fully invariant congruence, see Sect. 3.2. (4) For particular proper families of premises, some of the conditions (4.32)–(4.34) will simplify or hold trivially. Remark 4.28. Let us comment on the intuitive meaning of (4.32)–(4.34). Suppose Σ = Impl(M) for some L-algebra M. So, think of Σ(P i (t ≈ t )) as of a degree to which P i (t ≈ t ) is true in M. Then (4.32), called extensivity, says that the validity degree of P i (t ≈ t ) is at least as high as the degree to which t, t (recall that t, t stands for identity t ≈ t ) belongs to P . Roughly speaking, if t, t is in P then P i (t ≈ t ) is always true which is an obvious property. (4.33), called stability, says, roughly speaking, that if P i (t ≈ t ) is true then h(P ) i (h(t) ≈ h(t )) is true. Thinking of endomorphisms as of substitutions, (4.33) says that validity is preserved under substitutions. Finally, (4.34), called ∗ -monotony, says that a mapping sending P to ΣP obeys one of characteristic properties of an L∗ -closure operator, see Sect. 1.3. If P = LT (X)×T (X) then Lemma 1.94 yields that a mapping sending P to ΣP is an L∗ -closure operator. Note, however, that if P = LT (X)×T (X) , then our mapping sending P to ΣP is in fact only a partial L∗ -closure operator since it is a mapping of P to LT (X)×T (X) , not of LT (X)×T (X) to LT (X)×T (X) (consider, for instance, families of crisp premises). Nevertheless, (4.32)–(4.34) say that sending P to ΣP has closure properties and that Σ is closed under substitutions. To sum up, L∗ -implicational P-indexed systems of L-relations (congruences) seem to be promising candidates for being algebraic representatives of L∗ -implicational P-theories.


191

The following auxiliary lemma shows that (4.33) and (4.34) can be equivalently replaced by a single condition. Lemma 4.29. Suppose we have a P-indexed system SΣ of L-relations. Then SΣ is an L∗ -implicational P-indexed system of L-relations if and only if SΣ satisfies (4.32) together with ΣP (t, t ) ≤ ∗ ≤ → ΣQ h(t), h(t ) (4.36) s,s ∈T (X) P (s, s ) → ΣQ h(s), h(s ) for every P, Q ∈ P, t, t ∈ T (X), and every endomorphism h on T(X). Proof. “⇒”: Let us suppose (4.32), (4.33), and (4.34) hold. Take P, Q ∈ P, t, t ∈ T (X), and an endomorphism h : T(X) → T(X). Using (1.40), (4.9), and the adjointness property it follows that h(P )(r, r ) → ΣQ (r, r ) = h(s) = r P (s, s ) → ΣQ (r, r ) = =

h(s) = r h(s ) = r

h(s ) = r

P (s, s ) → ΣQ (r, r ) =

h(s) = r h(s ) = r

P (s, s ) → ΣQ h(s), h(s ) ,

for every r, r ∈ T (X). Thus, using (4.33), (4.35) we obtain ΣP (t, t ) ≤ Σh(P ) h(t), h(t ) ≤ ∗ ≤ → ΣQ h(t), h(t ) = r,r ∈T (X) h(P )(r, r ) → ΣQ (r, r ) ∗ P (s, s ) → ΣQ h(s), h(s ) → ΣQ h(t), h(t ) = = h(s) = r r,r ∈T (X) =

s,s ∈T (X)

h(s ) = r

∗ P (s, s ) → ΣQ h(s), h(s ) → ΣQ h(t), h(t ) .

Hence, the inequality (4.36) is satisfied. “⇐”: Assume that conditions (4.32), (4.36) hold. (4.33): Take any endomorphism h : T(X) → T(X) and put Q = h(P ). From Definition 4.1 it follows that Q ∈ P. Now, using (4.36) we obtain ΣP (t, t ) ≤ ∗ ≤ → Σh(P ) h(t), h(t ) . r,r ∈T (X) P (r, r ) → Σh(P ) h(r), h(r ) Moreover, applying (4.9), (4.32), we have P (r, r ) ≤ h(s) = h(r) P (s, s ) = h(P ) h(r), h(r ) ≤ Σh(P ) h(r), h(r ) , h(s ) = h(r )

i.e. P (r, r ) → Σh(P ) h(r), h(r ) = 1 for all r, r ∈ T (X). Now, (1.11) gives ΣP (t, t ) ≤ 1∗ → Σh(P ) h(t), h(t ) = Σh(P ) h(t), h(t ) , proving (4.33). (4.34): Applying (4.36) to h being the identical morphism (i.e. h(t) = t) we get (4.35), a condition equivalent to (4.34).

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4 Fuzzy Horn Logic

Remark 4.30. In the classical case, a concept of a semantically closed set of implications corresponds to so-called fully invariant closure operators and fully invariant closure systems in T (X) × T (X), see [97]. Recall that the condition of full invariance of a closure operator cl means that h(cl (P )) ⊆ cl (h(P )) for every P ⊆ T (X)×T (X). In case of implications with finite premises, semantically closed sets of implications correspond to algebraic (i.e. finitely generated) fully invariant closure operators. The following assertions show that if we restrict ourselves to P = LT (X)×T (X) , our L∗ -implicational P-indexed systems of L-relations can be seen as particular L∗ -closure operators which have properties analogous to the fully invariant closure operators known from the classical case (cf. Remark 4.28). Theorem 4.31. Let P = LT (X)×T (X) and let SΣ be an L∗ -implicational Pindexed system of L-relations. Then an operator cl on LT (X)×T (X) defined by cl (P ) = ΣP is an L∗ -closure operator and h cl (P ) ⊆ cl h(P ) (4.37) holds for every P ∈ LT (X)×T (X) , and every endomorphism h : T(X) → T(X). Proof. Due to Lemma 1.94, we only check (4.37). Take P ∈ LT (X)×T (X) and let h be an endomorphism on T(X). We have h(ΣP )(t, t ) = h(s) = t ΣP (s, s ) ≤ h(s) = t Σh(P ) (h(s), h(s )) = =

h(s ) = t h(s) = t h(s ) = t

h(s ) = t

Σh(P ) (t, t ) = Σh(P ) (t, t )

for all t, t ∈ T (X). Thus, h cl (P ) ⊆ cl h(P ) .

Theorem 4.32. Let P = LT (X)×T (X) and suppose ∗ is an implicational truth stresser. Let cl be an L∗ -closure operator on LT (X)×T (X) satisfying (4.37) for every P ∈ LT (X)×T (X) and every endomorphism h : T(X) → T(X). Then the P-indexed system SΣ = {ΣP | ΣP = cl (P )} is an L∗ -implicational P-indexed system of L-relations. Proof. (4.33): Using (4.37) we get ΣP (t, t ) = cl (P )(t, t ) ≤ h(s) = h(t) cl (P )(s, s ) = h cl (P ) (h(t), h(t )) ≤ h(s ) = h(t )

≤ cl h(P ) (h(t), h(t )) = Σh(P ) (h(t), h(t )) for all terms t, t ∈ T (X). The rest follows from Lemma 1.94.

Remark 4.33. Note that the correspondences described by Theorem 4.31 and Theorem 4.32 are in fact mutually inverse. Therefore, for P = LT (X)×T (X) (no restriction on premises) there is a natural bijective correspondence between L∗ -implicational P-indexed system of binary L-relations and fully invariant L∗ -closure operators in T (X), generalizing the ordinary case.


∗

∗

193

∗

We are now going to show that L∗ -implicational P-indexed systems of congruences are in one-to-one correspondence with L∗ -implicational P-theories, i.e. theories (composed of P-implications) of classes of L-algebras with respect to a given implicational truth stresser. For brevity,we adopt the following convention: if K is a class of L-algebras, we denote by “ M,v · · · ” the infimum “ M∈K v:X→M · · · ” which ranges over all L-algebras M ∈ K and all valuations v : X → M . Theorem 4.34. Let Σ be an L∗ -implicational P-theory. Then ΣP is a congruence on T(X) for every P ∈ P. Proof. Since Σ is an L∗ -implicational P-theory, there is a class K of L-algebras, such that Σ = Impl(K). That is, we have Σ P i (t ≈ t ) = !P i (t ≈ t )!K = M∈K !P i (t ≈ t )!M = = M∈K v:X→M !P i (t ≈ t )!M,v . Thus, denoting Σ P i (t ≈ t ) by M,v !P i (t ≈ t )!M,v , we check the conditions of Definition 2.11. Reflexivity and symmetry of ΣP follow from reflexivity and symmetry of every L-equality. For transitivity, let P ∈ P and t, t , t ∈ T (X). Using (1.34), (1.41), (1.18), properties of ≈M ’s, and the isotony of → in the second argument, we have ΣP (t, t ) ⊗ ΣP (t , t ) = = M,v !P i (t ≈ t )!M,v ⊗ M,v !P i (t ≈ t )!M,v ≤ ≤ M,v !P i (t ≈ t )!M,v ⊗ !P i (t ≈ t )!M,v = = M,v !P !M,v → !t ≈ t !M,v ⊗ !P !M,v → !t ≈ t !M,v ≤ ≤ M,v !P !M,v ⊗ !P !M,v → !t ≈ t !M,v ⊗ !t ≈ t !M,v = = M,v !P !M,v → !t ≈ t !M,v ⊗ !t ≈ t !M,v ≤ ΣP (t, t ) . Hence, ΣP is transitive. It suffices to check the compatibility with functions since ≈T(X) ⊆ ΣP holds trivially. Take P ∈ P, an n-ary f ∈ F , and terms t1 , t1 , . . . , tn , tn . Since n n M !ti !M,v ≤ i=1 !ti ≈ ti !M,v = i=1 !ti !M,v ≈ ≤ f M !t1 !M,v , . . . , !tn !M,v ≈M f M !t1 !M,v , . . . , !tn !M,v = = !f (t1 , . . . , tn )!M,v ≈M !f (t1 , . . . , tn )!M,v = = !f (t1 , . . . , tn ) ≈ f (t1 , . . . , tn )!M,v , we get

n ΣP (t1 , t1 ) ⊗ · · · ⊗ ΣP (tn , tn ) = i=1 M,v !P i (ti ≈ ti )!M,v ≤ n ≤ M,v i=1 !P i (ti ≈ ti )!M,v =

194

4 Fuzzy Horn Logic

!P !M,v → !ti ≈ ti !M,v ≤ n n ≤ M,v !P !M,v → i=1 !ti ≈ ti !M,v ≤ ≤ M,v !P !M,v → !f (t1 , . . . , tn ) ≈ f (t1 , . . . , tn )!M,v = = ΣP f (t1 , . . . , tn ), f (t1 , . . . , tn ) .

=

M,v

n

i=1

Hence, ΣP is compatible. Altogether, ΣP is a congruence for any P ∈ P.

Remark 4.35. In the proof of Theorem 4.34 we used (1.18), i.e. a∗ ⊗ a∗ = a∗ , which is required for implicational truth stressers. Without postulating (1.18), we are not able to prove that ΣP is transitive and compatible. When every ΣP is a congruence, we can easily define a class of models as factorizations of T(X), see Theorem 4.38. We need the following lemma. Lemma 4.36. Suppose P is a proper set of premises and ∗ is an implicational truth stresser. Then for every P, Q ∈ P, for every L-algebra M, and for every valuation v : X → M , we have ∗ ≤ s,s ∈T (X) P (s, s ) → M,v !Q i (s ≈ s )!M,v ≤ M,v !Q!M,v → !P !M,v . (4.38) Proof. First, we can use properties of ∧, ∨ together with (1.29), (1.39) to get s,s ∈T (X) P (s, s ) → M,v !Q i (s ≈ s )!M,v = = s,s ∈T (X) P (s, s ) → M,v !Q!M,v → !s ≈ s !M,v = = s,s ∈T (X) M,v P (s, s ) → !Q!M,v → !s ≈ s !M,v = = M,v s,s ∈T (X) !Q!M,v → P (s, s ) → !s ≈ s !M,v = = M,v !Q!M,v → s,s ∈T (X) P (s, s ) → !s ≈ s !M,v . Now using (1.17), (1.13), and (1.19), we have ∗ = s,s ∈T (X) P (s, s ) → M,v !Q i (s ≈ s )!M,v ∗ = = M,v !Q!M,v → s,s ∈T (X) P (s, s ) → !s ≈ s !M,v ∗ = M,v !Q!M,v → s,s ∈T (X) P (s, s ) → !s ≈ s !M,v ≤ ∗ ∗ ≤ M,v !Q!M,v → = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v ∗ = M,v !Q!M,v → = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v = M,v !Q!M,v → !P !M,v , proving the inequality (4.38).

Theorem 4.37. Let Σ be an L∗ -implicational P-theory. Then the corresponding SΣ is an L∗ -implicational P-indexed system of congruences.


195

Proof. Let us have a class K of L-algebras such that Σ = Impl(K). We have to check extensivity, stability and ∗ -monotony of SΣ . The rest follows from Theorem 4.34. (4.32): Suppose we have P ∈ P and terms t, t ∈ T (X). For every M ∈ K and a valuation v : X → M , we have P (t, t ) ≤ P (t, t ) → !t ≈ t !M,v → !t ≈ t !M,v , which follows from the fact that the inequality a ≤ (a → b) → b holds in every residuated lattice. Moreover, we can use (1.12) and the antitony of → in the first argument to state that ∗ P (t, t ) ≤ P (t, t ) → !t ≈ t !M,v → !t ≈ t !M,v . Hence, by (1.19), and by properties of , we obtain the following inequality, ∗ P (t, t ) ≤ s,s ∈T (X) P (s, s ) → !s ≈ s !M,v → !t ≈ t !M,v = ∗ = → !t ≈ t !M,v = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v = !P i (t ≈ t )!M,v , showing that P (t, t ) ≤ M,v !P i (t ≈ t )!M,v = ΣP (t, t ), i.e. P ⊆ ΣP . (4.33): First, we can use Lemma 3.18 to observe that for all terms r, r ∈ T (X), a truth degree !h(r) ≈ h(r )!M,v equals to !r ≈ r !M,w for some valuation w : X → M . Moreover, w is determined uniquely by the endomorphism h. Hence, it follows that ∗ → !t ≈ t !M,v ≤ ΣP (t, t ) = M,v s,s ∈T (X) P (s, s ) → !s ≈ s !M,v ∗ ≤ M,v → !h(t) ≈ h(t )!M,v s,s ∈T (X) P (s, s ) → !h(s) ≈ h(s )!M,v for any endomorphism h on T(X). Furthermore, s,s ∈T (X) P (s, s ) → !h(s) ≈ h(s )!M,v = = s,s ∈T (X) h(r) = h(s) P (r, r ) → !h(s) ≈ h(s )!M,v = =

h(r ) = h(s )

s,s ∈T (X)

h(r) = h(s) h(r ) = h(s )

h(P )(h(s), h(s )) → !h(s) ≈ h(s )!M,v = = s,s ∈T (X) h(P )(s, s ) → !s ≈ s !M,v . =

P (r, r ) → !h(s) ≈ h(s )!M,v =

s,s ∈T (X)

Putting both facts together, we obtain ∗ → ΣP (t, t ) ≤ M,v s,s ∈T (X) P (s, s ) → !h(s) ≈ h(s )!M,v → !h(t) ≈ h(t )!M,v = ∗ = M,v → !h(t) ≈ h(t )!M,v = s,s ∈T (X) h(P )(s, s ) → !s ≈ s !M,v = M,v !h(P ) i (h(t) ≈ h(t ))!M,v = Σh(P ) (h(t), h(t )) . (4.34): We can use a consequence of Lemma 4.36 together with (1.30), (1.41), and well-known properties of to prove the following inequality

196

4 Fuzzy Horn Logic

∗ ΣP (t, t ) ⊗ = s,s ∈T (X) P (s, s ) → ΣQ (s, s ) = ⊗ M,v !P !M,v → !t ≈ t !M,v ∗ ⊗ s,s ∈T (X) P (s, s ) → M,v !Q i (s ≈ s )!M,v ≤ ≤ ⊗ ≤ M,v !P !M,v → !t ≈ t !M,v M,v !Q!M,v → !P !M,v ≤ M,v !P !M,v → !t ≈ t !M,v ⊗ !Q!M,v → !P !M,v ≤ ≤ M,v !Q!M,v → !t ≈ t !M,v = ΣQ (t, t ) . The previous inequality is equivalent to (4.34). The proof is complete.

Now we turn our attention to the converse problem. Having given an L∗ implicational P-indexed system of congruences SΣ , we construct a suitable class of L-algebras K, such that Σ = Impl(K). Theorem 4.38. Let SΣ be an L∗ -implicational P-indexed system of congruences. Then Σ is an L∗ -implicational P-theory. Namely, Σ = Impl(K) for a class K defined by K = {T(X)/ΣP | P ∈ P}. Proof. We have to check that ΣP (t, t ) = !P i (t ≈ t )!K , where !P i (t ≈ t )!K = Q ∈ P !P i (t ≈ t )!T(X)/ΣQ ,v . v: X→T (X)/ΣQ

“≤”: Take T(X)/ΣQ ∈ K and v : X → T (X)/ΣQ . Lemma 3.21 yields that there is an endomorphism h on T(X), such that [h(s)]ΣQ = !s!T(X)/ΣQ ,v for each s ∈ T (X). Consequently, !s ≈ s !T(X)/ΣQ ,v = !s!T(X)/ΣQ ,v ≈T(X)/ΣQ !s !T(X)/ΣQ ,v = = [h(s)]ΣQ ≈T(X)/ΣQ [h(s )]ΣQ = ΣQ h(s), h(s ) for every s, s ∈ T (X). Using (4.36) we get ∗ ΣP (t, t ) ≤ → ΣQ h(t), h(t ) = s,s ∈T (X) P (s, s ) → ΣQ h(s), h(s ) ∗ = → !t ≈ t !T(X)/ΣQ ,v = s,s ∈T (X) P (s, s ) → !s ≈ s !T(X)/ΣQ ,v = !P i (t ≈ t )!T(X)/ΣQ ,v . That is, ΣP (t, t ) ≤ !P i (t ≈ t )!T(X)/ΣQ ,v holds for arbitrary ΣQ , and every valuation v : X → T (X)/ΣQ . Hence, we have proved the “≤” inequality. “≥”: For every P ∈ P and a valuation v : X → T (X)/ΣP , where v(x) = [x]ΣP for all x ∈ X, we have !t!T(X)/ΣP ,v = [t]ΣP for each t ∈ T (X) (easy proof by structural induction). Thus, it follows that ΣP (s, s ) = [s]ΣP ≈T(X)/ΣP [s ]ΣP = = !s!T(X)/ΣP ,v ≈T(X)/ΣP !s !T(X)/ΣP ,v = !s ≈ s !T(X)/ΣP ,v for every s, s ∈ T (X). Now we can use property (1.11) of obtain the desired inequality:

∗

and (4.32) to


197

ΣP (t, t ) = !t ≈ t !T(X)/ΣP ,v = 1∗ → !t ≈ t !T(X)/ΣP ,v = ∗ = → !t ≈ t !T(X)/ΣP ,v = s,s ∈ T (X) P (s, s ) → ΣP (s, s ) ∗ = → !t ≈ t !T(X)/ΣP ,v = s,s ∈ T (X) P (s, s ) → !s ≈ s !T(X)/ΣP ,v = !P i (t ≈ t )!T(X)/ΣP ,v ≥ Q ∈ P !P i (t ≈ t )!T(X)/ΣQ ,v . v: X→T (X)/ΣQ

Putting all together, we obtain Σ = Impl(K). A summary of the previous observations follows.

Corollary 4.39. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . Let P be a proper family of premises, Σ be an L-set of Pimplications. Then SΣ is an L∗ -implicational P-indexed system of congruences iff Σ is an L∗ -implicational P-theory. ∗

∗

∗

From this moment on, we focus on a semantic entailment from L-sets of Pimplications. First, we check that a system of all L∗ -implicational P-indexed systems of congruences is a closure system itself. Then, given an L-set Σ of P-implications, we introduce its semantic closure and show that a degree !P i (t ≈ t )!Σ of semantic entailment equals a degree to which P i (t ≈ t ) belongs to the semantic closure of Σ. Let N = {Si | i ∈ I} be a system of P-indexed L-relations. systems Si of(X) Si = Si,P ∈ LT (X)×T | P ∈ P . That is, each Si ∈ N is a P-indexed system We define a P-indexed i∈I Si P | P ∈ P of L-relations system i∈I Si = as the intersection N , i.e. (4.39) i∈I Si P = i∈I Si,P . for every P ∈ P. Hence, i∈I

Si

P

(t, t ) =

i∈I

Si,P (t, t )

for all t, t ∈ T (X) and P ∈ P. Theorem 4.40. Let S ∗ denote a system of all L∗ -implicational P-indexed systems of congruences. Then S ∗ is a closure system. Proof. We have to check, that S ∗ is non-empty and closed under arbitrary intersections. It is easy to observe that the P-indexed system of L-relations Smax = {SP | SP (t, t ) = 1 for every P ∈ P, t , t ∈ T (X)} satisfies conditions (4.32)–(4.34) trivially, and every SP ∈ Smax , P ∈ P is a congruence relation. Hence, Smax ∈ S ∗ and so S ∗ is non-empty. Let N = {Si ∈ S ∗ | i ∈ I}. We have to show that N = i∈I Si ∈ S ∗ . Theorem 2.15 yields that i∈I Si P = i∈I Si,P ∈ ConL (T(X)) for every

198

4 Fuzzy Horn Logic

P ∈ P, since Si,P is a congruence for each i ∈ I. Hence, it remains to check that i∈I Si satisfies conditions (4.32)–(4.34). t ) for all (4.32): Let P ∈ P, t, t ∈ T (X). We have i ∈ I P (t, t ) ≤ S i,P (t, S since every Si is extensive. Thus, P (t, t ) ≤ i∈I Si,P (t, t ) = i∈I i P (t, t ), i.e. (4.32) is satisfied. (4.33): We can proceed analogously, let P ∈ P, t,t ∈ T (X)and let h be an endomorphism on T(X). Thus, Si,P (t, t ) ≤ Si,h(P ) h(t), h(t ) for each i ∈ I. That is, i∈I Si P (t, t ) = i∈I Si,P (t, t ) ≤ ≤ i∈I Si,h(P ) h(t), h(t ) = i∈I Si h(P ) h(t), h(t ) . (4.34): Take any P, Q ∈ P, and terms t, t ∈ T (X). Taking into account (1.19) and ∗ -monotony of every Si , it follows that ∗ = i∈I Si P (t, t ) ⊗ s,s ∈T (X) P (s, s ) → i∈I Si Q (s, s ) ∗ = = i∈I Si,P (t, t ) ⊗ s,s ∈T (X) P (s, s ) → j∈I Sj,Q (s, s ) ∗ = i∈I Si,P (t, t ) ⊗ j∈I s,s ∈T (X) P (s, s ) → Sj,Q (s, s ) = ∗ = i∈I Si,P (t, t ) ⊗ j∈I ≤ s,s ∈T (X) P (s, s ) → Sj,Q (s, s ) ∗ ≤ i,j∈I Si,P (t, t ) ⊗ s,s ∈T (X) P (s, s ) → Sj,Q (s, s ) ≤ ∗ ≤ i∈I Si,P (t, t ) ⊗ s,s ∈T (X) P (s, s ) → Si,Q (s, s ) ≤ Sk,Q (t, t ) for each k ∈ I. Hence, ∗ ≤ i∈I Si P (t, t ) ⊗ s,s ∈T (X) P (s, s ) → i∈I Si Q (s, s ) ≤ k∈I Sk,Q (t, t ) = i∈I Si Q (t, t ) , verifying (4.35) which is equivalent to (4.34).

As a direct consequence of Theorem 4.40, we obtain the following corollary. Corollary 4.41. For every L-set Σ of P-implications, and every implicational ∗ truth stresser ∗ , there P-indexed system of con exists a least L -implicational gruences SΣ = ΣP | P ∈ P , such that ΣP ⊆ ΣP for every P ∈ P. Thus, we define a semantic closure as follows. Definition 4.42. Let L∗ be a complete residuated lattice with implicational truth stresser, Σ be an L-set of P-implications. Then SΣ = ΣP | P ∈ P (see Corollary 4.41) is called a semantic closure of SΣ . The corresponding L-set Σ of P-implications is called a semantic closure of Σ. Finally, recalling (Impl(Mod(Σ)))(P i (t ≈ t )) = !P i (t ≈ t )!Σ , the following assertion shows that Σ = Impl(Mod(Σ)).


Theorem 4.43. Suppose Σ is an L-set of P-implications, and cational truth stresser. Then

∗

199

is an impli-

!P i (t ≈ t )!Σ = ΣP (t, t )

(4.40)

for every P ∈ P, and for all terms t, t ∈ T (X). Proof. “≤”: Previous results yield that Σ can be closed to Σ . Since Σ ⊆ Σ , we have Mod(Σ ) ⊆ Mod(Σ). Furthermore, SΣ is an L∗-implicational P |Q ∈ indexed system of congruences, thus Theorem 4.38 yields T(X)/ΣQ P ⊆ Mod(Σ ) since ΣP (t, t ) ≤ !P i (t ≈ t )!T(X)/Σ for every P, Q ∈ P, Q and t, t ∈ T (X). Thus, we have !P i (t ≈ t )!Σ = M ∈ Mod(Σ) !P i (t ≈ t )!M ≤ ≤ M ∈ Mod(Σ ) !P i (t ≈ t )!M ≤ ≤ Q ∈ P !P i (t ≈ t )!T(X)/Σ = ΣP (t, t ) . Q

“≥”: Take an arbitrary L-algebra M ∈ Mod(Σ). That is, for every P-implication P i (t ≈ t ) we have Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!M , whence Σ ⊆ Impl({M}). Moreover, Impl({M}) is an L∗ -implicational Ptheory (defined by the one element class K = {M}), and so SImpl({M}) is an L∗ -implicational P-indexed system of congruences (due to Theorem 4.37), containing SΣ . As SΣ is the least one containing SΣ , we get SΣ ≤ SImpl({M}) . As a consequence, ΣP (t, t ) ≤ Impl({M}) (P i (t ≈ t )) = !P i (t ≈ t )!M . Since M ∈ Mod(Σ) is arbitrary, we obtain ΣP (t, t ) ≤ M ∈ Mod(Σ) !P i (t ≈ t )!M = !P i (t ≈ t )!Σ proving “≥”. Thus, !P i (t ≈ t )!Σ = ΣP (t, t ) for each P i (t ≈ t ).

Remark 4.44. Theorem 4.43 provides an algebraic characterization of the degree of semantic entailment from L-sets of P-implications. In Sect. 4.3, we are going to shows under what conditions the degree of semantic entailment has a syntactic characterization. ∗

∗

∗

We are now going to investigate a relationship between L∗ -implicational Ptheories and their special subtheories determined by restrictions on P. Namely, we will be interested in a restriction on finiteness of every P ∈ P. Recall that for any proper family P of premises we can consider its restriction on finite premises PFin . PFin -implications, called P-Horn clauses, are P-implications which have finite L-sets of premises. The following definition introduces particular subtheories resulting from L∗ -implicational P-theories by a restriction of finite premises.

200

4 Fuzzy Horn Logic

Definition 4.45. Let Σ be an L∗ -implicational P-theory. An L-set ΣFin of PHorn clauses, where ΣFin (P i (t ≈ t )) = Σ(P i (t ≈ t )) for each P-Horn clause P i (t ≈ t ), is called an L∗ -implicational Horn subtheory of Σ. Remark 4.46. (1) For an L∗ -implicational P-theory Σ, the corresponding L∗ implicational Horn subtheory ΣFin is an L∗ -implicational PFin -theory. (2) In case of finite premises, the equivalent formulation (4.35) of condition (4.34) simplifies. Namely, if P = PFin , (4.35) simplifies to n ∗ ΣP (t, t ) ⊗ ≤ ΣQ (t, t ) , (4.41) i=1 P (ti , ti ) → ΣQ (ti , ti ) for every t, t ∈ T (X), P, Q ∈ P such that Supp(P ) ⊆ {t1 , t1 , . . . , tn , tn }. Now, for any L∗ -implicational P-theory Σ we have an L∗ -implicational PFin -theory ΣFin which is a restriction of Σ on P-implications with finitely many premises. Conversely, we may ask a question, whether for a given proper family of premises P and an L∗ -implicational PFin -theory Γ there is an L∗ implicational P-theory Σ such that Γ = ΣFin . The subsequent theorems show that Σ exists for any Γ provided that P and L∗ satisfy additional conditions. Before presenting the results, recall that by a finite restriction of P ∈ P we mean any finite L-relation P ∈ LT (X)×T (X) , where P (t, t ) > 0 implies P (t, t ) = P (t, t ) for all t, t ∈ T (X). The following definition introduces the closedness under finite restrictions. Definition 4.47. For any proper family P of premises and P ∈ P let Fin(P ) denote the set of all finite restrictions of P . P is said to be closed under finite restrictions if Fin(P ) ⊆ P for each P ∈ P. The following characterization is restricted to Horn truth stressers. Theorem 4.48. Let L∗ be a complete residuated lattice with a Horn truth stresser ∗ , let Γ be an L∗ -implicational PFin -theory, where P is a proper family of premises closed under finite restrictions. Then putting ΣP = P ∈ Fin(P ) ΓP , for P ∈ P , (4.42) Σ is an L∗ -implicational P-theory. Moreover, Σ is the least L∗ -implicational P-theory with Γ = ΣFin . Proof. Using Theorem 4.38, we need to check conditions (4.32)–(4.34) of SΣ , the fact that every ΣP is a congruence, and the fact that Σ is the least one with Γ = ΣFin . ΣP is a congruence: This is easy to see since the system of congruences {ΓP | P ∈ Fin(P )} is directed. Indeed, for every P1 , P2 ∈ Fin(P ), we have P1 , P2 ⊆ P1 ∪ P2 , and P1 ∪ P2 ∈ Fin(P ), thus by extensivity of SΓ , it follows that P1 , P2 ⊆ ΓP1 ∪ P2 , whence, by (4.34) ΓP1 , ΓP2 ⊆ ΓP1 ∪ P2 . This idea general)} is a directed izes easily for finitely many congruences, i.e. {ΓP | P ∈ Fin(P system of congruences. Hence, Lemma 2.68 yields that ΣP = P ∈ Fin(P ) ΓP is a congruence for every P ∈ P.


