Goguen Categories: A Categorical Approach to L-fuzzy Relations (Trends in Logic, Volume 25)

GOGUEN CATEGORIES

TRENDS IN LOGIC Studia Logica Library VOLUME 25 Managing Editor Ryszard Wójcicki, Institute of Philosophy and Sociology, Polish Academy of Sciences, Warsaw, Poland Editors Vincent F. Hendricks, Department of Philosophy and Science Studies, Roskilde University, Denmark Daniele Mundici, Department of Mathematics “Ulisse Dini”, University of Florence, Italy Ewa Orłowska, National Institute of Telecommunications, Warsaw, Poland Krister Segerberg, Department of Philosophy, Uppsala University, Sweden Heinrich Wansing, Institute of Philosophy, Dresden University of Technology, Germany

SCOPE OF THE SERIES

Trends in Logic is a bookseries covering essentially the same area as the journal Studia Logica – that is, contemporary formal logic and its applications and relations to other disciplines. These include artificial intelligence, informatics, cognitive science, philosophy of science, and the philosophy of language. However, this list is not exhaustive, moreover, the range of applications, comparisons and sources of inspiration is open and evolves over time.

Volume Editor Ewa Orłowska

The titles published in this series are listed at the end of this volume.

GOGUEN CATEGORIES A Categorical Approach to L-fuzzy Relations

by

MICHAEL WINTER Brock University, St. Catharines, ON, Canada

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-6163-9 (HB) ISBN 978-1-4020-6164-6 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com

Printed on acid-free paper

All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.

To my family

Contents

INTRODUCTION

ix

1. SETS, RELATIONS, AND FUNCTIONS

1

2. LATTICES 2.1 Galois correspondences and residuated operations 2.2 Distributive lattices 2.3 Brouwerian lattices 2.4 Boolean algebras 2.5 Special elements 2.6 Fixed points 2.7 The complete Brouwerian lattice of antimorphisms 2.8 Filters 2.9 Lattice-ordered semigroups

5 13 17 18 20 22 23 25 31 40

3. L-FUZZY RELATIONS 3.1 Basic operations and properties 3.2 Crispness 3.3 Operations derived from lattice-ordered semigroups

43 43 48 52

4. CATEGORIES OF RELATIONS 4.1 Categories 4.2 Allegories 4.3 Distributive allegories 4.4 Division allegories 4.5 Dedekind categories 4.6 Relational constructions in Dedekind categories 4.7 The Dedekind category of antimorphisms 4.8 Scalars and crispness in Dedekind categories 4.9 Schr¨ oder categories 4.10 Formal languages of relational categories

55 55 57 63 65 68 74 75 79 85 86

vii

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5. CATEGORIES OF L-FUZZY RELATIONS 5.1 Arrow categories 5.2 The arrow category of antimorphisms 5.3 Arrow categories with cuts 5.4 The arrow category with cuts of antimorphisms 5.5 Goguen categories 5.6 The Goguen category of antimorphisms 5.7 Representation of Goguen categories 5.8 Boolean-based Goguen categories 5.9 Equations in Goguen categories 5.10 Operations derived from lattice-ordered semigroups

93 94 112 122 126 127 131 134 139 142 150

6. FUZZY CONTROLLERS IN GOGUEN CATEGORIES 6.1 The Mamdani approach to fuzzy controllers 6.2 Linguistic entities and variables 6.3 Fuzzification 6.4 The rule base 6.5 Decision module 6.6 Defuzzification 6.7 Proving properties of a controller 6.8 Discussion of the approach

169 169 170 176 176 179 179 182 194

INDEX

197

SYMBOLS

201

REFERENCES

203

INTRODUCTION

In a wide variety of problems one has to treat uncertain or incomplete information. Some kind of exact science is needed to describe and understand existing methods, and to develop new attempts. Especially in applications of computer science, this is a fundamental problem. To handle such information, Zadeh [44], and simultaneously Klaua [22, 23], introduced the concept of fuzzy sets and relations. In contrast to usual sets, fuzzy sets are characterized by a membership relation taking its values from the unit interval [0, 1] of the real numbers. After its introduction in 1965 the theory of fuzzy sets and relations was ranked to be some exotic field of research. The success during the past years even with consumer products involving fuzzy methods causes a rapidly growing interest of engineers and computer scientists in this field. Nevertheless, Goguen [12] generalized this concept in 1967 to L-fuzzy sets and relations for an arbitrary complete Brouwerian lattice L instead of the unit interval [0, 1] of the real numbers. He described one of his motivating examples as follows: A housewife faces a fairly typical optimization problem in her grocery shopping: she must select among all possible grocery bundles one that meets as well as several criteria of optimality, such as cost, nutritional value, quality, and variety. The partial ordering of the bundles is an intrinsic quality of this problem. (Goguen [12] 1967)

It seems to be unnatural – comparing apples to oranges – to describe the criteria of optimality by a linear ordering as the unit interval. Why should the nutritional value of a given product be described by 0.6 (instead of 0.65, or any other value from [0, 1]), and why should a product with a high nutritional value be better than a product with high quality since those criteria are usually incomparable? This observation has led to the theory of multiobjective or multicriteria optimization problems (cf. [13]). Instead of combining several criteria into a single number, and choosing the highest value, the concept of Pareto optimailty is used. In this approach the elements that are not dominated are taken for further considerations. Here an element x is said to dominate an element y if the value xi for each objective i is greater than or equal to the corresponding value yi of y. Traditional techniques of optimization and search have been applied ix

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in this area. Recently, even genetic algorithms have been used to search for multicriteria optima (e.g., [30, 31]). One important notion within fuzzy theory is 0-1 crispness. A 0-1 crisp set or relation is described by the property that their characteristic function supplies either the least element 0 or the greatest element 1 of the unit interval [0, 1] or more general the complete Brouwerian lattice L. The class of 0-1 crisp fuzzy sets or relations may be seen as the subclass of regular sets or relations within the fuzzy world. Especially in applications, this notion is fundamental. We want to demonstrate this by considering two examples. In fuzzy decision theory the basic problem is to select a specific element from a fuzzy set of alternatives. Therefore, several cuts are used [9, 24]. Basically, an α-cut of a fuzzy set M is a set N such that an element x is in N if, and only if, x is in M with a degree ≥ α. Analogously, an α-cut of a fuzzy relation R is a crisp relation S such that a pair of elements is related in S if, and only if, they are related in R with a degree ≥ α. Some variants of this notion may also be used. By definition, these cut operations are strongly connected to the notion of crispness. In particular, using the notion of crispness, one may define cut operations, and a cut operation naturally implies a notion of crispness. Another example might be the development of a fuzzy controller. Usually the output of the controller has to be a 0-1 crisp value since it is used to control some nonfuzzy physical or software system. Therefore, a procedure, called defuzzification, is applied to transform the fuzzy output into some 0-1 crisp value. This list of examples may be continued. The bottom line is that a convenient theory for L-fuzzy relations should be able to express the notion of crispness. Today, fuzzy theory as well as its application is usually formulated as a variation of set theory or some kind of many-valued logic (e.g., cf. [2, 14, 26]). Although many algebraic laws are developed, these formalizations are not algebraic themselves. But an algebraic description would have several advantages. Applications of fuzzy theory may be described by simple terms in this language. In this way, we get in some sense a denotational semantics of the application, and, hence, a mathematical theory to reason about notions as correctness. One may prove such properties using the calculus of the algebraic theory, which is quite often more or less equational. Furthermore, this denotational semantics may be used to get a prototype of the application. On the other hand, the calculus of binary relations has been investigated since the middle of the nineteenth century as an algebraic theory for logic and set theory [36, 37]. A first adequate development of such algebras was given by de Morgan and Peirce. Their work has been taken up and systematically extended by Schr¨ oder in [34]. More than 40 years later, Tarski started with the exhaustive study of relation algebras [35], and more generally, Boolean algebras with operators [17]. The papers above deal with relational algebras presented in their classical form. Elements of such algebras might be called quadratic or homogeneous; relations over a fixed universe. Usually a relation acts between two different

INTRODUCTION

xi

kinds of objects, e.g., between customers and products. Therefore, a variant of the theory of binary relations has evolved that treats relations as heterogeneous or rectangular . A convenient framework to describe such kind of typing is given by category theory [3, 10, 27, 28, 32, 33]. There are some attempts to extend the calculus of relations to the fuzzy world. In [21] the concept of fuzzy relation algebras was introduced as an algebraic formalization of fuzzy relations with sup-min composition. These algebras are equipped with a semiscalar multiplication, i.e., an operation mapping an element from [0, 1] and a fuzzy relation to a fuzzy relation. In the standard model this is done by componentwise multiplication of the real values. Fuzzy relation algebras and their categorical counterpart [11], so-called Zadeh categories, constitute a convenient algebraic theory for fuzzy relations. Using the semiscalar multiplication it is also possible to characterize 0-1 crisp relations. Unfortunately, there is no way to extend or modify this approach for L-fuzzy relations since for an arbitrary complete Brouwerian lattice such a semiscalar multiplication may not exist. Another approach is based on Dedekind categories and was introduced in [27]. It was shown that the class of L-fuzzy relations constitutes such a category. Unfortunately, the notion of 0-1 crispness causes some problems. Using the notion of scalar elements, i.e., a set of partial identities corresponding to the lattice L, several notions of crispness in an arbitrary Dedekind category were introduced in [11, 20]. It was shown that the notion of s-crispness as well as the notion of l-crispness coincides with 0-1 crispness under an assumption concerning the underlying lattice. This assumption is fulfilled by all linear orderings, e.g., the unit interval. Unfortunately, it was also shown that both classes of crisp relations are trivial if the underlying lattice is a Boolean lattice. Actually, it can be shown (Theorem 5.1) that the notion of 0-1 crispness cannot be formalized in the language of Dedekind categories, i.e., this theory is too weak to express this property. Therefore, an extended theory is needed: the theory of Goguen categories. In this book, we want to focus on Goguen categories introduced in [40] and some weaker structures as a convenient algebraic/categorical framework for Lfuzzy relations and their application in computer science. In particular, we are interested in the development process of fuzzy controllers using the method of Mamdani [25]. One major problem is to ensure totality of the controller, i.e., the controller should produce an output value for each input. If the controller is described by a relation R within a Goguen category, this property can be proved by showing I  R; R
, where I is the identity relation, ; is composition of relations, and R
is the converse of R. In most applications the controller is constructed by several components, which are combined using t-norms and t-conorms. The actual choice of the norms and their parameters is often done by experts using their experiences. Especially in complex applications, such a development process might easily lead to “holes” in the domain of the controller, i.e., to a partially defined controller. On the other hand, the relational description R of the controller can be parametric in those norms. From a generic

xii

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proof of R being total (which is necessarily parametric too) we can generate a set of conditions that have to be satisfied in order to ensure the totality of R. The expert may now select a convenient set of norms and parameters fulfilling these conditions. The controller generated is guaranteed to be total. We will give an example of the development process sketched above in Chapter 6. This book is organized as follows. In Chapters 1 and 2, we will introduce several mathematical notions as sets and lattices. The basic properties of Lfuzzy relations are investigated in Chapter 3. Afterwards, we will concentrate on the categorical description of relations, i.e., we will introduce several categories of relations in Chapter 4. Furthermore, their basic properties are proved, and their connections to L-fuzzy relations are studied. Chapter 5 is dedicated to Goguen categories and several weaker structures. We will prove some basic properties of those kinds of categories, focus on their representation theory, concentrate on derived connectives from a generalized notion of t-norms and t-conorms, and investigate the validity of equations in the substructure of crisp relations. In the last chapter we will give an applications of Goguen categories in computer science. We want to construct an L-fuzzy controller with respect to a given set of rules. This controller is not based on the unit interval. Furthermore, we will construct the controller without deciding in advance which norms and parameters should be used. From a generic proof of the totality of the controller we derive properties that can be used by an engineer to finally decide about those parameters. The writing of this book extended over almost 5 years. The early version grew out of the Habilitation thesis of the author in Munich, in 2003. In the following years several parts were revised and extended. In particular, Sections 5.1–5.4 were added in order to provide a more detailed overview of categories of L-fuzzy relations. The author would like to thank Gunther Schmidt and Yasuo Kawahara for their constant support during the Habilitation. Ivo D¨ untsch has to be thanked not only as a colleague, but also as a source of suggestions and advice. The RelMiCS (Relational Methods in Computer Science) community was not only a source of useful comments and criticism, but also of friendship. The first draft of this book was written in Munich. The author would like to thank the Department of Computer Science of the University of the Federal Armed Forces, Munich, Germany, for its support during this phase. The revision and the writing of the final version took place in St. Catharines. The author would like to thank the Department of Computer Science of Brock University, St. Catharines, Canada, for its support during the later phase. Last but not least, a special thanks goes to Ewa Orlowska, who suggested to publish the result of the Habilitation in a book.

1 SETS, RELATIONS, AND FUNCTIONS

Sets are fundamental in mathematics. In this chapter we briefly introduce the concepts and notations from set theory we will use throughout the book. We assume that the reader is familiar with the basic concepts of set theory. He may use some kind of naive set theory or a formal theory as ZF or ZFC [18], i.e., the Zermelo-Fraenkel axioms of set theory. As usual, we denote the fact that “x is an element of a set A” by x ∈ A. The set with no elements is called the empty set, and is denoted by ∅. If every element of a set A is also an element of the set B, we say A is a subset of B denoted by A ⊆ B. The set comprehension “the set of all elements of a set A fulfilling a predicate P” is denoted by {x ∈ A | P(x)}. If it is clear from the context or if it is insignificant which A is meant, we simply write {x | P(x)}. Union, intersection, and set difference are defined as usual: union intersection set difference

A∪B A∩B A\B

:= {x | x ∈ A or x ∈ B}, := {x | x ∈ A and x ∈ B}, := {x | x ∈ A and x ∈ B}.

The complement A of a set A in respect to a set B ⊇ A is just the set difference B \ A. The binary operations union and intersection may be generalized to an arbitrary set of sets as argument. Suppose Ai for i ∈ I are sets. Then,
Ai := {x | ∃i ∈ I : x ∈ Ai }, union i∈I  intersection Ai := {x | ∀i ∈ I : x ∈ Ai }. i∈I

1

2

GOGUEN CATEGORIES

The Cartesian product of two sets A and B is the set of all pairs (x, y) with x ∈ A and y ∈ B, and is denoted by A × B. The set of all subsets of A is called the power set of A, and is denoted by P(A). A binary relation R between two sets A and B is an element of P(A × B). A is called the source and B the target of R. If A = B, the relation R is also called an endorelation or homogeneous. To indicate that a binary relation R has source A and target B we usually write R : A → B. Apart from the set theoretic operations, we consider two further operations on binary relations. Let R be a relation between A and B and S between B and C. Then we define conversion composition

RT R◦S

:= {(y, x) | (x, y) ∈ R}, := {(x, z) | ∃y ∈ B : (x, y) ∈ R and (y, z) ∈ S}.

Due to the definition above, a composition Q ◦ R has to be read from the left to the right, i.e., first Q, and then R. We usually write R(x, y) instead of T (x, y) ∈ R. Notice that RT = R, and that composition is associative, i.e., for all relation Q : A → B, R : B → C and S : C → D we have (Q◦R)◦S = Q◦(R◦S). The identity relation IA on a set A is defined as the set {(x, x) | x ∈ A}. Then for all relations R : A → B we have R = IA ◦ R = R ◦ IB . The range or image ran(R) of a relation R : A → B is defined as the set {y ∈ B | ∃x ∈ A : R(x, y)}. Dually, the domain dom(R) of R is defined as the set {x ∈ A | ∃y ∈ B : R(x, y)}. Obviously, we have dom(R) = ran(RT ) and ran(R) = dom(RT ). A function f from A to B is a binary relation f : A → B which is univalent f (x, y1 ) and f (x, y2 ) implies y1 = y2 for all x ∈ A and y1 , y2 ∈ B, total for all x ∈ A there exists some y ∈ B so that f (x, y). Both properties may be expressed using the relational constructions. The first property is equivalent to f T ◦ f ⊆ IB , and the second to IA ⊆ f ◦ f T . The image of a function f : A → B will also be denoted by f (A). As indicated above, arbitrary binary relations are denoted by uppercase and functions by lowercase letters. If f is a function, we usually write f (x) to indicate the (necessarily unique) y so that f (x, y). The set of all functions from A to B will be denoted by A → B. A relation R : A → B is called (1) injective iff1 R(x1 , y) and R(x2 , y) implies x1 = x2 for all x1 , x2 ∈ A and y ∈ B, (2) surjective iff for all y ∈ B there exists some x ∈ A so that R(x, y), (3) bijective iff it is injective and surjective.

1 We

use the phrase “iff” as an abbreviation for “if and only if.”