201

P , extensivity of SΓ gives P = P ∈ Fin(P ) P ⊆ P ∈ Fin(P ) ΓP = ΣP ,

(4.32): Since P =

P ∈ Fin(P )

verifying (4.32) for every P ∈ P. (4.33): For P ∈ Fin(P ), P ∈ P, and an endomorphism h on T(X) we have h(P ) ⊆ h(P ) and h(P ) is finite, i.e. we can take P ∈ Fin(h(P )) such that Supp(h(P )) = Supp(P ). Obviously, h(P ) ⊆ P . Thus, using (4.32) and (4.34) of SΓ we have Γh(P ) ⊆ ΓP . Hence, ΣP (t, t ) = P ∈ Fin(P ) ΓP (t, t ) ≤ P ∈ Fin(P ) Γh(P ) h(t), h(t ) ≤ ≤ P ∈ Fin(h(P )) ΓP h(t), h(t ) = Σh(P ) h(t), h(t ) proves (4.33) for SΣ . (4.34): Let P ∈ Fin(P ) with Supp(P ) = {ti , ti | i = 1, . . . , k}, P ∈ P, and t, t ∈ T (X). Since SΓ satisfies (4.34) equivalently formulated by (4.35), we have k ∗ ≤ ΓQ (t, t ) ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) for every Q ∈ Fin(Q), Q ∈ P. Hence, k ∗ ΓP (t, t ) ⊗ Q ∈ Fin(Q) ≤ i=1 P (ti , ti ) → ΓQ (ti , ti ) ≤ Q ∈ Fin(Q) ΓQ (t, t ) = ΣQ (t, t ) . Using (1.19), (1.20) we get ∗ k ΓP (t, t ) ⊗ Q ∈ Fin(Q) i=1 P (ti , ti ) → ΓQ (ti , ti ) ≤ ΣQ (t, t ) .

(4.43)

Since {ΓQ | Q ∈ Fin(Q)} is a directed system, we have k ∗ = i=1 Q ∈ Fin(Q) P (ti , ti ) → ΓQ (ti , ti ) ∗ k = Q ,...,Q ∈ Fin(Q) i=1 P (ti , ti ) → ΓQi (ti , ti ) = 1 k ∗ k = Q ∈ Fin(Q) i=1 P (ti , ti ) → ΓQ (ti , ti ) . Using (1.19), (1.20), (1.21), (4.43), and the previous equality we further get k ∗ = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΣQ (ti , ti ) k ∗ = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΣQ (ti , ti ) = ∗ k = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → Q ∈ Fin(Q) ΓQ (ti , ti ) = ∗ k = ΓP (t, t ) ⊗ i=1 Q ∈ Fin(Q) P (ti , ti ) → ΓQ (ti , ti ) = ∗ k = ΓP (t, t ) ⊗ Q ∈ Fin(Q) i=1 P (ti , ti ) → ΓQ (ti , ti ) ≤ ΣQ (t, t ) . Note that (1.21) can be used properly because Fin(Q) is always a non-empty set (e.g. ∅T (X)×T (X) ∈ Fin(Q)). Moreover, for every P ∈ Fin(P ), we have P (s, s ) = P (s, s ) for every s, s ∈ Supp(P ). Thus, using monotony of ∗ , we have

202

4 Fuzzy Horn Logic

∗ P (s, s ) → ΣQ (s, s ) ≤ ∗ k ≤ ΣQ (t, t ) ≤ ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΣQ (ti , ti )

ΓP (t, t ) ⊗

s,s ∈ T (X)

for any P ∈ Fin(P ). Hence, we have ∗ ΣP (t, t ) ⊗ = s,s ∈ T (X) P (s, s ) → ΣQ (s, s ) ∗ = P ∈ Fin(P ) ΓP (t, t ) ⊗ ≤ ΣQ (t, t ) , s,s ∈ T (X) P (s, s ) → ΣQ (s, s ) verifying (4.34). Altogether, SΣ is an L∗ -implicational P-indexed system of congruences, thus the corresponding Σ is an L∗ -implicational P-theory by Theorem 4.38. Σ is the least one with Γ = ΣFin : First, we check Γ = ΣFin . Take any P ∈ PFin . For every P ∈ Fin(P ), using (4.32), (4.34), we have P ⊆ ΓP , and consequently ΓP ⊆ ΓP . On the other hand, ΓP ⊆ P ∈ Fin(P ) ΓP holds trivially due to the fact P ∈ Fin(P ). Hence, ΣP = P ∈ Fin(P ) ΓP = ΓP for every P ∈ PFin , i.e. Γ = ΣFin . Suppose Σ is an L∗ -implicational P-theory such that Γ = Σ Fin . Take any P ∈ P. We will show that ΣP ⊆ ΣP . If P ∈ PFin , we are done, since ΣP = ΓP = ΣP as we have shown in the previous paragraph. For P ∈ PFin , and arbitrary P ∈ Fin(P ), it follows that P ⊆ P . Using (4.32) and (4.34) ΣP ⊆ ΣP . Thus, ΣP = ΓP = ΣP ⊆ ΣP for for SΣ , we have P ⊆ ΣP , and all P ∈ Fin(P ). Hence, ΣP = P ∈ Fin(P ) ΓP ⊆ ΣP . That is, Σ is the least L∗ -implicational P-theory for which Γ = ΣFin . If L is a Noetherian residuated lattice, we can avoid the usage of (1.21): Theorem 4.49. Let L∗ be a Noetherian residuated lattice with an implicational truth stresser ∗ satisfying (1.20), Γ be an L∗ -implicational PFin -theory, where P is a proper family of premises closed under finite restrictions, ΣP (P ∈ P) be defined by (4.42). Then Σ is an L∗ -implicational P-theory. Moreover, Σ is the least L∗ -implicational P-theory with Γ = ΣFin . Proof. We show only the critical part where (1.21) is used. Since L is a Noetherian residuated lattice, for each i = 1, . . . , k, there are {Q1 , . . . , Qm } ⊆ Fin(Q) such that ΣQ (ti , ti ) = Q ∈ Fin(Q) ΓQ (ti , ti ) = ΓQ1 (ti , ti ) ∨ · · · ∨ ΓQm (ti , ti ) . Furthermore, {ΓQ | Q ∈ Fin(Q)} is directed due to (4.32) and (4.34). Hence, there is Qi ∈ Fin(Q) such that ΓQ1 ∪ · · · ∪ ΓQm ⊆ ΓQi . Thus, ΣQ (ti , ti ) ≤ ΓQi (ti , ti ), i.e. ΣQ (ti , ti ) = ΓQi (ti , ti ) because the “≥”-part follows from (4.42). Therefore, for each i = 1, . . . , k there is Qi ∈ Fin(Q) where ΣQ (ti , ti ) = ΓQi (ti , ti ). This further yields k ∗ k ∗ i=1 (P (ti , ti ) → ΣQ (ti , ti )) = i=1 (P (ti , ti ) → ΓQi (ti , ti )) ≤ k ∗ ≤ i=1 Q ∈ Fin(Q) (P (ti , ti ) → ΓQ (ti , ti )) . The rest follows from the proof of Theorem 4.48.


203

Remark 4.50. In the ordinary case [97], fully invariant algebraic closure systems of congruences form the algebraic counterparts of Horn theories. Theorem 4.48 shows a condition analogous to algebraicity in our framework. Thus, we could define algebraic P-indexed systems of congruences as those SΣ , where Σ satisfies (4.42). For P = LT (X)×T (X) , this could yield fully invariant algebraic L∗ -closure systems of congruences. For L = 2, this would further yield fully invariant algebraic closure systems of congruences – the classical case. ∗

∗

∗

Let us close this section with a remark on the role of L∗ in the interpretation of P-implications. Notice that the truth degree !P i (t ≈ t )!M,v depends on both ∗ and →. Using → is clear (we deal with implications). We saw that the use of ∗ naturally unifies two possible meanings of P i (t ≈ t ). Clearly, if ∗ is globalization then !P i (t ≈ t )!M,v does not depend on →. The effect of → is completely displaced by ∗ . Surprisingly, an analogy applies also to general implicational truth stresser satisfying (1.20). Since !P i (t ≈ t )!M,v equals a∗ → !t ≈ t !M,v for a = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v , we are interested in truth degrees of a∗ → b. For a∗ ≤ b we have a∗ → b = 1. If a∗ b, a∗ → b is the greatest element of {c ∈ L | a∗ ⊗ c ≤ b}. Due to (1.20), a∗ → b is the greatest element of {c ∈ L | a∗ ∧ c ≤ b}. That is, a∗ → b is the relative pseudocomplement of a∗ to b. For instance, when L is a chain, then a∗ → b = b for a∗ > b. Note that even a = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v is defined using →. However, the following theorem shows that a∗ = !P !M,v is not influenced by the definition of →. As a consequence, we get that the truth degree !P i (t ≈ t )!M,v is (for given M, v) fully determined by the lattice structure of L and the implicational truth stresser ∗ satisfying (1.20). Theorem 4.51. Let L1 = L, ∨, ∧, ⊗1 , →1 , 0, 1 , L2 = L, ∨, ∧, ⊗2 , →2 , 0, 1 be complete residuated lattices. Let ∗ be an implicational truth stresser satis∗ ∗ fying (1.20) for both L1 and L2 . Then (a →1 b) = (a →2 b) for all a, b ∈ L. Proof. For each a ∈ L, let H(a) = {c∗ | c ∈ L and c∗ ≤ a}. We claim that H(a →1 b) = H(a →2 b)

(4.44)

for all a, b ∈ L. Indeed, for any a, b, c ∈ L we have c∗ ∈ H(a →1 b) iff c∗ ≤ a →1 b iff a ⊗1 c∗ ≤ b by adjointness, iff a ⊗2 c∗ = a ∧ c∗ = a ⊗1 c∗ ≤ b by (1.20), iff c∗ ≤ a →2 b by adjointness, iff c∗ ∈ H(a →2 b). Hence, (4.44) holds ∗ true for all a, b ∈ L. Clearly, (a →1 b) ≤ (a →1 b) due to (1.12). Thus, (4.44) ∗ ∗ yields (a →1 b) ∈ H(a →1 b) = H(a →2 b). That is, (a →1 b) ≤ a →2 b. ∗ Analogously, we have (a →2 b) ≤ a →1 b. Now using monotony of ∗ together ∗ ∗∗ ∗ ∗ with (1.17) we have (a →1 b) = (a →1 b) ≤ (a →2 b) and (a →2 b) = ∗∗ ∗ ∗ ∗ (a →2 b) ≤ (a →1 b) . Hence, (a →1 b) = (a →2 b) for any a, b ∈ L.

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4 Fuzzy Horn Logic

Now an easy inspection of (4.16) and (4.17) shows that !P i (t ≈ t )!M,v is not influenced by the definition of →. This is an immediate consequence of previous observations. A summary follows. Corollary 4.52. Let L1 = L, ∨, ∧, ⊗1 , →1 , 0, 1 , L2 = L, ∨, ∧, ⊗2 , →2 , 0, 1 be complete residuated lattices. Let ∗ be an implicational truth stresser satisfying (1.20) for both L1 and L2 . Let M be an Li -algebra (for both i = 1, 2) and v be a valuation on M. Then, L∗

L∗

1 2 !P i (t ≈ t )!M,v = !P i (t ≈ t )!M,v

for every P-implication P i (t ≈ t ).

Remark 4.53. One should not be mislead. Even though the effect of residuum is eliminated as shown in Corollary 4.52, multiplication and residuum in L∗ are still relevant. For instance, compatibility (1.83) is based on ⊗. Generally, there are L1 -algebras which are not L2 -algebras and vice versa, i.e. semantic ∗ ∗ entailments given by !· · ·!L1 and !· · ·!L2 do not coincide.

4.3 Completeness of Fuzzy Horn Logic In this section we are interested in soundness and completeness of fuzzy Horn logic. The soundness and completeness theorems will be proved as follows: We define the notion of a deductive closure of an L-set of P-Horn clauses and then we show that a deductive closure equals a semantic closure, which has already been introduced in Sect. 4.2. Then we focus on the relationship of a syntactic (deductive) closure and a provability degree. We prove that fuzzy Horn logic is Pavelka-sound and in many important cases it is Pavelka-complete. On the other hand, we show that fuzzy Horn logic is not Pavelka-complete for arbitrary L, ∗ , and P. To demonstrate the influence of the choice of L, ∗ , and P on completeness, we will pay attention to particular fuzzy Horn logics which arise by restrictions to particular subclasses of structures of truth degrees (Noetherian residuated lattices, residuated lattices on the real unit interval), or to particular families of proper premises (unrestricted premises, K-valent premises, crisp premises). We start with general results which cover all the important cases. Until otherwise mentioned, P denotes a proper family of premises satisfying P = PFin , i.e. each P ∈ P is finite. For such a P, any P-implication is a P-Horn clause. An L-set of P-Horn clauses will be denoted by Γ . Definition 4.54. Suppose Γ is an L-set of P-Horn clauses. A deductive closure of SΓ is the least P-indexed system SΓ of L-relations satisfying ΓP ⊆ ΓP , ΓP (t, t) ΓP (t, t )

(4.45)

=1, ≤

ΓP (t , t)

(4.46) ,

(4.47)

4.3 Completeness of Fuzzy Horn Logic

ΓP (t, t )X ⊗ ΓP (t , t ) ≤ ΓP (t, t ) ,

(4.48)

ΓP (t, t )

≤

(4.49)

ΓP (t, t ) ∗ ΓQ (ti , ti )

≤

ΓP (s, s ) , ΓP (t, t ) , ΓP(x/r) t(x/r), t (x/r)

≤

ΓQ (t, t )

P (t, t ) ≤

ΓP (t, t )

⊗

k

205

i=1 P (ti , ti )

→

,

(4.50) , (4.51) (4.52)

for every P, Q ∈ P, Supp(P ) = {ti , ti | i = 1, . . . , k}, x ∈ X, terms t, t , t , r, s, s ∈ T (X), where s contains t as a subterm and s results from s by replacing one occurrence of t by t . The corresponding L-set Γ of P-Horn clauses is called a deductive closure of Γ . Remark 4.55. (1) Observe that for a Horn truth stresser ∗ and finite P ∈ P, condition (4.52) coincides with (4.41). Indeed, this is a consequence of (1.19) and (1.20). (2) The deductive closure Γ of an L-set Γ of P-Horn clauses always exists since the system of all P-indexed systems of L-relations satisfying conditions (4.45)–(4.52) is non-empty and closed under arbitrary intersections. The proof is analogous to that of Theorem 4.40 and therefore omitted. We now present an auxiliary lemma that will be used in the sequel. For the sake of brevity, we denote substitutions by τ , τ1 , τ2 , . . . , and so on. Furthermore, instead of writing (· · · ((tτ1 )τ2 ) · · · )τn , we simply write tτ1 τ2 · · · τn . Similarly, we denote (· · · ((P τ1 )τ2 ) · · · )τn by P τ1 τ2 · · · τn . Lemma 4.56. Suppose P is a proper family of premises, and let us have substitutions τ1 , . . . , τk . Let τ denote τ1 · · · τk . We have, P τ (t, t ) = sτ = t P (s, s ) (4.53) s τ =t

for all terms t, t ∈ T (X). Proof. We can prove the claim using induction on the number of substitutions. If k = 1, i.e. τ = τ1 , the claim follows directly from (4.28). Suppose that the claim holds for each k − 1 substitutions and let us denote τ = τ1 · · · τk−1 . Using the induction hypothesis, we have sτ = u P (s, s ) = P τ τk (t, t ) = uτk = t P τ (u, u ) = uτk = t s τ = u u τk = t u τk = t = sτ τk = t P (s, s ) s τ τk = t

proving (4.53).

The following assertion shows that the semantic closure of Γ and the deductive closure of Γ coincide provided that we consider implicational truth stressers which satisfy (1.20).

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4 Fuzzy Horn Logic

Theorem 4.57. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ satisfying (1.20). Then for any L-set Γ of P-Horn clauses we have Γ = Γ . Proof. “⊆”: It is sufficient to show that SΓ is an L∗ -implicational P-indexed system of congruences. Then Γ ⊆ Γ since Γ is the least L∗ -implicational P-indexed system of congruences containing Γ . First, we check that every ΓP (P ∈ P) is a congruence on T(X). Reflexivity, symmetry, and transitivity of ΓP follows from (4.46)–(4.48). For compatibility with functions, we repeatedly use (4.49). Namely, for t1 , s1 , . . . , tn , sn ∈ T (X) and any n-ary f ∈ F (4.49) yields ΓP (ti , si ) ≤ ΓP f (s1 , . . . , si−1 , ti , ti+1 , . . . , tn ), f (s1 , . . . , si−1 , si , ti+1 , . . . , tn ) for each i = 1, . . . , n. Recall that in each i-th step, we have applied (4.49) correctly, because f (s1 , . . . , si−1 , ti , ti+1 , . . . , tn ) contains ti as a subterm and f (s1 , . . . , si−1 , si , ti+1 , . . . , tn ) results from it by replacing ti by si . Hence, by (4.48) and monotony of ⊗, ΓP (t1 , s1 ) ⊗ · · · ⊗ ΓP (tn , sn ) ≤ n ≤ i=1 ΓP f (s1 , . . . , si−1 , ti , . . . , tn ), f (s1 , . . . , si , ti+1 , . . . , tn ) ≤ ≤ ΓP f (t1 , . . . , tn ), f (s1 , . . . , sn ) which is the desired compatibility with functions. Clearly, ≈T(X) ⊆ ΓP . Thus, every ΓP is a congruence on T(X). Now we check (4.32)–(4.34). (4.32): Trivial, because of (4.50). (4.33): Take t, t ∈ T (X), P ∈ P, where Supp(P ) = {ti , ti | i = 1, . . . , k}. Since P is finite, a set of variables X = var(P )∪var(t)∪var(t ) = {x1 , . . . , xn } is finite as well. For arbitrary endomorphism h on T(X) we can take a set variables Y = {y1 , . . . , yn } such that yi ∈ X , yi ∈ var(h(x)) for each i = 1, . . . , n, and each variable x ∈ X . That is, each yi is different from variables occurring in terms t1 , t1 , . . . , tn , tn , t, t , and yi does not occur in any endomorphic image h(x) of a variable x ∈ X . Since X is assumed to be denumerable, we can always pick such yi ’s. Put τ = (x1 /y1 ) · · · (xn /yn )(y1 /h(x1 )) · · · (yn /h(xn )). First of all, we will show that for a term r ∈ T (X ) we have rτ = h(r). Indeed, we have r(x1 , . . . , xn )(x1 /y1 ) · · · (xn /yn ) = r(y1 , . . . , yn ) since none of yi ’s is in X . Moreover, since none of yi ’s has an occurrence in any endomorphic image h(xi ), i = 1, . . . , n, we directly obtain rτ = r(y1 , . . . , yn )(y1 /h(x1 )) · · · (yn /h(xn )) = = r h(x1 ), . . . , h(xn ) = h r(x1 , . . . , xn ) = h(r) . Applying this fact to t, t ∈ T (X ) and using (4.51) we obtain


207

ΓP (t, t ) ≤ ΓP(x1 /y1 ) t(x1 /y1 ), t (x1 /y1 ) ≤ ≤ ΓP(x1 /y1 )(x2 /y2 ) t(x1 /y1 )(x2 /y2 ), t (x1 /y1 )(x2 /y2 ) ≤ · · · ≤ ≤ ΓPτ (tτ, t τ ) = ΓPτ h(t), h(t ) . Clearly, now it suffices to show that h(P ) = P τ . Observe that P (r, r ) > 0 implies r, r ∈ T (X ) (X consists of all the variables occurring in couples of terms which belong to P in some nonzero degree). Thus, using Lemma 4.56, and the fact rτ = h(r) for r ∈ T (X ), it follows that P τ (s, s ) = rτ = s P (r, r ) = h(r) = s P (r, r ) = h(P ) (s, s ) r τ =s r,r ∈T (X )

h(r ) = s r,r ∈T (X )

for every s, s ∈ T (X). Hence, h(P ) = P τ , and so ΓP (t, t ) ≤ Γh(P ) h(t), h(t ) , proving (4.33). (4.34): For finite P ∈ P, we can express (4.34) equivalently by (4.41). Hence, using (1.19), (1.20), and (4.52) we have k ∗ ΓP (t, t ) ⊗ = i=1 P (ti , ti ) → ΓQ (ti , ti ) ∗ k = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) = ∗ k = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) ≤ ΓQ (t, t ) for every P, Q ∈ P, Supp(P ) = {ti , ti | i = 1, . . . , k} and any terms t, t ∈ T (X). Altogether, SΓ is an L∗ -implicational P-indexed system of congruences, showing “⊆”. “⊇”: We check that SΓ satisfies conditions (4.45)–(4.52). Since SΓ is the least P-indexed system of L-relations satisfying (4.45)–(4.52), we obtain Γ ⊆ Γ . Equation (4.45) holds trivially. Since every ΓP (P ∈ P) is a congruence, conditions (4.46)–(4.48) are satisfied obviously. (4.49): This property will be proved using the compatibility of every ΓP with functions. Let us have terms t, t , s, s ∈ T (X), where s has an occurrence of t as a subterm and s is a term resulting from s by substitution of t by t . If s = f (t1 , . . . , tk−1 , t, tk+1 , . . . , tn ) and s = f (t1 , . . . , tk−1 , t , tk+1 , . . . , tn ), compatibility of ΓP with f ∈ F and ΓP (ti , ti ) = 1 yield k−1 n ΓP (t, t ) = i = 1 ΓP (ti , ti ) ⊗ ΓP (t, t ) ⊗ j = k+1 ΓP (tj , tj ) ≤ ≤ ΓP f (t1 , . . . , tk−1 , t, tk+1 , . . . , tn ), f (t1 , . . . , tk−1 , t , tk+1 , . . . , tn ) = = ΓP (s, s ) . This argument can be used to show ΓP (t, t ) ≤ ΓP (s, s ) even in general case (one can proceed by structural induction over the rank of s). (4.50): Holds trivially because of (4.32). (4.51): Take a substitution (x/r) and a mapping g : X → T (X) defined by g(x) = r and g(y) = y for each y ∈ X with y = x. As in Remark 4.21,

208

4 Fuzzy Horn Logic

g has a homomorphic extension h : T(X) → T(X), i.e. h = g . Evidently, h(t) = t(x/r) for all terms t ∈ T (X). Moreover, P (x/r) (t, t ) = s(x/r) = t P (s, s ) = h(s) = t P (s, s ) = h(P ) (t, t ) . s (x/r) = t

h(s ) = t

Thus, (4.33) gives

ΓP (t, t ) ≤ Γh(P ) h(t), h(t ) = ΓP (x/r) t(x/r), t (x/r)

which is the required inequality. (4.52): Since SΓ satisfies (4.34), we can use the equivalent formulation for finite sets of premises (4.41) together with (1.19), (1.20). Hence, ∗ k ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) = ∗ k = ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) = k ∗ = ΓP (t, t ) ⊗ ≤ ΓQ (t, t ) i=1 P (ti , ti ) → ΓQ (ti , ti ) holds for every P, Q ∈ P, where Supp(P ) = {ti , ti | i = 1, . . . , k}, and for all t, t ∈ T (X). The following lemma shows that given any complete residuated lattice with truth stresser, a provability degree of a P-Horn clause from an L-set Γ of P-Horn clauses is smaller than or equal to a degree to which the P-Horn clause belongs to a deductive closure of Γ . Lemma 4.58. Let Γ be an L-set of P-Horn clauses. Then |P i (t ≈ t )|Γ ≤ ΓP (t, t )

(4.54)

for every P ∈ P and t, t ∈ T (X). Proof. It is sufficient to check that for every member P i (t ≈ t ), a of an L∗ -weighted proof from Γ , we have a ≤ ΓP (t, t ). We proceed by induction over the length of an L∗ -weighted proof. Suppose P i (t ≈ t ), a is a member of an L∗ -weighted proof from Γ where a = Γ (P i (t ≈ t )) = ΓP (t, t ). Using (4.45), we have a = ΓP (t, t ) ≤ ΓP (t, t ). Otherwise, P i (t ≈ t ), a was derived using one of (Ref)–(Mon). We check “≤” for all of these rules separately. (Ref): If P i (t ≈ t), 1 was inferred by (Ref), we directly obtain 1 ≤ ΓP (t, t) = 1 due to (4.46). (Sym): If P i (t ≈ t), a was inferred from P i (t ≈ t ), a , then induction hypothesis together with (4.47) give a ≤ ΓP (t, t ) ≤ ΓP (t , t). (Tra): Suppose P i (t ≈ t ), a ⊗ b was inferred from P i (t ≈ t ), a and P i (t ≈ t ), b . By induction hypothesis, a ≤ ΓP (t, t ), b ≤ ΓP (t , t ). Hence, (4.48) yields a ⊗ b ≤ ΓP (t, t ) ⊗ ΓP (t , t ) ≤ ΓP (t, t ). (Rep): If P i (s ≈ s ), a was inferred by (Rep) from P i (t ≈ t ), a , induction hypothesis and (4.49) give a ≤ ΓP (t, t ) ≤ ΓP (s, s ). (Ext): Apply (4.50).


209

(Sub): If P (x/r) i (t(x/r) ≈ t (x/r)), a results from P i (t ≈ t ), a , induction hypothesis and (4.51) yield a ≤ ΓP (t, t ) ≤ ΓP (x/r) t(x/r), t (x/r) . (Mon): Suppose Q i (t ≈ t ), c was inferred from P i (t ≈ t ), b and Q i (ti ≈ ti ), ai ’s with Supp(P ) = {ti , ti | i = 1, . . . , k}. Thus, we have ∗ k c = b ⊗ i=1 P (ti , ti ) → ai . Using the induction hypothesis, we have b ≤ ΓP (t, t ), and ai ≤ ΓQ (ti , ti ) for each i = 1, . . . , k, from which we obtain ∗ k c = b ⊗ i=1 P (ti , ti ) → ai ≤ ∗ k ≤ ΓP (t, t ) ⊗ i=1 P (ti , ti ) → ΓQ (ti , ti ) ≤ ΓQ (t, t )

by (1.32), (1.56), and (4.52). We are now ready to prove soundness of fuzzy Horn logic.

Theorem 4.59 (soundness). Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ satisfying (1.20). Let Γ be an L-set of P-Horn clauses. Then |P i (t ≈ t )|Γ ≤ ΓP (t, t ) = ΓP (t, t ) = !P i (t ≈ t )!Γ

(4.55)

for every P ∈ P and t, t ∈ T (X). Proof. Follows from Theorem 4.43, Theorem 4.57, and Lemma 4.58.

Remark 4.60. Notice that Theorem 4.59 yields that for any complete residuated lattice L we have at least one Pavelka-sound fuzzy Horn logic because any L can be endowed with a globalization on L which is an implicational truth stresser satisfying (1.20). Hence, soundness of fuzzy Horn logic is established without any restrictions on P or L. ∗

∗

∗

We now turn our attention to completeness. Due to the previous observations, we are interested in equality |P i (t ≈ t )|Γ = ΓP (t, t ). We first prove some properties of the provability degree |· · ·|Γ . Lemma 4.61. Let Γ be an L-set of P-Horn clauses. For any P-Horn clause P i (t ≈ t ) put DP (t, t ) = |P i (t ≈ t )|Γ . Then D = {DP | P ∈ P} is a P-indexed system of L-relations satisfying (4.45)–(4.51). Proof. (4.45): Since P i (t ≈ t ), ΓP (t, t ) is an L∗ -weighted proof of length 1, we have DP (t, t ) = |P i (t ≈ t )|Γ ≥ ΓP (t, t ), verifying (4.45). (4.46): Evidently, DP (t, t) = 1 since P i (t ≈ t), 1 is a proof by (Ref). (4.47): Using (Sym), each proof δi1 , . . . , δini, P i (t ≈ t ), ai can be extended to a proof δi1 , . . . , δini ,

1: P i (t ≈ t ), ai , 2: P i (t ≈ t), ai .

proof of P i (t ≈ t ), ai by (Sym) on 1

210

4 Fuzzy Horn Logic

Hence, DP (t, t ) = |P i (t ≈ t )|Γ ≤ |P i (t ≈t)|Γ =DP (t , t). (4.48): Let DP (t, t ) = i∈I ai and DP (t , t ) = j∈J bj where for each i ∈ I and j ∈ J there are proofs

δi1 , . . . , δini, P i (t ≈ t ), ai

and

δj1 , . . . , δjnj, P i (t ≈ t ), bj .

Concatenating the proofs and using (Tra) we get a proof δi1 , . . . , δini ,

1: P i (t ≈ t ), ai ,

proof of P i (t ≈ t ), ai

δj1 , . . . , δjnj ,

2: P i (t ≈ t ), bj , 3: P i (t ≈ t ), ai ⊗ bj .

proof of P i (t ≈ t ), bj by (Tra) on 1, 2

Hence, P i (t ≈ t ) is provable in degree at least ai ⊗ bj and so DP (t, t ) ⊗ DP (t , t ) = i∈I ai ⊗ j∈J bj = i∈I j∈J (ai ⊗ bj ) ≤ ≤ |P i (t ≈ t )|Γ = DP (t, t ) . That is, D satisfies (4.48). (4.49): For a term s resulting from s by substitution of one occurrence of t in s by t , every proof δi1 , . . . , δini, P i (t ≈ t ), ai can be extended by (Rep) to a proof δi1 , . . . , δini ,

1: P i (t ≈ t ), ai , 2: P i (s ≈ s ), ai .

proof of P i (t ≈ t ), ai by (Rep) on 1

It follows that DP (t, t ) = |P i (t ≈ t )|Γ ≤ |P i (s ≈ s )|Γ = DP (s, s ). (4.50): P (t, t ) ≤ |P i (t ≈ t )|Γ = DP (t, t ) since P i (t ≈ t ), P (t, t ) is a proof of length 1. (4.51): Let DP (t, t ) = i∈I ai where for each i ∈ I there is a proof δi1 , . . . , δini, P i (t ≈ t ), ai . Using (Sub), such proofs can be extended to δi1 , . . . , δini ,

1: P i (t ≈ t ), ai ,

2: P (x/r) i t(x/r) ≈ t (x/r) , ai . Hence, DP (t, t ) ≤ DP (x/r) t(x/r), t (x/r) .

proof of P i (t ≈ t ), ai by (Sub) on 1

Theorem 4.62. Let L∗ be a complete residuated lattice with a truth stresser ∗ , P be a proper family of premises where each P ∈ P is finite and let ∗ ∗ P (s, s ) → i∈I bi = i∈I (P (s, s ) → bi ) (4.56) for any bi ∈ L (i ∈ I = ∅), P ∈ P, and s, s ∈ T (X). Then |P i (t ≈ t )|Γ = ΓP (t, t ) for every P ∈ P and t, t ∈ T (X).

(4.57)


211

Proof. “≤”: See Lemma 4.58. “≥”: Consider D = {DP | P ∈ P}, where DP (t, t ) = |P i (t ≈ t )|Γ for any P ∈ P, and t, t ∈ T (X). It suffices to show that D satisfies (4.45)–(4.52) from which SΓ ≤ D follows immediately by definition of Γ . Observing that SΓ ≤ D means ΓP (t, t ) ≤ |P i (t ≈ t )|Γ , then finishes the proof. By Lemma 4.61, D satisfies conditions (4.45)–(4.51); it remains to check (4.52). Let P, Q ∈ P and Supp(P ) = {tk , tk | k = 1, . . . , n}. For any t, t ∈ T (X), we can write DP (t, t ) = i∈I bi , where for each i ∈ I there is a proof δi1 , . . . , δini, P i (t ≈ t ), bi . Furthermore, we can write DQ (tk , tk ) = jk ∈Jk ak,jk (k = 1, . . . , n), where for each jk ∈ Jk , there is a proof δjk ,1 , . . . , δjk ,njk, Q i (tk ≈ tk ), ak,jk . Now, we can take any i ∈ I, j1 ∈ J1 , . . . , jn ∈ Jn , concatenate the proofs, and apply (Mon) to get a proof δj1 ,1 , . . . , δj1 ,nj1 , 1: Q i (t1 ≈ t1 ), a1,j1 , .. .. .. . . . δjn ,1 , . . . , δjn ,njn ,

n: Q i (tn ≈ tn ), an,jn ,

proof of Q i (t1 ≈ t1 ), a1,j1 .. . proof of Q i (tn ≈ tn ), an,jn

δi1 , . . . , δini , n+1: P i (t ≈ t ), bi , proof of P i (t ≈ t ), bi ∗ n . (Mon) on 1, . . . , n + 1 n+2: Q i (t ≈ t ), bi ⊗ k=1 P (tk , tk ) → ak,jk Hence, we have DQ (t, t ) = |Q i (t ≈ t )|Γ ≥ bi ⊗

n k=1

∗

(P (tk , tk ) → ak,jk ) .

Since Q i (tk ≈ tk ), ΓQ (tk , tk ) is a proof, every Jk is non-empty. Thus, applying (4.56) we get ∗ n DP (t, t ) ⊗ k=1 P (tk , tk ) → DQ (tk , tk ) = ∗ n = i∈I bi ⊗ k=1 P (tk , tk ) → jk ∈Jk ak,jk = ∗ n = i∈I bi ⊗ k=1 jk ∈Jk P (tk , tk ) → ak,jk = ∗ n = i∈I bi ⊗ j1 ∈J1 ,...,jn ∈Jn k=1 P (tk , tk ) → ak,jk = ∗ n bi ⊗ k=1 P (tk , tk ) → ak,jk ≤ DQ (t, t ) , = i∈I j1 ∈J1 ,...,jn ∈Jn

which gives (4.52). The proof is finished.

Recall that the notions of a semantic and syntactic entailment are determined by L∗ and P. In the sequel, we present results on completeness dependent on various families P of premises, truth stressers ∗ , and complete residuated lattices L.

212

4 Fuzzy Horn Logic

General Families of Premises In this section we focus on P-Horn clauses for general families of premises. That is, we do not assume any special properties of P. In order to prove completeness, we restrict ourselves to particular structures of truth degrees. Theorem 4.63 (completeness for Horn truth stressers). Let L∗ be a complete residuated lattice with a Horn truth stresser ∗ , P be a proper family of premises. Then for any L-set Γ of P-Horn clauses we have |P i (t ≈ t )|Γ = ΓP (t, t ) = ΓP (t, t ) = !P i (t ≈ t )!Γ

(4.58)

for any P-Horn clause P i (t ≈ t ). Proof. Equation (4.56) is satisfied because it is a particularization of (1.21). Therefore, (4.58) follows from Theorem 4.43, Theorem 4.57, and Theorem 4.62. If L is a Noetherian residuated lattice, we can prove completeness of fuzzy Horn logic for any implicational truth stresser satisfying (1.20). That is, in this particular case we can avoid (1.21) which was required in Theorem 4.63. For technical reasons we introduce an additional deduction rule: P i (t ≈ t ), a , P i (t ≈ t ), b (Sup) : , P i (t ≈ t ), a ∨ b where P ∈ P, a, b ∈ L, and t, t ∈ T (X). Let R∨ denote the deductive system which results from the original system R by adding (Sup). The following lemma shows that if L∗ is a Noetherian residuated lattice then each degree of R provability |· · ·|Γ ∨ is fully described by a single proof from R∨ . Lemma 4.64. Let L∗ be a Noetherian residuated lattice with a truth stresser ∗ . Then for any L-set Γ of P-Horn clauses, R Γ R∨ P i (t ≈ t ), |P i (t ≈ t )|Γ ∨ for any P-Horn clause P i (t ≈ t ). Proof. Recall that {a ∈ L | Γ R∨ P i (t ≈ t ), a } is non-empty. Since L is a Noetherian residuated lattice, there are truth degrees a1 , . . . , ak ∈ L such that R Γ R∨ P i (t ≈ t ), ai (i = 1, . . . , k), and |P i (t ≈ t )|Γ ∨ = a1 ∨ · · · ∨ ak . Thus, for each i = 1, . . . , k there is a proof δi,1 , . . . , δi,ni , P i (t ≈ t ), ai from Γ . We can concatenate these proofs and repeatedly use (Sup) to get a R proof of P i (t ≈ t ), a1 ∨ · · · ∨ ak = |P i (t ≈ t )|Γ ∨ from Γ . Theorem 4.65 (completeness for Noetherian residuated lattices). Let L∗ be a Noetherian residuated lattice with an implicational truth stresser ∗ satisfying (1.20), P be a proper family of premises. Then for any L-set Γ of P-Horn clauses we have R

|P i (t ≈ t )|Γ ∨ = !P i (t ≈ t )!Γ for any P-Horn clause P i (t ≈ t ).


213

Proof. Using Theorem 4.43 and Theorem 4.57, we only check that R

|P i (t ≈ t )|Γ ∨ = ΓP (t, t ) . “≤”: If P i (t ≈ t ), a ∨ b was inferred from weighted P-Horn clauses P i (t ≈ t ), a and P i (t ≈ t ), b using (Sup), then assuming a ≤ ΓP (t, t ) and b ≤ ΓP (t, t ), we have a ∨ b ≤ ΓP (t, t ). The rest follows from Lemma 4.58. R “≥”: Put DP (t, t ) = |P i (t ≈ t )|Γ ∨ for any P-Horn clause P i (t ≈ t ). We check that D = {DP | P ∈ PFin } satisfies (4.52). Suppose P, Q ∈ PFin and Supp(P ) = {tk , tk | k = 1, . . . , n}. Take t, t ∈ T (X). Let DP (t, t ) = b, and DQ (tk , tk ) = ak (k = 1, . . . , n). Lemma 4.64 yields that there are proofs δ1 , . . . , δn ,P i (t ≈ t ), b ,

and

δk,1 , . . . , δk,nk ,Q i (tk ≈ tk ), ak

for each k = 1, . . . , n. Concatenating the proofs and applying (Mon), we get ∗ n DP (t, t ) ⊗ k=1 P (tk , tk ) → DQ (tk , tk ) = ∗ n R = b ⊗ k=1 P (tk , tk ) → ak ≤ |Q i (t ≈ t )|Γ ∨ = DQ (t, t ) . Hence, D satisfies (4.52); Lemma 4.61 gives that D satisfies (4.45)–(4.52). This R yields ΓP (t, t ) ≤ DP (t, t ) = |P i (t ≈ t )|Γ ∨ . Example 4.66. The following are examples of L∗ for which we have a complete fuzzy Horn logic. (1) If L is a Noetherian residuated lattice then globalization defined on L is an implicational truth stresser satisfying (1.20) since 1 ⊗ a = 1 ∧ a, and 0 ⊗ a = 0 ∧ a. The usage of globalization as a truth stresser has an important influence on the deduction rule (Mon). If P (ti , ti ) ai for some i ∈ I, then the resulting formula Q i (t ≈ t ) is inferred in degree 0 (not interesting). On the other hand, when P (ti , ti ) ≤ ai for all i ∈ I, Q i (t ≈ t ) is inferred in degree b. To sum up, for ∗ being the globalization, (Mon) simplifies to (BMon) :

{Q i (ti ≈ ti ), ai ; i = 1, . . . , n} , P i (t ≈ t ), b Q i (t ≈ t ), b

if Supp(P ) = {t1 , t1 , . . . ,tn , tn }, and P (ti , ti ) ≤ ai for each i = 1, . . . , n. (2) Consider a subalgebra L of the standard product algebra with universe L = { 21n | n ∈ N0 } ∪ {0}. Since 1 if 21n ≤ 21m , 1 1 1 1 1 ⊗ = , → = n m m+n n m 1 2 2 2 2 2 otherwise , 2m−n L is closed under ⊗ and →. L is a Noetherian residuated lattice which can be endowed with globalization – a particular case of (1). (3) Take a Noetherian residuated lattice L such that ⊗ = ∧. That is, L is a complete Heyting algebra such that the lattice part of L is a Noetherian lattice. We can define ∗ by a∗ = a (a ∈ L). Trivially, ∗ is an implicational truth stresser; (1.20) is satisfied since ⊗ = ∧. (4) Consider L = { n1 | n ∈ N} ∪ {0} endowed with ∧ and ∨ being minimum and maximum, respectively; ⊗ = ∧, a → b = 1 if a ≤ b, a → b = b else. L =

214

4 Fuzzy Horn Logic

L, ∧, ∨, ⊗, →, 0, 1 is a Noetherian residuated lattice which is a subalgebra of the standard G¨ odel algebra. Such an L with globalization or identity is a particular case of (1) or (3), respectively. (5) Globalization and identity (on a Heyting algebra) can be seen as two extreme cases of implicational truth stressers satisfying (1.20). There are, of course, other examples of implicational truth stressers with this property. Suppose L is a finite BL-chain. Prelinearity and finiteness of L imply (1.59). Due to Theorem 1.52, we can define ∗ on L by (1.60), i.e. ∗ sends each element a ∈ L to the greatest idempotent of L which is less or equal to a. In addition to that, (1.20) follows from divisibility. Hence ∗ is a Horn truth stresser for L. (6) Take L1 from example (2) and L2 from example (4). Let ∗1 be globalization on L1 and ∗2 be identity on L2 . Let L∗ be L∗11 × L∗22 . Then ∗ is an implicational truth stresser on L which satisfies (1.20). Furthermore, L∗11 ×L∗22 is a Noetherian lattice: take {ai , bi ∈ L1 × L2 | i ∈ I} = ∅ since both L1 and lattices, there are a1 , . . . , am ∈ L1 , and b1 , . . . , bn ∈ L2 L2 are Noetherian such that i∈I ai = a1 ∨ · · · ∨ am , and i∈I bi = b1 ∨ · · · ∨ bn . Thus, for a finite set {ai , bj | i = 1, . . . , m and j = 1, . . . , n} we have i∈I ai , bi = i∈I ai , i∈I bi = = a1 ∨ · · · ∨ am , b1 ∨ · · · ∨ bn = a1 , b1 ∨ · · · ∨ am , bn . To sum up, L is a non-linear infinite Noetherian lattice with implicational truth stresser ∗ satisfying (1.20). Notice that ∗ is neither globalization nor identity. (7) Consider a non-linear residuated lattice from Example 1.9 (page 13), see Fig. 1.2. The left-most truth stresser in Fig. 1.3 (page 16) is an implicational truth stresser satisfying (1.20). Continuous Fuzzy Horn Logic In the previous section, we proved completeness over structures of truth degrees equipped with Horn truth stressers and over all Noetherian residuated lattices. These classes of structures of truth degrees do not include, in general, structures defined on the real unit interval. For instance, for L being the standard L ukasiewicz algebra, there is no Horn truth stresser. In addition to that, the following example shows that in case of L = [0, 1], fuzzy Horn logic is not Pavelka-style complete for general proper families of premises even if we use globalization as the truth stresser. In this section we are going to show that, considering only particular theories which obey some form of continuity, fuzzy Horn logic is Pavelka-style complete for any complete residuated lattice on [0, 1] given by a left-continuous t-norm. Example 4.67. Let L∗ be a complete residuated lattice on [0, 1] given by a leftcontinuous t-norm, ∗ be the globalization on [0, 1]. Take P = LT (X)×T (X) and consider an L-set Γ of P-Horn clauses such that


Γ (x ≈ y, 0.5 i (x ≈ y)) = 1 n Γ (∅ i (xn ≈ yn )) = 2n+1 Γ (· · ·) = 0

215

for fixed x, y ∈ X , for each n ∈ N , otherwise .

n Described verbally, Γ contains identities xn ≈ yn in degrees 2n+1 , and a single P-Horn clause with non-empty premises: x ≈ y, 0.5 i (x ≈ y) (this P-Horn clause belongs to Γ in degree 1, i.e. it fully belongs to Γ ). Take any model M ∈ Mod(Γ ). Clearly,

!x ≈ y!M = !xn ≈ yn !M ≥

n 2n+1

for each n ∈ N, i.e. !x ≈ y!M ≥ 0.5. Since !x ≈ y, 0.5 i (x ≈ y)!M = 1, for any valuation v : X → M we get ∗

!x ≈ y!M,v = 1 → !x ≈ y!M,v = (0.5 → !x ≈ y!M,v ) → !x ≈ y!M,v = = !x ≈ y, 0.5 i (x ≈ y)!M,v = 1 . Hence, !x ≈ y!M = 1 for any model M ∈ Mod(Γ ). This yields !x ≈ y!Γ = 1. Theorem 4.43 and Theorem 4.57 thus give Γ∅ (x, y) = 1. We now show that the provability degree |x ≈ y|Γ is strictly lower than 1. n (n ∈ N). On Evidently, |x ≈ y|Γ ≥ 0.5 due to (Sub) applied on xn ≈ yn , 2n+1 R the other hand, if Γ x ≈ y, a then a < 0.5. Indeed, if Γ R x ≈ y, a then there is a finite Γ ⊆ Γ such that Γ R x ≈ y, a because in any weighted proof from R one can use only finitely many weighted P-Horn clauses of the form Pi (t ≈ t ), Γ(P i (t ≈ t )) . For Γ we can consider an Lalgebra M = M, ≈M , F M with M = {a , b }; functions f M ∈ F M are m for some m ∈ N satisfying Γ (xn ≈ defined by f M (· · ·) = a ; a ≈M b = 2m+1 yn ) = 0 for all n ≥ m (m always exists since Γ is finite). Then !xn ≈ yn !M =

m 2m+1

≥ Γ (xn ≈ yn )

for each n ∈ N, and !x ≈ y, 0.5 i (x ≈ y)!M,v = 1 for any v because either m < 0.5 and so !x!M,v = !y!M,v , or !x ≈ y!M,v = 2m+1 ∗

!x ≈ y, 0.5 i (x ≈ y)!M,v = (0.5 → !x ≈ y!M,v ) → !x ≈ y!M,v = = (0.5 →

∗ m 2m+1 )

→ !x ≈ y!M,v =

= 0 → !x ≈ y!M,v = 1 because

∗

is the globalization on [0, 1]. Thus,

1 = !x ≈ y, 0.5 i (x ≈ y)!M ≥ Γ (x ≈ y, 0.5 i (x ≈ y)) . m As a consequence, M ∈ Mod(Γ ), and !x ≈ y!M = 2m+1 < 0.5. Since fuzzy Horn logic is sound, we get a ≤ |x ≈ y|Γ < 0.5. So, we have shown that if Γ R x ≈ y, a then a < 0.5. Using this observation, we readily obtain |x ≈ y|Γ = {a ∈ L | Γ R x ≈ y, a } ≤ {a ∈ L | a < 0.5} = 0.5 .

Therefore, |x ≈ y|Γ = 0.5 < Γ∅ (x, y) = !x ≈ y!Γ = 1. So, in this particular case, fuzzy Horn logic is Pavelka-sound but it is not Pavelka-complete.

216

4 Fuzzy Horn Logic

For technical reasons, we define a type of continuity of an L-set of P-Horn clauses using an arbitrary but fixed strict continuous Archimedean t-norm. Definition 4.68. Let L be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let " be a strict continuous Archimedean t-norm. For P ∈ LT (X)×T (X) and d ∈ [0, 1) we define d"P ∈ LT (X)×T (X) by (d"P )(r, r ) = d " P (r, r )

(4.59)

for all r, r ∈ T (X). Remark 4.69. (1) Note that in general, " has nothing to do with ⊗. We just use two conjunctions simultaneously: the left-continuous t-norm ⊗ determines the structure of truth degrees L while the strict continuous t-norm " determines the notion of "-continuity. (2) Note that if P i (t ≈ t ) is a P-Horn clause for P = LT (X)×T (X) then so is d"P i (t ≈ t ) because d"P ∈ LT (X)×T (X) is finite. On the other hand, there are proper families of premises such that for some P ∈ P and d ∈ [0, 1), we have d"P ∈ P. This is the case of e.g. families of crisp premises. From this moment on, we assume P = LT (X)×T (X) ; L is a complete residuated lattice on the real unit interval given by a left-continuous t-norm; ∗ is the globalization on [0, 1]. Definition 4.70. An L-set Γ of P-Horn clauses is called "-continuous if for each P-Horn clause P i (t ≈ t ) and each e ∈ [0, 1) there is d ∈ [0, 1) such that IF Γ R P i (t ≈ t ), b , THEN there is be ∈ [e " b, 1] such that Γ R d"P i (t ≈ t ), be . For "-continuous L-sets we have the following characterization. Theorem 4.71 (completeness of continuous FHL). Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1], " be a strict continuous Archimedean t-norm, P = LT (X)×T (X) . Then for any "-continuous L-set Γ of P-Horn clauses, |P i (t ≈ t )|Γ = !P i (t ≈ t )!Γ for each P-Horn clause P i (t ≈ t ). Proof. According to Theorem 4.59, we check that ΓP (t, t ) ≤ |P i (t ≈ t )|Γ . Put DP (t, t ) = |P i (t ≈ t )|Γ for every P ∈ P, and t, t ∈ T (X). We check that D = {DP | P ∈ P} satisfies (4.45)–(4.52). We focus only on (4.52), the rest follows from Lemma 4.61. Since ∗ is globalization, it suffices to show that IF P (s, s ) ≤ DQ (s, s )

for all s, s ∈ T (X) such that P (s, s ) > 0,

THEN DP (t, t ) ≤ DQ (t, t ) .


217

So, let P (s, s ) ≤ DQ (s, s ) for any s, s ∈ T (X) such that P (s, s ) > 0 and let Γ R P i (t ≈ t ), b . Since Γ is "-continuous, for each e ∈ [0, 1) there is d ∈ [0, 1) such that we have Γ R d"P i (t ≈ t ), be for some be ≥ e " b. Furthermore, if P (s, s ) > 0 then (d"P )(s, s ) < P (s, s ) ≤ DQ (s, s ) = |Q i (s ≈ s )|Γ . Therefore, for any s, s ∈ T (X) satisfying P (s, s ) > 0 we have Γ R Q i (s ≈ s ), as,s , where as,s ≥ (d"P )(s, s ). Thus, applying (Mon) on {Q i (s ≈ s ), as,s | P (s, s ) > 0} and d"P i (t ≈ t ), be , we get

Γ R Q i (t ≈ t ), be .

That is, for each e ∈ [0, 1), we have Γ R Q i (t ≈ t ), be where b " e ≤ be . So, for each e ∈ [0, 1), b " e ≤ |Q i (t ≈ t )|Γ = DQ (t, t ), i.e. b ≤ DQ (t, t ) by Lemma 1.35 (i). Hence, we immediately obtain DP (t, t ) ≤ DQ (t, t ). Example 4.72. (1) Let L be a residuated lattice on [0, 1] with ⊗ = ∧. One can prove by induction on the length of a proof that if for each d ∈ [0, 1) and P i (t ≈ t ) we have d " Γ (P i (t ≈ t )) ≤ Γ (d"P i (t ≈ t )), then Γ is "-continuous. Indeed, by induction on length of a proof we can prove IF Γ R P i (t ≈ t ), b , THEN there is bd ∈ [d " b, 1] such that Γ R d"P i (t ≈ t ), bd for each d ∈ [0, 1) from which the "-continuity of Γ follows immediately. Let P i (t ≈ t ), b be a member of a proof from Γ using R. If b = Γ (P i (t ≈ t )) then Γ R d"P i (t ≈ t ), bd for bd = Γ (d"P i (t ≈ t )) ≥ d " Γ (P i (t ≈ t )) = d " b , i.e. bd ∈ [d"b, 1]. The claim is routine to check if P i (t ≈ t ), b was inferred using (Ref), (Sym), (Rep), (Ext), or (Mon). If P i (t ≈ t ), b was inferred from P i (t ≈ s), a and P i (s ≈ t ), a by (Tra) then b = a⊗a ; by induction hypothesis, Γ R d"P i (t ≈ s), ad and Γ R d"P i (s ≈ t ), ad where ad ≥ d " a and ad ≥ d " a . Therefore, one can use (Tra) to conclude that Γ R d"P i (t ≈ t ), ad ⊗ ad where ad ⊗ ad = ad ∧ ad ≥ (d " a) ∧ (d " a ) = d " (a ∧ a ) = d " (a ⊗ a ) , i.e. ad ⊗ ad ∈ [d " (a ⊗ a ), 1]. If P i (t ≈ t ), b results from Q i (s ≈ s ), b by (Sub) then there is a substitution (x/r) such that P = Q(x/r), t = s(x/r), and t = s (x/r). Observe that (d"Q)(x/r) (t, t ) = s(x/r) = t (d"Q)(s, s ) = d " s(x/r) = t Q(s, s ) = s (x/r) = t

= d " Q(x/r)(t, t )

s (x/r) = t

218

4 Fuzzy Horn Logic

for any t, t ∈ T (X). Therefore, we have (d"Q)(x/r) = d"(Q(x/r)). By induction hypothesis, Γ R d"Q i (s ≈ s ), bd , where bd ∈ [d " b, 1]. Hence, using (Sub) we get Γ R (d"Q)(x/r) i (s(x/r) ≈ s (x/r)), bd from which it follows that Γ R d"(Q(x/r)) i (s(x/r) ≈ s (x/r)), bd . As a consequence, Γ is "-continuous. (2) If ⊗ = ∧ then we can use (1) to get that each L-set Γ of P-Horn clauses where Γ (P i (t ≈ t )) = 0 if P = ∅ is "-continuous. (3) Let F be a type which consists of a single binary function symbol f , Γ be an L-set of P-Horn clauses, where Γ (x ◦ z ≈ y ◦ z, a i x ≈ y) = a for each a ∈ [0, 1], and Γ (P i (t ≈ t )) = 0 otherwise. If ⊗ = ∧, (1) gives that Γ is "-continuous. Γ can be seen as a theory of grupoids (understood as L-algebras) satisfying a particular type of continuous cancellation. (4) There are, of course, L-sets of P-Horn clauses which are not "continuous. For instance, consider Γ such that Γ (x ≈ y, 0.5 i (x ≈ y)) = 1, and Γ (P i (t ≈ t )) = 0 otherwise. For e = 0.9 there is not any d ∈ [0, 1) such that Γ R x ≈ y, d " 0.5 i (x ≈ y), be for be ≥ e" 1 = 0.9, because for each d ∈ [0, 1) we can consider an L-algebra M = M, ≈M , ∅ (of the empty type) with M = {a , b } and a ≈M b = d " 0.5 which is a model of such Γ but for a valuation v : X → M with v(x) = a and v(y) = b we have !x ≈ y, d " 0.5 i (x ≈ y)!M,v = d " 0.5 0.9 = be . Therefore, the soundness of fuzzy Horn logic (see Theorem 4.59) gives |x ≈ y, d " 0.5 i (x ≈ y)|Σ < be , meaning that x ≈ y, d " 0.5 i (x ≈ y) is not provable from Γ using R in degree at least be . As a consequence, Γ is not "-continuous. Families with Restricted Premises So far, we were considering P-Horn clauses without additional restrictions on P (sometimes, we used P = LT (X)×T (X) ). From now on, we will restrict ourselves to P-Horn clauses with P being a particular (restricted) proper family of premises. Our motivation is basically to show that restricting ourselves to simpler families of premises we can obtain Pavelka-style complete fuzzy Horn logic for wider classes of structures of truth degrees. We will be interested mainly in families of crisp premises, i.e. families where every P ∈ P is crisp. Notice that crisp premises naturally correspond with premises of ordinary Horn clauses. Later on, we comment on a way to generalize this result for P-Horn clauses with K-valent premises. In the sequel, we let P be defined by P = {P ∈ LT (X)×T (X) | P is finite and crisp} . Recall that in such a case, the notion of an endomorphic image coincides with the classical one, i.e. h(P ) = {h(t), h(t ) | t, t ∈ P } for every P ∈ P and arbitrary endomorphism h on T(X). Since every P ∈ P is crisp, we either


219

have P (s, s ) = 1, or P (s, s ) = 0. Thus, the degree to which a P-Horn clause P i (t ≈ t ) is true in an L-algebra M under a valuation v simplifies to ∗ !P i (t ≈ t )!M,v = → !t ≈ t !M,v . s,s ∈P !s ≈ s !M,v Using Corollary 4.41, and Theorem 4.43, we can describe a semantic closure of SΓ , where Γ is any L-set of P-Horn clauses with crisp premises. As we have shown before, the semantic closure SΓ of SΓ is the least P-indexed system of congruences satisfying (4.32)–(4.34) such that SΓ ≤ SΓ . For crisp premises, conditions (4.32) and (4.34) simplify to t, t ∈ P implies ΓP (t, t ) = 1 , ∗ ≤ S(ΓP , ΓQ ) . s,s ∈P ΓQ (s, s ) Moreover, (4.61) is equivalent to ∗ ≤ ΓQ (t, t ) ΓP (t, t ) ⊗ s,s ∈P ΓQ (s, s )

(4.60) (4.61)

(4.62)

which is required to be true for any P, Q ∈ P and t, t ∈ T (X). Now we turn our attention to the notion of a provability degree. In the case of crisp premises, properties (4.50) and (4.52) of a deductive closure SΓ can be equivalently reformulated: t, t ∈ P implies ΓP (t, t ) = 1 , ∗ ΓP (t, t ) ⊗ s,s ∈P ΓQ (s, s ) ≤ ΓQ (t, t ) ,

(4.63) (4.64)

where P, Q ∈ P, and t, t ∈ T (X). In order to prove Γ = Γ , it is sufficient to assume L∗ to be a complete residuated lattice equipped with an implicational truth stresser ∗ satisfying (1.20), see Theorem 4.57. So, it remains to connect the deductive closure with degree of provability. Consider deductive system R as it has been introduced in Sect. 4.1. (Ext) and (Mon) simplify to (CExt) : P i (t ≈ t ), 1 (CMon) :

for every t, t ∈ P ,

{Q i (ti ≈ ti ), ai ; i = 1, . . . , n}, P i (t ≈ t ), b , Q i (t ≈ t ), b ⊗ a∗1 ⊗ · · · ⊗ a∗n

where P, Q are crisp sets of premises, P = {t1 , t1 , . . . , tn , tn }, t, t ∈ T (X), and a1 , . . . , an , b ∈ L. The following assertion establishes a connection between a deductive closure and a degree of provability. Theorem 4.73. Let Γ be an L-set of P-Horn clauses such that each P ∈ P is crisp. If ∗ a truth stresser satisfying (1.62) then |P i (t ≈ t )|Γ = ΓP (t, t )

(4.65)

for each P-Horn clause P i (t ≈ t ). Proof. “≤”: Apply Lemma 4.58. “≥”: Since each P ∈ P is crisp, P (s, s ) ∈ {0, 1} for any s, s ∈ T (X). If P (s, s ) = 0 then (4.56) is satisfied trivially. If P (s, s ) = 1, we get

220

4 Fuzzy Horn Logic

P (s, s ) →

i∈I

bi

∗

=

i∈I

bi

∗

=

i∈I

b∗i =

i∈I

∗

(P (s, s ) → bi )

on account of (1.62). Now apply Theorem 4.62.