SETS, RELATIONS, AND FUNCTIONS

3

Obviously, a relation is injective iff RT is univalent, surjective iff RT is total, and bijective iff RT is a function. A bijective function is also called a bijection. For historical reasons, the converse of a bijection f is denoted by f −1 . Notice that we have f −1 (f (x)) = x and f (f −1 (y)) = y for all bijections f : A → B and x ∈ A, y ∈ B. The cartesian product construction is associative up to a bijection, i.e., the function αA,B,C : (A × B) × C → A × (B × C) defined by αA,B,C ((a, b), c) = (a, (b, c)) is bijective for arbitrary sets A, B, and C. We define n-ary products by iterating binary products. Due to the associativity this is well-defined. Given an n-ary function f : A1 × · · · × An → B we will use the extended set comprehension scheme {f (x1 , . . . , xn ) | x1 ∈ A1 , . . . , xn ∈ An } as an abbreviation for {y | ∃x1 ∈ A1 · · · ∃xn ∈ An : y = f (x1 , . . . , xn )}. The concept of a Cartesian product of sets may be further  generalized using functions. Let Ai for i ∈ I be sets. The I-indexed product Ai of the sets i∈I
Ai so that f (i) ∈ Ai Ai is defined as the set of all functions f from I to i∈I

for all i ∈ I. For a finite set I = {1, . . . , n} we get the usual n-ary product of if Ai = Aj =: A for all i, j ∈ I, i.e., all components of the A1 , . . . , An . Notice,  Ai is just I → A. product are equal, i∈I

We introduce some notations for commonly known sets: B set of Boolean values {t, f} or {0, 1} (t = 1 = ˆ true, f = 0 = ˆ false), N set of the natural numbers {0, 1, 2, . . .}, N∞ set of the natural numbers with an additional greatest element ∞, R set of real numbers, [0, 1] = {x ∈ R | 0 ≤ x ≤ 1} unit interval of real numbers. The concept of a homomorphism between structured sets, i.e., sets with some operations and/or relations defined on them, is usually somewhat informal. One may obtain a formal definition using the theory of universal algebras. In this book a homomorphism is a function reflecting the structure of the corresponding sets. For example, a homomorphism between the semigroups (G1 , +1 , 01 ) and (G2 , +2 , 02 ) is a function f : G1 → G2 respecting the group operation + and the neutral element 0, i.e., f (x +1 y) = f (x) +2 f (y) for all x, y ∈ G1 and f (01 ) = 02 . As usual, a bijective homomorphism f so that f −1 is also a homomorphism is called an isomorphism. In this situation the source and the target of f are called isomorphic.

2 LATTICES

In this chapter we want to introduce basic concepts from lattice theory we will need throughout this book. For a comprehensive introduction to this theory we refer to [4, 16, 29]. A very natural concept is a (partially) ordered set (or poset). Elements of such a set may be related to each other by notion of “being smaller or equal.” Formally, a poset is a set P with a binary relation ≤ on it so that reflexive transitive antisymmetric

x ≤ x for all x ∈ P , if x ≤ y and y ≤ z, then x ≤ z for all x, y, z ∈ P , if x ≤ y and y ≤ x, then x = y for all x, y ∈ P .

A poset P is called linear iff x ≤ y or y ≤ x holds for all x, y ∈ P . The set of real numbers R with the usual ordering is a linear poset. Obviously, the unit interval [0, 1] of the real numbers is also a linear poset. The power set P(A) of a set A with more than one element together with set-inclusion ⊆ is a standard example of a nonlinear poset. Finite posets are often visualized by their Hasse diagrams. A Hasse diagram is a drawing of the transitive reduction of the partial order, i.e., each element of the set is represented as a vertex, and the order relationship by upward oriented edges. The graph has an edge from x to y if x ≤ y and x = y, and there is no z with z = x and z = y such that x ≤ z ≤ y. In this case, y is said to cover x, or y is an immediate successor of x. Figure 2.1 shows the Hasse diagram of 5

6

GOGUEN CATEGORIES

{a}

{a, b} GG v GG v v v II II I

∅

Figure 2.1.

v vv vv

3 {b}

2 1

Two Hasse diagrams.

the powerset of {a, b} and the linear order of the subset {1, 2, 3} of the natural numbers. If we define x  y iff y ≤ x for a poset P , we will obtain a new poset, called the reversed ordering on P . A function f between two posets P1 and P2 is called monotone iff x ≤ y implies f (x) ≤ f (y). Notice that the symbol ≤ in the previous property refers to different partial orderings, namely to that of P1 and P2 , respectively. Monotone functions are the homomorphisms between posets as introduced in the previous ≤ chapter. We will denote the set of monotone functions from P1 to P2 by P1 → P2 . Dually, an antitone function f between two posets P1 and P2 is a function so that x ≤ y implies f (y) ≤ f (x) for all x, y ∈ P1 . Therefore, an antitone function is a monotone function from P1 to the reversed ordering on P2 . The ≥ set of antitone functions from P1 to P2 is denoted by P1 → P2 . Since monotone and antitone functions are defined dually, many properties hold for both sets of functions. Therefore, we will refer to either the set of monotone or the set of ∗ antitone functions by P1 → P2 . Consider the powerset P(A) of the set A = {a, b}, the linear poset P = {1, 2, 3, 4} and the function f : P(A) → P visualized in Figure 2.2. This function is monotone. Furthermore, f is also bijective, but its converse f −1 : P → P(A) is not monotone. Indeed, we have 2 ≤ 3 but f −1 (2) = {b}
{a} = f −1 (3). The cartesian product of posets can be ordered componentwise. Figure 2.3 shows the Hasse diagram of the product of the posets from Figure 2.1.

gggg3 4 ggggg g g g g gggg {a, b} eeee2 3 G GGeeeeeeeeeee v v G e v e veeeeeee f /2 {b} {a} e II v v II I vvv /1 ∅ Figure 2.2.

A monotone function.

LATTICES

7

({a, b}, 3) :: ::   :   ({a, b}, 2) :::  ::   ::: ::    ::  :   ::   :  ({a}, 3)  ({a, b}, 1) : ({b}, 3) LL :: : :: rrrr:::  LLLL  r :   r:  ({a}, 2)  (∅, 3) ::: ({b}, 2) LL : rrr:::  LLLL  rrr ({a}, 1) (∅, 2) ({b}, 1) LLL r r LLL r rrr (∅, 1) Figure 2.3.

Hasse diagram of product of the posets from Figure 2.1.

Theorem 2.1 Let Pi be posets for i ∈ I. Then



Pi together with the relation

i∈I

f ≤ g ⇐⇒ ∀i ∈ I : f (i) ≤ g(i) is again a poset. The proof of the theorem above is an easy exercise and, therefore, omitted. Notice that n-ary products and the set of all functions from I to a poset P are special cases of the previous theorem. An upper bound of a subset M of a poset P is an element u ∈ P so that x ≤ u for all x ∈ M , and dually a lower bound of M is an element l ∈ P so that l ≤ x for all x ∈ M . A greatest element of M is an element of M , which is also an upper bound of M . Dually, a least element of M is defined as an element of M , which is also a lower bound of M . It is easy to verify that greatest and least elements are unique (if they exist) so that we can refer to them as the greatest and the least element of M . The least upper bound of M is the least element of the set of upper bounds of M , and the greatest lower bound of M is the greatest element of the set of lower bounds of M .1 Figure 2.4 shows two posets with a subset M and its lower bounds. Notice that the second order has exactly one additional element. As a consequence, the first ordering does not provide a greatest lower bound of M whereas the second does. Definition 2.2 A lower semilattice L is a poset so that every pair of elements x and y has a greatest lower bound or meet denoted by x∧y. It is called complete 1 The least upper bound and the greatest lower bound of M may not exist. But, if they exist, they are unique since they are defined as greatest resp. least elements.

8

GOGUEN CATEGORIES

=< =< ?> •7 • • • M 77  :; 89 89 777  :; 77  77    77  77  =< ?> 7  77  / 7  •7 greatest lower bound of M 7  777  777   77 77   77 =< 77  ?>    • • • • lower bounds of M 89 :; 89 :; ?>

Figure 2.4.

Example of lower bounds.

 iff every subset M = ∅ of L has a greatest lower bound denoted by M . A subset lower subsemilattice L of a (complete) lower semilattice  L is called a (complete)  for all x, y ∈ L we have x∧y ∈ L resp. of L iff L is closed under ∧ resp. , i.e.,    for all subsets M = ∅ of L we have M ∈ L . Dually, an upper semilattice L is a poset so that every pair of elements x and y has a least upper bound or join denoted by x ∨ y. It is called  complete iff every subset M = ∅ of L has a least upper bound denoted by M . A subset L of a (complete) upper semilattice L is subsemilattice of L iff L is closed under ∨ resp.  called a (complete) upper    , i.e., for  all x, y ∈ L we have x ∨ y ∈ L resp. for all subsets M = ∅ of L we have M ∈ L .  Notice that a complete lower semilattice has a least element  0 = L. Dually, a complete upper semilattice has a greatest element 1 = L. The posets of Figure 2.1 and 2.3 are lower semilattices as well as upper semilattices, and the posets of Figure 2.4 are not. Another example of a complete lattice is the powerset of a set A. In this example, the greatest lower bound of a set of subsets of A is given by the intersection, and the least upper bound by the union of the sets. An element x of a lower semilattice L with least element 0 is called linear iff x ∧ y = 0 implies y = 0 for all y ∈ L. Notice that iff L has at least two elements 0 is not linear. In a linear poset every nonzero element is linear. The converse is, in general, not true. In the nonlinear lower semilattice {0, a, b, c} induced by Figure 2.5 all nonzero elements are linear. c

b >> >> a 0 Figure 2.5.

A semilattice so that all nonzero elements are linear.

9

LATTICES

The binary operations ∧ and ∨ of the corresponding semilattices fulfill the following properties: idempotent x ∧ x = x, commutative x ∧ y = y ∧ x, associative x ∧ (y ∧ z) = (x ∧ y) ∧ z, consistent x ∧ y = x ⇔ x ≤ y, monotone ∀y ≤ z : x ∧ y ≤ x ∧ z,

x ∨ x = x, x ∨ y = y ∨ x, x ∨ (y ∨ z) = (x ∨ y) ∨ z, x ∨ y = y ⇔ x ≤ y, x ∨ y ≤ x ∨ z.

The proof of the properties above is an easy exercise, and, therefore, omitted. Furthermore, the formulas above may be used to get an algebraic definition of semilattices, e.g., every set with a binary operation ∧, which is idempotent, commutative, and associative is a lower semilattice by defining x ≤ y iff x∧y=x. A monotone function f between two upper/lower semilattices L1 and L2 fulfills f (x ∧ y) ≤ f (x) ∧ f (y) resp. f (x) ∨ f (y) ≤ f (x ∨ y) since f (x ∧ y) ≤ f (x) and f (x ∧ y) ≤ f (y) resp. f (x) ≤ f (x ∨ y) and f (y) ≤ f (x ∨ y). Dually, for an antitone function g we have g(x∨y) ≤ g(x)∧g(y) resp. g(x)∨g(y) ≤ g(x∧y). If semilattices, the more general inequalities L1and L2 arecomplete lower/upper   f (x) resp. f (x) ≤ f ( M ) for all ∅ = M ⊆ L1 follow. f( M) ≤ x∈M x∈M    f (x) and f (x) ≤ Dually, for an antitone function we have f ( M ) ≤ x∈M x∈M  f ( M ) for all ∅ = M ⊆ L1 . Theorem2.3 Let Li be (complete) lower/upper semilattices for i ∈ I. Then Li together with either the operation the poset i∈i

(f ∧ g)(i) :=f (i) ∧ g(i), or

(f ∨ g)(i) := f (i) ∨ g(i)

is again a (complete) lower/upper semilattice, respectively. Furthermore, for ∗ a poset P and a (complete) lower/upper semilattice L the posets P → L are (complete) lower/upper subsemiattices of P → L, respectively. ≤

Proof. We just prove that P → L is a lower subsemilattice of P → L for a poset P and a lower semilattice L. Recall, that P → L is a special case of a product of semilattices. Suppose f and g are monotone and x ≤ y. Then we have (f ∧ g)(x) = f (x) ∧ g(x) ≤ f (y) ∧ g(y) = (f ∧ g)(y) by the monotonicity of ∧, and, hence, f ∧ g monotone.




A function f between two lower semilattices L1 and L2 fulfilling f (x ∧ y) = f (x) ∧ f (y) is called a lower semilattice homomorphism. Upper semilattice homomorphisms are defined dually. Since x ≤ y iff x ∧ y = x, resp. x ≤ y iff x ∨ y = y, all lower/upper semilattice homomorphisms are monotone. With respect to antitone functions we call a function f between a lower semilattice L1 and an upper semilattice L2 with f (x ∧ y) = f (x) ∨ f (y) a lower

10

GOGUEN CATEGORIES

co-semilattice homomorphism. Again, upper co-semilattice homomorphisms are defined dually. A similar argument shows that these functions are antitone. A complete lower/upper semilattice homomorphism f between two complete lower/upper semilattices L1 and L2 is a function fulfilling      f (x) resp. f f (x) f M = M = x∈M

x∈M

for all subsets M = ∅ of L1 . Obviously, every complete lower/upper semilattice homomorphism is a lower/upper semilattice homomorphism. Complete upper semilattice homomorphisms are also called continuous. Complete lower/upper co-semilattice homomorphisms are defined similarly. They have to fulfill      f (x) resp. f M = f (x). f M = x∈M

x∈M

The function f from Figure 2.2 is neither a lower nor an upper semilattice homomorphism since f ({a} ∩ {b}) = f (∅) = 1 = 2 = 2 ∧ 3 = f ({a} ∧ f ({b}) and f ({a} ∪ {b}) = f ({a, b}) = 4 = 3 = 2 ∨ 3 = f ({a} ∨ f ({b}). Definition 2.4 A lattice L is a poset that is a lower and an upper semilattice. A subset L of L is a called a sublattice of L iff L is a lower and an upper subsemilattice of L. Before we define complete lattices we want to prove the following lemma: Lemma 2.5 Let L be a lattice. Then we have the following: (1) If L has a greatest element 1 and is complete as a lower semilattice, then it is complete as an upper semilattice. (2) If L has a least element 0 and is complete as an upper semilattice, then it is complete as a lower semilattice. Proof. We just prove (1) since (2) is shown dually. Suppose M is a subset of L.  Let N be the set of upper bounds of M . Then N = ∅ since 1 ∈ N so that x := N is well-defined. Furthermore, x is an upper bound of M since it is a meet of upper bounds of M . Last but not least, by definition it is the least upper bound of M .
The previous lemma motivates the following definition of complete lattices: Definition 2.6 A lattice L is called complete iff it is either a complete lower semilattice with a greatest element or a complete upper semilattice with a least element. A subset L of L is called a complete sublattice of L iff it is a complete lower and a complete upper subsemilattice.

LATTICES

11

Notice that complete lattices have aleast element 0 and a greatest element    1. They fulfill 0 = L = ∅ and 1 = L = ∅. The binary operations ∧ and ∨ fulfill the following absorption laws: x ∧ (y ∨ x) = x

and

x ∨ (y ∧ x) = x

for all x and y. Again, the proof is an easy exercise, and, therefore, omitted. The standard example of a complete lattice is the powerset P(A) of a set A with the usual operations ∩ and ∪. Furthermore, the unit interval [0, 1] is also a complete lattice with minimum as meet and maximum as join. The real numbers itself do not constitute a complete lattice since there is no least and greatest element. Nevertheless, they form a lattice. Theorem 2.7 Let Li for i ∈ I be (complete) lattices. Then the poset



Li is

i∈I

again a (complete) lattice. Furthermore, for a poset P and a (complete) lattice ∗ L the posets P → L are (complete) sublattices of P → L. Again, the proof of the theorem above is omitted. A lattice homomorphism is a function that is a lower and an upper semilattice homomorphism. In the case of completeness the defining equations are also required for M = ∅, i.e., a complete lattice homomorphism is a function f : L1 → L2 so that      f (x), and f f (x) f M = M = x∈M

x∈M

  for all subsets of L1 . Notice that we have f (1) = f ( ∅) = ∅ = 1 and  M f (0) = f ( ∅) = ∅ = 0. Again, every complete lattice homomorphism is a lattice homomorphism. Later on, complete upper co-semilattice homomorphisms with f (0) = 1 will play an important role. As an abbreviation we call them antimorphisms. Notice that we could equivalently extend the defining equation for complete upper co-semilattice homomorphisms to the case M = ∅ instead of requiring f (0) = 1. Sometimes, we are interested in subsets of complete lattices, which are given as an image of a special class of functions. Definition 2.8 Let f : P → P be an endofunction on a poset P . Suppose f fulfills (monotone) f is monotone, (idempotent) f (f (x)) = f (x) for all x ∈ P . Then f is called a closure operation iff x ≤ f (x) (extensive), and a kernel or coclosure operation iff f (x) ≤ x (contractive) for all x ∈ P . The following properties are stated for closure operations. By duality, a version of the corresponding properties for kernel operations is obvious.