Summarizing previous observations, we get the following Corollary 4.74 (completeness for crisp premises). Let L∗ be a complete residuated lattice equipped with an implicational truth stresser ∗ satisfying (1.20), (1.62), P be a proper family of premises such that each P ∈ P is crisp. For every L-set Γ of P-Horn clauses, |P i (t ≈ t )|Γ = ΓP (t, t ) = ΓP (t, t ) = !P i (t ≈ t )!Γ for any P-Horn clause P i (t ≈ t )

Example 4.75. The following are examples of L∗ for which we have a complete fuzzy Horn logic with crisp premises. (1) Let L be any complete residuated lattice such that 1 (the greatest element of L) is ∨-irreducible. Then, globalization defined on a complete residuated lattice with ∨-irreducible 1 is an implicational truth stresser satisfying (1.20) and (1.62), see Example 1.57 (2). (2) For any complete residuated lattice L we can consider L ⊕ 2. In fact, L ⊕ 2 results from L by equipping it with a new top element, see Fig. 1.6, which is then ∨-irreducible. That is, L ⊕ 2 with globalization is a particular case of (1). The new complete residuated lattice L ⊕ 2 is “rather similar” to the starting one. (3) Complete residuated lattices with implicational truth stressers satisfying (1.20) and (1.62) can be ordinally added yielding new structures with nontrivial implicational truth stressers satisfying (1.20) and (1.62), see Theorem 1.60 and Fig. 1.7. Remark 4.76. Let us discuss some epistemic impacts of adding a new top element to a complete residuated lattice. First, we may ask about the nature of the structures with ∨-irreducible greatest element. The top element of a complete residuated lattice represents full truth, while other elements can be thought of as degrees of partial truths, but not the full truth. From this point of view, it might be not natural to allow the fully true statement to be obtained from partially true statements. On the level of the structure of truth degrees, this corresponds to ∨-irreducibility of the greatest element, i.e. 1 cannot be obtained as a supremum of truth degrees which differ from 1. From the point of view of Pavelka-style logic, if we use L with ∨-irreducible 1 as the structure of truth degrees, we have to find a weighted proof of ϕ, 1 in order to show that the provability degree of ϕ equals 1. In other words, |ϕ|R Γ = 1 implies that there is a weighted proof of ϕ, 1 from Γ using R. Remark 4.77. Obviously, if each P ∈ P is crisp and if Γ is a crisp set of P-Horn clauses then Γ can be seen as an ordinary set of the classical Horn clauses. Furthermore, (Ref)–(Mon) collapse to the deduction rules of the classical Horn

4.4 Fuzzy Equational Logic Revisited

221

logic [97]. It is then easily seen that Γ R P i (t ≈ t ), a yields a ∈ {0, 1} and, more importantly, |P i (t ≈ t )|Γ = 1 iff P i (t ≈ t ) is provable from Γ in the classical sense, see [97]. So, analogously as in the comment on page 146, in case of crisp sets of P-Horn clauses with crisp premises, the notion of graded provability can be fully described by the classical one. ∗

∗

∗

The idea of having crisp premises can be naturally generalized. Let L∗ be a complete residuated lattice with implicational truth stresser satisfying (1.20). Take K ⊆ L such that {0, 1} ⊆ K. A proper family of premises P can be called K-valent if {P (s, s ) | s, s ∈ T (X)} ⊆ K for each P ∈ P. Obviously, if P is {0, 1}-valent then each P ∈ P is crisp. Now, if L∗ satisfies (1.21) for any clause due to a ∈ K, then |P i (t ≈ t )|Γ = !P i (t ≈ t )!Γ for any P-Horn Theorem 4.43, Theorem 4.57, and Theorem 4.62. For instance, i∈I (Li ⊕ 2) on (see Fig. 1.7), equipped with ∗ which is an ordinal sum of globalizations (i ∈ I), satisfies (1.21) for any a ∈ K, where K = {00 } ∪ 1i | i ∈ I . Li ⊕ 2 So, for i∈I (Li ⊕ 2) we have Pavelka-style complete fuzzy Horn logic which uses P-Horn clauses with K-valent premises.

4.4 Fuzzy Equational Logic Revisited The main aim of this section is to look again at fuzzy equational logic which has been introduced in Chap. 3. We are going to show how the results of fuzzy Horn logic generalize the results of fuzzy equational logic. First, we show that the completeness of fuzzy equational logic can be easily derived from our results on fuzzy Horn logic. Recall that if we deal with identities, we can put P = {∅}. In this case, P-implications are exactly formulas of the form ∅ i (t ≈ t ), i.e. P-implications with empty premises. We have !∅ i (t ≈ t )!M,v = !t ≈ t !M,v for any M and v. Hence, from the point of view of interpretation of formulas, we can identify t ≈ t with ∅ i (t ≈ t ), see Remark 4.10 (4). In Sect. 3.2, we introduced fully invariant congruences as algebraic counterparts of semantically closed L-sets of identities. Now, in the context of fuzzy Horn logic, we can consider SΣ , which is a one-element system SΣ = {Σ∅ }, where Σ∅ = Σ. Hence, due to Corollary 4.41 and 4.43, Theorem the semantic closure of SΣ is a one-element system SΣ = Σ∅ , where Σ∅ is the least congruence on T(X) which contains Σ and SΣ satisfies (4.32)– (4.34). Equation (4.32), i.e. ∅ ⊆ Σ∅ , is a trivial property. Equation (4.34) holds trivially as well because S(Σ∅ , Σ∅ ) = 1. Thus, the only one nontrivial property of SΣ is that of stability, i.e. Σ∅ (t, t ) ≤ Σ∅ (h(t), h(t )), which is but the full-invariance of Σ∅ . To sum up, Σ∅ is the least fully invariant congruence on T(X) which contains Σ. Hence, up to a slightly different formalism, the representations of semantic entailment from Sect. 3.2 and Sect. 4.2 coincide if P = {∅}. As a consequence, !t ≈ t !Σ = !t ≈ t !T(X)/Σ . As to the notion ∅

222

4 Fuzzy Horn Logic

of provability, deduction rules (ERef)–(ESub), introduced in Sect. 3.1, are instances of (Ref)–(Sub) for P = ∅. Furthermore, for P = {∅}, both rules (Ext) and (Mon) can be omitted. Namely, (Ext) yields t ≈ t , 0 and (Mon) derives t ≈ t , b from t ≈ t , b . It is thus easily seen that the degree of provability of t ≈ t from an L-set of identities as defined in Sect. 3.1 coincides with the degree of provability of ∅ i (t ≈ t ) from the L-set Σ of P-implications where Σ (∅ i (t, t )) = Σ(t, t ). Analogously as for semantic closure, P = {∅} leads to a simplified notion of a deductive closure Σ which in fact coincides with that presented in Sect. 3.2. Lemma 4.58 gives |t ≈ t |Σ ≤ Σ∅ (t, t ). The converse inequality follows from Theorem 4.62 because in case of P = ∅, (4.56) is satisfied trivially. We thus have |t ≈ t |Σ = Σ∅ (t, t ) for any complete residuated lattice L because any L can be equipped with an implicational truth stresser satisfying (1.20), e.g. globalization on L. Finally, |t ≈ t |Σ = Σ∅ (t, t ) = !t ≈ t !T(X)/Σ = !t ≈ t !Σ ∅

is a consequence of Theorem 4.57. This is just the same completeness result as presented in Sect. 3.2. Another way of looking at fuzzy equational logic inside fuzzy Horn logic is the following. Consider any proper family P with ∅ ∈ P, and let Σ be an L-set of P-Horn clauses such that for any ∅ = P ∈ P we have Σ(P i (t ≈ t )) = 0. Thus, Σ can be seen as an L-set of identities. Using Theorem 3.31, |t ≈ t |Σ = !t ≈ t !Σ . Soundness of fuzzy Horn logic gives R

|∅ i (t ≈ t )|Σ ≤ !∅ i (t ≈ t )!Σ = !t ≈ t !Σ = |t ≈ t |Σ , where R denotes the deductive system of fuzzy Horn logic. Since R contains all instances of (ERef)–(ESub) for P = ∅, we get |∅ i (t ≈ t )|R Σ ≥ |t ≈ t |Σ . R So, we have |∅ i (t ≈ t )|Σ = |t ≈ t |Σ which has the following interpretation: the degree to which t ≈ t is provable using the original deductive system of fuzzy equational logic equals to the degree to which t ≈ t is provable using the deductive system of fuzzy Horn logic. From this point of view, fuzzy Horn logic is a kind of a conservative extension of fuzzy equational logic – if we focus on provability (using rules of fuzzy Horn logic) of identities from Lsets of identities, we cannot infer more that can be inferred by the deductive system of fuzzy equational logic.

4.5 Implicationally Defined Classes of L-Algebras In this and further sections we turn our attention to the problem of closure properties of implicationally defined classes of L-algebras. Our development is motivated by the following question: “How does a choice of L, ∗ , and P affect closure properties of the model classes?” Such a question in nontrivial from the very beginning. We already know that a generalization of Birkhoff’s variety theorem, which was presented in Sect. 3.4, can be established for any complete residuated lattice taken as the structure of truth degrees. If we take

4.5 Implicationally Defined Classes of L-Algebras

223

into consideration P-implications as our basic formulas, the situation gets more complicated. In the classical case, closure properties of implicational classes are influenced by the structure (complexity) of premises of the implications in question. It is not surprising that in fuzzy setting the structure of premises will play an analogous role. We are therefore going to investigate closure properties in dependence on particular proper families of premises. However, unlike the classical case, we have to take into account also a structure L of truth degrees equipped with a truth stresser ∗ . The subsequent development will show how the choice of L∗ and P influences the closure properties of model classes and under what conditions the appropriate closure properties imply the definability by L-sets of P-implications. In the sequel, we will use the notion of a P-implication which has been introduced in Sect. 4.1, see Definition 4.4. Recall that the interpretation of a P-implication is always considered with respect to a particular truth stresser ∗ for L, see Definition 4.9. We start by investigating closure properties of implicationally defined classes of L-algebras. Note again that in the ordinary case, closure properties depend on the complexity of sets of premises we take into consideration. For instance, universal Horn classes (determined by formulas with finite sets of premises) are known to be closed under direct limits while general implicational classes (determined by generalized formulas with infinite sets premises) are not, see [97]. In what follows, we present an analogy to those closure properties in dependence on a given proper family of premises. Lemma 4.78. Let L∗ be a complete residuated lattice with a truth stresser ∗ . For M ∼ = N we have !P i (t ≈ t )!M = !P i (t ≈ t )!N

(4.66)

for every P-implication P i (t ≈ t ). Proof. Let M ∼ = N. Thus, there is an isomorphism h : M → N. For any valuation v in M we can take valuation v ◦ h in N. We have !t ≈ t !M,v = !t ≈ t !N,v◦h for all t, t ∈ T (X). Conversely, for any valuation v in N we can take v ◦ h−1 so that !t ≈ t !N,v = !t ≈ t !M,v◦h−1 . This immediately gives !P i (t ≈ t )!M = v:X→M !P i (t ≈ t )!M,v = = v:X→N !P i (t ≈ t )!N,v = !P i (t ≈ t )!N ,

proving (4.66).

Lemma 4.79. Let L∗ be a complete residuated lattice with a truth stresser ∗ . If N is a subalgebra of an L-algebra M then !P i (t ≈ t )!M ≤ !P i (t ≈ t )!N for every P-implication P i (t ≈ t ).

(4.67)

224

4 Fuzzy Horn Logic

Proof. Since N ∈ Sub(M), every valuation v in N is a valuation in M as well. Moreover, Lemma 3.46 yields !P i (t ≈ t )!M,v = !P i (t ≈ t )!N,v for every valuation v : X → N . Thus, !P i (t ≈ t )!M = v:X→M !P i (t ≈ t )!M,v ≤ ≤ v:X→N !P i (t ≈ t )!M,v = = v:X→N !P i (t ≈ t )!N,v = !P i (t ≈ t )!N ,

which is the desired inequality.

∗ Lemma 4.80. Let LQ be a complete residuated lattice with an implicational ∗ truth stresser . Let i∈I Mi be the direct product of {Mi | i ∈ I}. Then

!P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!Q

(4.68)

i∈I Mi

for every P-implication P i (t ≈ t ). Proof. First, using Lemma 3.46 (iii), for every valuation v of X in and arbitrary terms r, r ∈ T (X) it follows that

Q

i∈I Mi

Q

!r ≈ r !Q Mi ,v = !r!Q Mi ,v ≈ i∈I Mi !r !Q Mi ,v = i∈I i∈I i∈I Mi Q Q (i) ≈ !r ! (i) = = i∈I !r! i∈I Mi ,v i∈I Mi ,v = i∈I !r!Mi ,v◦πi ≈Mi !r !Mi ,v◦πi = i∈I !r ≈ r !Mi ,v◦πi . Thus, using this fact together with property (1.19) of implicational truth stressers and (1.39), we have i∈I !P !Mi ,v◦πi = ∗ = i∈I = s,s ∈T (X) P (s, s ) → !s ≈ s !Mi ,v◦πi ∗ = = i∈I s,s ∈T (X) P (s, s ) → !s ≈ s !Mi ,v◦πi ∗ = = s,s ∈T (X) i∈I P (s, s ) → !s ≈ s !Mi ,v◦πi ∗ = = s,s ∈T (X) P (s, s ) → i∈I !s ≈ s !Mi ,v◦πi ∗ Q = = !P !Q Mi ,v . s,s ∈T (X) P (s, s ) → !s ≈ s ! Mi ,v i∈I

i∈I

Now, Q it is essential to observe that the i-th projection v ◦ πi of a valuation v : for any valuation vi : X → Mi X → i∈I M i is a valuation in Mi . Conversely, Q there is at least one valuation v : X → i∈I M i such that v ◦ πi = vi . Finally, using (1.44) it follows that !P i (t ≈ t )!{Mi | i∈I} = i∈I !P i (t ≈ t )!Mi = = i∈I vi :X→Mi !P i (t ≈ t )!Mi ,vi = = v:X→Q Mi i∈I !P i (t ≈ t )!Mi ,v◦πi = i∈I = v:X→Q Mi i∈I !P !Mi ,v◦πi → !t ≈ t !Mi ,v◦πi ≤ i∈I ≤ v:X→Q Mi i∈I !P !Mi ,v◦πi → i∈I !t ≈ t !Mi ,v◦πi = i∈I


= =

Q v:X→ i∈I Mi Q v:X→ i∈I Mi

!P !Q

i∈I Mi ,v

→ !t ≈ t !Q

!P i (t ≈ t )!Q

i∈I Mi ,v

i∈I Mi ,v

225

=

= !P i (t ≈ t )!Q

i∈I Mi

,

i.e. we have proved (4.68).

Hence, we have the following consequence for complete residuated lattices equipped with implicational truth stressers. Corollary 4.81. Let L∗ be a complete residuated lattice with an implicational Q truth stresser ∗ . Let M ∈ Sub( i∈I Mi ) be a subdirect product of a family {Mi | i ∈ I} of L-algebras. Then !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!M

(4.69)

for every P-implication P i (t ≈ t ).

Recall that abstract classes of L-algebras are classes closed under isomorphic images, see Definition 2.137 on page 127. Theorem 4.82. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For an L-set Σ of P-implications, Mod(Σ) is an abstract class of L-algebras closed under the formations of subalgebras and direct products. Proof. For any M ∈ I(Mod(Σ)) there is M ∈ Mod(Σ) such that M ∼ = M . By (4.66), Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!M = !P i (t ≈ t )!M for every P-implication P i (t ≈ t ). Thus, Mod(Σ) is closed under isomorphic images, i.e. Mod(Σ) is an abstract class of L-algebras. “S(Mod(Σ)) ⊆ Mod(Σ)”: Let N ∈ S(Mod(Σ)), i.e. there is M ∈ Mod(Σ) where N ∈ Sub(M). Using (4.67) we have Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!M ≤ !P i (t ≈ t )!N for all P-implications P i (t ≈ t ), whence N ∈QMod(Σ). “P(Mod(Σ)) ⊆ Mod(Σ)”: Analogously, for i∈I Mi ∈ P(Mod(Σ)), where {Mi | i ∈ I} ⊆ Mod(Σ), we can use (4.68) to get Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!Q Mi i∈I Q for all P-implications P i (t ≈ t ), i.e. i∈I Mi ∈ Mod(Σ).

Theorem 4.83. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For a class K of L-algebras we have Impl(K) = Impl(I(K)) = Impl(S(K)) = Impl(P(K)) . Proof. Put Σ = Impl(K). Theorem 4.82 yields O(Mod(Σ)) ⊆ Mod(Σ) for O being any of I, S, or P. Since O(K) ⊆ O(Mod(Σ)) and Σ = Impl(Mod(Σ)), Impl(K) = Σ = Impl(Mod(Σ)) ⊆ Impl(O(Mod(Σ))) ⊆ Impl(O(K)) . Conversely, K ⊆ O(K) for O being I, S, or IP. Thus, Impl(O(K)) ⊆ Impl(K) for O being I, S, or IP. Furthermore, Impl(P(K)) = Impl(IP(K)) ⊆ Impl(K).

226

4 Fuzzy Horn Logic

In much the same way as in the ordinary case, one may easily show that P-implicational classes are not closed under (arbitrary) homomorphic images. We will demonstrate this later on. Lemma 4.84. Let L∗ be a complete residuated lattice with a truth stresser ∗ . Let i∈I Mi be a direct union of a directed family {Mi | i ∈ I} of L-algebras. Then !P i (t ≈ t )!{Mi |i∈I} ≤ !P i (t ≈ t )!

i∈I

(4.70)

Mi

for every P-finitary implication P i (t ≈ t ). Proof. Since P i (t ≈ t ) is supposed to be a P-finitary implication, Y = var(P ) ∪ var(t) ∪ var(t ) is a finite set. Thus, for every valuation v of X in i∈I Mi , we canconsider a finite set M = {v(x) | x ∈ Y }. Moreover, M is a finite subset of i∈I Mi , i.e. there is some index k ∈ I such that M ⊆ Mk . Hence the restriction vY of v on Y can be thought of as a valuation in Mk ∈ Sub( i∈I Mi ). Thus, Lemma 3.46 yields !r ≈ r !

i∈I

Mi ,vY

= !r ≈ r !Mk ,vY

for every r, r ∈ Supp(P ) ∪ {t, t }. Furthermore, !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!Mk ,vY = = !P i (t ≈ t )!

i∈I

Mi ,vY

= !P i (t ≈ t )!

As a consequence, !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!

i∈I

i∈I

Mi ,v

.

Mi .

Theorem 4.85. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For an L-set Σ of P-finitary implications, Mod(Σ) is an abstract class of L-algebras closed under the formations of subalgebras, direct products, and direct unions. Proof. Let i∈I Mi ∈ U(Mod(Σ)), where {Mi ∈ Mod(Σ) | i ∈ I} is a directed family of L-algebras. Now, using (4.70) it follows that Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!{Mi |i∈I} ≤ !P i (t ≈ t )! Mi , i∈I for every P-finitary implication P i (t ≈ t ). Hence, i∈I Mi ∈ Mod(Σ). The rest follows from Theorem 4.82. Theorem 4.86. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For a class K of L-algebras we have Implω(K) = Implω(I(K)) = Implω(S(K)) = Implω(P(K)) = Implω(U(K)) . Proof. Since K ⊆ U(K), we have Implω(U(K)) ⊆ Implω(K). Conversely, Theorem 4.85 yields U(K) ⊆ U(Mod(Implω(K))) ⊆ Mod(Implω(K)) This gives Implω(K) ⊆ Implω(U(K)). The rest follows from Theorem 4.83.


227

Universal Horn classes of ordinary algebras are closed under direct limits and reduced products. In the sequel we present analogous closure properties of P-Horn classes of L-algebras. However, the situation is not so straightforward as in the classical case. On the one hand, we show that P-Horn classes are closed under direct limits of direct families and safe reduced products. On the other hand, we show that in general P-Horn classes are not closed under direct limits and reduced products of arbitrary families of L-algebras. Lemma 4.87. Let L∗ be a complete residuated lattice with a truth stresser ∗ . Let lim Mi be a direct limit of a direct family {Mi | i ∈ I} of L-algebras. Then !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!lim Mi

(4.71)

for every P-Horn clause P i (t ≈ t ).

Proof. Let us have a valuation v : X → i∈I Mi /θ∞ and let P i (t ≈ t ) be a P-Horn clause, where Supp(P ) = {tm , tm | m = 1, . . . , n}. Take Y = var(P ) ∪ var(t) ∪ var(t ) and consider a restriction vY of v on Y and the homomorphic extension vY : T(Y ) → lim Mi . Since Y is finite, T(Y ) is finitely presented. Due to Theorem 2.108 there is an index k ∈ I and a morphism g : T(Y ) → Mk such that vY = g ◦ hk . Let gY denote a restriction of g on Y . We have !r!lim Mi ,v = !r!lim Mi ,vY = vY (r) = hk (g(r)) = hk !r!Mk ,gY for any r ∈ T (Y ). Moreover for r, r ∈ T (Y ) it follows that !r ≈ r !lim Mi ,v = !r!lim Mi ,v ≈lim Mi !r !lim Mi ,v = = hk !r!Mk ,gY ≈lim Mi hk !r !Mk ,gY = = !r!Mk ,gY θ∞ ≈lim Mi !r !Mk ,gY θ∞ = = θ∞ !r!Mk ,gY , !r !Mk ,gY . Taking into account previous observations and Remark 2.95, for every r, r ∈ T (Y ) there is an index l ∈ I, k ≤ l such that !r ≈ r !lim Mi ,v = θ∞ !r!Mk ,gY , !r !Mk ,gY = = hkl !r!Mk ,gY ≈Ml hkl !r !Mk ,gY = = !r!Ml ,gY ◦hkl ≈Ml !r !Ml ,gY ◦hkl = !r ≈ r !Ml ,gY ◦hkl . The previous idea yields that there are indices j0 , j1 , . . . , jn ≥ k such that !tm ≈ tm !lim Mi ,v = !tm ≈ tm !Mj

!t ≈ t !lim Mi ,v = !t ≈ t !Mj

m,gY

,gY ◦hkj0

0

◦hkjm

for each m = 1, . . . , n,

.

Moreover, I is directed, i.e. there is an index j ∈ I with j0 , j1 , . . . , jn ≤ j. Using (2.25) it follows that !tm ≈ tm !lim Mi ,v = !tm ≈ tm !Mj ,(gY ◦hkj

!t ≈ t !lim Mi ,v = !t ≈ t !Mj ,(gY ◦hkj

0

m )◦hjm j

)◦hj0 j

.

for each m = 1, . . . , n,

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4 Fuzzy Horn Logic

Now observe that for each m = 0, . . . , n we have (gY ◦ hkjm ) ◦ hjm j = gY ◦ (hkjm ◦ hjm j ) = gY ◦ hkj . Denoting gY ◦ hkj by w, we get !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!Mj ,w = !P i (t ≈ t )!lim Mi ,v . Hence, !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!lim Mi .

Theorem 4.88. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For an L-set Σ of P-Horn clauses, Mod(Σ) is an abstract class of L-algebras closed under the formations of subalgebras, direct products, direct unions, direct limits, and safe reduced products. Proof. The closedness under I, S, P, and U follows from Theorem 4.85. Let lim Mi be the direct limit of direct family {Mi ∈ Mod(Σ) | i ∈ I}. Equation (4.71) gives Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!lim Mi , for every P-Horn clause P i (t ≈ t ). Hence, lim Mi ∈ Mod(Σ). Take a filer F which is safe w.r.t {Mi ∈ Mod(Σ) | i ∈ I}. Using Theorem 4.82, we have MZ ∈ P(Mod(Σ)) ⊆ Mod(Σ) for every Z ∈ F (see definition of Q MZ on page 123). Furthermore, Theorem 2.127 and Theorem 2.132 yield F Mi ∼ = lim MZ , where {MZ ∈ Mod(Σ) | Z ∈ F } is a direct family of L-algebras. Thus, Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!{MZ | Z∈F } ≤ ≤ !P i (t ≈ t )!lim MZ = !P i (t ≈ t )!Q Mi F Q is a consequence of (4.66), (4.68), and (4.71). Hence, F Mi ∈ Mod(Σ).

Theorem 4.89. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . For a class K of L-algebras we have Horn(K) = Horn(I(K)) = Horn(S(K)) = Horn(P(K)) = = Horn(U(K)) = Horn(L(K)) = Horn(PR (K)) . Proof. Follows from Theorem 4.86 and Theorem 4.88 using analogous arguments as in the proof of Theorem 4.86. Remark 4.90. Theorem 4.88 and Theorem 4.89 are formulated for implicational truth stressers, although they can be reformulated for ordinary truth stressers satisfying (1.19) since it is the only nontrivial property of implicational truth stressers being used in the proofs. Theorem 4.88 showed that P-Horn classes are closed under direct limits of direct families and safe reduced products. A natural question arises whether PHorn classes are closed also under arbitrary direct limits and reduced products. The following examples show a negative answer: general P-Horn classes are not closed under direct limits of weak direct families and arbitrary reduced products.


229

Fig. 4.1. L-equality from Example 4.91 and Example 4.92

Example 4.91. Take L∗ , where L is a structure of truth degrees on the unit interval [0, 1], and ∗ is globalization. Let F = {f1 , f2 , g1 , g2 } be a type of L-algebras such that f1 , f2 , g1 , g2 are nullary function symbols (constants). Consider a proper family of premises P = LT (∅)×T (∅) and P-Horn clause f1 ≈ f2 , 1 i g1 ≈ g2 .

(4.72)

Moreover, let Σ be an L-set of P-Horn clauses such that Supp(Σ) = {f1 ≈ f2 , 1 i g1 ≈ g2 } . Thus, Mod(Σ) is a P-Horn class of L-algebras. Now we introduce a weak direct family of L-algebras. Let I, ≤ with I = [0, 1) be a directed index set and let Mi = Mi , ≈Mi , f1Mi , f2Mi , g1Mi , g2Mi , i ∈ I be L-algebras of type F such that Mi = {ai , ai , bi , bi }, f1Mi = ai , f2Mi = ai , g1Mi = bi , g2Mi = bi , and ≈Mi is an L-equality defined in Fig. 4.1 (left). In addition to that, we define a family {hij : Mi → Mj | i ≤ j} of morphisms by hij (ai ) = aj , hij (ai ) = aj , hij (bi ) = bj , hij (bi ) = bj . It is immediate that this defines a weak direct family of L-algebras which is not a direct family. Clearly, !f1 ≈ f2 !Mi = i < 1 for every i ∈ I, thus we have !f1 ≈ f2 , 1 i g1 ≈ g2 !Mi = 1 .