12

GOGUEN CATEGORIES

hh;386 7 hmhmhmphmpwmpwmpw >>> h h h pw hh >> hhhh mmmpp w > hhhh mmmmpmppwpwww h h h mmpppp www 4 > m 5 7 h> m >> 6 >> >>mmmm ppp ww >> w m p > > m w >> >> mm pp>>pp ww m m w w pp mmm h3 h3 h3 5> 4> 6 hhhh 1h>hhh 2hhhh 3 >> >> hhhhhhhhhhhhh>h>h>hhh h h > >> h >> > hhhhh>h> hhhhhh hhhhhh hhhh hhhhh hhhhh h h3 3 2 1> hhhh 0 >> hhhh h h >> h hh > hhhh hhhh h 0 Figure 2.6.

7> >> >> > 1> 2 >> >> > 0

3

A closure operation and the lattice of closed elements.

A closure operation induces a substructure of the given lattice. Figure 2.6 shows a closure operation and the lower subsemilattice of closed elements. Notice that this structure is, in fact, a lattice (cf. Theorem 2.10) but not a sublattice. The join of the elements 1 and 2 in the original lattice is 4 whereas their join among the closed elements is 7. Lemma 2.9 Let L be a complete lattice and f a closure operation. Then f (L) is a complete lower subsemilattice. Proof. Let M be a subset of f (L). Then for every y ∈ M there is an x so that y = f (x). This implies f (y) = f (f (x)) = f (x) = y since f is idempotent. We conclude that    f (x) f is monotone f M ≤ x∈M

=



≤f

M 

 M ,

computation above f extensive

   which shows f ( M ) = M , and, hence, M ∈ f (L).




Since f is extensive 1 ∈ f (L). The previous lemma together with Lemma 2.5 imply that f (L) is a complete lattice. The next theorem gives us a little more insight into the structure, i.e., the join operation, of this lattice. Theorem 2.10 If L is a complete lattice and f a closure operation, then the image f (L) of f together with ∧ and x ∨· y := f (x ∨ y) is a complete lattice.  Proof. By Lemma 2.9 it remains to show that f ( M ) is the least upper bound of M in f (L). Since f is monotone it is an upper bound. Suppose y is an element of f (L) and  another upper  bound of M . Then there is an x with f (x) = y, and we have M ≤ y since M is the least upper bound of M in L.  We get f ( M ) ≤ f (y) = f (f (x)) = f (x) = y, and, hence, the assertion.


LATTICES

13

Last but not least, we want to prove a lemma about isomorphisms. Some properties of f −1 follow from the corresponding properties of f and vice versa. This gives us some flexibility in proving properties of bijective functions. Lemma 2.11 Let L1 and L2 be (complete) lattices and f : L1 → L2 a bijection so that (1) either f or f −1 is a (complete) lower semilattice homomorphism, and (2) either f or f −1 is a (complete) upper semilattice homomorphism. Then f is an isomorphism. Proof. We just prove that f −1 is an upper semilattice homomorphism implies that f is as well. The other properties are shown analogously. Consider the computation f (x ∨ y) = f (f −1 (f (x)) ∨ f −1 (f (y))) = f (f

−1

f bijective f −1 is an upper semilattice homomorphism

(f (x) ∨ f (y)))

= f (x) ∨ f (y).

f bijective

In the case of completeness the assertion is proved similarly. 2.1




GALOIS CORRESPONDENCES AND RESIDUATED OPERATIONS

We now describe a fundamental construction, which arises from pairs of functions between posets. This notion originates from a revolutionary method introduced by Galois. He related every field extension to a group in order to study properties of the field extension by studying the properties of the related group. We shall have nothing further to say about this connection, called the Galois correspondence. Our attention is devoted to connections between functions in a much broader setting. Definition 2.12 Let P1 and P2 be posets. A pair of functions f : P1 → P2 and g : P2 → P1 is called a Galois correspondence between P1 and P2 iff f (x) ≤ y

⇐⇒

x ≤ g(y)

for all x ∈ P1 and y ∈ P2 . f is called the lower and g the upper adjoint of the Galois correspondence. Notice that the definition of a Galois correspondence found in the literature [4] often requires y ≤ f (x) ⇐⇒ x ≤ g(y). Our notion is in fact a Galois correspondence between P1 and the reversed ordering on P2 in the sense above. Since all Galois correspondences considered in this book are of this special form we use this slightly modified definition.

14

GOGUEN CATEGORIES

We will see later that Galois correspondences and resituated operations (cf. Definition 2.16) arise naturally in certain lattices so that we do not provide an explicit example. Lemma 2.13 Let P1 and P2 be posets and (f, g) a Galois correspondence between P1 and P2 . Then we have (1) x ≤ g(f (x)) and f (g(y)) ≤ y for all x ∈ P1 and y ∈ P2 , (2) f and g are monotone, (3) f (x) = f (g(f (x))) and g(y) = g(f (g(y))) for all x ∈ P1 and y ∈ P2 , (4) f ◦ g, i.e., the operation x → g(f (x)), is a closure and g ◦ f , i.e., the operation x → f (g(x)), is a kernel operation. Proof. (1) By definition x ≤ g(f (x)) is equivalent to f (x) ≤ f (x), and f (g(y)) ≤ y is equivalent to g(y) ≤ g(y). (2) Suppose x1 ≤ x2 . Then by (1) we have x1 ≤ x2 ≤ g(f (x2 )), and, hence, f (x1 ) ≤ f (x2 ). The second assertion is shown analogously. (3) We just show the first assertion. The inclusion ≥ follows from (1), and ≤ from (1) and (2). (4) Again, we just show the first assertion. f ◦ g is monotone since f and g are. Furthermore, it is extensive by (1) and idempotent by (3).
Property (2) can be strengthened if P1 and P2 are lattices. Lemma 2.14 Let (f, g) be a Galois correspondence between two lattices. Then f is an upper and g a lower semilattice homomorphism. Proof. The first assertion follows from f (x ∨ y) ≤ u ⇔ x ∨ y ≤ g(u) ⇔ x ≤ g(u) and y ≤ g(u)

Galois correspondence

⇔ f (x) ≤ u and f (y) ≤ u ⇔ f (x) ∨ f (y) ≤ u,

Galois correspondence

and the second follows analogously.




The previous lemma motivates an interesting characterization of Galois correspondences within complete lattices. The corresponding properties provide also necessary and sufficient conditions for the existence of an upper or lower adjoint of a given function.

LATTICES

15

Theorem 2.15 Let L1 and L2 be complete lattices, and let f : L1 → L2 and g : L2 → L1 be functions. Then the following properties are equivalent: (1) (f, g) is a Galois correspondence, (2) f is a complete upper semilattice homomorphism and g(y) = {u ∈ L1 | f (u) ≤ y}, (3) g is a complete lower semilattice homomorphism and  f (x) = {v ∈ L2 | x ≤ g(v)}.   f (x) Proof. (1)⇒(2): Let M be a subset of L1 . The inclusion f ( M ) ≥ x∈M  f (x). Then we is trivial since f is monotone by Lemma 2.13 (2). Let u := x∈M  have f (y) ≤ u, and, hence,  y ≤ g(u)  for all y ∈ M . This implies M ≤ g(u), f (x). The second assertion follows from which is equivalent to f ( M ) ≤ x∈M   {u ∈ L1 | f (u) ≤ y} = {u ∈ L1 | u ≤ g(y)} = g(y). (2)⇒(1): We have to show f (x) ≤ y ⇐⇒ x ≤ {u ∈ L1 | f (u) ≤ y}.  Suppose f (x) ≤y. Then we  immediately conclude that x ≤ {u ∈ L1 | f (u) ≤y}. On the other hand, x ≤ {u ∈ L1 | f (u) ≤ y} implies   f (x) ≤ f {u ∈ L1 | f (u) ≤ y} = {f (u) ∈ L1 | f (u) ≤ y} ≤ y since f is an upper semilattice homomorphism. (1)⇔(3) is shown analogously.




The notion of a pair of residuated operations is a slight generalization of the notion of a Galois correspondence. Definition 2.16 Let P1 , P2 and P3 be posets. A triple of functions f : P1 × P2 → P3 , gl : P3 ×P2 → P1 and gr : P1 ×P3 → P2 is called a triple of residuated operations iff f (u, v) ≤ w

⇐⇒

u ≤ gl (w, v)

⇐⇒

v ≤ gr (u, w)

for all u ∈ P1 , v ∈ P2 and w ∈ P3 . Obviously, if f, gl and gr constitute a triple of residuated operations, then for u ∈ P1 and v ∈ P2 the pair of functions hu (x) := f (u, x) and ku (y) := gr (u, y) resp. lv (x) := f (x, v) and mv (y) := gl (y, v) are Galois correspondences. Therefore, we call f the lower, gl the upper left and gr the upper right adjoint of the triple of residuated operations. Furthermore, for all w ∈ P3 the pair of

16

GOGUEN CATEGORIES

functions rw (x) := gl (w, x) and sw (y) := gr (y, w) constitutes a Galois correspondence between P2 and the reversed order on P1 . These observations give us the following corollary: Corollary 2.17 Let P1 , P2 and P3 be posets and (f, gl , gr ) a triple of residuated operations. Then we have (1) v ≤ gr (u, f (u, v)) and f (u, gr (u, w)) ≤ w for all u ∈ P1 , v ∈ P2 and w ∈ P3 , (2) u ≤ gl (f (u, v), v) and f (gl (w, v), v) ≤ w for all u ∈ P1 , v ∈ P2 and w ∈ P3 , (3) v ≤ gr (gl (w, v), w) and u ≤ gl (w, gr (u, w)) for all u ∈ P1 , v ∈ P2 and w ∈ P3 , (4) f is monotone in both arguments, (5) gl is monotone in the first and antitone in the second argument, (6) gr is antitone in the first and monotone in the second argument. Obviously, there is a version of Lemma 2.14 for residuated operations. Corollary 2.18 Let (f, gl , gr ) be a triple of residuated operations between lattices. Then f is an upper semilattice homomorphism in both arguments, gl is a lower semilattice homomorphism in the first and an upper co-semilattice homomorphism in the second argument and gr is an upper co-semilattice homomorphism in the first and a lower semilattice homomorphism in the second argument. As above, we obtain an interesting characterization of residuated triples within complete lattices. Corollary 2.19 Let L1 , L2 and L3 be complete lattices, and let f : L1 × L2 → L3 , gl : L3 × L2 → L1 and gr : L1 × L3 → L2 be functions. Then the following properties are equivalent: (1) (f, gl , gr ) is a triple of residuated operations, (2) f is a complete upper semilattice homomorphism in both arguments,  gl (w, v) = {u ∈ L1 | f (u, v) ≤ w} and gr (u, w) = {v ∈ L2 | f (u, v) ≤ w}, (3) gl is a complete lower semilattice homomorphism in the first and a complete upper co-semilattice homomorphism in the second argument,  f (u, v) =  {w ∈ L3 | u ≤ gl (w, v)} and gr (u, w) = {v ∈ L2 | u ≤ gl (w, v)}, (4) gr is a complete upper co-semilattice homomorphism in the first and a complete lower semilattice homomorphism in the second argument,  f (u, v) = {w ∈ L3 | u ≤ gr (u, w)} and gl (u, w) = {u ∈ L1 | v ≤ gr (u, w)}. If f is commutative, i.e., P1 = P2 and f (u, v) = f (v, u) for all u, v ∈ P1 , then gl (w, v) = gr (v, w) =: g(v, w) and we call g just the upper adjoint.

LATTICES

2.2

17

DISTRIBUTIVE LATTICES

In the next three chapters, we want to focus on special classes of lattices. The first class we are interested in is the class of distributive lattices. First of all, there is a weak version of distributivity, which is valid in all lattices. Lemma 2.20 Let L be a lattice. Then we have for all elements u, v, w ∈ L (1) u ∧ (v ∨ w) ≥ (u ∧ v) ∨ (u ∧ w), (2) u ∨ (v ∧ w) ≤ (u ∨ v) ∧ (u ∨ w). Proof. We just prove (1). Obviously, u ∧ v ≤ u and u ∧ v ≤ v ≤ v ∨ w, and, hence, u ∧ v ≤ u ∧ (v ∨ w). Analogously, we get u ∧ w ≤ u ∧ (v ∨ w), which implies the assertion.
An important class of lattices is defined by the law of distributivity, i.e., where the inequalities of the previous lemma are replaced by equalities. Lemma 2.21 In any lattice L, the following identities are equivalent: (1) u ∧ (v ∨ w) = (u ∧ v) ∨ (u ∧ w) for all u, v, w ∈ L, (2) u ∨ (v ∧ w) = (u ∨ v) ∧ (u ∨ w) for all u, v, w ∈ L. Proof. We prove that (1) implies (2). The converse implication is shown analogously. For all u, v, w ∈ L, we have (u ∨ v) ∧ (u ∨ w) = ((u ∨ v) ∧ u) ∨ ((u ∨ v) ∧ w) = u ∨ (w ∧ (u ∨ v)) = u ∨ ((w ∧ u) ∨ (w ∧ v))

by (1) absorption and commutative law by (1)

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

associative law absorption law




It is easy to verify that the identities of the previous lemma do not hold in all lattices. Definition 2.22 A lattice L is called distributive iff Lemma 2.21(1) (or equivalently Lemma 2.21 (2)) holds. A completelattice iscalled completely upward(x ∧ y) and completely distributive iff it is distributive and x ∧ M = y∈M   downward-distributive iff it is distributive and x ∨ M = (x ∨ y) holds y∈M

for all x ∈ L and M ⊆ L. A completely distributive lattice is a lattice that is completely upward- and downward-distributive. Notice that the generalized distribution laws given in the previous definition are not equivalent.

18

GOGUEN CATEGORIES

• ~ // ~ // ~ // • /

•@ • @@ ~~ ~ •

Figure 2.7.

• ~ @@@ ~ ~ • •@ • @@ ~~ ~ •

Two nondistributive lattices.

Figure 2.7 shows two nondistributive lattices. It can be shown that those lattices actually characterize the class of nondistributive lattices. However, the weak version of distributivity (cf. Lemma 2.20) is valid, of course. Completely upward-distributive lattices are also called complete Brouwerian lattices or complete Heyting algebras. Usually, Brouwerian lattices or Heyting algebras are defined via relative pseudo-complements. Later on, we will establish the connection between completely upward-distributive and complete Brouwerian lattices. Notice that with a distributive complete lattice we refer to a lattice, which is distributive and complete but neither necessarily completely upward- nor downward-distributive. The power set of a set as well as the unit interval [0, 1] is a completely distributive lattice. Theorem 2.23 Let Li for  i ∈ I be (completely upward-/downward-) distribuLi . tive lattices. Then so is i∈I

Again, the proof of the theorem above is an easy exercise, and, therefore, omitted. ∗ Notice that the (complete) sublattices P → L for a poset P and a (completely upward-/downward-) distributive lattice L are again (completely upward-/ downward-) distributive. This follows immediately from the fact that the required properties are true in P → L. 2.3

BROUWERIAN LATTICES

Another important class of lattices is motivated by the possibility to use complements or negation, i.e., elements similar to the complement A of a set A. First, we want to consider a weak version of relative complements. Definition 2.24 Let L be a lattice and x, y ∈ L. A relative pseudo-complement y:x of x in y is an element so that u ≤ y:x ⇐⇒ x ∧ u ≤ y for all u ∈ L. A lattice in which for every pair of elements the relative pseudocomplement exist is called a Brouwerian lattice or a Heyting algebra. Notice that ∧ and : induces a triple of residuated operations since ∧ is commutative. Furthermore, any Brouwerian lattice has a greatest element.

LATTICES

19

Consider x:x for an element x. Then y ≤ x:x iff x ∧ y ≤ x, which is true for all elements y. A Brouwerian lattice homomorphism f is a lattice homomorphism between Brouwerian lattices so that f (y:x) = f (y):f (x) for all x and y. As mentioned above, completely upward-distributive lattices are also called complete Brouwerian lattices or complete Heyting algebras. The reason for that is given by the next theorem. Theorem 2.25 (1) Every Brouwerian lattice is distributive. (2) A complete lattice is a Brouwerian lattice iff it is completely upwarddistributive. Proof. (1) For elements u, v, w ∈ L define x := (u ∧ v) ∨ (u ∧ w), and consider x:u. Since u ∧ v ≤ x and u ∧ w ≤ x we have v ≤ x:u and w ≤ x:u. We conclude that u ∧ (v ∨ w) ≤ u ∧ (x:u) ≤ x = (u ∧ v) ∨ (u ∧ w). By Lemma 2.20 the assertion follows immediately. (2) If L is a complete Brouwerian lattice, we have u∧v ≤w

⇐⇒

v ≤ w:u

⇐⇒

u ≤ w:v.

Therefore, ∧ and : induce a triple of residuated operations. By Corollary 2.19 (1)⇒(2) ∧ is a complete upper semilattice homomorphism, and, hence, L completely upward-distributive. For the other implication, assume∧ is a complete upper semilattice homomorphism. Then we define w:u := {v ∈ L | u ∧ v ≤ w}. Corollary 2.19 (2)⇒(1) implies the assertion.
The previous theorem also shows that every finite distributive lattice is actually a Brouwerian lattice. Figure 2.8 gives an example of such a lattice. If we want to compute the relative pseudo-complement of 2 in 1, we have to consider the inclusion 2 ∧ x ≤ 1. Since 2 ∧ 0 = 2 ∧ 1 = 2 ∧ 3 = 0 and 2 ∧ 2 = 2 ∧ 4 = 2 ∧ 5 = 2 ≤ 1 the inclusion is satisfied for the elements from the set {0, 1, 3}. The join of those elements is 3 so that we obtain 1:2 = 3. Unfortunately, not every complete lattice homomorphism between complete Brouwerian lattices is a complete Brouwerian lattice homomorphism but we 5
===

3= 4 ==

===
2 1= ==


0 Figure 2.8.

A finite distributive lattice.

20

GOGUEN CATEGORIES

have 

 {u ∈ L1 | x ∧ u ≤ y}

f (y:x) = f = {f (u) | x ∧ u ≤ y})

≤ {f (u) | f (x) ∧ f (u) ≤ f (y)}

= ≤



f complete lattice homomorphism f lower semilattice homomorphism

{v ∈ f (L1 ) | f (x) ∧ v ≤ f (y)}

{v ∈ L2 | f (x) ∧ v ≤ f (y)}

f (L1 ) ⊆ L2

= f (y):f (x). The computation above also shows that if f is bijective, it is indeed a Brouwerian lattice homomorphism. Furthermore, consider the computation f −1 (y:x) = f −1 ((f (f −1 (x))):(f (f −1 (y))) = f −1 (f (f −1 (x):f −1 (y))) =f

−1

(x):f

−1

f homomorphism

(y).

This implies that the notions of complete lattice isomorphisms and complete Brouwerian lattice isomorphisms are equivalent. If there is a least element 0, the relative pseudo-complement 0:x of x in 0 is called the pseudo-complement of x, and denoted by ¬x. Notice that by definition an element x of a Brouwerian lattice is linear iff ¬x = 0. 2.4

BOOLEAN ALGEBRAS

The unit interval [0, 1] is a Brouwerian lattice since it is completely upwarddistributive. In this lattice we have ¬x = 0, and, hence, x ∨ ¬x = x for all elements x = 0. This shows that a Brouwerian lattice need not be complemented, i.e., need not be a Boolean algebra. Another example is the lattice from Figure 2.8. In this lattice we have ¬1 = 2 but 1 ∨ ¬1 = 1 ∨ 2 = 4 = 5. Definition 2.26 A Brouwerian lattice L with least element 0 and a greatest element 1 is called a Boolean algebra iff x ∨ ¬x = 1 for all elements x ∈ L. The element ¬x is called the complement of x, and denoted by x. Notice that in a Boolean algebra y:x = x∨y since x∧(x∨y) = (x∧x)∨(x∧y) = x∧y ≤ y and x∧u ≤ y implies u ≤ x∨u = (x∨x)∧(x∨u) = x∨(x∧u) ≤ x∨y. Furthermore, x is the unique element u so that x ∧ u = 0 and x ∨ u = 1 since

LATTICES

21

we have u=u∧1 = u ∧ (x ∨ x) = (u ∧ x) ∨ (u ∧ x) = (u ∧ x) ∨ (x ∧ x) = (u ∨ x) ∧ x = x.

distributivity x∧u=0=x∧x distributivity x∨u=1

The set B of Boolean values with f ≤ t is a simple example of a (complete) Boolean algebra. Furthermore, the power set of a set A also forms a complete Boolean algebra with B as complement of B. Any nontrivial linear ordering, i.e., a poset with more than two elements, is not a Boolean algebra. Theorem 2.27 (DeMorgan) In every Boolean algebra L we have for all elements x, y ∈ L (1) x ∧ y = x ∨ y, (2) x ∨ y = x ∧ y. Proof. We just prove (1). We immediately conclude that (x ∧ y) ∧ (x ∨ y) = (x ∧ y ∧ x) ∨ (x ∧ y ∧ y)

distributivity

= 0, (x ∧ y) ∨ (x ∨ y) = (x ∨ x ∨ y) ∧ (y ∨ x ∨ y)

distributivity

= 1,


and, hence, the assertion.

As mentioned above, the Boolean algebra of a power set of a set A is completely distributive. This is an application of the following theorem: Theorem 2.28 Every complete Boolean algebra is completely distributive. Proof. Suppose L is a  completeBoolean algebra. By Theorem 2.25 (2) it (x ∨ y) for all x ∈ L and M ⊆ L. Since remains to show that x ∨ M = y∈M  ∨ is monotone the inclusion ≤ is trivial. Let u := (x ∨ y). Then we have u ≤ x ∨ y for all y ∈ M . This implies u ∧ x ≤ (x ∨ y) ∧ x = (x ∧ x) ∨ (y ∧ x) =y∧x ≤ y.

y∈M

distributivity

22

GOGUEN CATEGORIES

We conclude that u ∧ x ≤



M , and, hence,

u≤x∨u = (x ∨ u) ∧ (x ∨ x) = x ∨ (u ∧ x)  ≤ x ∨ M.

distributivity


Now, Theorem 2.27 may be generalized for complete Boolean algebras. Theorem 2.29 (DeMorgan) In every complete Boolean algebra L we have for all subsets M ⊆ L   x, M= (1) x∈M

(2)



M=



x.

x∈M

In the proof of Theorem 2.27 the distribution laws are essential. The theorem above may be shown analogously using the generalized distribution laws. 2.5

SPECIAL ELEMENTS

The prime numbers are of special interest within the natural numbers. They are characterized by the fact that they cannot be generated by other elements using the binary operation of multiplication. In a lattice we may have an analogous situation. Definition 2.30 An element x = 0 of a lattice L with least element 0 is called irreducible iff y ∨ z = x implies x = y or x = z. Consider again the lattice from Figure 2.8. The irreducible elements are given by {1, 2, 3}. The elements 4 is not irreducible since 4 = 1 ∨ 2. In a complete lattice we may switch to a more restricted notion of irreducibility. Definition 2.31 An element x of a complete lattice  L is called completely irreducible iff for all subsets of M ⊆ L the property M = x implies x ∈ M .  Notice that 0 is not completely irreducible since 0 = ∅. Furthermore, every completely irreducible element is irreducible but not vice versa. The element 1 of the unit interval [0, 1] is irreducible since if 1 is the maximum of two elements x and y, then 1 is either x or y. On the other hand, 1 is not included in the set {1 − n1 | n ∈ N \ {0}} ⊆ [0, 1], but it is its least upper bound. Definition 2.32 An element a = 0 of a lattice L with least element 0 is called an atom of L iff x ≤ a implies x = a or x = 0. The set of all atoms of

LATTICES

23

L isdenoted by At(L). Furthermore, a complete lattice is called atomic iff x = {a ∈ At(L) | a ≤ x} holds for all x ∈ L. In the power set of a set A the singleton sets {x} with x ∈ A are the atoms. Furthermore, every upper neighbor of 0, i.e., every element x so that x = 0, and there is no element y = 0 and y = x with 0 ≤ y ≤ x, is an atom. Therefore, every finite lattice with at least two elements has at least one atom. Obviously, every atom is completely irreducible but not vice versa. Let N∞ be the complete linear lattice of natural numbers with an additional greatest element ∞. Then all elements except 0 and ∞ are completely irreducible, but 1 is the unique atom of N∞ . Nevertheless, the observation above implies that every finite lattice has at least one completely irreducible element. 2.6

FIXED POINTS

We will describe several operations on lattices using fixed points of suitable endofunctions. Furthermore, later on we will use an induction principle to conclude properties of such fixed points, called the principle of fixed point induction. The proof of the existence of a fixed point given in this chapter is somewhat more complex than the usual one. The reason is that we obtain the principle of fixed point induction as an application of the notions to be introduced. Definition 2.33 Let f : P → P be an endofunction on a poset P and a ∈ P . Then an element µf (a) ∈ P is called the least fixed point of f greater or equal to a iff f (µf (a)) = µf (a) and for all elements b ∈ P the properties f (b) = b and a ≤ b imply µf (a) ≤ b. µf (a) does not exist in general. Therefore, in the rest of this section we suppose that L is a complete lattice, f : L → L is a monotone endofunction, and a is an element of L with a ≤ f (a). Definition 2.34 A subset M of L is called an f, a-closed set iff (1) a ∈ M , (2) if x ∈ M , then so is f (x),  (3) if ∅ = N ⊆ M , then N ∈ M . Obviously, L is an f, a-closed set. Lemma 2.35 The intersection of f, a-closed sets is again f, a-closed. Proof. Suppose X is a set of f, a-closed sets. Then we obtain the following:  (1) a ∈ X since a is in every element of X .  (2) If x ∈ X , then x ∈ M for all M ∈ X , which implies f (x) ∈ M for all M , and, hence, f (x) ∈ X .

24

GOGUEN CATEGORIES

 (3) Suppose ∅ = N ⊆ X . Then∅ = N ⊆ M for all M ∈ X . This implies  N ∈ M , and, hence, N ∈ X .  The computation above shows that X is f, a-closed.
 Now, let Xf,a := {M | M is f, a-closed}. From the previous lemma we conclude that Xf,a is f, a-closed. Theorem 2.36 (Fixed point Theorem) Let L be a complete lattice, f : L → L a monotone endofunction and  a an element of L with a ≤ f (a). Then µf (a) exists, and we have µf (a) = Xf,a .  Proof. Let c := Xf,a and E := {x | x ≤ f (x) and a ≤ x}. We want to show that E is f, a-closed. (1) a ∈ E since a ≤ a and a ≤ f (a) by the assumption. (2) Suppose x ∈ E. Then we have f (x) ≤ f (f (x)) and a ≤ f (a) ≤ f (x) since f is monotone.  (3) If ∅ = N ⊆ E, then a ≤ N is trivial, and we have N≤ f (y) since y ∈ E y∈N

≤f



 N .

f monotone

c ∈ E, which implies c ≤ f (c). Since f (c) We conclude Xf,a ⊆ E, and, hence,  is an element of Xf,a and c = Xf,a we get f (c) ≤ c, and, hence, that c is a fixed point of f greater or equal to a. It remains to show that c is indeed the least fixed point with that property. Suppose b is another fixed point of f greater or equal to a. Let B := {x | a ≤ x ≤ b}. Again, we want to show that B is f, a-closed. (1) a ∈ B is trivial. (2) If x ∈ B, then a ≤ f (a) ≤ f (x) ≤ f (b) = b since f is monotone and b is a fixed point of f .   (3) Suppose ∅ = N ⊆ B. Then a ≤ N ≤ b, and, hence, N ∈ B. We conclude that Xf,a ⊆ B, and, hence, c ∈ B, which is equivalent to c ≤ b.
As mentioned above, we are interested in an induction principle in order to prove properties of µf (a). Unfortunately, the principle of fixed point induction is not valid for arbitrary predicates. Definition 2.37 A predicate P on a complete lattice L is called admissible (or continuous)iff for every nonempty set M ⊆ L the property P(x) for all x ∈ M implies P( M ).

LATTICES

25

Notice that we may use n-ary predicates by taking the n-ary product of the underlying lattices. Theorem 2.38 (Fixed point Induction) Let L be a complete lattice, f : L → L a monotone endofunction, a an element of L with a ≤ f (a), and P be an admissible predicate on L. Then the properties Base case: P(a), Induction step: P(b) implies P(f (b)) for all b ∈ L, imply P(µf (a)). Proof. Let MP := {x | P(x)}. Then MP is an f, a-closed set since property (1) corresponds to the base case, (2) to the induction step, and (3) to the
continuity of P. We conclude Xf,a ⊆ MP , and, hence, µf (a) ∈ MP . Notice that the principle of fixed point induction is equivalent to a special case of transfinite induction on the class of ordinal numbers. Since we did not introduce ordinal numbers we just want to sketch this connection. For a monotone endofunction f on a complete lattice L and an element a with a ≤ f (a) one may define f 0 (a) := a, f x+1 (a) := f (f x (a)) f x (a) f y (a) :=

for successor ordinals x + 1, for limiting ordinals y.

x i} yi := zi ∪ {(0, j) | j ≥ 0} ∪ {(j, 0) | j ≥ 0}



y2 C CC C



s

z2

{{ {{

y0 = BB N × N BB z0 {{ { { z1

m y∞ =h z∞ = ∅ Figure 5.5.

The relations yi and zi . ↓

but the dual property (↓ ) Q↑  R↓ = (Q↑  R) is not. The main reason is that the theory of complete Brouwerian lattices, which serves as a foundation  of arrow categories, is also asymmetric. The infinite distribution law x∧ yi = i∈I    yi = i∈I (x ∨ yi ) might not. i∈I (x ∧ yi ) is valid but its dual version x ∨ i∈I

Based on that observation we construct an example where (↓ ) fails to hold. Example 5.52 For each k ∈ N let be Rk := {R ⊆ N × N | ∀i, j ≥ k : iRj}, i.e., the set of all binary relations on N such that all elements greater or equal to k are related. It is not hard to verify that the following properties: (1) if R ∈ Rk , then so is RT , (2) if R1 ∈ Rk1 and R2 ∈ Rk2 , then R1 ∩ R2 ∈ Rmax(k1 ,k2 ) , (3) if R1 ∈ Rk1 and R2 ∈ Rk2 , then R1 ◦ R2 ∈ Rmax(k1 ,k2 ) , (4) if Ri ∈ Rki for all i ∈ I and k = min{ki | i ∈ I}, then




Ri ∈ Rk ,

i∈I

The properties above show that R∞ := {∅} ∪




Rk together with ∩, ∪, .T and

k∈N

◦ is a simple Dedekind category. Notice that in this category infinite meets and the residuals do not coincide with the corresponding operation on P(N × N). Let yi , zi with i ∈ N be the relations defined in Figure 5.5. The relations above show that the dual version of the infinite distribution law is not valid. Furthermore, the infinite meet y∞ of the yi ’s does not correspond to their set-theoretic intersection – the relation {(0, j) | j ≥ 0}∪{(j, 0) | j ≥ 0}. ∞ Now, consider the Goguen category RN ∞ , i.e., the antimorphisms from the natural numbers extended by a greatest element to R∞ . Furthermore, let be f := z˙0 and g(k) := yk for all k ∈ N∞ . Obviously f and g are antimorphisms. Since (f  g)(k) = f (k)  g(k)  z0  yk = N × N = y0

CATEGORIES OF L-FUZZY RELATIONS

141

for all k ∈ N∞ we have f · g = y˙0 . This implies ↓

(f ↑ · g) = (f · g)

↓

f is crisp

↓

see above

= y˙0 = y˙0 .

y˙0 is crisp

˙ = y∞ for k = 0, and, hence, f ↑ · g ↓ = On the other hand, g ↓ (k) = g(∞) ˙ = z˙0 .  z˙0 · y∞ We have already shown in Lemma 5.7 (2) that in a Boolean arrow category (↓ ) is valid. In the remainder of this section we want to show that the theory of Boolean-based Goguen categories is already strong enough to prove this property. Recall that a Boolean-based Goguen category is Goguen category G so that oder category. the substructure of crisp relations G ↑ is a Schr¨ Due to our pseudo-representation theorem (Theorem 5.50 it is sufficient to show (↓ ) for antimorphisms. Theorem 5.53 Let RL be a Boolean-based Goguen category. Then we have ↓ f ↑ · g ↓ = (f ↑ · g) for all f, g in RL [A, B]. Proof. First of all, we show that Q˙  g is an antimorphism, and, hence, that ˙ Q · g = Q˙  g holds. Assume x = 0 and consider the following computation: ϕ(Q˙  g)(x) =



(Q˙  g)(y)



definition ϕ

M =x y∈M

=



˙ (Q(y)  g(y))



M =x y∈M

=



˙ (Q(y)  g(y))



x = 0

M =x 0 =y∈M

=



(Q  g(y))



M =x 0 =y∈M

=





Q



M =x

=





 g(y)

Theorem 2.28

0 =y∈M

(Q  g(x))

g antimorphism

M =x

= Q  g(x) ˙ = Q(x)  g(x) = (Q˙  g)(x).

x = 0

142

GOGUEN CATEGORIES

For x = 0 we have ϕ(Q˙  g)(x) =



(Q˙  g)(y)



definition ϕ

M =0 y∈M

= (Q˙  g)(x).

since M = ∅ or M = {0}

So, we have shown that ϕ(Q˙  g) = Q˙  g, which implies that Q˙  g is an antimorphism. For x = 0 we have ↓

(Q˙  g) (x) = (Q˙  g)(1) ˙ = Q(1)  g(1) ˙ = Q(x)  g ↓ (x) = (Q˙  g ↓ )(x)

definition .↓ definition .↓

↓ ↓ and for x = 0 obviously (Q˙  g) (x) =  AB = (Q˙  g ↓ )(x) so that (Q˙  g) = Q˙  g ↓ follows.  f (x). Then the assertion follows from Now, let Q := x =0 ↓

(f ↑ · g) = (Q˙ · g)

↓

= (Q˙  g) = Q˙  g ↓

↓

= Q˙ ↑ · g ↓ ↑

↓

= f · g .

definition f ↑ see above see above see above definition f ↑




Finally, we obtain the following: Corollary 5.54 Let G be a Boolean-based Goguen category. Then we have ↓ Q↑  R↓ = (Q↑  R) for all Q, R : A → B. 5.9