(4.73)

Hence, Mi ∈ Mod(Σ) for all i ∈ I. On the other hand, for the direct limit lim Mi it follows that θ∞ (ai , ai ) = 1, and θ∞ (bi , bi ) = 0. That is, f1lim Mi = [ai ]θ∞ = [ai ]θ∞ = f2lim Mi , g1lim Mi = [bi ]θ∞ = [bi ]θ∞ = g2lim Mi , i.e. f1lim Mi ≈lim Mi f2lim Mi = 1 while g1lim Mi ≈lim Mi g2lim Mi = 0. Thus, !f1 ≈ f2 , 1 i g1 ≈ g2 !lim Mi = !g1 ≈ g2 !lim Mi = 0 . As a consequence, lim Mi ∈ Mod(Σ) since Σ(f1 ≈ f2 , 1 i g1 ≈ g2 ) > 0. In other words, a P-Horn class Mod(Σ) is not closed under direct limits of arbitrary weak direct families. Example 4.92. Take L∗ such that L = [0, 1], and ∗ is globalization. Consider the same type of L-algebras and Σ as in Example 4.91. Let N be an index set and for every i ∈ N let Mi = Mi , ≈Mi , f1Mi , f2Mi , g1Mi , g2Mi be defined the same way as in Example 4.91 except that for ai , ai we put ai ≈Mi ai =

230

4 Fuzzy Horn Logic

ai ≈Mi ai = 1 − 1i , see Fig. 4.1 (right). Clearly, Q Mi ∈ Mod(Σ) for each i ∈ N. Let F be the Fréchet filter over N. Put M = i∈N Mi . For any X ∈ F , [[f1M ≈ f2M ]]X = i∈X f1M (i) ≈Mi f2M (i) = = i∈X f1Mi ≈Mi f2Mi = i∈X ai ≈Mi ai < 1 . Thus, F is not safe w.r.t. {Mi | i ∈ N} since θF (f1M , f2M ) = X∈F [[f1M ≈ f2M ]]X = 1 . As a consequence, !f1 ≈ f2 , 1 i g1 ≈ g2 !Q Mi = !g1 ≈ g2 !Q Mi = θF (g1M , g2M ) = F F = X∈F [[g1M ≈ g2M ]]X = X∈F i∈X bi ≈Mi bi = 0 , Q showing F Mi ∈ Mod(Σ). In certain cases, P-Horn classes are closed even under weak direct limits and arbitrary reduced products. Obviously, this is the case if L is a Noetherian residuated lattice since then closedness under direct limits (safe reduced products) implies closedness under weak direct limits (arbitrary reduced products), see Remark 2.93 and Remark 2.130 (3). Previous examples showed that considering a complete residuated lattice on [0, 1] given by left-continuous t-norm, there are P-Horn classes which are not closed under such constructions. In what follows we introduce P-Horn classes which obey some form of continuity. For these classes we can prove the closedness under the constructions in question even if we work with structures of truth degrees on the real unit interval. Definition 4.93. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. Let " be a strict continuous Archimedean t-norm. Let P = LT (X)×T (X) , K be a class of L-algebras, P i (t ≈ t ) be a P-Horn clause. K satisfies P i (t ≈ t ) "-continuously if for each e ∈ [0, 1) there is d ∈ [0, 1) such that e " !P i (t ≈ t )!K ≤ !d"P i (t ≈ t )!K .

(4.74)

K is called -continuous if K satisfies all P-Horn clauses "-continuously. If K is a P-Horn class which is "-continuous then K is called a "-continuous P-Horn class. Theorem 4.94. Let L∗ be a complete residuated lattice on [0, 1] given by ⊗. Let ∗ be the globalization on [0, 1]. If K is a "-continuous P-Horn class for some ", then K is an abstract class of L-algebras closed under subalgebras, direct products, direct unions, direct limits of weak direct families, and arbitrary reduced products.


231

Proof. Let K be a "-continuous P-Horn class. The closedness under I, S, P, and U follows from Theorem 4.88. We now show that K is closed under arbitrary reduced products. Take a family {Mi ∈ K | i ∈ I} of L-algebras and a filter F over I. Since K = Mod(Σ), it suffices to check !P i (t ≈ t )!Q Mi ≥ !P i (t ≈ t )!K for F each P i (t ≈ t ). Assuming !t ≈ t !Q Mi ,w < !P i (t ≈ t )!K for a valuaF Q Q tion w : X → i∈I M i /θF , we show that !P i (t ≈ t )! F Mi ,w = 1. So, let !t ≈ t !Q Mi ,w < !P i (t ≈ t )!K . Using Lemma 1.35 (ii), there is e ∈ [0, 1) F such that !t ≈ t !Q Mi ,w < e " !P i (t ≈ t )!K . Since K is "-continuous, we F have !t ≈ t !Q Mi ,w < e " !P i (t ≈ t )!K ≤ !d"P i (t ≈ t )!K for some F d ∈ [0, 1). We prove the following claim: there are s, s ∈ T (X) with P (s, s ) = 0 such that for each Z ∈ F we have (4.75) i∈Z !s ≈ s !Mi ,vi < (d"P )(s, s ) , where vi : X → Mi is a valuation defined by vi (x) = v(x)(i), and v : X → Q M i is a mapping with v(x) ∈ w(x) for all x ∈ X. This claim can be i∈I proved by contradiction: assumethat for all s, s ∈ T (X) with P (s, s ) = 0 there is Zs,s ∈ F such that i∈Zs,s !s ≈ s !Mi ,vi ≥ (d"P )(s, s ). Since P is finite, for Z = P (s,s )=0 Zs,s we have Z ∈ F . As a consequence, i∈Z !s ≈ s !Mi ,vi ≥ (d"P )(s, s ) for all s, s ∈ T (X) with P (s, s ) = 0, i.e. (d"P )(s, s ) ≤ !s ≈ s !Mi ,vi for all s, s ∈ T (X), and any i ∈ Z . Therefore, one can conclude i∈Z !t ≈ t !Mi ,vi = i∈Z !d"P i (t ≈ t )!Mi ,vi ≥ ≥ !d"P i (t ≈ t )!K ≥ e " !P i (t ≈ t )!K . This immediately gives !t ≈ t !Q

F Mi ,w

=

Z∈F

i∈Z

!t ≈ t !Mi ,vi ≥ e " !P i (t ≈ t )!K

which violates !t ≈ t !Q Mi ,w < e " !P i (t ≈ t )!K . Hence, (4.75) is true. F Using this claim, we get !s ≈ s !Q Mi ,w = Z∈F i∈Z !s ≈ s !Mi ,vi ≤ (d"P )(s, s ) < P (s, s ) F

showing that P (s, s ) !s ≈ s !Q Mi ,w , i.e. !P i (t ≈ t )!Q Mi ,w = 1. We F F have shown that K is closed under reduced products. Consider a weak direct family {Mi ∈QK | i ∈ I}. Due to Theorem 2.128, lim Mi is isomorphic to a subalgebra of F Mi . Since K is an abstract class closed under subalgebras and reduced products, we get that lim Mi ∈ K, proving that K is closed under direct limits of weak direct families. Remark 4.95. Recall the P-Horn classes Mod(Σ) introduced in Example 4.91 and Example 4.92. Neither of these two classes is "-continuous due to Theorem 4.94. Examples of "-continuous classes will be shown in a further section.

232

4 Fuzzy Horn Logic

∗

∗

∗

In the ordinary case, validity of an implication can be expressed using the notion of injectivity. Namely, an algebra M is injective w.r.t. an implication P i (t ≈ t ) iff P i (t ≈ t ) is valid in M. This criterion is known as the Banaschewski-Herrlich criterion, see [97]. Such a criterion can be used to prove that a P-Horn class is closed under L. In what follows, we present an analogy to Banaschewski-Herrlich criterion. Definition 4.96. Let L∗ be a complete residuated lattice with a truth stresser. For a weighted P-implication P i (t ≈ t ), a we put Q = P ∪ {a/t, t }. Let hP Q : T(X)/θ(P ) → T(X)/θ(Q) be a morphism defined by hP Q [t]θ(P ) = [t]θ(Q) (4.76) for all t ∈ T (X). An L-algebra M is called injective w.r.t. P i (t ≈ t ), a if for every morphism h : T(X)/θ(P ) → M, there exists a morphism g : T(X)/θ(Q) → M such that h = hP Q ◦ g. Remark 4.97. (1) For L-set Q introduced in the previous definition we have P (t, t ) ∨ a if s = t, and s = t , Q(s, s ) = (4.77) otherwise . P (s, s ) for every s, s ∈ T (X). (2) Due to Lemma 2.43 mapping hP Q sending elements of T (X)/θ(P ) to T (X)/θ(Q) defined by hP Q [t]θ(P ) = [t]θ(Q) for all t ∈ T (X) is a well-defined morphism. The injectivity of M w.r.t. P i (t ≈ t ), a is depicted in Fig. 4.2. Theorem 4.98. Let L∗ be a complete residuated lattice with a truth stresser and let P i (t ≈ t ), a be a weighted P-implication.

∗

(i) If a ≤ !P i (t ≈ t )!M , then M is injective w.r.t. P i (t ≈ t ), a . (ii) If ∗ is globalization and M is injective w.r.t. P i (t ≈ t ), a , then a ≤ !P i (t ≈ t )!M . Proof. (i): By the assumption, a ≤ !P i (t ≈ t )!M,v holds for every valuation v : X → M . It suffices to check that M is injective w.r.t. weighted P-implication P i (t ≈ t ), a .

Fig. 4.2. Injectivity of M w.r.t. P i (t ≈ t ), a


233

Fig. 4.3. Scheme for the proof of Theorem 4.98

Consider a morphism h : T(X)/θ(P ) → M and a valuation v : X → M , where v(x) = h [x]θ(P ) for each x ∈ X. Hence, for a homomorphic extension v of v we have v = hθ(P ) ◦ h. Furthermore, !s ≈ s !M,v = v (s) ≈M v (s ) = θv (s, s ) = θhθ(P ) ◦h (s, s ) for all terms s, s ∈ T (X). Therefore, P (s, s ) ≤ θ(P )(s, s ) = [s]θ(P ) ≈T(X)/θ(P ) [s ]θ(P ) ≤ ≤ h [s]θ(P ) ≈M h [s ]θ(P ) = θhθ(P ) ◦h (s, s ) = θv (s, s ) . for all s, s ∈ T (X). By (1.11), a ≤ !P i (t ≈ t )!M,v = ∗ → !t ≈ t !M,v = = s,s ∈T (X) P (s, s ) → !s ≈ s !M,v ∗ = → θv (t, t ) = s,s ∈T (X) P (s, s ) → θv (s, s ) = 1∗ → θv (t, t ) = 1 → θv (t, t ) = θv (t, t ) . We thus have P (t, t ) ≤ θv (t, t ) and a ≤ θv (t, t ), i.e. Q(t, t ) = P (t, t ) ∨ a ≤ θv (t, t ). Since θ(Q) ∈ ConL (T(X)) is generated by Q, it readily follows that θ(Q) ⊆ θv . Now, Lemma 2.44 yields that there is a morphism g : T(X)/θ(Q) → M such that v = hθ(Q) ◦ g, see Fig. 4.3. Hence, hθ(P ) ◦ h = hθ(Q) ◦ g, that is hθ(P ) ◦ h = (hθ(P ) ◦ hP Q ) ◦ g. Surjectivity of hθ(P ) implies h = hP Q ◦ g. Hence, M is injective w.r.t. P i (t ≈ t ), a . (ii): Let ∗ be globalization and let M be injective w.r.t. P i (t ≈ t ), a . We have to show a ≤ !P i (t ≈ t )!M,v for every v. Take v : X → M . If there are terms s, s ∈ T (X) such that P (s, s ) !s ≈ s !M,v , then !P !M,v = 0, i.e. a ≤ !P i (t ≈ t )!M,v = 1. If P (s, s ) ≤ !s ≈ s !M,v for all s, s ∈ T (X), we have !P !M,v = 1. Hence, we check a ≤ !t ≈ t !M,v . For the homomorphic extension v of v we have P ⊆ θv . That is, θ(P ) ⊆ θv . Furthermore, from Lemma 2.44 it follows that there is a morphism g : T(X)/θ(P ) → M such that v = hθ(P ) ◦ g . Since M is injective w.r.t. P i (t ≈ t ), a , there is a morphism g : T(X)/θ(Q) → M with g = hP Q ◦ g. Thus, v = hθ(P ) ◦ g = hθ(P ) ◦ hP Q ◦ g = hθ(Q) ◦ g . As a consequence,

a ≤ θ(Q)(t, t ) = [t]θ(Q) ≈T(X)/θ(Q) [t ]θ(Q) ≤ g [t]θ(Q) ≈M g [t]θ(Q) = = v (t) ≈M v (t ) = !t!M,v ≈M !t !M,v = !t ≈ t !M,v .

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4 Fuzzy Horn Logic

By (1.11), we obtain a ≤ !t ≈ t !M,v = 1∗ → !t ≈ t !M,v = !P i (t ≈ t )!M,v showing a ≤ !P i (t ≈ t )!M .

Remark 4.99. (1) For a globalization, Theorem 4.98 gives an “if and only if” criterion for a P-implication to be true in M in degree at least a. (2) For P = ∅, Theorem 4.98 also gives an “if and only if” criterion because the truth stresser does not influence the degree !∅ i (t ≈ t )!M,v which equals to !t ≈ t !M,v . Thus, for any identity t ≈ t we have a ≤ !t ≈ t !M iff M is injective w.r.t. ∅ i (t ≈ t ), a . (3) Theorem 4.98 can be used to prove that, considering a globalization as the truth stresser on L, !P i (t ≈ t )!{Mi | i∈I} ≤ !P i (t ≈ t )!lim Mi is true for every direct family {Mi | i ∈ I} of L-algebras (this is already covered by Theorem 4.71): Put a = !P i (t ≈ t )!{Mi | i∈I} , i.e. a ≤ !P i (t ≈ t )!Mi for all i ∈ I. That is, every Mi is injective w.r.t. P i (t ≈ t ), a . It remains to show that lim Mi is injective w.r.t. P i (t ≈ t ), a as well. Consider a morphism h : T(X)/θ(P ) → lim Mi . Since P i (t ≈ t ) is a P-Horn clause, T(X)/θ(P ), where X = var(P ) ∪ var(t) ∪ var(t ) is a finitely presented L-algebra. Due to Theorem 2.108, for some index k ∈ I the mapping h factorizes through some component of lim Mi , i.e. h = h ◦ hk , where h : T(X)/θ(P ) → Mk is a morphism, see Fig. 4.4. By assumption, Mk is injective w.r.t. P i (t ≈ t ), a , thus there is a morphism g : T(X)/θ(Q) → Mk such that h = hP Q ◦ g. As a consequence, h = h ◦ hk = hP Q ◦ (g ◦ hk ), i.e. g ◦ hk is the desired morphism. Thus, lim Mi is injective with respect to P i (t ≈ t ), a . To sum up, this section has shown that every Mod(Σ) is an abstract class of L-algebras closed under subalgebras and direct products. In addition to that, for certain P-implicational classes with additional constraints on their families of premises, the model classes are closed under direct unions, (weak) direct limits, and (safe) reduced products. In the following sections we show that the converse assertions to those of Theorem 4.82, Theorem 4.85, Theorem 4.88, and Theorem 4.94 are true as well. That is, we show that closedness of an abstract class K under the corresponding operators implies that K is a Pimplicational (P-finitary implicational, P-Horn, "-continuous P-Horn) class.

Fig. 4.4. Injectivity of lim Mi with respect to P i (t ≈ t ), a

4.6 Sur-Reflections and Sur-Reflective Classes

r

R

235

K

h

M h

N

Fig. 4.5. Sur-reflection of M in K

4.6 Sur-Reflections and Sur-Reflective Classes In this section, we characterize P-implicational classes as abstract classes of L-algebras closed under subalgebras and direct products. We start by surreflections which play an analogous role to that of K-free L-algebras. Later on, we will show that every abstract class of L-algebras admits sur-reflections if and only if it satisfies the desired closure properties. The last result in this section establishes a connection between P-implicational classes and the so-called sur-reflective classes. Definition 4.100. Let K be an abstract class of L-algebras of type F , M be an L-algebra of type F . A morphism r : M → R, where R ∈ K, is called a reflection of M in K, if for every morphism h : M → N, where N ∈ K, there exists uniquely determined morphism h : R → N such that h = r ◦ h . Moreover, if r is an epimorphism (surjective morphism), then r is called a sur-reflection of M in K, see Fig. 4.5. An abstract class K of L-algebras of type F is called sur-reflective if every L-algebra M of type F admits a sur-reflection r : M → R in K. Example 4.101. Let K be a an abstract class of L-algebras. If FK (X) ∈ K then the natural mapping hθK (X) : T(X) → FK (X) is a sur-reflection of the term L-algebra T(X) in K. Indeed, hθK (X) is an epimorphism. Moreover, for every morphism h : T(X) → N, where N ∈ K, we can consider a mapping g : X → N defined by g(x) = h(x) for all x ∈ X. Due to Theorem 2.139 and Theorem 3.39, g has a uniquely determined homomorphic extension g : FK (X) → N. Hence, h(t) = g [t]θK (X) = (hθK (X) ◦g )(t) holds for all t ∈ T (X), i.e. h = hθK (X) ◦g . To sum up, hθK (X): T(X) → FK (X) is a sur-reflection of T(X) in K. Theorem 4.102. Let K be an abstract class of L-algebras and let r1 : M → R1 , r2 : M → R2 be sur-reflections of M in K. Then R1 ∼ = R2 . Proof. Since R1 , R2 ∈ K by the definition of sur-reflections, there are uniquely determined morphisms r1 : R2 → R1 , r2 : R1 → R2 such that r1 = r2 ◦r1 , and r2 = r1 ◦ r2 . Thus, r1 = (r1 ◦ r2 ) ◦ r1 , and r2 = (r2 ◦ r1 ) ◦ r2 . As a consequence, r2 ◦ r1 = idR1 , r1 ◦ r2 = idR2 . Hence, using Theorem 2.30 we readily obtain R1 ∼ = R2 .

236

4 Fuzzy Horn Logic

Remark 4.103. According to Theorem 4.102, a sur-reflection of M in K is determined up to an isomorphism. This observation enables us to denote a sur-reflection of M in K by rM : M → RK (M). Moreover, when considering sur-reflections, we sometimes omit the surjective mapping rM and use the term “sur-reflection” for RK (M) instead. In such a case we assume that rM is the corresponding mapping. A sur-reflection of M in an abstract class K can be thought of as the greatest image of M in K. The notion of a greatest image can be defined as follows. An L-algebra M ∈ K is said to be the greatest image of M in K if there is an epimorphism h : M → M and for every epimorphism g : M → N, N ∈ K we have θh ⊆ θg . Obviously, M ∼ = M/θh , and M/θh is a “greater factor L-algebra” than M/θg ∼ = N. That is, the definition of the greatest image corresponds well to the intuition. The following theorem characterizes the relationship between sur-reflections and greatest images in more detail. Theorem 4.104. Let K be an abstract class of L-algebras such that S(K) ⊆ K. Then an epimorphism r : M → R is a sur-reflection of M in K iff R is the greatest image of M in K. Proof. “⇒”: Let r : M → R be a sur-reflection of M in K. That is, for every epimorphism g : M → N there is a morphism g : R → N such that g = r ◦ g . Thus, θr ⊆ θr◦g = θg , i.e. R is the greatest image of M in K. “⇐”: Let R be the greatest image of M in K. Take arbitrary N ∈ K and a morphism h : M → N. Note that h is not supposed to be surjective. On the other hand, Theorem 2.35 yields h = h ◦ g, where h : M → M/θh is an epimorphism, and g : M/θh → N is an embedding. Since K is closed under subalgebras, we have M/θh ∈ IS(K) = K. Moreover, R is supposed to be the greatest image, i.e. θr ⊆ θh for some epimorphism r : M → R. Since R∼ = M/θr , from Lemma 2.44 it follows that there is a uniquely determined morphism g : R → M/θh such that h = r ◦ g , see Fig. 4.6. Therefore, h = h ◦ g = (r ◦ g ) ◦ g = r ◦ (g ◦ g). Altogether, r : M → R is a sur-reflection of M in K. Theorem 4.105. An abstract class of L-algebras is sur-reflective iff it is closed under the formations of subalgebras and direct products.

Fig. 4.6. Sur-reflection as the greatest image in K


237

Proof. “⇒”: Let K be a sur-reflective class of L-algebras. It remains to check closedness under S and P. Take M ∈ K, and N ∈ Sub(M). We show N ∈ K by checking that N is isomorphic to its sur-reflection RK (N) with rN : N → RK (N). Consider an embedding h : N → M. Then h = rN ◦ h for some morphism h : RK (N) → M. Since h is an embedding, we have

a ≈N b ≤ rN (a ) ≈RK (N) rN (b ) ≤ h (rN (a )) ≈M h (rN (b )) = = h(a ) ≈M h(b ) = a ≈N b . for every a , b ∈ N . Thus, rN is an embedding. Since rN is a sur-reflection it is also an epimorphism. Hence, N ∼ = RK (N) ∈ K proving Q N ∈ K, i.e. S(K) ⊆ K. | i ∈ I} ⊆ K. We will show Take a family {M i Q Q Q i∈I Mi ∈ K by proving Q ∼ R M ( M ), where r : M → R ( = i K i i K i∈I i∈I i∈I i∈I Mi ) is a sur-reflection Q Q an epimorphism. of i∈I Mi in K. Every projection πj : i∈I Mi → Mj is Q Hence, for every j ∈ I there exists a morphism pj : RK ( i∈I Mi ) → Mj suchQthat πj = Q r ◦ pj . Theorem 2.51 yields that there is a morphism h : RK ( i∈I Mi ) → i∈I Mi such that h◦πj = pj for every j ∈ I. Thus, r◦h◦πj = r ◦ pj = πj for every j ∈ I and so r ◦ h = idQi∈I Mi . Now, we have Q

a≈

i∈I Mi

b ≤ r(a ) ≈RK (

Q

Q

i∈I Mi )

r(b ) ≤

Q

≤ h(r(a )) ≈ h(r(b )) = a ≈ i∈I Mi b Q Q Q for all Q a , b ∈ i∈I M i . Hence, r : i∈I Mi → RK ( i∈I Mi ) is an isomorphism, i.e. i∈I Mi ∈ K. “⇐”: Suppose K is an abstract class of L-algebras of type F and let S(K) ⊆ K and P(K) ⊆ K. We show that every L-algebra M of type F has a surreflection in K. Let i∈I Mi

HK (M) = {θ ∈ ConL (M) | M/θ ∈ K} .

(4.78)

HK (M) is non-empty since K is closed under P. Putting, Q PK (M) = θ∈HK (M) M/θ ,

(4.79)

P(K) ⊆ K implies PK (M) ∈ K. For a family {hθ : M → M/θ | θ ∈ HK (M)} of natural morphisms we can apply Theorem 2.51 to get a uniquely determined morphism p : M → PK (M) with p ◦ πθ = hθ . Finally, from Theorem 2.35 it follows that p = r ◦ s, where r : M → R is an epimorphism, and s : R → PK (M) is an embedding, i.e. R ∈ IS(K) ⊆ K. We claim that r : M → R is a sur-reflection of M in K. Take a morphism h : M → N where N ∈ K. Using Theorem 2.35 we have h = hθh ◦ g, where hθh : M → M/θh is a natural morphism, and g : M/θh → N is an embedding, see Fig. 4.7. Thus, M/θh ∈ K, i.e. θh ∈ HK (M). We have, h = hθh ◦ g = p ◦ πθh ◦ g = r ◦ s ◦ πθh ◦ g .

Hence, for h : R → N being s ◦ πθh ◦ g we have h = r ◦ h . Since r is surjective, the uniqueness of h is immediate. Altogether, r : M → R is a sur-reflection of M in K, i.e. K is sur-reflective.

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4 Fuzzy Horn Logic

Fig. 4.7. Construction of a sur-reflection of M in K

Remark 4.106. In fact, the construction presented in Theorem 3.40 can be thought of as a special case to that one presented in “⇐” of Theorem 4.105. Indeed, for M being T(X), we have HK (M) = ΦK (X), and PK (M) stands for Q θ∈ΦK (X) T(X)/θ. Using this we obtain FK (X) ∈ K. Theorem 4.107. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ . Then every P-implicational class is sur-reflective.

Proof. Consequence of Theorem 4.82 and Theorem 4.105. ∗

∗

∗

Now we present a closure characterization of P-implicational classes in terms of closedness under S and P. Unlike the ordinary case, we do not have an “if and only if” criterion for any P and any L∗ . On the other hand, the results presented below show the importance of P-implications with P = LT (X)×T (X) as well as the importance of complete residuated lattices with globalization. For an L-algebra M = M, ≈M , F M we can consider a set X of variables with |X| = |M |. For the sake of convenience, we can assume X = M . Then T(M ) is a term L-algebra of type F . The terms of type F over M are denoted by a, b, f (a1 , . . . , an ), and so on while the elements of M are denoted by a , b , f M (a1 , . . . , an ). Evidently, for the identical mapping idM : M → M and the corresponding homomorphic extension idM : T(M ) → M we have idM (f (a1 , . . . , an )) = f M (a1 , . . . , an ) .

(4.80)

For technical reasons we introduce particular L-sets of P-implications: Definition 4.108. Suppose K is a sur-reflective class of L-algebras of type F . For every L-algebra M of type F let PM = LT (M )×T (M ) . Moreover, we define an L-set PM ∈ PM of premises by PM (s, s ) = idM (s) ≈M idM (s )

(4.81)

K for all terms s, s ∈ T (M ). A PM -theory of M over K is an L-set ΣM of PM -implications defined by


K P i (t ≈ t ) = ΣM

rM (idM (t)) ≈RK (M) rM (idM (t )) 0

239

if P = PM , otherwise , (4.82)

where rM : M → RK (M) is a sur-reflection of M in K. Lemma 4.109. Let L∗ be a complete residuated lattice with globalization. Let K ) K be a sur-reflective class of L-algebras of type F . Then K = M Mod(ΣM where M ranges over all L-algebras of type F . K K Proof. “K ⊆ M Mod(ΣM )”: It suffices to show K ⊆ Mod(ΣM ) for every L-algebra M of type F , i.e. to check that K ΣM (P i (t ≈ t )) ≤ !P i (t ≈ t )!N

holds for every PM -implication P i (t ≈ t ) and every N ∈ K. So, let us have N ∈ K, and let v : M → N be a valuation of M in N with its homomorphic extension v : T(M ) → N. Theorem 2.35 yields that v = g ◦ g , where g : T(M ) → T(M )/θv is a natural morphism and g : T(M )/θv → N is an embedding. Let P i (t ≈ t ) be a PM -implication such that P = PM . If P (s, s ) v (s) ≈N v (s ) for some s, s ∈ T (M ), then obviously !P i (t ≈ t )!N,v = 1. Thus, let P (s, s ) ≤ v (s) ≈N v (s ) for all s, s ∈ T (M ). Consider a mapping h : M → T (M )/θv defined by h(a ) = [a]θ for every v a ∈ M . Since idM (s) ≈M idM (s ) = P (s, s ) ≤ v (s) ≈N v (s ) , for all s, s ∈ T (M ), it follows that

a ≈M b = idM (a) ≈M idM (b) ≤ ≤ v (a) ≈N v (b) = θv (a, b) = [a]θ

v

≈T(M )/θv [b]θ

v

for all a , b ∈ M . That is, h is an ≈-morphism. Consider an n-ary f M ∈ F M and a1 , . . . , an ∈ M . Putting f M (a1 , . . . , an ) = b , we have 1 = f M (a1 , . . . , an ) ≈M b = idM (f (a1 , . . . , an )) ≈M idM (b) ≤ ≤ [f (a1 , . . . , an )]θ

v

≈T(M )/θv [b]θ

v

.

Then, [f (a1 , . . . , an )]θ = [b]θ . This further gives v v M h f (a1 , . . . , an ) = h(b ) = [b]θ = [f (a1 , . . . , an )]θ = v v T(M )/θv T(M )/θv [a1 ]θ , . . . , [an ]θ = f h(a1 ), . . . , h(an ) . =f v

v

Altogether, h : M → T(M )/θv is a morphism. Clearly, g = idM ◦ h. Since K is sur-reflective, we have h ◦ g = rM ◦ h , where h : RK (M) → N, see Fig. 4.8. Thus, v = g ◦ g = idM ◦ h ◦ g = idM ◦ rM ◦ h , which implies

240

4 Fuzzy Horn Logic

Fig. 4.8. Scheme for the proof of Lemma 4.109

K ΣM (P i (t ≈ t )) = rM idM (t) ≈RK (M) rM idM (t ) ≤ ≤ h rM idM (t) ≈N h rM idM (t ) = = v (t) ≈N v (t ) = !t ≈ t !N,v = !P i (t ≈ t )!N,v . K K (P i (t ≈ t )) ≤ !P i(t ≈ t )!N , i.e. N ∈ Mod(ΣM ). Therefore,ΣM K K “K ⊇ M Mod(ΣM )”: Let N ∈ M Mod(ΣM ). It suffices to show that since then N ∼ the sur-reflection rN : N → RK = RK (N), (N) is an embedding, K K ). Hence, i.e. N ∈ K. Evidently, N ∈ M Mod(ΣM ) implies N ∈ Mod(ΣN we can consider a valuation idN : N → N and its homomorphic extension K ), it follows that idN : T(N ) → N. Taking into account N ∈ Mod(ΣN RK (N) K rN idN (t) ≈ rN idN (t ) = ΣN (PN i (t ≈ t )) ≤

≤ !PN i (t ≈ t )!N,idN = !t ≈ t !N,idN = idN (t) ≈N idN (t ) . Thus, rN : N → RK (N) is an embedding, i.e. N ∈ K.

Now we face the following problem. Given a sur-reflective class K, we K have shown that there is a class of L-sets ΣM of PM -implications such that K K = M Mod(ΣM ). For every M we use a separate proper family of premises PM . In addition to that, we deal with a proper class of L-sets since M ranges over a proper class of all L-algebras of type F . Thus, Lemma 4.109 itself does not yield that K is a P-implicational class. In the ordinary case, the above-mentioned problem of definability by sets of implications has been solved by J. Ad´ amek, see [2]. In what follows, we adopt Ad´ amek’s approach to get the desired result. The key point of [2] is that one can show that every sur-reflective class (in [2] called a quasivariety) is definable by a set of implications assuming the so-called Vopˇenka’s Principle. Moreover, it has been shown that considering the negation of Vopˇenka’s Principle, there is always a sur-reflective class which cannot be defined by implications. For our purpose, we use the following principle concerning classes of L-algebras. Principle 4.110. Given any proper class K of L-algebras of the same type, there are distinct L-algebras M, N ∈ K such that M can be embedded into N. Theorem 4.111. Vopˇenka’s Principle implies Principle 4.110.