EQUATIONS IN GOGUEN CATEGORIES

A Goguen category may provide some relational constructions as products, sums, or splittings (cf. Section 4.6). Such a construction is given by an object together with a set of relations fulfilling some equations. One may expect that the pairing Q; π
 R; ρ
of two crisp relations Q and R is crisp too. If the projections are crisp, then this is the case since the class of crisp relations is closed under the usual relational operations. Especially in applications, such a property seems to be essential. For example, if the input domain of a fuzzy controller is a product with noncrisp projection, there would be a fuzzification, which is not an integral part of the controller. This fuzzification arises from the specific choice of the product. In this case, reasoning about the controller using

CATEGORIES OF L-FUZZY RELATIONS

143

a suitable relation of a Goguen category seems to be impossible or at least difficult. Unfortunately, there may exist such noncrisp projections. Consider the Boolean lattice B4 := {0, a, b, 1} and the following relations represented as B4 -valued matrices: ⎛

⎛ ⎛ ⎞ ⎞ 1 0 1 0 1 0 ⎜ a b ⎟ ⎜ b a ⎟ ⎜ 1 0 ⎜ ⎜ ⎜ ⎟ ⎟ π1 := ⎝ , ρ1 := ⎝ , π2 := ⎝ b a ⎠ a b ⎠ 0 1 0 1 0 1 0 1

⎛

⎞

1 ⎜ ⎟ ⎟ , ρ2 := ⎜ 0 ⎝ 1 ⎠ 0

⎞ 0 1 ⎟ ⎟. 0 ⎠ 1

Both pairs (π1 , ρ1 ) and (π2 , ρ2 ) constitute a product of two copies of a set with two elements, i.e., they fulfill the equations required for a product. The second pair of relations is the usual pair of crisp projections. The first pair is not crisp. But, this example also indicates that in B4 -Rel for any pair of projections there is also a crisp version of the corresponding product, i.e., there are crisp relations between the same objects fulfilling the equations required for a product. In our example, one may require without loss of generality that the projections are crisp. In this chapter, we will investigate when the validity of a set of equations S within a Goguen category G implies the validity of S in G ↑ . The application to products, sums and splittings of crisp relations then is obvious. If one of these constructions exists in G, then it exists in G ↑ , i.e., the corresponding relations may be chosen to be crisp. Throughout this chapter L is supposed to be a proper lattice, i.e., a lattice with at least one complete prime filter. We start with a lemma on the function ϑ introduced in Section 2.8. Lemma 5.55 Let L be a complete Brouwerian lattice, and R be a Dedekind ≥ ≥ category. For all antitone functions f : L → R[A, B] and g : L → R[B, C] we have (1) ϑ(f ; g) = ϑ(f ); ϑ(g),




(2) ϑ(f
) = ϑ(f ) .

Proof. (1) First, we show that



(f (x); g(x)) = (

M⊆ψ(x)



M⊆ψ(x)

f (x)); (



g(x)).

M⊆ψ(x)

The inclusion  is trivial. Suppose M ⊆ ψ(x) and M ⊆ ψ(y). Then M ⊆ ψ(x) ∩ ψ(y) = ψ(x ∧ y). Furthermore, f (x); g(y)  f (x ∧ y); g(x ∧ y)

144

GOGUEN CATEGORIES

since f and g are antitone. We conclude that     f (x)); g(x) ( M⊆ψ(x)

=

M⊆ψ(x)

⎛



(f (x); ⎝

M⊆ψ(x)

=



g(x)⎠

M⊆ψ(x)





⎞

(f (x); g(y))

M⊆ψ(x) M⊆ψ(y)





(f (x ∧ y); g(x ∧ y))

computation above

M⊆ψ(x∧y)

=



(f (x); g(x)).

M⊆ψ(x)

Now, consider the following computation:  (ϑ(f ; g))(M) = (f ; g)(x)

definition ϑ

M⊆ψ(x)



=

(f (x); g(x))

M⊆ψ(x)

⎛

⎞ ⎛



=⎝

f (x)⎠ ; ⎝

M⊆ψ(x)

definition of ; 

⎞ g(x)⎠

computation above

M⊆ψ(x)

= ϑ(f )(M); ϑ(g)(M) = (ϑ(f ); ϑ(g))(M). (2) The assertion follows immediately from  f
(x) (ϑ(f
))(M) =

definition ϑ definition of ;

definition ϑ

M⊆ψ(x)



=




(f (x))

M⊆ψ(x)

⎛ =⎝



definition




⎞
f (x)⎠

M⊆ψ(x)

= (ϑ(f )(M))





= (ϑ(f ) )(M).

definition ϑ definition







In the next lemma we have summarized the essential properties of the validity of equations within several categories introduced so far. Lemma 5.56 Let RL be a Goguen category, t be a term in the language of distributive allegories with greatest elements, σ be an environment over RL and σ  be an environment over RL ≥ . Then we have

CATEGORIES OF L-FUZZY RELATIONS

145

(1) VRL (t)(σ) = τ (VRL (t)(σ)), ≥



(2) ϑ(VRL (t)(σ )) = VRP(FL ) (t)(σ  ) where σ  (a) := σ  (a) for all object vari≥

≥

ables a and σ  (r : a → b) := ϑ(σ  (r : a → b)) for all relational variables r : a → b,

(3) VRL (t)(σ  )(x) = VR (t)(σ  ) where σ  (a) := σ  (a) for all object variables a ≥ and σ  (r : a → b) := σ  (r : a → b)(x) for all relational variables r : a → b, (4) if σ(r : a → b) is crisp for all r : a → b ∈ RV(t), then VRL (t)(σ) is crisp and we have VRL (t)(σ) = R˙ with R := VRL (t)(σ)(x) for an arbitrary ≥ x = 0. Proof. (1) We prove the assertion by structural induction. t=⊥ ⊥ab : We immediately conclude that ˙ σ(a)σ(b) ⊥ab )(σ) = ⊥ ⊥ VRL (⊥ = τ (⊥ ⊥σ(a)σ(b) ) = τ (VRL (⊥ ⊥ab )(σ)). ≥

definition VRL ˙ σ(a)σ(b) definition ⊥ ⊥ definition VRL

≥

t ∈ {Ia ,  ab } : is shown analogously. t = r : a → b: We immediately conclude that VRL (r : a → b)(σ) = σ(r : a → b)

definition VRL

= τ (σ(x)) = τ (VRL (r : a → b)(σ)).

σ(r : a → b) antimorphism definition VRL

≥

≥

t = t1  t2 : Again, we immediately conclude that VRL (t1  t2 )(σ) = VRL (t1 )(σ)  VRL (t2 )(σ) = τ (VRL (t1 )(σ))  τ (VRL (t2 )(σ))

definition VRL induction hypothesis

= τ (VRL (t1 )(σ)  VRL (t2 )(σ))

Lemma 2.46 (2)

= τ (VRL (t1  t2 )(σ)).

definition VRL

≥

≥

≥

≥

≥

≥

t = t1  t2 : Consider the following computation: VRL (t1  t2 )(σ) = VRL (t1 )(σ) · VRL (t2 )(σ)

definition VRL

= τ (VRL (t1 )(σ)  VRL (t2 )(σ)) = τ (VRL (t1 )(σ)  VRL (t2 )(σ))

definition · induction hypothesis

= τ (VRL (t1  t2 )(σ)).

definition VRL

≥ ≥

≥

≥

146

GOGUEN CATEGORIES

t = t1 ; t2 : Again, consider the following computation: VRL (t1 ; t2 )(σ) = VRL (t1 )(σ) ··, VRL (t2 )(σ) = τ (VRL (t1 )(σ); VRL (t2 )(σ)) = τ (VRL (t1 )(σ); VRL (t2 )(σ)) ≥

definition VRL definition ··, induction hypothesis

≥

definition VRL

= τ (VRL (t1 ; t2 )(σ)). ≥

≥

t = t
1 : Last but not least, the following computation shows the assertion: VRL (t
1 )(σ) = (VRL (t1 )(σ))




= τ (VRL (t1 )(σ)))

definition VRL


induction hypothesis

≥




= τ ((VRL (t1 )(σ)) )

Lemma 4.32 (2)

= τ (VRL (t
1 )(σ)).

definition VRL

≥

≥

≥

(2) Again, the assertion is proved by structural induction. t=⊥ ⊥ab : We immediately conclude that ϑ(VRL (⊥ ⊥ab )(σ  ))(M) ≥ VRL (⊥ ⊥ab )(σ  )(x) = ≥

definition ϑ()

M⊆ψ(x)

=⊥ ⊥σ (a)σ (b)

definition VRL

≥



= VRP(FL ) (⊥ ⊥ab )(σ )(M).

definition VRP(FL ) ,

≥

≥

σ  (a) = σ  (a) and σ  (b) = σ  (b) ab } : is shown analogously. t ∈ {Ia ,  t = r : a → b: follows immediately from the required property of σ  . t = t1  t2 : We immediately conclude that ϑ(VRL (t1  t2 )(σ  )) ≥

= ϑ((VRL (t1 )(σ  )  VRL (t2 )(σ  ))) ≥

≥





definition VRL

≥

= ϑ((VRL (t1 )(σ ))  ϑ(VRL (t2 )(σ )))

Lemma 2.63 (3)

= VRP(FL ) (t1 )(σ  )  VRP(FL ) (t2 )(σ  )

induction hypothesis

= VRP(FL ) (t1  t2 )(σ  ).

definition VRP(FL )

≥

≥

≥

≥

≥

≥

t ∈ {t1  t2 , t1 ; t2 , t
1 }: is shown analogously by applying Lemma 2.63 (2), Lemma 5.55 (1) and Lemma 5.55 (2), respectively.

CATEGORIES OF L-FUZZY RELATIONS

147

(3) The assertion is trivial since all operations and constants are defined componentwise. (4) We define an environment over R by σx (a) := σ(a) σx (r : a → b) := σ(r : a → b)(x)

for all object variables a, for all relational variables r : a → b.

Since crisp relations in RL are of the form Q˙ for a suitable relation Q in R we conclude that σx (r : a → b) = σy (r : a → b) for all r : a → b ∈ RV(t) and x = 0 and y = 0. Theorem 4.53 implies VR (t)(σx ) = VR (t)(σy ), and, hence, VRL (t)(σ)(x) = VRL (t)(σ)(y) by (3) of this lemma. ≥ ≥ With other words VRL (t)(σ) is apart from 0 the constant function returning ≥ ˙ which shows the assertion using (1) R. Consequently, τ (VRL (t)(σ)) = R, ≥




of this lemma. Now, we are ready to prove our main theorem in this chapter.

Theorem 5.57 Let L be a proper Brouwerian lattice with 0 = 1, R a simple AB for all objects A and B, S = {ti1 = ti2 | i ∈ Dedekind category with ⊥ ⊥AB =  I} be a set of equations in the language of distributive allegories with greatest elements, V = {r1 : a1 → b1 , . . . , rn : an → bn } a set of relational variables, σ an environment over RL such that σ(r : a → b) is crisp for all relational variables r : a → b ∈ RV(S) \ V , and f1 , . . . , fn be elements of RL such that RL |=σ[f1 /r1 :a1 →b1 ,...,fn /rn :an →bn ] ti1 = ti2  for all i ∈ I. Then for Uj := fj (x) with 1 ≤ j ≤ n and an arbitrary F ∈ FL we have for all i ∈ I

x∈F

RL |=σ[U˙ 1 /r1 :a1 →b1 ,...,U˙ n /rn :an →bn ] ti1 = ti2 . Proof. For brevity, let σ ˜ := σ[f1 /r1 : a1 → b1 , . . . , fn /rn : an → bn ], σ ˜  (a) := σ(a) σ (r : a → b)) σ ˜  (r : a → b) := ϑ(˜   σ ˜ (a) := σ ˜ (a) = σ(a) 



˜ (r : a → b)({F }) σ ˜ (r : a → b) := σ

for all object variables a, for all relational variables r : a → b, for all object variables a, for all relational variables r : a → b

and hij := VRL (tij )(˜ σ ) for j = 1, 2 and i ∈ I. Then we have ≥

σ ) = VRL (ti2 )(˜ σ) VRL (ti1 )(˜ ⇔ τ (hi1 ) = τ (hi2 ) ⇒ ϑ(τ (hi1 )) = ϑ(τ (hi2 ))

Lemma 5.56 (1)

148

GOGUEN CATEGORIES

⇒ τ (ϑ(τ (hi1 ))) = τ (ϑ(τ (hi2 ))) ⇔ τ (ϑ(hi )1 ) = τ (ϑ(hi )2 ) ⇔

τ (VRP(FL ) (ti1 )(˜ σ  )) ≥

=

Lemma 2.64

τ (VRP(FL ) (ti2 )(˜ σ  ))

Lemma 5.56 (2)

≥

σ  ))({F }) = τ (VRP(FL ) (ti2 )(˜ σ  ))({F }) ⇒ τ (VRP(FL ) (ti1 )(˜ ≥

≥

σ  ))({F }) = (VRP(FL ) (ti2 )(˜ σ  ))({F }) ⇔ (VRP(FL ) (ti1 )(˜

Lemma 2.63 (4)

σ  ) = VR (ti2 )(˜ σ  ). ⇔ VR (ti1 )(˜

Lemma 5.56 (3)

≥

≥

Now, define δ := σ[U˙ 1 /r1 : a1 → b1 , . . . , U˙ n /rn : an → bn ] and δ  (a) := δ(a) = σ(a) 

δ (r : a → b) := δ(r : a → b)(x)

for all object variables a, for all relational variables r : a → b

First of all, we have  = VRL (ti2 )(δ)(0) VRL (ti1 )(δ)(0) =  since both functions are antimorphisms. Suppose x = 0. Then δ(r : a → b) is crisp for all relational variables r : a → b ∈ RV(S) and Lemma 5.56 (4) implies that VRL (tij )(δ) is crisp and VRL (tij )(δ)(x) = VRL (tij )(δ)(x) since x = 0. ≥ Furthermore, for a relational variable r : a → b ∈ RV(S) \ V we have ˜  (r : a → b)({F }) σ ˜  (r : a → b) = σ = ϑ(˜ σ (r : a → b))({F })  = σ ˜ (r : a → b)(y)

definition σ ˜  definition σ ˜ Lemma 2.63 (4)

y∈F

=



σ(r : a → b)(y)

y∈F

= σ(r : a → b)(x) = δ(r : a → b)(x) = δ  (r : a → b)

r : a → b ∈ RV(S) \ V and definition σ ˜ σ(r : a → b) is crisp, 0 ∈ F and x = 0 r : a → b ∈ RV(S) \ V and definition δ definition δ 

and for ri : ai → bi ∈ R with i ∈ I ˜  (ri : ai → bi )({F }) σ ˜  (ri : ai → bi ) = σ = ϑ(˜ σ (ri : ai → bi ))({F }) = ϑ(fi )({F })  fi (y) = y∈F

definition σ ˜  definition σ ˜ ri : ai → bi ∈ V Lemma 2.63 (4)

CATEGORIES OF L-FUZZY RELATIONS

= Ui = U˙ i (x)

149

definition Ui

= δ(ri : ai → bi )(x) = δ  (ri : ai → bi )

x = 0 definition δ and ri : ai → bi ∈ R definition δ 

such that σ ˜  and δ  are equal for all relational variables from RV(S). We obtain VRL (tij )(δ)(x) = VRL (tij )(δ)(x)

see above

≥

= =

VR (tij )(δ  ) VR (tij )(˜ σ  ).

Lemma 5.56 (3) Theorem 4.53 and computation above

Finally, we have σ  ) VRL (ti1 )(δ)(x) = VR (ti1 )(˜ σ  ) = VR (ti2 )(˜ = VRL (ti2 )(δ)(x), and, hence, RL |=σ[U˙ 1 /r1 :a1 →b1 ,...,U˙ n /rn :an →bn ] ti1 = ti2 .




Using the pseudo-representation functor PG the previous theorem may be transferred to arbitrary Goguen categories. For a relation R fulfilling a set of equations S we may define the corresponding antimorphism by f (α) :=  ↓ f (α) = PG (R)(α) = (αA \R) . By the previous theorem we obtain U = α∈F  ↓ (αA \R) , and, hence, a crisp relation Q fulfilling S by α∈F

Q := PG−1 (U˙ )  = (βA ; U˙ (β))

definition PG−1

β∈Sc[G]



=

(βA ; U˙ (β))

⊥ ⊥ =β∈Sc[G]



=





βA ;

⊥ ⊥ =β∈Sc[G]

⎛

=⎝





↓

definition U˙

(αA \R)

α∈F

⎞  βA ⎠ ;

⊥ ⊥ =β∈Sc[G]

=







 ↓

(αA \R)

α∈F ↓

(αA \R) .

α∈F

Compare the last expression with representation of R↑ by its cuts. Our relation Q is computed similar by restricting the scalars to elements of the complete prime filter F .

150

GOGUEN CATEGORIES

We aim at the following corollary: Corollary 5.58 Let G be a Goguen category with a proper underlying lattice Sc[G], S = {ti1 = ti2 | i ∈ I} be a set of equations in the language of distributive allegories with greatest elements, V = {r1 : a1 → b1 , . . . , rn : an → bn } a set of relational variables, σ an environment over G such that σ(r : a → b) is crisp for all relational variables r : a → b ∈ RV(S) \ V , and R1 , . . . , Rn be relations such that G |=σ[R1 /r1 :a1 →b1 ,...,Rn /rn :an →bn ] ti1 = ti2  ↓ for all i ∈ I. Then for Qj := (ασ(aj ) \Rj ) with 1 ≤ j ≤ n and an arbitrary α∈F

F ∈ FSc[G] we have for all i ∈ I

G |=σ[Q1 /r1 :a1 →b1 ,...,Qn /rn :an →bn ] ti1 = ti2 . Since products, sums, and splittings induced by crisp symmetric idempotents are defined by equations in the sense of the previous corollary, we may require in a Goguen category with a proper underlying lattice Sc[G] without loss of generality that the related relations are crisp. Unfortunately, we were just able to prove Theorem 5.57 for Goguen categories RL with a proper lattice L. On the other hand, we did not find a counterexample to this theorem in general. This observation may motivate the following conjecture: Conjecture 5.59 The existential part of Theorem 5.57 is true for all Goguen categories of the form RL . Obviously, if the previous conjecture is proved, the result could be transferred to arbitrary Goguen categories in a similar way. 5.10 OPERATIONS DERIVED FROM LATTICE-ORDERED SEMIGROUPS In Chapter 3 we have defined some ∗-based operations for a loos (L, ∗, e, z) on L-fuzzy relations. Now, we want to investigate an abstract counterpart of these operations within an arbitrary Goguen category and a loos (Sc[G], ∗, , ζ) on the scalars of G. Throughout this section, unless otherwise stated, let G be a Goguen category and ∗ an operation such that (Sc[G], ∗, , ζ) is a loos. Furthermore, suppose ⊗ is a binary operation on relations such that (1) ⊗ is defined for all pairs of relations from G[A, A] for all objects A and its value is within G[B, B] for a suitable B and if Q ⊗ R is defined for Q : A → B and R : C → D, then ⊗ is defined for all pairs of relations from G[A, B] and G[C, D] and its value is within G[E, F ] for suitable E and F , (2) if Q ⊗ R is defined for Q : A → B and R : C → D and within G[E, F ], ⊥AB ⊗ R = ⊥ ⊥EF , then Q ⊗ ⊥ ⊥CD = ⊥

CATEGORIES OF L-FUZZY RELATIONS

151

CD is defined and within G[E, F ], then  AB ⊗  CD =  EF , (3) if  AB ⊗  (4) ⊗ distributes over arbitrary unions in both arguments, i.e., for all relations Q, Qi , R, Ri with i ∈ I we have         Q⊗ Ri = (Q ⊗ Ri ) and Qi ⊗ R = (Qi ⊗ R) i∈I

i∈I

i∈I

i∈I

whenever the application of ⊗ is defined, (5) for all α, β ∈ Sc[G] and relations Q : A → B, R : C → D such that Q ⊗ R is defined and within G[E, F ] we have (αE  βE ); (Q ⊗ R) = (αA ; Q) ⊗ (βC ; R), (6) ⊗ is closed on G ↑ , i.e., for all crisp relations Q, R such that Q ⊗ R is defined Q ⊗ R is crisp. Notice that  and ; satisfy the properties above. (1),(2), and (4) follow immediately from the definition of a Dedekind category. For meet property (3) is trivial and for composition it follows from the fact that G is uniform. Property (6) is true since the crisp relations are closed under the relational operations. Finally, property (5) is shown as follows: AB  Q  R (αA  βA ); (Q  R) = (αA  βA );  AB  βA ;  AB  Q  R = αA ;  = αA ; Q  βA ; R (αA  βA ); Q; S = αA ; βA ; Q; S = αA ; Q; βB ; S

Lemma 4.24 (2) Lemma 4.24 (3) Lemma 4.24 (2), Lemma 4.9 (3) Lemma 5.14

Now, we may define the ∗-based operation Q ⊗∗ R as follows: It is defined via the cut representation of Q and R. Definition 5.60 Let Q : A → B, R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. Then we define  ↓ ↓ (α ∗ β)E ; ((αA \Q) ⊗ (βC \R) ). Q ⊗∗ R := α,β∈Sc[G]

Notice that (α ∗ β)E in the definition above denotes the corresponding scalar α ∗ β ∈ Sc[G] on the object E. Furthermore, by our convention on the notion of scalars we have (α  β)A = αA  βA and (α  β)A = αA  βA . Since ⊗ and the left residual in the second argument are monotone the operation ⊗∗ is also monotone in both arguments. For L-fuzzy relations the lattice L and the lattice of scalar elements are isomorphic. Therefore, the operation ∗ may be considered as to be defined on L x∗y and (αx ∗αy )A denote or on the scalar elements. If we identify these lattices, αA

152

GOGUEN CATEGORIES

the same scalar on the set A. Theorem 3.7 shows that for L-fuzzy relations and ⊗ ∈ {∩, ◦} the definitions given in Chapter 3 are special cases of the abstract definition above. Now, we want to give an alternative definition of Q ⊗∗ R. Lemma 5.61 Let Q : A → B, R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. Then we have ⎛ ⎞   ⎜ ↓ ↓ ⎟ αE ; ⎝ ((βA \Q) ⊗ (γC \R) )⎠ . Q ⊗∗ R = β,γ∈Sc[G] β∗γ α

α∈Sc[G]

Proof. This may be obtained from  ↓ ↓ Q ⊗∗ R = (α ∗ β)E ; ((αA \Q) ⊗ (βC \R) ) α,β∈Sc[G]

=



⎛

⎛ =

⎞ 

⎜ αE ; ⎝

α∈Sc[G]



definition ⊗∗

↓ ↓ ⎟ ((βA \Q) ⊗ (γC \R) )⎠

β,γ∈Sc[G] β∗γ=α



⎜ αE ; ⎝

⎞ ↓ ↓ ⎟ ((βA \Q) ⊗ (γC \R) )⎠ ,

β,γ∈Sc[G] β∗γ α

α∈Sc[G]

where the last equality is shown as follows: The inclusion  is trivial and  is ↓ ↓ ↓ ↓ implied by αE ; ((βA \Q) ⊗ (γC \R) )  (β ∗ γ)E ; ((βA \Q) ⊗ (γC \R) ) for all β ∗ γ  α in Sc[G].
There is a connection between the previous lemma and the α-cut representation of Q ⊗∗ R. Define f : Sc[G] → G ↑ [E, F ] by  ↓ ↓ f (α) := ((βA \Q) ⊗ (γC \R) ). β,γ∈Sc[G] β∗γ α

From the following computation with α  α  ↓ ↓ f (α ) = ((βA \Q) ⊗ (γC \R) )

definition f

β,γ∈Sc[G] β∗γ α





↓

↓

((βA \Q) ⊗ (γC \R) )

α  α  β ∗ γ

β,γ∈Sc[G] β∗γ α

= f (α)

definition f

we conclude that f is antitone. By Lemma 5.49 (2) and the previous lemma  Q ⊗∗ R = (αE ; τ (f )(α)) α∈Sc[G] ↓

follows. From Lemma 5.45 we conclude τ (f )(α) = (αE \Q ⊗∗ R) .

CATEGORIES OF L-FUZZY RELATIONS

153

⊥ and Unfortunately, ⊗∗ is not an operation on scalars since for  = I, ζ = ⊥ ⊗ =; we have 

⊥AA = αA ;∗ ⊥

↓

↓

↓

↓

((γ ∗ δ)A ; (γA \αA ) ; (δA \⊥ ⊥AA ) )

definition ;∗

γ,δ∈Sc[G]



=

((γ ∗ δ)A ; (γA \αA ) ; (δA \⊥ ⊥AA ) )

since ζ = ⊥ ⊥

γ,δ∈Sc[G] γ=⊥ ⊥



=

↓

((γ ∗ δ)A ; (δA \⊥ ⊥AA ) )

Lemma 5.20

((γ ∗ δ)A ;  AA )

definition residual

γ,δ∈Sc[G] ⊥ ⊥=γ α



=

γ,δ∈Sc[G] ⊥ ⊥=γ α

⎛ ⎜ =⎝

⎞



⎟ (γ ∗ δ)A ⎠ ;  AA

γ,δ∈Sc[G] ⊥ ⊥=γ α

= (α ∗ I)A ;  AA = αA ;  AA .

∗ monotone =I

Therefore, we define a new operation for scalars by ˜ ∗ βA := ((αA ;  αA ⊗ AA ) ⊗∗ (βA ;  AA ))  IB . ˜ ∗ is well-defined. By Property (1) of ⊗ the operation ⊗ Lemma 5.62 Let α and β be scalars such that αA ⊗ βA is defined and within G[B, B]. Then we have AA ) ⊗∗ (βA ;  AA ) = (α ∗ β)B ;  BB , (1) (αA ;  (2) (αA ;  AA ) ⊗∗ (βA ;  AA ) = ((αA ;  AA ) ⊗∗ (βA ;  AA ));  BB . Proof. ↓

↓

AA )) ⊗ (δA \(βA ;  AA )) . Then we im(1) For brevity, let R := (γA \(αA ;  mediately conclude that AA ) ⊗∗ (βA ;  AA ) (αA ;   = ((γ ∗ δ)B ; R)

definition ⊗∗

γ,δ∈Sc[G]

=



γ α δ β

((γ ∗ δ)B ; ( AA ⊗  AA ))

Lemma 5.20

154

GOGUEN CATEGORIES

=



((γ ∗ δ)B ;  BB )

Property (3) of ⊗

γ α δ β

⎛

⎞  ⎜ ⎟ =⎝ (γ ∗ δ)B ⎠ ;  BB γ α δ β

= (α ∗ β)B ;  BB

monotonicity of ∗

(2) The computation (αA ;  AA ) ⊗∗ (βA ;  AA ) = (α ∗ β)B ;  BB

(1) of this lemma Lemma 4.21 (2)

BB ;  BB = (α ∗ β)B ;  = (αA ;  AA ) ⊗∗ (βA ;  AA );  BB

(1) of this lemma


shows the assertion.

˜ ∗ we may compare ∗ with the corresponding ∗-based operation as Using ⊗ follows: Lemma 5.63 Let α and β be scalars such that αA ⊗ βA is defined and within ˜ ∗ βA = (α ∗ β)B . G[B, B]. Then αA ⊗ Proof. We immediately conclude that ˜ ∗ βA = ((αA ;  AA ) ⊗∗ (βA ;  AA ))  IB αA ⊗ = (α ∗ β)B ;  BB  IB = (α ∗ β)B .

˜∗ definition ⊗ Lemma 5.62 (1) Lemma 4.24 (1)




By Lemma 2.66 (2)  is the strongest t-norm-like operation. This leads to the following property of the corresponding ∗-based operation: Lemma 5.64 Suppose ⊗ is defined on G[A, B] and G[C, D] and its value is within G[E, F ]. Then we have ⊗∗ = ⊗ iff ∗ = , i.e., Q ⊗∗ R = Q ⊗ R for all Q : A → B and R : C → D iff ∗ = . Proof. The implication ⇐ follows from Q ⊗ R  =

↓

↓

((α  β)E ; ((αA \Q) ⊗ (βC \R) ))

α,β∈Sc[G]

definition ⊗∗

CATEGORIES OF L-FUZZY RELATIONS



=

↓

155

↓

((αE  βE ); ((αA \Q) ⊗ (βC \R) ))

α,β∈Sc[G]



=

↓

α,β∈Sc[G]

⎛

=⎝

↓

((αA ; (αA \Q) ) ⊗ (βC ; (βC \R) ))



⎞

⎛

α∈Sc[G]

⎞



↓ (αA ; (αA \Q) )⎠ ⊗ ⎝

Property (5) of ⊗ ↓

(βC ; (βC \R) )⎠

Property (4) of ⊗

β∈Sc[G]

= Q ⊗ R.

Theorem 5.35

The computation ˜ ∗ βA (α ∗ β)B = αA ⊗ = ((αA ;  AA ) ⊗∗ (βA ;  AA ))  IB

Lemma 5.63 ˜∗ definition ⊗

= ((αA ;  AA ) ⊗ (βA ;  AA ))  IB = ((αA ;  AA ) ⊗ (βA ;  AA ))  IB ˜ = αA ⊗ βA

assumption as shown above ˜ definition ⊗ Lemma 5.63

= (α  β)B




proves the other implication.

Since  is the weakest t-conorm-like operation by Lemma 2.67 (2) we get a similar result in the special case ⊗ = . Lemma 5.65 We have ∗ =  iff ∗ = , i.e., Q ∗ R = Q  R for all Q, R : A → B iff ∗ = . Proof. The implication ⇐ follows from Q  R  =

↓

↓

((α  β)A ; ((αA \Q)  (βA \R) ))

definition 

α,β∈Sc[G]



=

↓

↓

((αA  βA ); ((αA \Q)  (βA \R) ))

α,β∈Sc[G]



=

↓

↓

(αA ; ((αA \Q)  (βA \R) )

α,β∈Sc[G] ↓

↓

 βA ; ((αA \Q)  (βA \R) ))  ↓ ↓  (αA ; (αA \Q)  βA ; (βA \R) ) α,β∈Sc[G]

⎛

=⎝



⎞

⎛

(αA ; (αA \Q) )⎠  ⎝

α∈Sc[G]

=QR

↓



⎞ ↓

(βA ; (βA \R) )⎠

β∈Sc[G]

Theorem 5.35

156

GOGUEN CATEGORIES

⎛ =⎝

⎞



↓

(αA ; ((αA \Q)   AB ))⎠

α∈Sc[G]

⎛



 ⎝ ⎛ =⎝

Theorem 5.35 ⎞

↓

(βA ; ( AB  (βA \R) ))⎠

β∈Sc[G]



⎞

↓ ↓ ((αA  ⊥ ⊥AA ); ((αA \Q)  (⊥ ⊥AA \R) ))⎠

α∈Sc[G]

⎛



 ⎝ 



definition residual ⎞

↓ ↓ ((⊥ ⊥AA  βA ); ((⊥ ⊥AA \Q)  (βA \R) ))⎠

β∈Sc[G] ↓

↓

((αA  βA ); ((αA \Q)  (βA \R) ))

α,β∈Sc[G]

=



↓

↓

((α  β)A ; ((αA \Q)  (βA \R) ))

α,β∈Sc[G]

= Q  R.

definition 

The computation ˜ ∗ βA (α ∗ β)A = αA  = ((αA ;  AA ) ∗ (βA ;  AA ))  IA

Lemma 5.63

= ((αA ;  AA )  (βA ;  AA ))  IA AA  IA )  (βA ;  AA  IA ) = (αA ;  = αA  βA = (α  β)A

definition ∗ assumption Lemma 4.24 (1) see above


proves the other implication.

Suppose (L, ∗, 1, 0) is a loos and R is a crisp L-fuzzy relation. Then we have (Q ∗ R)(x, y) = Q(x, y) ∗ R(x, y) = Q(x, y)  R(x, y) = (Q  R)(x, y). The next lemma shows that this property is true in general. Lemma 5.66 Let  = I, and Q : A → B and R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. If Q or R is crisp, then we have Q ⊗∗ R = Q ⊗ R. Proof. Suppose R is crisp. The computation  ↓ ↓ ((α ∗ β)E ; ((αA \Q) ⊗ (βA \R) )) Q ⊗∗ R =

definition ⊗∗

α,β∈Sc[G]

=



α∈Sc[G] β= ⊥ ⊥

↓

↓

((α ∗ β)E ; ((αA \Q) ⊗ (βC \R) ))

Lemma 2.66 (1)

CATEGORIES OF L-FUZZY RELATIONS

=



↓

((α ∗ β)E ; ((αA \Q) ⊗ R))

α∈Sc[G] β= ⊥ ⊥

=

⎛⎛



⎝⎝

=

Lemma 5.12 (1)

⎞

⎞ ↓

(α ∗ β)E ⎠ ; ((αA \Q) ⊗ R)⎠

β = ⊥ ⊥

α∈Sc[G]





157

↓

((α ∗ I)E ; ((αA \Q) ⊗ R))

∗ is monotone

α∈Sc[G]

=



↓

(αE ; ((αA \Q) ⊗ R))

=I

α∈Sc[G]

=



↓

((αE  IE ); ((αA \Q) ⊗ R))

α∈Sc[G]

=



↓

((αA ; (α\Q) ) ⊗ R)

α∈Sc[G]

⎛



=⎝

Property (5) of ⊗

⎞ ↓

(αA ; (αA \Q) )⎠ ⊗ R

Property (4) of ⊗

α∈Sc[G]

=Q⊗R

Theorem 5.35

shows the assertion. The second assertion is shown analogously.




Lemma 2.66 may also be lifted to the ∗-based operations as follows: Lemma 5.67 Let  = I, and Q : A → B and R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. Then we have ⊥CD = ⊥ ⊥AB ⊗∗ R = ⊥ ⊥EF , (1) Q ⊗∗ ⊥ AB = Q and  CD ∗ R = R, (2) Q ∗  (3) Q ⊗ R  Q ⊗∗ R  Q ⊗ R. Proof. (1) The assertion follows immediately from ⊥CD = Q ⊗ ⊥ ⊥CD Q ⊗∗ ⊥ =⊥ ⊥EF .