241

Proof. Let K be a proper class of L-algebras of type F . We use K to construct a proper class Kc of ordinary first-order structures corresponding to L-algebras from K and then apply Vopˇenka’s Principle for K to show that there are distinct L-algebras M, N ∈ K and an embedding h : M → N. M∈ Let R = {≈a | a ∈ L} be a set of binary relation symbols. For each K, we can consider a first-order structure Mc = Mc , RMc , F Mc of type R, F, σ , where Mc = M , F Mc = F M (i.e. the functional parts of M and Mc M Mc M c coincide), and each ≈M is an a , i.e. u ≈a v iff u ≈ v ≥ a. a-cut ofM≈ M a | u ≈a c v (u , v ∈ M ). Clearly, Kc is a Thus, we have u ≈ v = proper class of first-order structures. Hence, by Vopˇenka’s Principle, there are Mc , Nc ∈ Kc such that Mc can be (isomorphically) embedded into Nc . That is, there is a mapping h : Mc → Nc such that h f Mc (u1 , . . . , un ) = Nc Nc c h(u1 ), . . . , h(un ) (f ∈ F , u1 , . . . , un ∈ Mc ) and u ≈M f a v iff h(u ) ≈a h(v ) (u , v ∈ Mc , a ∈ L). As a consequence, N c c a | h(u ) ≈N u ≈M v = a | u ≈M a v = a h(v ) = h(u ) ≈ h(v ) , showing that h is an embedding of L-algebras M, N.

In Sect. 2.5, we introduced κ-direct unions as particular direct unions. We are going to use this construction to delimit cardinalities of sets of variables. The following definition introduces the closedness under κ-direct unions. Definition 4.112. An abstract class K of L-algebras is said to be closed under κκ-direct unions, if for every κ-directed family {Mi ∈ K | i ∈ I} we have i∈I Mi ∈ K. Lemma 4.113. Suppose K is an abstract class of L-algebras which is closed under κ-direct unions. Then K is closed under κ -direct unions for all κ > κ. Proof. Let {Mi ∈ K | i ∈ I} be a κ -directed family. Then for every J ⊆ I, |J| < κ we have |J| < κ , i.e. there is some i ∈ I such that Mj ∈ Sub(Mi ) for all j ∈ J. Therefore, {Mi ∈ K | i ∈ I} is a κ-directed family. Since K is supposed to be closed under κ-direct unions, we have i∈I Mi ∈ K, i.e. K is closed under κ -direct unions. Remark 4.114. Since ω is the least infinite cardinal, a class being closed under ω-direct unions is closed under arbitrary κ-direct unions. In other words, if U(K) ⊆ K, then K is closed under κ-direct unions for any κ. Lemma 4.115. Let L∗ be a complete residuated lattice with globalization. Suppose K is an abstract class of L-algebras, κ is an infinite cardinal, P = LT (X)×T (X) , |X| = κ. Then K = Mod(Σ) for some L-set Σ of Pκ implications iff K is a sur-reflective class which is closed under κ-direct unions. Proof. “⇒”: Let K = Mod(Σ) for some L-set Σ of Pκ -implications. From Theorem 4.82 and Theorem 4.105 it follows that K is a sur-reflective class. Thus, it suffices to show that K is closed under κ-direct unions.

242

4 Fuzzy Horn Logic

Let {Mi ∈ K | i ∈ I} be a κ-directed family. We have to show that Σ(P i (t ≈ t )) ≤ !P i (t ≈ t )! κ

i∈I

Mi

for every Pκ -implication P i (t ≈ t ). Take a valuation v : X → i∈I Mi and κ its homomorphic extension v : T(X) → i∈I Mi . Put Y = var(P ) ∪ var(t) ∪ var(t ). For each x ∈ Y we can choose an index ix ∈ I such that v(x) ∈ Mix . Let us have an index set J = {ix | v(x) ∈ Mix and x ∈ Y }. Since |Y | < κ, it follows that |J| < κ, i.e. there is i ∈ I such that v(x) ∈ Mi for each x ∈ Y . Consequently,

!P i (t ≈ t )! κ Mi ,v = !P i (t ≈ t )!Mi ,v . i∈I κ This further gives i∈I Mi ∈ Mod(Σ) = K. “⇐”: Let K be a sur-reflective class which is closed under κ-direct unions. Put Σ = Implκ(K). We claim that K = Mod(Implκ(K)). Trivially, K ⊆ Mod(Implκ(K)). Thus, it remains to check the converse inequality. Doing so, it is sufficient to show that every κ-generated L-algebra from Mod(Implκ(K)) belongs to K. Indeed, due to Theorem 2.87, every M ∈ Mod(Implκ(K)) is isomorphic to a κ-direct union of {[M ]M | M ⊆ M, |M | < κ} ⊆ Mod(Implκ(K)) and K is assumed to be closed under κ-direct unions. So, let us have a κ-generated M ∈ Mod(Implκ(K)). Since K is surreflective, M has a sur-reflection rM : M → RK (M) in K. We will show that rM is an embedding. By contradiction, suppose there are b , b ∈ M such that b ≈M b rM (b ) ≈RK (M) rM (b ). Let M such that |M | < κ, denote the set of generators of M. For a subset of variables Y ⊆ X such that |Y | = |M | we can consider a surjective valuation v : Y → M and its surjective homomorphic extension v : T(Y ) → M, see Theorem 2.81 and Theorem 2.141. Define an L-set P ∈ LT (X)×T (X) by v (s) ≈M v (s ) for s, s ∈ T (Y ) , P (s, s ) = 0 otherwise . Since |Y | < κ, it follows that P ∈ Pκ . The surjectivity of v yields that there are terms t, t ∈ T (Y ), where v (t) = b and v (t ) = b . Hence, !P i (t ≈ t )!M,v = !t ≈ t !M,v = v (t) ≈M v (t ) = = b ≈M b rM (b ) ≈RK (M) rM (b ) . Since M ∈ Mod(Implκ(K)), we have (Implκ(K))(P i (t ≈ t )) rM (b ) ≈RK (M) rM (b ) . Thus, there is an L-algebra N ∈ K and a valuation w : Y → N , where P (s, s ) ≤ !s ≈ s !N,w holds for all terms s, s ∈ T (Y ), and !t ≈ t !N,w rM (b ) ≈RK (M) rM (b ). On the other hand, we clearly have θv ⊆ θw . Thus, from Lemma 2.44 it follows that there is a morphism g : M → N such that w = v ◦g, see Fig. 4.9. Since N ∈ K, there is a morphism g : RK (M) → N, where g = rM ◦ g . As a consequence, w = v ◦ rM ◦ g . Moreover,


v T(Y ) w

M

rM

g N

243

RK (M) g

Fig. 4.9. Scheme for the proof of Lemma 4.115

!t ≈ t !N,w = w (t) ≈N w (t ) = g (rM (v (t))) ≈N g (rM (v (t ))) = = g (rM (b )) ≈N g (rM (b )) ≥ rM (b ) ≈RK (M) rM (b ) which is a contradiction. Altogether, K = Mod(Σ) for Σ = Implκ(K).

Remark 4.116. Note that the “⇐” part of the original proof of Lemma 4.115 for ordinary algebras (see [2]) differs from that one presented above. In [2], the author presents a direct construction of an algebra in K, while we proceed by contradiction and use properties of sur-reflections. From the viewpoint of theory of L-algebras, the original construction pertains only to trivial ≈M ’s and is thus not applicable for general L-algebras. Analogously as for L-sets of P-implications and accordingly to the relationship between L-set of P-implications and sets of weighted P-implications we denote the class of all models of a class Σ of weighted P-implications by Mod(Σ), i.e. Mod(Σ) = {M | a ≤ !ϕ!M for each ϕ, a ∈ Σ} .

(4.83)

Lemma 4.117. Let L∗ be a complete residuated lattice with globalization, K be a sur-reflective class of L-algebras of type F . Then there is a class Σ of weighted implications with weighted premises such that K ) = Mod(Σ) , K = M Mod(ΣM where M ranges over all L-algebras of type F . K i (t ≈ t )) = a Proof. Let Σ be a class, where P i (t ≈ t ), a ∈ Σ iff ΣM (P K for some L-algebra M of type F . We will check M Mod(ΣM ) = Mod(Σ). The rest follows from Lemma 4.109. K K ), i.e. ΣM (P i (t ≈ t )) ≤ !P i (t ≈ t )!N “⊆”: Let N ∈ M Mod(ΣM for every L-algebra M of type F and every PM -implication P i (t ≈ t ). Thus, a ≤ !P i (t ≈ t )!N for every P i (t ≈ t ), a ∈ Σ, i.e. N ∈ Mod(Σ). “⊇”: Let N ∈ Mod(Σ). For every L-algebra M of type F and arbitrary PM -implication P i (t ≈ t ) we have P i (t ≈ t ), a ∈ Σ, where a = K ΣM (P i (t ≈ t )). Since N ∈ Mod(Σ), it follows thata ≤ !P i (t ≈ t )!N . K K ) for every M of type F , i.e. N ∈ M Mod(ΣM ). Thus, N ∈ Mod(ΣM

Theorem 4.118. Let L∗ be a complete residuated lattice with globalization. Assuming Principle 4.110, for every sur-reflective class K of L-algebras there exists an L-set Σ of P-implications such that K = Mod(Σ).

244

4 Fuzzy Horn Logic

Proof. We will show by contradiction that if K were not definable by any L-set of P-implications then Principle 4.110 would be violated. Thus, assume K to be a sur-reflective class such that K = Mod(Σ) for every L-set Σ of Pimplications, where P is an arbitrary proper family of premises. Lemma 4.117 yields that K is definable by a class Σ of weighted implications with weighted premises. For every infinite cardinal κ we can consider a proper family of premises Pκ such that P = LT (X)×T (X) , |X| = κ. Clearly, for every Pκ there is only a set of weighted Pκ -implications in Σ. Moreover, from Lemma 4.115 it follows that K cannot be closed under κ-direct unions. That is, for every infinite cardinal κ there is an L-algebra Mκ ∈ K such that Mκ is a κ-direct union of some κ-directed family {Mκ,i ∈ K | i ∈ Iκ }. Let us define an ordinal sequence of Lalgebras formed of such Mκ ’s. Put N0 = Mω . For every ordinal α let Nα be Mκ such that κ > | β |Nβ | for every β < α. Thus, for β < α there cannot be an injective mapping sending elements of Nα to Nβ . Since every embedding is injective, Nα cannot be embedded into Nβ . On the other hand, Nβ cannot be embedded into Nα either. Indeed, suppose h : Nβ → Nα is an embedding. Since Nα is a κ-direct union and |Nβ | < κ by definition, for every a ∈ Nβ there is an index ia ∈ Iκ such that h(a ) ∈ Mκ,ia . Moreover, |{ia | a ∈ Nβ }| < κ, i.e., there is some k ∈ Iκ such that h(Nβ ) ⊆ Mκ,k . Thus, Nβ ∈ K is a subalgebra of Mκ,k ∈ K, which is a contradiction. To sum up, the class of all Nα ’s is proper and there is no Nα which can be embedded into another Nβ (for α = β), i.e. Principle 4.110 is violated. Remark 4.119. In ordinary case, assuming the negation of Vopˇenka’s Principle, there are sur-reflective classes of algebras definable only by proper classes of implications, see [2]. This immediately applies in fuzzy case for L being 2. The following theorem summarizes previous observations. Theorem 4.120. Let L∗ be a complete residuated lattice with globalization. Assuming Principle 4.110, for any abstract class K of L-algebras the following are equivalent: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii)

K is a P-implicational class, K is closed under S and P, K = SP(K), K = SP(K ) for some abstract class K of L-algebras, K is a sur-reflective class, K = Mod(Σ) for some class Σ of weighted implications, K = Mod(Impl(K)), K = Mod(Impl(K )) for some abstract class K of L-algebras.

4.7 Semivarieties

245

Proof. “(i) ⇒ (ii)”: Consequence of Theorem 4.82. “(ii) ⇒ (iii)”: Clearly, SP(K) = S(K) = K. “(iii) ⇒ (iv)”: Trivial. “(iv) ⇒ (v)”: Evidently, S(K) = SSP(K ) = SP(K ) = K. Analogously, we have P(K) = PSP(K ) ⊆ SPP(K ) = SP(K ) = K. Now apply Theorem 4.105. “(v) ⇒ (vi)”: Apply Lemma 4.117. “(vi) ⇒ (vii)”: Clearly, Mod(Σ) is sur-reflective since it is an abstract class closed under S, P (routine to check as in Theorem 4.82). From Theorem 4.118 it follows that K = Mod(Σ ) for some L-set Σ of Pκ -implications. Hence, using (1.99) on Mod(Σ ) we obtain K = Mod(Σ ) = Mod(Impl(Mod(Σ ))) = Mod(Impl(K)) . “(vii) ⇒ (viii)”: Trivial. “(viii) ⇒ (i)”: By definition.

From this moment on, we shall assume Principle 4.110 when necessary. Theorem 4.121. Let L∗ be a complete residuated lattice with an implicational truth stresser ∗ , P be any proper family of premises. Then if a class K of Lalgebras is definable by P -implications whose interpretation is given by ∗ then K is definable by P-implications whose interpretation is given by globalization on L, and P = LT (X)×T (X) for some infinite X. Proof. Take L∗ with an implicational truth stresser ∗ . Let K be definable by P -implications whose interpretation is given by ∗ . Thus, there is an L-set Σ ∗ of P -implications such that K = ModL (Σ), i.e. M ∈ K iff ∗

!P i (t ≈ t )!L M ≥ Σ(P i (t ≈ t ))

for any P -implication P i (t ≈ t ). Theorem 4.82 yields that K is an abstract class of L-algebras closed under S and P. By Theorem 4.105, K is sur-reflective. Finally, Theorem 4.118 gives that for some sufficiently large X there is an L-set Σ of Pκ -implications whose interpretation is given by globalization and P = LT (X)×T (X) such that K = Mod(Σ ). Hence, K is definable by P-implications (observe that Pκ ⊆ P) whose interpretation is given by globalization. Remark 4.122. Theorem 4.121 showed the importance of globalization and proper families of premises of the form P = LT (X)×T (X) . The assertion says that the definability by P-implications with P = LT (X)×T (X) and globalization is, in fact, the most general one.

4.7 Semivarieties The aim of this section is to investigate relationship between semivarieties of L-algebras and P-finitary implicational classes of L-algebras. The correspondence between semivarieties and P-finitary implicational classes can be

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easily obtained from the previous results. We start with definition of a semivariety and characterization of semivarieties by a single closure operator. Our approach to semivarieties is based on that one presented in [97]. Definition 4.123. A class K of L-algebras is called a semivariety if it is an abstract class closed under formations of subalgebras, direct products, and direct unions. For an abstract class K of L-algebras let VS (K) denote the semivariety generated by K. Remark 4.124. By Theorem 4.105, semivarieties are sur-reflective classes which are closed under direct unions. Lemma 4.125. Let K be an abstract class of L-algebras of type F and let M be an L-algebra of type F . Then M ∈ US(K) iff every finitely generated subalgebra of M belongs to S(K). Proof. “⇒”: Let M ∈ US(K), i.e. M is a direct union of {Mi ∈ S(K) | i ∈ I}. Take a finitely generated subalgebra N = [{a1 , . . . , an }]M . Clearly, there is an index i ∈ I such that {a1 , . . . , an } ⊆ Mi . Therefore, N ∈ Sub(Mi ). Hence, N ∈ SS(K) = S(K), i.e. every finitely generated subalgebra of M is in S(K). The “⇐”-part of the claim follows from Corollary 2.88. Lemma 4.126. For class operators U, S we have SU(K) ⊆ US(K) , UUS(K) ⊆ US(K)

(4.84) (4.85)

for every abstract class K of L-algebras. Moreover, US is a closure operator on abstract classes of L-algebras. Proof. (4.84): Let N ∈ SU(K), i.e. N ∈ Sub( i∈I Mi ) for some directed family {Mi ∈ K | i ∈ I}. A family {[Mi ∩ N ]M | i ∈ I} of L-algebras is also directed because for any i, j ∈ I there is k ≥ i, j such that Mi , Mj ∈ Sub(Mk ). Thus, [Mi ∩ N ]M , [Mj ∩ N ]M ∈ Sub([Mk ∩ N ]M ). Clearly, N is a direct union of {[Mi ∩ N ]M | i ∈ I}. Hence, N ∈ US(K). (4.85): Take arbitrary M ∈ UUS(K). That is, M is a direct union of {Mi ∈ US(K) | i ∈ I}. Due to Lemma 4.125 it suffices to show that all finitely generated subalgebras of M belong to S(K). Let N = [{a1 , . . . , an }]M . Clearly, there is some index i ∈ I such that N ∈ Sub(Mi ). Moreover, N is finitely generated and Mi ∈ US(K), i.e Lemma 4.125 yields N ∈ S(K). Therefore, M ∈ US(K). Finally, USUS(K) ⊆ UUSS(K) ⊆ UUS(K) ⊆ US(K), thus it follows that US is idempotent. As a consequence, US is a closure operator (extensivity and monotony of US is obvious) on abstract classes of L-algebras. Lemma 4.127. For class operators P, U we have PU(K) ⊆ UP(K) for every abstract class K of L-algebras.

4.7 Semivarieties

247

Q

Proof. Let us have M ∈ PU(K), i.e. M is a direct product i∈I Mi , where Mi ∈ U(K) for all i ∈ I. Thus, for every i ∈ I there is a directed index set Ji , ≤i and a directed family {Mi,j ∈ K | j ∈ Ji } such that Mi is j∈Ji Mi,j . We Q will construct a suitable directed family of direct products. First, put J = i∈I Ji . Moreover, J can be equipped with a partial order. For every j1 , j2 ∈ J we put j1 ≤ j2 iff j1 (i) ≤i j2 (i) for all i ∈ I. Clearly, J, ≤ is a partially ordered set. Furthermore, since every Ji , ≤i is directed, for j1 , j2 ∈ J there is some j ∈ J such that j1 (i), j2 (i) ≤i j(i) for all i ∈ I, thus j1 , j2 ≤ j. Altogether, Q J, ≤ is a directed index set. Let Nj denote i∈I Mi,j(i) for every j ∈ J. We claim that {Nj | j ∈ J} is a directed family of L-algebras from P(K). Indeed, for j1 ≤ j2 we have Mi,j1 (i) ∈ Sub(Mi,j2 (i) ) for all i ∈ I, i.e. there is an embedding h : Nj1 → Nj2 defined by h(a )(i) = a (i). That is, Nj1 ∈ Sub(Nj2 ). It follows from the ordinary case that the skeletons ske(M), ske(N) coincide (it is easy to check). Hence, it suffices to show that ≈M , ≈N coincide as well. For a , b ∈ M there is an index j ∈ J such that a (i), b (i) ∈ Mi,j(i) for all i ∈ I. That is, a ≈M b = i∈I a (i) ≈Mi b (i) = = i∈I a (i) ≈Mi,j(i) b (i) = a ≈Nj b = a ≈N b . Hence, M ∈ UP(K).

Lemma 4.128. A class operator USP is a closure operator on abstract classes of L-algebras. Proof. Monotony and extensivity of USP are evident. It suffices to check idempotency. Thus, let K be an abstract class of L-algebras. Using Lemma 4.126 and Lemma 4.127 we have USPUSP(K) ⊆ USUPSP(K) ⊆ USUSPP(K) = USPP(K) = USP(K) since K is an abstract class of L-algebras. That is, USP is idempotent.

Theorem 4.129. For the class operators VS and USP we have VS (K) = USP(K)

(4.86)

for every abstract class K of L-algebras. Proof. “⊆”: Let K be an abstract class. We check that USP(K) is closed under U, S, and P. Lemma 4.126 gives U(USP(K)) ⊆ USP(K), S(USP(K)) ⊆ USSP(K) ⊆ USP(K). Furthermore, Lemma 4.127 yields P(USP(K)) ⊆ UPSP(K) ⊆ USPP(K) = USP(K) . “⊇”: Clearly, USP(K) ⊆ USPVS (K) = USVS (K) = UVS (K) = VS (K). Altogether, USP(K) is the semivariety generated by K.

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The following characterization shows that semivarieties are in one-toone correspondence with P-finitary implicational classes the interpretation of which is determined by globalization. In fact, Theorem 4.130 can be thought of as a special case of Lemma 4.115. Theorem 4.130. Let L∗ be a complete residuated lattice with globalization, K be an abstract class of L-algebras. Then K is a P-finitary implicational class for some proper family of premises P iff K is a semivariety. Proof. “⇒”: Consequence of Theorem 4.85. “⇐”: Since K is supposed to be a semivariety, K is closed under direct unions and thus under κ-direct unions for κ being ω. Hence, put κ = ω and apply Lemma 4.115. Then clearly K = Mod(Σ), where Σ is an L-set of P-finitary implications, P = LT (X)×T (X) , and X is a denumerable set of variables. A summary follows. Theorem 4.131. Suppose L∗ is a complete residuated lattice with globalization. Then for any abstract class K of L-algebras the following are equivalent: (i) K is a P-finitary implicational class, (ii) K is a P-implicational class closed under U, (iii) K is closed under U, S, and P, (iv) K is a sur-reflective class closed under arbitrary κ-direct unions, (v) K = USP(K), (vi) K = USP(K ) for some abstract class K of L-algebras, (vii) K is a semivariety, (viii) K = Mod(Implω(K)), (ix) K = Mod(Implω(K )) for some abstract class K of L-algebras. Proof. “(i) ⇒ (ii)”: Every P-finitary implicational class is a P-implicational class, the rest follows from Theorem 4.85. “(ii) ⇒ (iii)”: Follows from Theorem 4.82. “(iii) ⇒ (iv)”: Consequence of Theorem 4.105 and Lemma 4.113. “(iv) ⇒ (v)”: Apply Lemma 2.84 (ii) and Theorem 4.129 for K = VS (K). “(v) ⇒ (vi)”: Trivial. “(vi) ⇒ (vii)”: Consequence of Theorem 4.129. “(vii) ⇒ (viii)”: Using Theorem 4.130, we have K = Mod(Σ), where Σ is an L-set of P-finitary implications. Hence, using (1.99), K = Mod(Implω(K)). “(viii) ⇒ (ix)”: Trivial. “(ix) ⇒ (i)” Trivial, by definition. Remark 4.132. Analogously as in Theorem 4.121, one can show that the definability by P-finitary implications with P = LT (X)×T (X) (X being denumerable) and globalization is the most general one. This is a direct consequence of Theorem 4.85, Theorem 4.105, and Theorem 4.130.

4.8 Quasivarieties

249

4.8 Quasivarieties In this section we concentrate on relationships between P-Horn classes of Lalgebras and quasivarieties, i.e. particular classes of L-algebras with desired closure properties. Recall that in the previous section we study implicational classes of L-algebras that were defined by implications with finitely many variables. However, the number of identities in premises of such implications was not limited in any way. In the sequel, we will look closer at implications with finite premises. Before we begin, let us recall an obstacle we observed earlier. In Sect. 4.5, we showed that P-Horn classes are not closed under arbitrary direct limits and reduced products in general. This observation opens a question what properties of quasivarieties we should actually require. In what follows we present three reasonable ways to deal with quasivarieties in fuzzy setting. Realizing that we have three independent generalizations of a quasivariety which all coincide in the classical case, we can claim that there is perhaps no “the right” notion of a quasivariety of L-algebras. We are going to use the following definition which will be common to all the subsequent approaches. Definition 4.133. A class K of L-algebras is called a quasivariety if it is an abstract class closed under formations of subalgebras, direct products, and direct limits of direct families of L-algebras from K. For an abstract class K of L-algebras let VQ (K) denote the quasivariety generated by K. Remark 4.134. From Theorem 4.105 and Lemma 4.144 it follows that quasivarieties of L-algebras are sur-reflective classes of L-algebras which are closed under direct limits of direct families. Basic Quasivarieties We now focus on classes of L-algebras which are closed on direct limits of direct families and safe reduced products. For technical reasons, several assertions will be restricted only to a particular subclass of complete residuated lattices with truth stressers. Since every quasivariety K is a sur-reflective class, we can consider a surreflection of M in K. We will use mainly sur-reflections of finitely presented Lalgebras. Thus, we will adopt the following convention. For a finitely presented L-algebra T(X)/θ(R) let RK (X, R) denote the sur-reflection of T(X)/θ(R) in K, the corresponding epimorphism will be usually denoted simply by r. In the ordinary case, sur-reflections of finitely presented algebras play an important role in theory of quasivarieties since every quasivariety can be reconstructed by direct limits of such sur-reflections. In what follows we focus on this phenomenon in fuzzy setting. Definition 4.135. Let K be an abstract class of L-algebras of type F . An L-algebra M of type F is said to satisfy the QF condition w.r.t. K, if

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Fig. 4.10. M satisfies the QF condition w.r.t. K

every morphism h : T(X)/θ(R) → M, where T(X)/θ(R) is a finitely presented L-algebra, factorizes through RSP(K) (X, R), i.e. h = r ◦ h for a sur-reflection r : T(X)/θ(R) → RSP(K) (X, R) of T(X)/θ(R) in SP(K) and a morphism h sending elements of RSP(K) (X, R) to M. The situation is depicted in Fig. 4.10. Lemma 4.136. Let L be a Noetherian residuated lattice, K be an abstract class of L-algebras, M be an L-algebra satisfying QF w.r.t. K. Then M ∼ = R, where R is a direct limit of a direct family which consists of sur-reflections RSP(K) (X, R) of certain finitely presented L-algebras. Proof. Let M be an L-algebra satisfying QF w.r.t. K. Theorem 2.107 yields that, under the notation used in its proof, M is isomorphic to a direct limit lim T(Yi )/θ(Si ) of a direct family {hij : T(Yi )/θ(Si ) → T(Yj )/θ(Sj ) | i ≤ j} of finitely presented L-algebras. Let {hi : T(Yi )/θ(Si ) → lim T(Yi )/θ(Si ) | i ∈ I} denote the associated limitcone. We consider a family ri : T(Yi )/θ(Si ) → RSP(K) (Yi , Si ) | i ∈ I of surreflections. For each composed morphism (hij ◦ rj ) : T(Yi )/θ(Si ) → RSP(K) (Yj , Sj ) there exists a morphism gij : RSP(K) (Yi , Si ) → RSP(K) (Yj , Sj ) such that hij ◦ rj = ri ◦ gij . (4.87) We claim that I, ≤ , RSP(K) (Yi , Si ) | i ∈ I together with such gij ’s is a weak direct family of L-algebras. Indeed, for i ≤ j ≤ k it follows that

ri ◦ gik = hik ◦ rk = (hij ◦ hjk ) ◦ rk = hij ◦ (hjk ◦ rk ) = hij ◦ (rj ◦ gjk ) = = (hij ◦ rj ) ◦ gjk = (ri ◦ gij ) ◦ gjk = ri ◦ (gij ◦ gjk ) . Since ri is surjective, we have gik = gij ◦ gjk , i.e. ri ◦ gii = hii ◦ ri = ri for every i ∈ I. Thus, every gii is an identity mapping. Hence, RSP(K) (Yi , Si ) | i ∈ I with gij ’s is a weak direct family. Since L is a Noetherian residuated lattice,

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251

Fig. 4.11. Morphism from lim T(Yi )/θ(Si ) to lim RSP(K) (Yi , Si )

this family is in fact a direct family, see Remark 2.93 on page 104. Now it suffices to check that M ∼ = lim RSP(K) (Yi , Si ). Let gi : RSP(K) (Yi , Si ) → lim RSP(K) (Yi , Si ) | i ∈ I be the limit cone of lim RSP(K) (Yi , Si ). We have, ri ◦ gi = ri ◦ (gij ◦ gj ) = (ri ◦ gij ) ◦ gj = (hij ◦ rj ) ◦ gj = hij ◦ (rj ◦ gj ) . Recall that {hi : T(Yi )/θ(Si ) → lim T(Yi )/θ(Si ) | i ∈ I} satisfies DLP with respect to {T(Yi )/θ(Si ) | i ∈ I}. Hence, from definition of DLP it follows that there is a unique morphism g : lim T(Yi )/θ(Si ) → lim RSP(K) (Yi , Si ) such that ri ◦ gi = hi ◦ g for every i ∈ I, see Fig. 4.11. On the other hand, M is supposed to satisfy QF w.r.t. K. That is, every hi : T(Yi )/θ(Si ) → lim T(Yi )/θ(Si ) factorizes through RSP(K) (Yi , Si ), i.e. there is a morphism gi : RSP(K) (Yi , Si ) → lim T(Yi )/θ(Si ) such that hi = ri ◦ gi for every i ∈ I. Moreover, for every gi we have ri ◦ (gij ◦ gj ) = (hij ◦ rj ) ◦ gj = hij ◦ hj = hi = ri ◦ gi . Hence, the surjectivity of ri implies gi = gij ◦ gj . Since the family of all gi ’s satisfies DLP with respect to RSP(K) (Yi , Si ) | i ∈ I , there is morphism g : lim RSP(K) (Yi , Si ) → lim T(Yi )/θ(Si ) such that gi = gi ◦ g for every i ∈ I, see Fig. 4.12. Now, observe that hi = ri ◦ gi = ri ◦ gi ◦ g = hi ◦ (g ◦ g ) , ri ◦ gi = hi ◦ g = ri ◦ gi ◦ g = ri ◦ (gi ◦ g ) ◦ g = (ri ◦ gi ) ◦ (g ◦ g) for each i ∈ I. Therefore, g ◦g = idlim T(Yi )/θ(Si ) and g ◦g = idlim RSP(K) (Yi ,Si ) . Hence, M ∼ = lim RSP(K) (Yi , Si ) by Theorem 2.30. = lim T(Yi )/θ(Si ) ∼

Fig. 4.12. Morphism from lim RSP(K) (Yi , Si ) to lim T(Yi )/θ(Si )

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Remark 4.137. The previous assertion is formulated for Noetherian residuated lattices. This is due to the fact that general weak direct families lack important properties: Theorem 2.107 yields that every L-algebra is isomorphic to a direct limit of a direct family of finitely presented L-algebras. However, for general L, there is no evidence that sur-reflections of those finitely presented algebras form a direct family. Restricting ourselves on Noetherian residuated lattices eliminates this obstacle. Every quasivariety K is a sur-reflective class by definition, thus if L is a Noetherian residuated lattice, and M ∈ K, then M can be thought of as a direct limit of some sur-reflections RK (X, R) ∈ K of finitely presented Lalgebras. This is an immediate consequence of Lemma 4.136 since M ∈ K satisfies QF w.r.t K trivially due to the sur-reflectivity of K. In other words, every quasivariety is then uniquely defined by sur-reflections of all finitely presented L-algebras. We sum up this observation by the following corollary. Corollary 4.138. Let L be a Noetherian residuated lattice, K be a quasivariety of L-algebras. Then for any M ∈ K we have M ∼ = R, where R is a direct limit of a direct family of certain sur-reflections RK (X, R) of finitely presented algebras. Hence, given a class K = {RK (X, R) | X, R are finite} it follows that K = IL(K ). The following assertion presents an “if and only if” condition for M to be in LSP(K) over an arbitrary Noetherian residuated lattice taken as the structure of truth degrees. Lemma 4.139. Let L be a complete residuated lattice, K be an abstract class of L-algebras of type F , M be an L-algebra of type F . Then (i) if M ∈ LSP(K), then M satisfies QF w.r.t. K; (ii) if L is a Noetherian residuated lattice and M satisfies QF w.r.t. K, then M ∈ LSP(K). Proof. (i): Let M ∈ LSP(K). Then M can be identified with a direct limit of a direct family {Mi ∈ SP(K) | i ∈ I} of L-algebras. Let hij : Mi → Mj (i ≤ j) be the corresponding morphisms. Take a morphism h : T(X)/θ(R) → lim Mi , where T(X)/θ(R) is finitely presented. Theorem 2.108 gives h = g ◦ hk for some k ∈ I and a morphism g : T(X)/θ(R) → Mk . Since Mk ∈ SP(K), we have g = r ◦ g , where r : T(X)/θ(R) → RSP(K) (X, R) is a sur-reflection of T(X)/θ(R) in SP(K), and g : RSP(K) (X, R) → Mk is a morphism. Thus, h = g ◦ hk = (r ◦ g ) ◦ hk = r ◦ (g ◦ hk ), i.e. M satisfies QF w.r.t. K. (ii): Clearly, Lemma 4.136 yields that M ∼ = R, where R is a direct limit of L-algebras from SP(K). Hence, M ∈ LSP(K). In what follows we present a single closure operator LSP on abstract classes of L-algebras and we will prove that LSP(K) is the quasivariety generated by K. Note that due to the limitation of Lemma 4.139 we restrict ourselves only to Noetherian residuated lattices. Also note that the proof of Lemma 4.140 is based on its crisp counterpart, see [97].