Lemma 5.66 Property (2) of ⊗

The second assertion is shown analogously. (2) Consider the computation AB = Q   AB Q ∗  = Q.

Lemma 5.66

Again, the second assertion is shown analogously. (3) follows immediately from Lemma 2.66 and Lemma 5.64.




158

GOGUEN CATEGORIES

Replacing I by ⊥ ⊥ we obtain a slightly modified dual version of the previous two lemmata. Lemma 5.68 Let  = ⊥ ⊥, and Q : A → B and R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. If Q or R is crisp, then Q ⊗∗ R = AB ⊗ R). (Q ⊗  CD )  ( Proof. Suppose R is crisp. Consider the computations  ↓ ((α ∗ ⊥ ⊥)E ; ((αA \Q) ⊗  CD )) α∈Sc[G]

=



↓

(αE ; ((αA \Q) ⊗  CD ))

Lemma 2.67 (1)

α∈Sc[G]

=



↓

((αE  IE ); ((αA \Q) ⊗  CD ))

α∈Sc[G]

=



↓

((αA ; (αA \Q) ) ⊗  CD )

α∈Sc[G]

⎛



=⎝

Property (5) of ⊗

⎞ ↓

(αA ; (αA \Q) )⎠ ⊗  CD

Property (4) of ⊗

α∈Sc[G]

= Q⊗ CD ,  ↓ ((α ∗ β)E ; ((αA \Q) ⊗ R)) α∈Sc[G] β= ⊥ ⊥

=

⎛⎛



=

⎞ ↓

(α ∗ β)E ⎠ ; ((αA \Q) ⊗ R)⎠

β = ⊥ ⊥

α∈Sc[G]



⎞



⎝⎝

Theorem 5.35

↓

(α ∗ I)E ; ((αA \Q) ⊗ R)

∗ is monotone

α∈Sc[G]

=



↓

((αA \Q) ⊗ R)

α∈Sc[G]

⎛

=⎝

Lemma 2.67 (1)

⎞



↓ (αA \Q) ⎠ ⊗ R

Property (4) of ⊗

α∈Sc[G] ↓

= ((⊥ ⊥AA \Q) ) ⊗ R

Corollary 4.15 (4)

= ↓ AB ⊗ R = AB ⊗ R.

definition residual

Together this implies  ↓ ↓ Q ⊗∗ R = ((α ∗ β)E ; ((αA \Q) ⊗ (βC \R) )) α,β∈Sc[G]

=



↓

↓

((α ∗ ⊥ ⊥)E ; ((αA \Q) ⊗ (⊥ ⊥CC \R) ))

α∈Sc[G]

definition ⊗∗

CATEGORIES OF L-FUZZY RELATIONS

 =





↓

159

↓

((α ∗ β)E ; ((αA \Q) ⊗ (βC \R) ))

α∈Sc[G] β= ⊥ ⊥

↓

((α ∗ ⊥ ⊥)E ; ((αA \Q) ⊗  CD ))

definition residual

α∈Sc[G]





↓

((α ∗ β)E ; ((αA \Q) ⊗ R))

Lemma 5.12 (1)

α∈Sc[G] β= ⊥ ⊥

= (Q ⊗  CD )  ( AB ⊗ R).

see above


The second assertion is shown analogously.

The next lemma is again a lifting of a corresponding result (Lemma 2.67) for loos’s. Lemma 5.69 Let  = ⊥ ⊥, and Q : A → B and R : C → D be relations such that Q ⊗ R is defined and within G[E, F ]. Then we have CD =  AB ⊗∗ R =  EF , (1) Q ⊗∗  (2) Q ∗ ⊥ ⊥AB = Q and ⊥ ⊥CD ∗ R = R, (3) Q ⊗ R  Q ⊗∗ R  Q ⊗ R. Proof. (1) The assertion follows immediately from Q ⊗∗  CD = (Q ⊗  CD )  ( AB ⊗  CD ) EF = (Q ⊗  CD )   = EF .

Lemma 5.68 Property (3) of ⊗

The second assertion is shown analogously. (2) Consider the computation Q ∗ ⊥ ⊥AB = (Q   AB )  ( AB  ⊥ ⊥AB ) = Q.

Lemma 5.68

Again, the second assertion is shown analogously. (3) follows immediately from Lemma 2.67.




In the rest of this section we want to prove some basic properties of the ∗-based operations and the structures induced by them. We start with the following theorem: Theorem 5.70 Let ⊗ be commutative. Then the operation ⊗∗ is commutative iff (Sc[G], ∗, , ζ) is a commutative loos.

160

GOGUEN CATEGORIES

Proof. The implication ⇐ follows from  ↓ ↓ ((α ∗ β)E ; ((αA \Q) ⊗ (βC \R) )) Q ⊗∗ R =

definition ⊗∗

α,β∈Sc[G]



↓

↓

α,β∈Sc[G]

⊗ and ∗ commutative

= R ⊗∗ Q.

definition ⊗∗

=

((β ∗ α)E ; ((βC \R) ⊗ (αA \Q) )

The computation ˜ ∗ βA (α ∗ β)B = αA ⊗ = ((αA ;  AA ) ⊗∗ (βA ;  AA ))  IB

Lemma 5.63 ˜∗ definition ⊗

= ((βA ;  AA ) ⊗∗ (αA ;  AA ))  IB ˜ = βA ⊗∗ αA

assumption ˜∗ definition ⊗

= (β ∗ α)B

Lemma 5.63


shows the other implication.

Next, we want to focus on associativity. Therefore, we need the following technical lemma: Lemma 5.71 Let (Sc[G], ∗, , ζ) be a cloos, g : Sc[G] → G ↑ [A, B] be antitone, h : Sc[G] → G ↑ [C, D] an antimorphism and ⊗ defined on G[A, B] and G[C, D]. Furthermore, let   (τ (g)(β) ⊗ h(γ)), f¯(α) := (g(β) ⊗ h(γ)). f (α) := β,γ∈Sc[G] β∗γ α

β,γ∈Sc[G] β∗γ α

Then we have τ (f ) = τ (f¯). Proof. The inclusion  is trivial since g  τ (g) and ⊗,  and τ are monotone. Obviously, f¯ is antitone and by Property (6) of ⊗ a function from Sc[G] to G ↑ [E, F ] for suitable objects E and F . Therefore, we may prove the other inclusion by fixed point induction. We define a predicate  (k(β) ⊗ h(γ))  l(α). P(k, l) : ⇐⇒ ∀α ∈ Sc[G] : β,γ∈Sc[G] β∗γ α

This predicate is admissible since P(ki , li ) for all i ∈ I implies    ∀α ∈ Sc[G] : (ki (β) ⊗ h(γ))  li (α) i∈I β∗γα

⇔ ∀α ∈ Sc[G] :

  β∗γα i∈I

i∈I

(ki (β) ⊗ h(γ)) 





i∈I

 li

(α)

definition 

CATEGORIES OF L-FUZZY RELATIONS





⇔ ∀α ∈ Sc[G] :

β∗γα



⇔ ∀α ∈ Sc[G] :  ⇔P

ki ,

i∈I









 ⊗ h(γ)

ki (β)

i∈I

 ki



 

(β) ⊗ h(γ)





 



(α)

li

i∈I



i∈I

β∗γα





Prop. (4) of ⊗

 li

(α)

definition 

i∈I

.

li

161

definition P

i∈I

The base case P(g, f¯) is trivial. In order to prove the induction step, we want to show the following property:     (k(δ) ⊗ h(γ))  (k(µ) ⊗ h(ν)). (∗) N α η∈N µ∗νη

β∗γα M =β δ∈M



Suppose β ∗ γ  α and M = β. If M = ∅, the left side of (∗) equals ⊥ ⊥EF and the inclusion is trivial. Therefore, let M = ∅. Then we define Nγ := {δ ∗ γ | δ ∈ M } and conclude that   Nγ = (δ ∗ γ) definition Nγ δ∈M



  M ∗γ

∗ is continuous and M = ∅  =β∗γ M =β  α.  Furthermore, we have k(δ) ⊗ h(γ)  (k(µ) ⊗ h(ν)) for all δ ∈ M since µ∗νδ∗γ  δ ∗ γ  δ ∗ γ. This implies (k(δ) ⊗ h(γ))  (k(µ) ⊗ h(ν)), and, =

η∈Nγ µ∗νη

δ∈M

hence, (∗). Now, suppose P(k, l). Then we conclude for α ∈ Sc[G]  (ϕ(k)(β) ⊗ h(γ)) β∗γα

=



⎛⎛ ⎝⎝

=









 k(δ)

β∗γα M =β



⎞

k(δ)⎠ ⊗ h(γ)⎠

M =β δ∈M

β∗γα



⎞



definition ϕ

 ⊗ h(γ)

Property (4) of ⊗

δ∈M

(k(δ) ⊗ h(γ))

⊗ monotone

(k(µ) ⊗ h(ν))

by (∗)

β∗γα M =β δ∈M







N α η∈N µ∗νη





l(η)

induction hypothesis

N α η∈N

= ϕ(l)(α),

Lemma 2.42

162

GOGUEN CATEGORIES

and, hence, P(ϕ(k), ϕ(l)). The principle of fixed point induction gives us f 
τ (f¯), which implies τ (f )  τ 2 (f¯) = τ (f¯). Notice that a version of the previous lemma, where h(γ) and g(β) resp. τ (g)(β) are exchanged, may also be proved. Lemma 5.72 Let (Sc[G], ∗1 , 1 , ζ1 ) and (Sc[G], ∗2 , 2 , ζ2 ) be clooses. Furthermore, suppose Q : A → B, R : C → D and S : E → F and Q ⊗ R : G → H, R ⊗ S : I → J and (Q ⊗ R) ⊗ S, Q ⊗ (R ⊗ S) : K → L. Then we have (1) (Q ⊗∗1 R) ⊗∗2 S  =

↓

↓

↓

(((α∗1 β)∗2 γ)K ; (((αA \Q) ⊗(βC \R) )⊗(γE \S) )),

α,β,γ∈Sc[G]

(2) Q ⊗∗1 (R ⊗∗2 S)  =

↓

↓

↓

((α∗1 (β∗2 γ))K ; ((αA \Q) ⊗((βC \R) ⊗(γE \S) ))).

α,β,γ∈Sc[G]

Proof. (1) Define the following functions:  ↓ ↓ ((βA \Q) ⊗ (γC \R) ), g(α) := β,γ∈Sc[G] β∗1 γ α

↓

h(α) := (αE \S) ,  ↓ f (α) := ((βG \(Q ⊗∗1 R)) ⊗ h(γ)), β,γ∈Sc[G] β∗2 γ α



f¯(α) :=

(g(β) ⊗ h(γ)).

β,γ∈Sc[G] β∗2 γ α

f, f¯ and g are antitone and by Property (6) of ⊗ functions from Sc[G] to crisp relations. Furthermore, h is an antimorphism, and we have f (α) =



↓

((βG \(Q ⊗∗1 R)) ⊗ h(γ))

β∗2 γα



= =

⎛

⎜⎝ ⎝ βG \ ⎝

β∗2 γα



⎛⎛



definition f ⎞⎞↓

⎞

⎟ (δG ; g(δ))⎠⎠ ⊗ h(γ)⎠

Lemma 5.61

δ∈Sc[G]

(τ (g)(βG ) ⊗ h(γ))

Lemma 5.49 (1),

β∗2 γα

f¯(α) =



(g(β) ⊗ h(γ))

β∗2 γα

definition f¯

163

CATEGORIES OF L-FUZZY RELATIONS

⎛⎛



=

⎝⎝

β∗2 γα

↓

⎞

↓

((µA \Q) ⊗ (νC \R) )⎠ ⊗ h(γ)⎠

definition g

µ∗1 νβ





=

⎞



↓

↓

↓

↓

(((µA \Q) ⊗ (νC \R) ) ⊗ h(γ))

Prop. (4) of ⊗

β∗2 γα µ∗1 νβ





=

↓

(((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) )

definition h

β∗2 γα µ∗1 νβ



=

↓

↓

↓

(((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) ),

(µ∗1 ν)∗2 γα



(αK ; f¯(α))

α∈Sc[G]

=

⎛





⎛

α∈Sc[G]

=

↓

↓

↓

(((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) )⎠⎠

(µ∗1 ν)∗2 γα

⎛



⎝αK ; ⎝



⎞⎞



⎝αK ; ⎝

α∈Sc[G]

=

⎛

⎞⎞ ↓

↓

↓

(((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) )⎠⎠

(µ∗1 ν)∗2 γ=α ↓

↓

↓

(((α ∗1 β) ∗2 γ)K ; (((αA \Q) ⊗ (βC \R) ) ⊗ (γE \S) )),

α,β,γ∈Sc[G]

where the second equality is shown as follows: The inclusion  is trivial and  is implied by ↓

↓

↓

αK ; (((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) ) ↓

↓

↓

 ((µ ∗1 ν) ∗2 γ); (((µA \Q) ⊗ (νC \R) ) ⊗ (γE \S) ) for all (µ ∗1 ν) ∗2 γ  α in Sc[G]. Finally, we have  (αK ; f (α)) Lemma 5.61 (Q ⊗∗1 R) ⊗∗2 S = α∈Sc[G]



=

(αK ; τ (f )(α))

Lemma 5.49 (2)

α∈Sc[G]



=

(αK ; τ (f¯)(α))

α∈Sc[G]



=

(αK ; f¯(α))

Lemma 5.71 and the first computation above Lemma 5.49 (2),

α∈Sc[G]

such that the assertion follows from the computations above. (2) is shown analogously.




Now, we are ready to state our theorem about associativity of ⊗∗ . Theorem 5.73 Let ⊗ be associative, and (Sc[G], ∗, , ζ) be complete. Then ⊗∗ is associative iff (Sc[G], ∗, , ζ) is a losg.

164

GOGUEN CATEGORIES

Proof. The implication ⇐ follows immediately from Lemma 5.72. Suppose ⊗∗ is associative. Then ⊗ maps pairs of relations from G[A, A] to G[A, A] and we have ˜ ∗ βA );  AA (αA ⊗ = (((αA ;  AA ) ⊗∗ (βA ;  AA ))  IA );  BB

˜∗ definition ⊗

= (((αA ;  AA ) ⊗∗ (βA ;  AA ));  AA  IA );  AA AA ) ⊗∗ (βA ;  AA ));  AA   AA = ((αA ;  AA ) ⊗∗ (βA ;  AA ). = (αA ; 

Lemma 5.62 (2) Lemma 4.23 (1) Lemma 5.62 (2)

Then for all α, β, γ ∈ Sc[G] we conclude that ((α ∗ β) ∗ γ)A ˜ ∗ γA = (α ∗ β)A ⊗ ˜ ∗ γA ˜ ∗ βA )⊗ = (αA ⊗ ˜ ∗ βA );  AA ) ⊗∗ (γA ;  AA ))  IA = (((αA ⊗

Lemma 5.63 Lemma 5.63 ˜∗ definition ⊗

= ((αA ;  AA ) ⊗∗ (βA ;  AA )) ⊗∗ (γA ;  AA ))  IA .

see above

The equality (α ∗ (β ∗ γ))A = ((αA ;  AA ) ⊗∗ ((βA ;  AA ) ⊗∗ (γA ;  AA )))  IA follows analogously.
If ⊗ is composition, one may ask for the categorical structure induced by ;∗ . The answer is given in the next theorem. Theorem 5.74 Let (Sc[G], ∗, , ζ) be complete and suppose that there is an object A in G such that  AA = IA . Then G together with composition ;∗ and identity morphisms  is a category iff (Sc[G], ∗, , ζ) is a losg with ζ = ⊥ ⊥. Proof. First of all, we have  ↓ ↓ ((α ∗ β)A ; (αA \Q) ; (βB \B ) ) Q;∗ B = α,β∈Sc[G]

⎛ ⎜ =⎝

⎞



↓ ↓ ⎟ ((α ∗ β)A ; (αA \Q) ; (βB \B ) )⎠

α,β∈Sc[G] β=⊥ ⊥

⎛

 ⎝ ⎛ ⎜ =⎝



⎞ ↓ ((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠

α∈Sc[G]



↓ ⎟ ((α ∗ β)A ; (αA \Q) )⎠

 ⎝



Lemma 5.20 ⎞

↓

((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠

α∈Sc[G]

definition residual

⎞

α,β∈Sc[G] ⊥ ⊥=β 

⎛

definition ;∗

CATEGORIES OF L-FUZZY RELATIONS

⎛

⎛⎛



=⎝

⎝⎝

 ⎝ ⎛ =⎝

⎞ (α ∗ β)A ⎠ ; (αA \Q)



⎞

((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠ ⎞

↓ ((α ∗ )A ; (αA \Q) )⎠

∗ monotone

α∈Sc[G]

⎛

 ⎝ ⎛ =⎝

⎞



↓

((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠

α∈Sc[G]



⎞

↓ (αA ; (αA \Q) )⎠

 neutral element

α∈Sc[G]

⎛

 ⎝ ⎛

⎞



↓

((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠

α∈Sc[G]

=Q⎝

↓ ⎠⎠

↓

α∈Sc[G]



⎞⎞

⊥ ⊥ =β

α∈Sc[G]

⎛



165



⎞ ↓

((α ∗ ⊥ ⊥)A ; (αA \Q) ;  BB )⎠ .