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Lemma 4.140. Let L be a Noetherian residuated lattice. Then LLSP(K) ⊆ LSP(K) , SLSP(K) ⊆ LSP(K) ,

(4.88) (4.89)

PLSP(K) ⊆ LSP(K)

(4.90)

for every abstract class K of L-algebras. Proof. (4.88): Let M ∈ LLSP(K), i.e. M can be identified with a direct limit lim Mi of a direct family {Mi ∈ LSP(K) | i ∈ I}. We will show that M satisfies QF w.r.t. K since then M ∈ LSP(K) by Lemma 4.139 (ii). Theorem 2.108 yields that every morphism h : T(X)/θ(R) → lim Mi from finitely presented L-algebra T(X)/θ(R) factorizes through some Mk , k ∈ I, i.e. there is a morphism g : T(X)/θ(R) → Mk such that h = g ◦ hk , where hk : Mk → lim Mi belongs to the limit cone associated with the direct limit lim Mi . Moreover, Mk ∈ LSP(K), i.e. there exists a morphism g : RSP(K) (X, R) → Mk such that g = r ◦ g , where r : T(X)/θ(R) → RSP(K) (X, R) is a sur-reflection of T(X)/θ(R) in SP(K). Thus, we have h = g ◦ hk = r ◦ (g ◦ hk ). That is, M satisfies QF w.r.t. K. (4.89): Let us have M ∈ SLSP(K), i.e. there is some N ∈ LSP(K) such that M ∈ Sub(N). We will show that M satisfies QF w.r.t. K. Let k : M → N denote the inclusion of M to N. That is, k(a ) = a for all a ∈ M . Clearly, k is an embedding. Let h : T(X)/θ(R) → M be a morphism from a finitely presented Lalgebra T(X)/θ(R). We define g : T(X)/θ(R) → N by putting g = h◦k. Since N ∈ LSP(K), it follows that g = r ◦ g , where r : T(X)/θ(R) → RSP(K) (X, R) is a sur-reflection of T(X)/θ(R) in SP(K), and g : RSP(K) (X, R) → N is a morphism. Due to Theorem 2.35 morphisms h, g can be expressed as h = h1 ◦ h2 , g = g1 ◦ g2 , where h1 : T(X)/θ(R) → M , g1 : RSP(K) (X, R) → N are epimorphisms, and h2 : M → M, g2 : N → N are embeddings, the situation is depicted in Fig. 4.13. Now it follows that g = h ◦ k = (h1 ◦ h2 ) ◦ k = h1 ◦ (h2 ◦ k), g = r ◦ g = r ◦ (g1 ◦ g2 ) = (r ◦ g1 ) ◦ g2 .

Fig. 4.13. SLSP(K) ⊆ LSP(K)

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Observe that h1 , r ◦ g1 are epimorphisms and h2 ◦ k, g2 are embeddings. Thus, Theorem 2.36 yields that there is a morphism d : N → M such that h1 = (r ◦ g1 ) ◦ d, and g2 = d ◦ (h2 ◦ k). Hence, h = h1 ◦ h2 = r ◦ (g1 ◦ d ◦ h2 ), i.e. h factorizes through the sur-reflection RSP(K) (X, R). Therefore, M satisfies QF w.r.t. K. Q (4.90): Let M ∈ PLSP(K), i.e. M is a direct product i∈I Mi , where Mi ∈ LSP(K) for all i ∈ I. We will show that Q M satisfies QF w.r.t. K. Take a morphism h : T(X)/θ(R) → i∈I Mi from a finitely presented L-algebra T(X)/θ(R). h induces a family {hi : T(X)/θ(R) → Mi | hi = h ◦ πi , i ∈ I} of morphisms. Since Mi ∈ LSP(K), it follows that for every index i ∈ I there exists a morphism gi : RSP(K) (X, R) → Mi such that hi = r ◦ gi , where r : T(X)/θ(R) → RSP(K) (X, R) is a sur-reflection of T(X)/θ(R) in SP(K). of morphisms we Furthermore, by Theorem 2.51 on the family {gi | i ∈ I} Q can conclude that there is a morphism g : RSP(K) (X, R) → i∈I Mi such that gi = g ◦πi for all i ∈ I. Now it readily follows that h◦πi = hi = r◦gi = r◦g ◦πi holds for all i ∈ I. Hence, h = r ◦ g, i.e. M satisfies QF w.r.t. K. The following theorem generalizes the result of T. Fujiwara, see [41]. Theorem 4.141. Let L be a Noetherian residuated lattice. Then VQ (K) = LSP(K)

(4.91)

for every abstract class K of L-algebras. Proof. “⊆”: Consequence of Lemma 4.140. “⊇”: Clearly, LSP(K) ⊆ LSPVQ (K) = LSVQ (K) = LVQ (K) = VQ (K).

We now give the first characterization of quasivarieties as P-Horn classes: Theorem 4.142. Let L∗ be a Noetherian residuated lattice with globalization. If K is a quasivariety then K = Mod(Horn(K)). Proof. Let P = LT (Y )×T (Y ) , where Y is a denumerable set of variables. K ⊆ Mod(Horn(K)) is obvious. It remains to show the converse inclusion. For the sake of brevity, put K = Mod(Horn(K)). From Theorem 4.88 it follows that K is a quasivariety. Thus, every M ∈ K is a isomorphic to direct limit of some sur-reflections RK (X, R) of finitely presented L-algebras, see Corollary 4.138. Hence, it suffices to show that every sur-reflection RK (X, R) of a finitely presented L-algebra T(X)/θ(R) belongs to K. Then clearly, since K is an abstract class closed under L, we obtain M ∈ K. Let T(X)/θ(R), where X and R are finite (without loss of generality, we can assume X ⊆ Y ), and let r : T(X)/θ(R) → RK (X, R), r : T(X)/θ(R) → RK (X, R) be the sur-reflections of T(X)/θ(R) in K, K . Since K ⊆ K it follows that RK (X, R) ∈ K . As a consequence, the sur-reflection r : T(X)/θ(R) → RK (X, R) factorizes through RK (X, R), i.e. there is a

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Fig. 4.14. Factorization of a sur-reflection in K

morphism g : RK (X, R) → RK (X, R) such that r = r ◦ g . The situation is depicted in Fig. 4.14. We finish the proof by demonstrating that g is an embedding, since then RK (X, R) ∈ IS(K) = K. We proceed by contradiction. So, let there be elements b , b such that

b ≈RK (X,R) b g (b ) ≈RK (X,R) g (b ) . Since both X and R are finite, we introduce a finite P ∈ LT (X)×T (X) by θ(R)(s, s ) for R(s, s ) > 0 , P (s, s ) = 0 otherwise . Define a valuation v of X in RK (X, R) by v(x) = r [x]θ(R) for every x ∈ X. Hence, for the homomorphic extension v : T(X) → RK (X, R) we have v (s) = r [s]θ(R) for all terms s ∈ T (X). Clearly, P (s, s ) ≤ θ(R)(s, s ) = [s]θ(R) ≈T(X)/θ(R) [s ]θ(R) ≤ ≤ r [s]θ(R) ≈RK (X,R) r [s ]θ(R) = = v (s) ≈RK (X,R) v (s ) = !s ≈ s !RK (X,R),v holds for alls, s ∈T (X). Since r is surjective, there are terms t, t ∈ T (X) such that r [t]θ(R) = b , and r [t ]θ(R) = b . Thus, !P i (t ≈ t )!RK (X,R),v = !t ≈ t !RK (X,R),v = v (t) ≈RK (X,R) v (t ) = = r [t]θ(R) ≈RK (X,R) r [t ]θ(R) = = b ≈RK (X,R) b g (b ) ≈RK (X,R) g (b ) . Therefore, !P i (t ≈ t )!RK (X,R) g (b ) ≈RK (X,R) g (b ). In addition to that, RK (X, R) ∈ K = Mod(Horn(K)) yields Horn(K) (P i (t ≈ t )) g (b ) ≈RK (X,R) g (b ) . Hence, there is N ∈ K, and w : X → N such that P (s, s ) ≤ !s ≈ s !N,w for all s, s ∈ T (X) but !t ≈ t !N,w g (b ) ≈RK (X,R) g (b ). Observe that for the homomorphic extension w : T(X) → N we have R ⊆ P ⊆ θw , and thus θ(R) ⊆ θw .

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4 Fuzzy Horn Logic

Fig. 4.15. Scheme for the proof of Theorem 4.142

Furthermore, we can apply Lemma 2.44 on θ(R) ⊆ θw to conclude that there is a morphism h : T(X)/θ(R) → N with w = hθ(R) ◦ h. Since N ∈ K, h factorizes through RK (X, R), i.e. there is a morphism h : RK (X, R) → N such that h = r ◦ h , see Fig 4.15. Now it readily follows that w = hθ(R) ◦ h = hθ(R) ◦ r ◦ h = hθ(R) ◦ r ◦ g ◦ h . Thus, we have !t ≈ t !N,w = w (t) ≈N w (t ) = = (hθ(R) ◦ r ◦ g ◦ h )(t) ≈N (hθ(R) ◦ r ◦ g ◦ h )(t ) = = (r ◦ g ◦ h ) [t]θ(R) ≈N (r ◦ g ◦ h ) [t ]θ(R) ≥ ≥ (r ◦ g ) [t]θ(R) ≈RK (X,R) (r ◦ g ) [t ]θ(R) = = g (b ) ≈RK (X,R) g (b ) which contradicts !t ≈ t !N,w g (b ) ≈RK (X,R) g (b ). Hence, g is an em bedding, RK (X, R) ∈ K. Remark 4.143. The proof of Theorem 4.142 differs from the corresponding one presented in [97] and it is not just for technical reasons that naturally appear in fuzzy approach (e.g. weighted premises, general structures of truth degrees). In [97], it is claimed that if M is isomorphic to a direct limit lim T(Yi )/θ(Si ) of finitely presented algebras, and if M ∈ Mod(Horn(K)), then every T(Yi )/θ(Si ) ∈ Mod(Horn(K)). This is not true as it is demonstrated by the following counterexample. Let us have a term algebra T(X), where X is finite. Clearly, T(X) is finitely presented. Take M ∈ K, where K is a variety such that T(X) ∈ K. Moreover, algebra M is isomorphic to a direct limit lim T(Yi )/θ(Si ) of a direct family {T(Yi )/θ(Si ) | i ∈ I} of finitely presented algebras. We can assume T(X) ∈ {T(Yi )/θ(Si ) | i ∈ I} (if T(X) ∈ {T(Yi )/θ(Si ) | i ∈ I}, T(X) can be added to {T(Yi )/θ(Si ) | i ∈ I} using morphisms gi : T(X) → T(Yi )/θ(Si ) defined by gi (t) = [t]θ(Si ) for every i ∈ I with X ⊆ Yi – this can be made without loss of generality, see Theorem 2.107). Using the argument from [97], one can conclude {T(Yi )/θ(Si ) | i ∈ I} ⊆ Mod(Horn(K)) = K. That is, the term algebra T(X) would be a member of K – a contradiction.

4.8 Quasivarieties

257

The following assertion shows that assuming a Noetherian residuated lattice as the structure of truth degrees quasivarieties can be equivalently defined using reduced products instead of direct limits. Lemma 4.144. Let L be a Noetherian residuated lattice, K be a class of Lalgebras. K is a quasivariety iff K is an abstract class which is closed under formations of subalgebras and safe reduced products. Proof. The “⇒”-part follows from Theorem 2.127 and Theorem 2.132 because K is closed under isomorphic images, direct products, and direct limits. For the “⇐”-part, observe that K is closed under direct products which are particular safe reduced products. Furthermore, since K is closed under isomorphic images, subalgebras, and reduced products, Theorem 2.128 gives that K is closed under direct limits of direct families of L-algebras. A summary follows. Theorem 4.145. Let L∗ be a Noetherian residuated lattice with globalization. Then for any abstract class K of L-algebras the following are equivalent: (i) K is a P-Horn class, (ii) K is a P-finitary implicational class closed under L, (iii) K is a P-implicational class closed under L, (iv) K is closed under L, S, and P, (v) K is closed under PR and S, (vi) K is closed under PR , L, U, S, and P, (vii) K = LSP(K), (viii) K = LSP(K ) for some abstract class K of L-algebras, (ix) K is a quasivariety, (x) K = Mod(Horn(K)), (xi) K = Mod(Horn(K )) for some abstract class K of L-algebras. Proof. “(i) ⇒ (ii)”: Every P-Horn class is a P-finitary implicational class, the rest follows from Theorem 4.88. “(ii) ⇒ (iii)”: Trivial, by definition. “(iii) ⇒ (iv)”: By Theorem 4.85. “(iv) ⇒ (v)”: Apply Lemma 4.144. “(v) ⇒ (vi)”: Consequence of Lemma 4.144 and Theorem 2.117. “(vi) ⇒ (vii)”: By Theorem 4.141. “(vii) ⇒ (viii)”: Trivial. “(viii) ⇒ (ix)”: By Theorem 4.141. “(ix) ⇒ (x)”: By Theorem 4.142. “(x) ⇒ (xi)”: Trivial. “(xi) ⇒ (i)”: By definition.

258

4 Fuzzy Horn Logic

Remark 4.146. It is well known that in the ordinary case, the collections of all varieties, quasivarieties, semivarieties, and sur-reflective classes are pairwise distinct. This applies to the fuzzy case as well. Namely, suppose L∗ is a complete residuated lattice with globalization. Let Σ be an L-set of P-implications given by Σ = {x ≈ y, a i x ≈ y, 1 | a ∈ L, a = 0} . Evidently, Mod(Σ) consists of all L-algebras (of the given type) with crisp L-equalities. Thus, Mod(Σ) is a quasivariety – the closedness under direct limits of (weak) direct families follows from the crispness of L-equalities. Furthermore, K is not a variety since Mod(Σ) is not closed under homomorphic images (an image of a crisp L-equality need not to be crisp). To see that quasivarieties and semivarieties of L-algebras are distinct, take an ordinary semivariety K which is not a quasivariety (such K exists, see [97]). Consider a class K of L-algebras defined by K = N | M ∈ K, ske(N) = M, and ≈N is crisp . That is, K results from K so that each M ∈ K uniquely corresponds with N ∈ K which results from M by equipping M with the crisp L-equality. K is a semivariety of L-algebras which is not a quasivariety (observe that K is closed under any of I, S, P, U, L iff K is closed under the corresponding crisp operator). In a similar way one can get a sur-reflective class of L-algebras which is not a semivariety. In the classical universal algebra, universal Horn classes are usually characterized as abstract classes closed under subalgebras and reduced products. An analogy of this assertion in fuzzy setting is already covered by Theorem 4.145. The following theorem shows the nontrivial part of that assertion even without invoking any connection to direct limits. Theorem 4.147. Let L∗ be a residuated lattice with globalization, K be an abstract class of L-algebras which is closed under subalgebras and safe reduced products. If every filter F is safe, then K = Mod(Σ) for an L-set Σ of P-Horn clauses, where P = LT (X)×T (X) . Proof. Observe that K is closed under direct products since Q is safe Q F = {I} with respect to any family {Mi | i ∈ I} of L-algebras, and F Mi ∼ = i∈I Mi . As a consequence, K is a sur-reflective class. Thus, K = Mod(Impl(K)) due to Theorem 4.120. We claim that K = Mod(Σ), where Σ = Horn(K). Trivially, K ⊆ Mod(Horn(K)), i.e. we have to check the converse inclusion. We will proceed by contradiction. Let M ∈ Mod(Horn(K)) and M ∈ Mod(Impl(K)). That is, !P i (t ≈ t )!M (Impl(K))(P i (t ≈ t ))

for some P i (t ≈ t ). Such a P i (t ≈ t ) induces a family {P i (t ≈ t ) | P ∈ Fin(P )}

(4.92)

4.8 Quasivarieties

259

of P-Horn clauses, where Fin(P ) denotes the set of all finite restrictions of P . Take any P ∈ Fin(P ). Clearly, !P i (t ≈ t )!M ≤ !P i (t ≈ t )!M . Thus, from (4.92) it follows that !P i (t ≈ t )!M (Impl(K))(P i (t ≈ t )). Since M ∈ Mod(Horn(K)), we have (Horn(K))(P i (t ≈ t )) (Impl(K))(P i (t ≈ t )) .

(4.93)

That is, for every P ∈ Fin(P ) there is an L-algebra NP ∈ K and a valuation vP of X in NP such that P (s, s ) ≤ !s ≈ s !NP ,vP for all terms s, s ∈ T (X) and !t ≈ t !NP ,vP (Impl(K))(P i (t ≈ t )). In the following, we construct certain safe reduced product of family {NP | P ∈ Fin(P )} to obtain a contradiction. Let us introduce a proper filter over Fin(P ). First, we can consider a family Ps,s = {P ∈ Fin(P ) | P (s, s ) = P (s, s )} for every s, s ∈ T (X). Evidently, ∅ = Ps,s ⊆ Fin(P ) for all s, s ∈ T (X). Put J = {Ps,s | s, s ∈ T (X)}. Obviously, for every s1 , s1 , . . . , sn , sn ∈ T (X) there is a finite restriction P ∈ Fin(P ) such that P (si ,si ) = P (si , si ) for all i = 1, . . . , n. As a consequence, Psi ,si | i = 1, . . . , n = ∅ showing that J has the finite intersection property. This enables us to define a proper filter F over Fin(P ) to be the filter generated by J. v: Family Q {vP : X → NP | P ∈ Fin(P )} of valuations induces a valuation X → P ∈Fin(P ) NP such that v(x)(P ) = vP (x) for all x ∈ X, P ∈ Fin(P ). Q Now Corollary 3.47 yields that there is a valuation w of X in F NP such that w(x) = [v(x)]θF for every x ∈ X. Thus, for s, s ∈ T (X) we have !s ≈ s !Q

Q

F NP ,w

Q

= w (s) ≈ F NP w (s ) = [v (s)]θF ≈ F NP [v (s )]θF = = θF v (s), v (s ) = Z∈F [[v (s) ≈ v (s )]]Z = = Z∈F P ∈Z v (s)(P ) ≈NP v (s )(P ) = = Z∈F P ∈Z vP (s) ≈NP vP (s ) .

Recall that P (s, s ) ≤ !s ≈ s !NP ,vP = vP (s) ≈NP vP (s ) holds for every P ∈ Fin(P ) and all s, s ∈ T (X). Since Ps,s ∈ F , we get !s ≈ s !Q NP ,w ≥ [[v (s) ≈ v (s )]]Ps,s = P ∈Ps,s vP (s) ≈NP vP (s ) ≥ F ≥ P ∈Ps,s P (s, s ) = P ∈Ps,s P (s, s ) = P (s, s ) , i.e. P (s, s ) ≤ !s ≈ s !Q NP ,w for all terms s, s ∈ T (X). Moreover, since F F is safe by the assumption, we have !t ≈ t !Q NP ,w = [[v (t) ≈ v (t )]]Z0 = P ∈Z0 vP (t) ≈NP vP (t ) F

for some Z0 ∈ F . In addition to that, vP (t) ≈NP vP (t ) = !t ≈ t !NP ,vP (Impl(K))(P i (t ≈ t )) for all P ∈ Z0 . Putting previous facts together, we have

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4 Fuzzy Horn Logic

!P i (t ≈ t )!Q

F NP ,w

= !t ≈ t !Q

F NP ,w

=

P ∈Z0

!t ≈ t !NP ,vP

(Impl(K))(P i (t ≈ t )) .

(4.94)

to K which is supposed to be closed under safe Since every NP belongs Q reduced products, F NP belongs to K. As a consequence, !P i (t ≈ t )!Q

F NP ,w

≥ (Impl(K))(P i (t ≈ t ))

which contradicts (4.94).

Remark 4.148. Note that an observation on the importance of globalization and unrestricted proper families of premises, which is an analog to those of Theorem 4.121 and Remark 4.132, pertains to quasivarieties as well. Continuous Quasivarieties So far, the characterization of quasivarieties was restricted to Noetherian residuated lattices. The reason for doing so was purely technical: L∗ being a Noetherian lattice ensured the desired closure properties. We now turn our attention to quasivarieties which are closed under arbitrary reduced products (direct limits of weak direct families). We shall be motivated by Theorem 4.94 showing that "-continuous P-Horn classes satisfy such closure properties. In the sequel we prove the converse assertion to that given by Theorem 4.94: any quasivariety closed under arbitrary reduced products (or direct limits) is a "-continuous P-Horn classes provided that we use complete residuated lattices on [0, 1] given by left-continuous t-norms. Unless otherwise mentioned, L is a complete residuated lattice on [0, 1] given by left-continuous t-norm, ∗ is the globalization on [0, 1], and " is a strict continuous Archimedean t-norm. Furthermore, P = LT (X)×T (X) where X is a denumerable set of variables. Theorem 4.149. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. If K is a class of L-algebras which is closed under isomorphic images, subalgebras, and arbitrary reduced products, then K is a P-Horn class. Proof. Take a denumerable X and let P = LT (X)×T (X) . Put Σ = Horn(K). Obviously, K ⊆ Mod(Σ). We check the converse inclusion. Take M ∈ Mod(Σ) and let M (the universe of M) be the set of variables. If M is trivial then M ∈ K because K is closed under P. So let M be a nontrivial L-algebra. Put I = {P i (t ≈ t ), c | !P i (t ≈ t )!M,idM < c} . The nontriviality of M yields I = ∅. Since M ∈ Mod(Horn(K)), for each P i (t ≈ t ), c ∈ I, abbreviated by ϕ, c , there is M ϕ,c ∈ K and a valuation v ϕ,c : M → M ϕ,c with P (s, s ) ≤ !s ≈ s !Mϕ,c ,vϕ,c

(4.95)

4.8 Quasivarieties

261

for all s, s ∈ T (M ), and !t ≈ t !Mϕ,c ,vϕ,c < c. For r, r ∈ T (M ), consider Zr,r ⊆ I defined by Zr,r = P i (t ≈ t ), c ∈ I | P (r, r ) = !r ≈ r !M,idM . (4.96) Clearly, Zr,r = ∅ and {Zr,r | r, r ∈ T (M )} has the finite intersection prop erty. Q Let F be the proper filter over I generated by {Zr,r | r, r ∈ T (M )}. Let M be the reduced product of {M | ϕ, c ∈ I} modulo F and let ϕ,c ϕ,c F Q valuation induced by v ϕ,c ’s. That is, w: M → ϕ,c ∈I M ϕ,c /θF be the Q w(x) = [v(x)]θF , where v : M → ϕ,c ∈I M ϕ,c such that v(x)(ϕ, c ) = v ϕ,c (x) for all x ∈ M , and ϕ, c ∈ I. We show that w is an embedding. One can easily check that w is compatible with functions: for any n-ary f M , and arbitrary a1 , . . . , an ∈ M , f (a1 , . . . , an ) ∈ T (M ), i.e. w f M (a1 , . . . , an ) = w f (a1 , . . . , an ) = Q = v f (a1 , . . . , an ) θ = f ϕ,c∈I Mϕ,c v (a1 ), . . . , v (an ) θ = F F Q Q Mϕ,c Mϕ,c F F [v (a1 )]θF , . . . , [v (an )]θF = f w(a1 ), . . . , w(an ) = =f Q = f F Mϕ,c w(a1 ), . . . , w(an ) . We now prove that w is an ≈-morphism: for a , b ∈ M , consider Za,b ∈ F , i.e. Za,b = P i (t ≈ t ), c ∈ I | P (a, b) = a ≈M b is an instance of (4.96). Using (4.95) we get a ≈M b ≤ !a ≈ b!Mϕ,c ,vϕ,c for each ϕ, c ∈ Za,b . Thus, a ≈M b ≤ ϕ,c ∈Za,b !a ≈ b!Mϕ,c ,vϕ,c , yielding

a ≈M b ≤

Z∈F

ϕ,c ∈Z

!a ≈ b!Mϕ,c ,vϕ,c = = !a ≈ b!Q

F Mϕ,c ,w

Q

= w(a ) ≈

F Mϕ,c

w(b ) .

Hence, w is a morphism. Now for a = b take c > a ≈M b . Then we get !a ≈ b!M,idM < c. So, for Zr,r ∈ F we have that if P i (t ≈ t ), d ∈ Zr,r , then P i (a ≈ b), c ∈ Zr,r . Since F is generated by {Zr,r | r, r ∈ T (M )}, for each Z ∈ F there is P such that P i (a ≈ b), c ∈ Z. Therefore, ϕ,d ∈Z !a ≈ b!Mϕ,d ,vϕ,d < c . Thus, ϕ,d ∈Z !a ≈ b!Mϕ,d ,vϕ,d ≤ a ≈M b for all Z ∈ F because c was chosen arbitrarily. Now we have Q

w(a ) ≈

F Mϕ,c

w(b ) = !a ≈ b!Q Mϕ,c ,w = F = Z∈F ϕ,c ∈Z !a ≈ b!Mϕ,c ,vϕ,c ≤ a ≈M b .

Altogether, w is an Q embedding. Thus, M is isomorphic to a subalgebra of the reduced product F M ϕ,c , i.e. M ∈ ISPR (K) ⊆ K.

262

4 Fuzzy Horn Logic

Theorem 4.150. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. If K is a class of L-algebras which is closed under reduced products then K is "-continuous for any strict continuous Archimedean t-norm ". Proof. By contradiction, let there be P i (t ≈ t ) and e ∈ [0, 1) such that for all d ∈ [0, 1) we have e " !P i (t ≈ t )!K > !d"P i (t ≈ t )!K . Therefore, for each d ∈ [0, 1) there is Md ∈ K and vd : X → Md with e " !P i (t ≈ t )!K > !d"P i (t ≈ t )!Md ,vd . Thus, (d"P )(s, s ) ≤ !s ≈ s !Md ,vd for all s, s ∈ T (X), and e " !P i (t ≈ t )!K > !t ≈ t !Md ,vd . Put I = Q1 − n1 | n ∈ N and let F be the Fréchet filter over I. For the reduced product FQMd of {M d | d ∈ I ⊆ [0, 1)} modulo F one can consider a valuation Q M w: X → d /θF such that w(x) = [v(x)]θF , where v : X → d∈I d∈I Md is the induced mapping satisfying v(x)(d) = vd (x) for all x ∈ X, and d ∈ I. Since " is a continuous t-norm, !s ≈ s !Q Md ,w = Z∈F d∈Z !s ≈ s !Md ,vd ≥ Z∈F d∈Z d"P (s, s ) = F = Z∈F P (s, s ) " d∈Z d = P (s, s ) " Z∈F d∈Z d = = P (s, s ) " 1 = P (s, s )

for all s, s ∈ T (X). In addition to that, !t ≈ t !Q Md ,w = Z∈F d∈Z !t ≈ t !Md ,vd ≤ F

≤ e " !P i (t ≈ t )!K < !P i (t ≈ t )!K . Hence, !P i (t ≈ t )!Q Md < !P i (t ≈ t )!K which contradicts the fact F Q that F Md ∈ K. The following theorem characterizes "-continuous P-Horn classes of Lalgebras as particular quasivarieties of L-algebras which are closed under arbitrary reduced products (direct limits). Theorem 4.151. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. Then for any abstract class K the following are equivalent: (i) K is a "-continuous P-Horn class, (ii) K is closed under S, and arbitrary reduced products, (iii) K is closed under S, P, and direct limits of weak direct families. Proof. “(i) ⇒ (ii)” and “(i) ⇒ (iii)”: Apply Theorem 4.94. “(ii) ⇒ (i)”: By Theorem 4.149 and Theorem 4.150. “(iii) ⇒ (ii)”: Consequence of Theorem 2.127.

4.8 Quasivarieties

263

Remark 4.152. As a consequence of Theorem 4.151 we get that a P-Horn class is "1 -continuous iff it is "2 -continuous. In other words, the notion of "continuity of P-Horn classes does not depend on the chosen ". ∗

∗

∗

Recall that in Sect. 4.3 we introduced a notion of a "-continuity of theories (i.e., L-sets of P-Horn clauses). We now look in more detail at the relationship between "-continuous P-Horn theories (see Definition 4.70 on page 216) and "-continuous P-Horn classes (Definition 4.93). Lemma 4.153. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. Let Σ be an L-set of P-Horn clauses. (i) If Σ is "-continuous, then Mod(Σ) is "-continuous. (ii) If Mod(Σ) is "-continuous, then Horn(Mod(Σ)) is "-continuous. Proof. (i): Let Σ be "-continuous and take a P-Horn clause P i (t ≈ t ). For each e ∈ [0, 1) there is d ∈ [0, 1) such that if Σ R P i (t ≈ t ), b for b ∈ [0, 1], then Σ R d"P i (t ≈ t ), be for be ≥ b " e. This immediately gives e " |P i (t ≈ t )|Σ ≤ |d"P i (t ≈ t )|Σ , thus e " !P i (t ≈ t )!Σ ≤ !d"P i (t ≈ t )!Σ by Theorem 4.71, i.e. e " !P i (t ≈ t )!Mod(Σ) ≤ !d"P i (t ≈ t )!Mod(Σ) . That is, Mod(Σ) is "-continuous. (ii): Let Mod(Σ) be "-continuous. Take P i (t ≈ t ) and e ∈ [0, 1). Using the "-continuity of Mod(Σ), we can take d ∈ [0, 1) such that e " !P i (t ≈ t )!Mod(Σ) ≤ !d"P i (t ≈ t )!Mod(Σ) . Let Horn(Mod(Σ)) R P i (t ≈ t ), b . We have b ≤ |P i (t ≈ t )|Horn(Mod(Σ)) ≤ !P i (t ≈ t )!Horn(Mod(Σ)) = = !P i (t ≈ t )!Mod(Σ) by the soundness of FHL and using Mod(Σ) = Mod(Horn(Mod(Σ))). Moreover, the "-continuity of Mod(Σ) yields e " b ≤ e " !P i (t ≈ t )!Mod(Σ) ≤ !d"P i (t ≈ t )!Mod(Σ) = = Horn(Mod(Σ))(d"P i (t ≈ t )) . Put be = Horn(Mod(Σ))(d"P i (t ≈ t )). Thus, d"P i (t ≈ t ), be is provable from Horn(Mod(Σ)) using R, i.e. Horn(Mod(Σ)) is "-continuous. The following assertion summarizes the relationship of "-continuous PHorn classes and "-continuous P-Horn theories. Theorem 4.154. Let L∗ be a complete residuated lattice on [0, 1] given by a left-continuous t-norm ⊗. Let ∗ be the globalization on [0, 1]. Let Σ be an L-set of P-Horn clauses. The following are equivalent.

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4 Fuzzy Horn Logic

(i) (ii) (iii) (iv)

Σ is "-continuous, Σ is "-continuous, Mod(Σ) is "-continuous, Mod(Σ) is closed under arbitrary reduced products.

Proof. “(i) ⇔ (ii)”: See Theorem 4.57. “(ii) ⇒ (iii)”: Apply Theorem 4.43 and Lemma 4.153 (i). “(iii) ⇒ (ii)”: Follows from Theorem 4.43 and Lemma 4.153 (ii). “(iii) ⇔ (iv)”: Consequence of Theorem 4.82 and Theorem 4.151.

Remark 4.155. As an immediate consequence of Theorem 4.154 we get that the "-continuity of the semantic closure of Σ does not depend on the chosen ". Crisply Generated Quasivarieties We now present a characterization of quasivarieties closed under an additional operator of fuzzification which has already been introduced in Sect. 3.5. Throughout the rest of this section, P denotes a proper family of crisp premises. That is, P is defined by P = {P ∈ LT (X)×T (X) | P is crisp}

(4.97)

where X is a denumerable set of variables. In what follows we will be interested in crisp L-sets of P-Horn clauses, i.e. crisp L-sets of Horn clauses with crisp premises, and show that model classes of such L-sets are exactly the quasivarieties closed under fuzzification. This result will be established for any complete residuated lattice taken as the structure of truth degrees. The following auxiliary lemma is a generalization of Lemma 3.60. Lemma 4.156. Let L∗ be a complete residuated lattice with globalization, P be defined by (4.97), Σ be a crisp L-set of P-Horn clauses. Then for any Lalgebras M, N with ske(M) = ske(N) we have M ∈ Mod(Σ) iff N ∈ Mod(Σ). Proof. Take M and N such that ske(M) = ske(N). Using the same arguments as in the proof of Lemma 3.60, we have !r ≈ r !M,v = 1 iff !r ≈ r !N,v = 1 for every r, r ∈ T (X). Since ∗ is globalization, for any P-Horn clause P i (t ≈ t ), !P i (t ≈ t )!M,v = 1 iff IF P (s, s ) ≤ !s ≈ s !M,v for all s, s ∈ T (X), THEN !t ≈ t !M,v = 1 iff IF P (s, s ) = 1 implies !s ≈ s !M,v = 1, THEN !t ≈ t !M,v = 1 iff IF P (s, s ) = 1 implies !s ≈ s !N,v = 1, THEN !t ≈ t !N,v = 1 iff IF P (s, s ) ≤ !s ≈ s !N,v for all s, s ∈ T (X), THEN !t ≈ t !N,v = 1 iff !P i (t ≈ t )!N,v = 1. Therefore, M ∈ Mod(Σ) iff for each P-Horn clause P i (t ≈ t ) such that Σ(P i (t ≈ t )) = 1 we have !P i (t ≈ t )!M,v = 1 for any v iff for each P i (t ≈ t ) such that Σ(P i (t ≈ t )) = 1 we have !P i (t ≈ t )!N,v = 1 for any v iff N ∈ Mod(Σ).

4.8 Quasivarieties

265

The following assertion characterizes quasivarieties which are closed under fuzzification as the classes of L-algebras defined by crisp L-sets of P-Horn clauses with crisp premises. Theorem 4.157. Let L∗ be a complete residuated lattice with globalization, P be defined by (4.97), K be a class of L-algebras. Then K is a quasivariety which is closed under F iff there is a crisp L-set Σ of P-Horn clauses such that K = Mod(Σ). Proof. “⇒”: Suppose K is closed under I, S, P, L, and F. Take Kc ⊆ K where Kc = {M ∈ K | ≈M is crisp}. Observe that Kc is a quasivariety. Indeed, since Kc contains only L-algebras with crisp L-equalities and K is a quasivariety, we have that Kc is closed under I, S, P, and L, i.e. Kc is a quasivariety. Furthermore, any weak direct family of L-algebras taken from Kc is a direct family, i.e. Kc is closed under arbitrary direct limits. Furthermore, consider a class K2 of 2-algebras where M ∈ K2 iff there is an L-algebra N ∈ Kc such that ske(N) = ske(M). Obviously, K2 is but a class of 2-algebras resulting from Kc so that the crisp L-equalities (on M ) have been replaced by 2-equalities (on M ). Hence, K2 is a quasivariety of 2-algebras. Since 2 endowed with ∗ (i.e., 0∗ = 0, 1∗ = 1) is a Noetherian residuated lattice with globalization, from Theorem 4.145 it follows that there is a 2-set Σ2 of P2 -Horn clauses such that P2 = 2T (X)×T (X) and K2 = Mod(Σ2 ). Since P2 -Horn clauses can be identified with P-Horn clauses where P is defined by (4.97), we can introduce a crisp L-set Σ of P-Horn clauses by 0 if Σ2 (P i (t ≈ t )) = 0 , Σ(P i (t ≈ t )) = 1 if Σ2 (P i (t ≈ t )) = 1 , for every P-Horn clause P i (t ≈ t ). Now we check that K = Mod(Σ). Take any L-algebra M. Consider an L-algebra Mc with ske(M) = ske(Mc ), and crisp ≈Mc . Furthermore, let M2 denote the 2-algebra with ske(M) = ske(M2 ). Since K is closed under F, we have M ∈ K iff Mc ∈ Kc . Obviously, Mc ∈ Kc iff M2 ∈ K2 = Mod(Σ2 ). Since ske(M2 ) = ske(Mc ) and Σ is crisp, we have that M2 ∈ Mod(Σ2 ) iff Mc ∈ Mod(Σ). Finally, since ske(Mc ) = ske(M), we get Mc ∈ Mod(Σ) iff M ∈ Mod(Σ) by Lemma 4.156. So, for every M it follows that M ∈ K iff M ∈ Mod(Σ), i.e. K = Mod(Σ). “⇐”: Let Σ be a crisp L-set of P-Horn clauses such that K = Mod(Σ). Theorem 4.88 yields that K is closed under I, S, P, and L. Thus, K is a quasivariety by Definition 4.133. It remains to check that F(K) ⊆ K. Take M ∈ F(K). Thus, there is N ∈ K such that ske(M) = ske(N). Lemma 4.156 gives M ∈ K. Altogether, K is a quasivariety closed under F. Remark 4.158. Note that Theorem 4.157 gives a characterization of particular quasivarieties (crisply generated quasivarieties) for any complete residuated lattice with globalization taken as the structure of truth degrees. This is an important distinction from characterizations given by Theorem 4.145 and Theorem 4.151. On the other hand, quasivarieties closed under fuzzification are

266

4 Fuzzy Horn Logic

relatively simple extensions of the classical quasivarieties because they can be reconstructed from classical quasivarieties (quasivarieties of 2-algebras) by means of fuzzification.

4.9 Bibliographical Remarks There are numerous results on properties of implicationally defined classes of algebras, see e.g. [5, 26, 41, 73], proofs from implicational theories have been studied as well, see e.g. [80]. A survey on implications in the context of universal algebra can be found in monograph [97]. This chapter is based mainly on [17, 18, 91]. Further results can be found in [90, 93], see also [15]. The present chapter shows an approach to Horn logic in Pavelka style. Note that this is not the only approach possible. For instance, one might develop Horn logic in a setting of H´ ajek’s Basic Logic [49]. This seems to be an open problem.

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Table of Notation

Sets and Structures A, B, . . . , U, . . . a, b , . . . , u, . . . ∈ ∅ A⊆B A ∩ B, A ∪ B A−B u1 , . . . , un U1 × · · · × Un Un r ⊆ U1 × · · · × Un r−1 f:A→B f ◦g U 2 Q i∈I Ui πj N Z Q R N0 α, . . . , κ U/E ≤ A, inf A A, sup A R, F, σ σ(s) M, M, RM , F M M, F M

sets elements of sets membership in set empty set set A is subset of set B intersection and union of sets A and B difference of sets A and B ordered n-tupple of elements direct product of sets U1 , . . . , Un direct product U × · · · × U (n times) r is relation between sets U1 , . . . , Un relation inverse to r f is mapping of set A to set B composition of mappings f and g set of all subsets of U direct product of system of sets Ui Q j-th projection of i∈I Ui to Uj positive integers integers rationals reals non-negative integers ordinal and cardinal numbers factor set of set U by equivalence relation E partial order infimum of A supremum of A type of structure arity of relation/function symbol structure algebra

1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 3 3 3 4 4 6 6 6 6

274

Table of Notation

Structures of Truth Degrees ||ϕ|| a, b, c, . . . L L ⊗ → ↔ ¬ an ∗

∗ L i∈I

Li

truth degree of formula ϕ truth degrees set of truth degrees complete residuated lattice truth function of conjunction (multiplication) truth function of implication (residuum) truth function of equivalence (biresiduum) truth function of negation n-th power of a truth stresser complete residuated lattice with truth stresser ordinal sum of Li ’s

7 7 7 11 11 11 13 13 13 14 14 21

L-set A in universe U elements of universe support set of A definition of L-set A a-cut of A set of all L-sets in U empty L-set in U full L-set in U binary L-relation in U inverse L-relation of R ◦-composition of R1 and R2 subsethood degree of A in B equality (similarity) degree of A and B crisp subsethood relation class of L-equivalence θ set U with L-equality ≈ kernel of ≈-morphism h factor set with fuzzy equality

33 33 34 34 34 35 35 35 35 35 35 35 35 36 39 44 45 49

Fuzzy Sets A: U → L a, b , c, . . . Supp(A) A = {A(u1 )/u1 , . . . a A LU , LU ∅U 1U R: U × U → L R−1 R1 ◦ R2 S(A, B) A≈B A⊆B [ u ]θ U, ≈ θh U/θ, ≈U/θ

Pavelka-style Fuzzy Logic Fml S E, E , . . . E(ϕ), ||ϕ||E T, T , . . . ||ϕ||S T , ||ϕ||T R = Rsyn , Rsem Rsyn Rsem T A,R ϕ, a , |ϕ|T |ϕ|A,R T ϕ, a Md(T ) Th(K)

set Fml of all formulas L-semantics for Fml evaluations truth degree of ϕ ∈ Fml in E ∈ S theories over Fml and L degree to which ϕ semantically follows from T deduction rule for Fml and L syntactic part of R semantic part of R ϕ, a is provable from T degree of provability of ϕ from T L-weighted formula set of all models of T theory of K

50 50 50 50 52 53 53 53 53 54 54 55 56 56

Table of Notation

275

Algebras with Fuzzy Equalities ≈, F, σ ≈ σ(f ) M, M , N, . . . M, M , N, . . . ≈M , ≈M , ≈N , . . . F M, F M , F N, . . . ske(M) fM Sub(M) [N ]M θ EqL (M ) ConL (M) θ(R) θ(a/b , c ) M/θ, M/1θ M/θ [ a ]θ h: M → N ∼N M= idM , idM h(M ) h−1 (N ) h(M) hθ : M → M/θ φ/θ Nθ θ|N [θ1 , θ2 ] Q i∈I Mi πi M1 × M2 θ, θ∗ x, x , y, y , . . . X, X , Y, Y , . . . t, t , s, s , . . . T (X) |t|x t(x1 , . . . , xk ) var(t) tM T(X) h I, ≤ Mi i∈I κ i∈I Mi θ∞

type of algebra with fuzzy equality equality symbol arity of function symbol f L-algebras universes of L-algebras M, M , N, . . . relational parts (L-equalities) of M, M , N, . . . functional parts of M, M , N, . . . skeleton of M σ(f )-ary function f M ∈ F M collection of all subuniverses of M subuniverse of M generated by N congruence L-relation set of all L-equivalences on M set of all congruences on M congruence generated by R principal congruence factor set of M by 1 θ factor L-algebra of M by θ congruence class containing a morphism from M to N isomorphic L-algebras identity mapping on M (universe of M) image of M inverse image of N subalgebra induced by h(M ) natural mapping (morphism) congruence on M/θ induced by φ ∈ ConL (M) with θ ⊆ φ union of certain congruence classes restriction of θ to N interval of congruences direct product of L-algebras Mi i-th projection direct product of M1 , M2 pair of factor congruences variables sets of variables terms set of all terms in variables X number of occurrences of x in t term t in variables x1 , . . . , xk set of variables occurring in t term function term L-algebra over X homomorphic extension of h partially ordered index set direct union of L-algebras Mi κ-direct union of L-algebras Mi L-relation induced by a weak direct family

60 60 60 60 60 60 60 60 60 68 68 69 69 69 73 73 74 74 74 76 77 77 77 77 77 78 80 81 81 82 83 84 85 86 91 91 91 91 91 92 92 92 93 93 98 99 99 104

276

Table of Notation

lim Mi i∈I Mi /θ∞ X, R M θ M [[a ≈ b ]]X θQF F Mi MX MX hXY K, K , . . . H(K) I(K) S(K) P(K) PS (K) U(K) L(K) PR (K) HS, SP, HPH, . . . O, O , O1 , O2 , . . .

direct limit of L-algebras Mi support set of lim Mi abbreviation Q for T(X)/θ(R) subset of i∈I M i congruence on M L-algebra generated by M degree of equality over X congruence induced by filter F reduced product Q of L-algebras Mi modulo F abbreviation for i∈X Mi support set of MX morphism MX to MY classes of L-algebras homomorphic images of K isomorphic images of K subalgebras from K direct products of families from K subdirect products of families from K direct unions of directed families from K direct limits of direct families from K safe reduced products of families from K composed class operators class operators

106 106 110 115 115 115 121 122 122 123 123 123 127 127 127 127 127 127 127 127 127 127 127

Fuzzy Equational Logic

t ≈ t v, v , w, w , . . . tM,v t ≈ t M,v t ≈ t M (x/r) t(x/r) (ERef) (ESym) (ETra) (ERep) (ESub) t ≈ t K t ≈ t Σ t ≈ t , a Σ A,R t ≈ t , a |t ≈ t |R Σ , |t ≈ t |Σ V(K) ΦK (X) θK (X) x X FK (X)

identity valuations value of t in M under v truth degree of t ≈ t in M under v truth degree of t ≈ t in M substitution term resulting from t by substitution of r for x equational rule of reflexivity equational rule of symmetry equational rule of transitivity equational rule of replacement equational rule of substitution truth degree of t ≈ t in K degree to which t ≈ t semantically follows from Σ weighted identity t ≈ t , a is provable from Σ using R degree of provability of t ≈ t from Σ variety generated by K set of all φ ∈ ConL (T(X)) such that T(X)/φ ∈ IS(K) intersection of congruences from ΦK (X) abbreviation for [x]θK (X) generators of FK (X) K-free L-algebra over X

141 141 141 141 142 143 143 143 143 143 143 143 144 144 145 145 145 154 156 156 156 156 156

Table of Notation

277

Fuzzy Horn Logic h(P ) endomorphic image of P ∈ LT (X)×T (X) var(P ) set of variables occurring in P ∈ P P, Q, . . . proper families of premises L-sets of premises P, P , Q, Q , . . . restriction of P induced by a ∈ L Pa P-implication P i (t ≈ t ) P i (t ≈ t ), a weighted P-implication Horn restriction of P PFin finitary restriction of P Pω restriction of P by infinite cardinal κ Pκ truth degree of premises P ∈ P in M under v P M,v P i (t ≈ t )M,v truth degree of P i (t ≈ t ) in M under v truth degree of ϕ, a in M under v ϕ, a M,v Mod(Σ) model class of Σ Impl(K) theory of K (L-set of P-implications) ∗ model class of Σ (explicit L∗ ) ModL (Σ) theory of K (explicit P) ImplP(K) ∗ (K) theory of K (explicit P and L∗ ) ImplL P abbreviation for ImplPκ(K) Implκ(K) Horn(K) abbreviation for ImplPFin(K) P i (t ≈ t )K truth degree of P i (t ≈ t ) in K P i (t ≈ t )Σ degree to which P i (t ≈ t ) semantically follows from Σ ϕ, a is provable from Γ using R Γ R ϕ, a R degree of provability of P i (t ≈ t ) from Γ using R |P i (t ≈ t )|Σ P (x/r) L-relation resulting from a substitution (x/r) (Ref) rule of reflexivity (Sym) rule of symmetry (Tra) rule of transitivity (Rep) rule of replacement (Ext) rule of extensivity (Sub) rule of substitution (Mon) rule of monotony Σ, Γ, . . . L-sets of P-implications P-indexed systems of L-relations SP , SP , . . . S, S , . . . L-set of P-implications corresponding to S ΣS P-indexed system of L-relations corresponding to Σ SΣ member of SΣ , SΓ , . . . ΣP , ΓP , . . . N system of P-indexed systems S intersection i i∈I of Si ’s member of i∈I Si determined by P ∈ P i∈I Si P greatest P-indexed system of L-relations Smax semantic closures of SΣ , SΓ SΣ , SΓ , . . . members of SΣ , SΓ , . . . determined by P ∈ P ΣP , ΓP , . . . semantic closures of Σ, Γ, . . . Σ , Γ , . . . Horn restriction of Σ ΣFin deductive closure of SΓ SΓ member of SΓ ΓP

175 175 175 175 176 177 177 178 178 178 179 179 179 181 181 181 181 181 181 181 181 181 182 183 184 185 185 185 185 185 185 185 189 189 189 189 189 197 197 197 197 198 198 198 200 204 204

278

Table of Notation

Γ τ, τ1 , τ2 , . . . (Sup) dP (CExt) (CMon) hP Q RK (M) rM HK (M) PK (M) PM PM K ΣM VS (K) VQ (K) RK (X, R)

deductive closure of Γ (composed) substitutions rule of supremum strict continuous Archimedean t-norm particular L-set of premises crisp version of extensivity crisp version of monotony morphism induced by P i (t ≈ t ), a sur-reflection of M in K (L-algebra) sur-reflection of M in K (morphism) set of all θ ∈ ConL (M) such that M/θ ∈ K direct product of M/θ over all θ ∈ HK (M) proper family of premises LT (M )×T (M ) particular member of PM PM -theory of M over K semivariety generated by K quasivariety generated by K sur-reflection of T(X)/θ(R) in K

205 205 212 216 216 219 219 232 236 236 237 237 238 238 238 246 249 249

Index

a-cut 34 abstract class 127 logic 56 additive generator of a continuous Archimedean t-norm 27 adjoint pair 11 adjointness 9, 11 algebra 6 with L-equality 60 arity 2, 6, 60 atomic inequality 129 automorphism 77 Banaschewski-Herrlich criterion bijection 2 binary L-relation 35 biresiduum 13 BL-algebra 14 Boolean algebra 14 bound lower 4 upper 4 class closed under O 127 of L-equivalence relation 39 of equivalence relation 3 operator 127 closure operator 5 system 5 compact element 4 compatibility 44, 60

232

complete lattice 4 residuated lattice 10 with truth stresser 14 completeness 212 in Pavelka’s sense 56 ◦-composition 35 composition of mappings 2 congruence 7, 69 axiom 44 class 74 -continuity 216, 230 -continuous class 230 theory 216 crisp L-set 35 premises 218 deduction rule 53 deductive closure 150, 204 system 184 degree of provability 54, 145, 183 of semantic entailment 53 of semantic entailment in fuzzy equational logic 144 of semantic entailment in fuzzy Horn logic 181 denumerable set 3 diagonal fill-in 79 direct

280

Index

family 103 indecomposability limit 106 property 107 product 1, 7, 83 union 99 directed family 98 index set 98 DLP 107

86

element 1 embedding 76 empty L-set 35 endomorphic image 175 endomorphism 77 epimorphism 76 ≈-morphism 45 equality axioms 44 symbol 60 equation implication 173 equational class 145 class of a theory 145 theory 145 theory of a class 145 equicontinuous function 129 satisfaction of atomic inequalities 129 equivalence 3 extensivity 185, 190, 219 extent 65 factor algebra 7 congruence 86 L-algebra 74 set 3 with fuzzy equality 49 filter 3, 121 Fréchet filter 3 proper 3 safe 125 trivial 3 finitary implication 173, 178 premises 176

restriction 178 finite L-set 34 premises 176 restriction 35, 200 set 2 finitely presented L-algebra formula L-weighted 55 truth-weighted 55 weighted 55 free generators 93, 128 L-algebra 156 full L-set 35 fully invariant L∗ -closure operator 192 closure operator 192 closure system 192 congruence 147, 190 function 2 antitone 3 characteristic 2 isotone 3 left-continuous 22 monotone 3 non-decreasing 3 non-increasing 3 symbol 60 fuzzification operator 164 fuzzy concept lattice 65 congruence 132 equality 37 equational logic 140 equivalence 37 Horn logic 171 logic abstract 50 Pavelka-style 50 partition 40 relation 35 set 33 G-algebra 14 G¨ odel algebra 14 Galois connection 5 graded equality 35

110

Index subsethood 35 truth 7 greatest element 4 image 236 Hasse diagram 3 Heyting algebra 14 homomorphism 76 Horn clause 173, 178 restriction 178 subtheory 200 truth stresser 17 idempotency 127 identity 141, 173 L-weighted 145 mapping 77 truth-weighted 145 weighted 145 image 77 implication 173 between atomic inequalities 129 L-weighted 176 truth-weighted 176 weighted 176 implicational truth stresser 16 infimum 4 infinite set 3 injection 2 injectivity 232 intent 65 interior operator 5 intersection 1 inverse L-relation 35 relation 2 isomorphic L-algebras 77 isomorphism 77 theorem 78, 81 κ-direct union 99 κ-directed family 98 K-valent premises 218 kernel 45 L-algebra 60 L∗ -closure operator

49

281

L-equality 37 L-equivalence 37 L∗ -implicational Horn subtheory 200 P-theory 182 P-theory of a class 182 P-indexed system 190 L-partition 40 L-relation 35 L-semantics 50 L-set 33 L-weighted proof 54, 145, 182 language of fuzzy equational logic 140 of fuzzy Horn logic 173 lattice 4, 6 algebraic 4 bounded 7 compactly generated 4 Noetherian 4 residuated 11 complete 10 least element 4 length 54, 145, 182 limit cone 107 linear order 4 logic complete (in Pavelka’s sense) 56 sound (in Pavelka’s sense) 56 lower cone 4 mapping 2 bijective 2 injective 2 surjective 2 maximal element 4 membership function 34 metric 42 algebra 129 generalized 42 minimal element 3 model 52 of a theory of fuzzy equational logic 144 of a theory of fuzzy Horn logic 181 monoid 6 monomorphism 76 monotony 185 ∗ -monotony 190, 219

282

Index

morphism 7, 76 multiplication 11 MV-algebra 14 n-ary term 92 natural mapping 78 morphism 78 negation 13 Noetherian lattice 4 non-trivial L-algebra 68 occurrence 91 open problem 135–137, 170, 266 operation 2 with L-sets 36 ordered n-tuple 1 partial order 3 partially ordered set 3 partition 3 fuzzy 40 P-finitary implication 178 implicational class 182 P-Horn class 182 clause 178 Π-algebra 14 P-implication 176 P-implicational class 182 P-indexed system 189 power in residuated lattice 13 premise 171, 175 presentation 110 principal congruence 73 product algebra 14 projection map 84 proof (L-weighted) 54, 145, 182 proper family 175 provability 54, 145, 182 pseudoinverse 27 pseudometric 42 generalized 42 P-theory 238 QF condition 249 quasivariety 173, 249 reduced product

122

reflection 235 relation 2 antisymmetric 3 reflexive 3 symmetric 3 transitive 3 relative pseudocomplement residuated lattice 11 residuum 11 restriction 35, 68, 81

14, 203

safe filter 125 reduced product 125 satisfaction of atomic inequalities 129 semantic closure 149, 198 semivariety 173, 246 separation 87 set 1 set with L-equality 44 similarity 37 singleton 34 skeleton 60 soundness (in Pavelka’s sense) 56 stability 190 structure 6 of truth degrees 8 subalgebra 7, 68 κ-generated 68 finitely generated 68 subdirect embedding 88 irreducibility 88 product 88 representation 89 subset 1, 36 subsethood 35 substitution 143, 185 subterm 142 subuniverse 7, 68 support 34 supremum 4 sur-reflection 235 sur-reflective class 173, 235 surjection 2 t-norm 12, 22 Archimedean 26 continuous 25

Index left-continuous 22 strict 26 strictly monotone 26 term 91 L-algebra 93 function 92 theory 52, 144, 181 trivial L-algebra 68 truth degree 7, 141, 142 degree of implication 178 stresser 14 Horn 17 implicational 16 tuple 1

type

6, 60

ultrafilter 3 UMP 128 union 1 universal algebra 6 universal mapping property universe 1, 33 upper cone 4 valuation 141 value of term 141 variable 91 variety 154, 173 weak direct family

103

128

283

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