Theorem 5.35

α∈Sc[G]

Analogously, we get A ;∗ Q = Q  (



↓

((⊥ ⊥ ∗ α)A ;  AA ; (αB \Q) )). Using

α∈Sc[G]

Theorem 5.73, we know that for ⇐ it is sufficient to show that  is the identity. This follows for ζ = ⊥ ⊥ immediately from the computation above. Again, using Theorem 5.73 it is sufficient for ⇒ to show that ζ = ⊥ ⊥. Suppose  is the identity and A is an object with  AA = IA . Then we have γA = γA ;∗ A ⎛ = γA  ⎝ ⎛ = γA  ⎝ ⎛ = γA  ⎝



⎞ ↓

((α ∗ ⊥ ⊥)A ; (αA \γA ) ;  AA )⎠

α∈Sc[G]

 αγ



computation above

⎞

((α ∗ ⊥ ⊥)A ;  AA )⎠

Lemma 5.20

⎞ AA (α ∗ ⊥ ⊥)A ⎠ ; 

αγ

= γA  (γ ∗ ⊥ ⊥)A ;  AA ,

∗ monotone

AA  γA . From Lemma 5.19 we conclude which is equivalent to (γ ∗ ⊥ ⊥)A ;  ⊥AA since  AA = ⊥ ⊥AA and  AA  IA . (⊥ ⊥ ∗ γ)A = ⊥ ⊥AA is that (γ ∗ ⊥ ⊥)A = ⊥ shown analogously such that ζ = ⊥ ⊥ follows.


166

GOGUEN CATEGORIES

Since the converse operation is well-behaved we get the following theorem:


Theorem 5.75 Let (Sc[G], ∗, , ζ) be a closg. Then (Q;∗ R) all Q : A → B and R : B → C.

= R
;∗ Q
for

Proof. The computation ⎛


(Q;∗ R) = ⎝

⎞




↓

↓

((α ∗ β)A ; (αA \Q) ; (βB \R) )⎠

definition ;∗

α,β∈Sc[G]

=



↓


((βB \R)

; (αA \Q)

↓





; (α ∗ β)A )

α,β∈Sc[G]

=



↓


((βB \R)

; (αA \Q)

↓


; (α ∗ β)A )

(α ∗ β)A partial identity

α,β∈Sc[G]

=



↓


((α ∗ β)C ; (βB \R)

; (αA \Q)

↓


)

Lemma 5.14

((α ∗ β)C ; (βC \R
) ; (αB \Q
) )

Lemma 5.15

α,β∈Sc[G]

=



↓

↓

α,β∈Sc[G]

= R
;∗ Q


definition ;∗


shows the assertion. Finally, we will focus on continuity of ⊗∗ . Theorem 5.76 Let (Sc[G], ∗, , ζ) be a cloos. Then we have 



 Qi

⊗∗ R =

i∈I



 (Qi ⊗∗ R)

Q ⊗∗

and

i∈I



i∈I

 Ri

=



(Q ⊗∗ Ri )

i∈I

for all Q, Qi , R, Ri with i ∈ I whenever the application of ⊗∗ is defined. 

Proof. Define g(α) :=

↓

(αA \Qi ) . Then g is antitone and a function from

i∈I

Sc[G] to G ↑ [A, B]. Furthermore, we have 

Qi =

i∈I





↓

(αA ; (αA \Qi ) )

i∈I α∈Sc[G]

=





αA ;

α∈Sc[G]

=







 (αA \Qi )

↓

i∈I

(αA ; g(α)),

α∈Sc[G]

Theorem 5.35

definition g

167

CATEGORIES OF L-FUZZY RELATIONS

and, hence, (βA \(



↓

Qi )) = τ (g)(β) for all β ∈ Sc[G] by Lemma 5.49 (2).

i∈I

Furthermore, let 

f (α) :=

↓



and f¯(α) :=

(τ (g)(β) ⊗ (γ\R) )

β∗γα

↓

(g(β) ⊗ (γ\R) ).

β∗γα

Then we conclude that    Qi

⊗∗ R

i∈I

=

 α∈Sc[G]

=







αE ;



((βA \(

β∗γα





αE ;

α∈Sc[G]

=

 





↓

Qi )) ⊗ (γC \R)↓ )

i∈I

Lemma 5.61



(τ (g)(β) ⊗ (γC \R)↓ )

computation above

β∗γα

(αE ; f (α))

definition f

(αE ; τ (f )(α))

Lemma 5.49 (2)

(αE ; τ (f¯)(α))

Lemma 5.71

(αE ; f¯(α))

Lemma 5.49 (2)

α∈Sc[G]

=



α∈Sc[G]

=



α∈Sc[G]

=



α∈Sc[G]

=

 α∈Sc[G]

=

 α∈Sc[G]

=



 αE ;



αE ;



αE ;





definition f¯

(g(β) ⊗ (γC \R) )





 



 ↓

(βA \Qi )

 ↓

⊗ (γC \R)

i∈I





definition g

 ↓

↓

↓

↓

((β\Qi ) ⊗ (γ\R) )

β∗γα i∈I

αE ;

i∈I α∈Sc[G]

=

 ↓

β∗γα



 



β∗γα



α∈Sc[G]

=



Property (4) of ⊗

 ((β\Qi ) ⊗ (γ\R) )

β∗γα

(Qi ⊗∗ R).

Lemma 5.61

i∈I

The second equality is shown analogously.




The previous theorem and Corollary 2.19 show that ⊗∗ is a lower adjoint of a triple of residuated operations.

168

GOGUEN CATEGORIES

Corollary 5.77 Let (Sc[G], ∗, , ζ) be a cloos. Then there are operations ⊗ ∗, ⊗ and syQ (., .) such that ⊗ ∗ ∗ Q ⊗∗ X  R ⇐⇒ X  Q ⊗ ∗ R, Y ⊗∗ S  R ⇐⇒ Y  R ⊗ ∗ S and

Q ⊗∗ X  R and X ⊗∗ R
 Q
⇐⇒ X  syQ⊗ ∗ (Q, R),

whenever the application of ⊗∗ is defined. If ⊗ is composition, we usually omit the superscript ⊗ resp. ;. The previous corollary shows that an inclusion Q;∗ X  R has a greatest solution in X, namely Q ∗ R. Furthermore, the equation Q;∗ X = R has a solution (X = Q ∗ R) iff Q;∗ (Q ∗ R) = R. ⊗ ⊗ Notice that we have Q ⊗ ∗ R = Q\R, R ∗ S = R/S and syQ∗ (Q, R) = syQ(Q, R) if ⊗ = ; and ∗ =  since ; = ; by Lemma 5.64.

6 FUZZY CONTROLLERS IN GOGUEN CATEGORIES

Following our mathematical investigation on the theory of Goguen categories, we now want to focus on an applications in computer science. Throughout this chapter, we will use the notations of Goguen categories even if the relations are concrete L-fuzzy relations. Furthermore, ∗ is considered to be an operation from a closg (Sc[G], ∗, I, ζ) unless otherwise stated. 6.1

THE MAMDANI APPROACH TO FUZZY CONTROLLERS

We want to show that a fuzzy controller may be described by a simple term in the language of Goguen categories. This may be used in at least two ways. First of all, we get in some sense a denotational semantics of fuzzy controllers, and, hence, a mathematical theory to reason about notions like correctness versus a given specification, safety properties, and so on. One may prove such properties of a controller using the calculus of Goguen categories developed so far. Furthermore, a system, which is able to compute relational terms, may be used to obtain a prototype of the controller. There are two suitable systems, the Relview system developed at the Christian-Albrechts-University of Kiel and the Rath system developed at the University of the Federal Armed Forces Munich [19]. The latter system is in fact a library of Haskell modules that allows to explore relational structures as relation algebras, Dedekind categories, allegories, and so on by providing different means to construct such algebras and to compute relational expressions in a given one. In the diploma thesis by Triebsess [38] certain modules containing the theory of Goguen categories 169

170

GOGUEN CATEGORIES

Fuzzy controller Fuzzy values

6 ? - Decision module

- Fuzzification

Crisp values Figure 6.1.

Rule base

- Defuzzification

Process 

Components of a fuzzy controller.

were developed. Furthermore, a special module allows the user to construct and execute L-fuzzy controllers in a given Goguen category. We want to concentrate on the method of Mamdani [25] for constructing a fuzzy controller summarized in Figure 6.1. In this approach a fuzzy controller consists of a rule base, a decision module, a fuzzification, and a defuzzification. Notice that our fuzzy controllers are not limited to use coefficients from the unit interval [0, 1] of the real numbers. In fact, our running example throughout this chapter will be a controller with values from a nonlinear ordering. 6.2

LINGUISTIC ENTITIES AND VARIABLES

A fuzzy controller is usually formulated using linguistic entities, i.e., abstract notions represented by common words from every day language (like: “extremely high speed”, “hot water”, “very heavy rain”, etc.). Variables ranging over those abstract entities are called linguistic variables. They are understood as variables over suitable L-fuzzy sets. As usual in the theory of relations we describe a subset of A by a relation M : I → A, where I is a unit of the corresponding relational category. In the case of Goguen categories, such a relation is an abstract notion of an L-fuzzy subset as required for the interpretation of linguistic variables. As indicated above, linguistic entities are often built up from two components. First of all, there is a basic notion of an abstract entity as “high speed.” This basic notion may be modified by an adverb as “very” or “extremely.” On a suitable level of abstraction, these adverbs may be seen as linguistic modifiers, i.e., functions mapping L-fuzzy sets to L-fuzzy sets. We will study orderingbased, weakening, and intensifying modifiers (cf. [6, 7]). Suppose E is an ordering relation, i.e., IA  E (E reflexive), E; E  E (E transitive), and E  E
= IA (E antisymmetric). The well-known concept and its relational description of upper and/or lower bounds of a subset M : I → A (cf. [32]) may be used to model the linguistic modifiers “greater than” and “less

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

1.0

171

L L less than average

0.8 0.6 0.4 0.2 0.0 0

2

4

L L  greater than average L L L L L L L L average L L L L L L 6 8 10 12 14 16

Figure 6.2.

Ordering-based modifiers.

than” with respect to a semigroup operation ∗. We define greater than∗ (E, M ) less than∗ (E, M )

:= :=




(E
∗ M ) ,
(E ∗ M ) .

Example 6.1 We want to illustrate the definitions above within fuzzy sets using L  ukasiewicz t-norm tL (x, y) := max{0, x + y − 1} for ∗. Suppose that the mean score of a test is 8. The fuzzy set “average” in the universe of scores [0, 16] may be represented by a triangular fuzzy set shown in Figure 6.2. Let E be the natural (crisp) ordering on [0, 16]. Then we get the fuzzy set “greater than average” as indicated in the same figure. Notice that this set is greatest solution X of average
;tL X  E. Therefore, the degree of membership x of 9 has to be 0.5. If z is a value with 9 < z ≤ 10, then E(z, 9) = 0, and, hence, x + y with y the degree of membership of z in “average” has to be less or equal to 1.  Another interesting class of modifiers originates from relations that model approximate equalities. Such a relation Ξ should be reflexive IA  Ξ and symmetric Ξ
 Ξ. In a first attempt one would also require transitivity. This may lead to a counterintuitive result. To illustrate this, we recall an example given in [8]: In everyday life we usually do not feel a difference in temperature between 0◦ and 1◦ , neither between 1◦ and 2◦ , between 35◦ and 36◦ and so on. For a human being, those degrees are approximately equal. Obviously, the corresponding relation is not transitive since otherwise all temperatures would be approximately equal, which is a counter-intuitive result.

Such approximate equality may be used to model weakening and intensifying modifiers as “extremely”, “very”, “more or less”, and “roughly”. The following intuitive inclusions should be guaranteed: extremely(M ) ⊆ very(M ) ⊆ M ⊆ more or less(M ) ⊆ roughly(M )

172

GOGUEN CATEGORIES

for all L-fuzzy sets M . We define extremely∗ (Ξ, M ) := (M ∗ Ξ) ∗ Ξ, very∗ (Ξ, M ) := M ∗ Ξ, more or less∗ (Ξ, M ) := M ;∗ Ξ, roughly∗ (Ξ, M ) := (M ;∗ Ξ);∗ Ξ. or more generally very1∗ (Ξ, M ) := very∗ (Ξ, M ), i veryi+1 ∗ (Ξ, M ) := very∗ (Ξ, very (Ξ, M )),

roughly1∗ (Ξ, M ) := more or less∗ (Ξ, M ), i roughlyi+1 ∗ (Ξ, M ) := more or less∗ (Ξ, roughly (Ξ, M ))

for all i ≥ 1. Lemma 6.2 Suppose Ξ : A → A is a reflexive relation and define Ξ1 := Ξ and Ξi+1 := Ξi ;∗ Ξ. Then we have (1) if ;∗ is associative, then veryi∗ (Ξ, M ) = very∗ (Ξi , M ) and roughlyi∗ (Ξ, M ) = more or less∗ (Ξi , M ), i (2) veryi+1 ∗ (Ξ, M )  very∗ (Ξ, M )  M for all i ≥ 1,

(3) M  roughlyi∗ (Ξ, M )  roughlyi+1 ∗ (Ξ, M ) for all i ≥ 1. Proof. (1) The second assertion is trivial and the first one follows from X  very∗ (Ξ, very(Ξ, M )) ⇔ X  (M ∗ Ξ) ∗ Ξ ⇔ X;∗ Ξ  M ∗ Ξ ⇔ (X;∗ Ξ);∗ Ξ  M ⇔ X;∗ Ξ  M 2

definition very residuated operation residuated operation ;∗ associative

⇔ X  M ∗ Ξ

residuated operation

⇔ X  very∗ (Ξ , M ).

definition very

2

2

(2) The assertion follows from very∗ (Ξ, M )  M . This is shown by X  very∗ (Ξ, M ) ⇔ X  M ∗ Ξ ⇔ X;∗ Ξ  M ⇒ X  X;∗ Ξ  M.

definition very residuated operation ;∗ is monotonic and IA  Ξ

(3) follows immediately from the reflexivity of Ξ.




FUZZY CONTROLLERS IN GOGUEN CATEGORIES

173

1.0

L L L L L L L L  greater than average less than average L L L L 0.6 L L L very less -L L than average L L L 0.4 L L L  average L L less L extremely  L L 0.2 than average L L L L L L L L 0.0 0 2 4 6 8 10 12 14 16 0.8

Figure 6.3.

Weakening and intensifying modifiers.

The previous lemma gives another reason not to require that Ξ is transitive. In that case all veryi∗ resp. roughlyi∗ would be equal. Example 6.3 Again, we want to illustrate the definitions above within fuzzy sets and the L  ukasiewicz t-norm. We define Ξ(x, y) := min(1, max(0, 1.2 − 0.1|x − y|)). Intuitively, two values are considered as equal (with degree 1) if they are within a range less or equal to 2. The result is shown in Figure 6.3.  The interpretation of the linguistic entities now consists of a suitable choice of L-fuzzy subsets of the input resp. output domain A, i.e., by relations Q : I → A, for any linguistic entity that occurs within the controller. The abstract set of linguistic entities itself is modelled within an arbitrary Goguen category by a suitable disjoint union of several copies of a unit I. Last but not least, the whole interpretation Lin for the input resp. Lout for the output is given by the corresponding disjoint union of the relations above. For example, if A is the set of temperatures and the input domain of the controller, the set of linguistic entities is given by the set {very cold, cold, medium, hot, very hot} and their interpretation by relations Qi : I → A for 1 ≤ i ≤ 5 then we have ⎛ ⎞ 
(ιi ; Qi )⎠ : I + I + I + I + I → A, Lin := ⎝ 1≤i≤5

where ιi for 1 ≤ i ≤ 5 are the crisp injections from I to I + I + I + I + I. Notice that it is essential that the injections are crisp. If the injections are noncrisp, there would be a fuzzification, which is not an integral part of the definition of the controller. This fuzzification arises from the specific choice of the sum. In this case, reasoning about the controller using the term above seems to be impossible or at least difficult. Example 6.4 As a running example in this chapter we want to construct a temperature controller. It has to control the temperature in a room with two

174

GOGUEN CATEGORIES

different colonies of bacteria. These colonies need a slightly different temperature for optimal growth. The first one needs about 24◦ and the second one about 25.5◦ . Both colonies will die if the temperature is less than 5◦ or greater than 45◦ for a while. The input of the controller are temperatures taken from the left-closed/right-open interval A := [0.0 . . . 50.0[. The output is a value from B := {−20, −19, −18, . . . , 18, 19, 20}, the adjusting values of the heating, or a special signal “AL” to indicate an alert. Therefore, the controller will be a relation C : A → B+I. The linguistic entities for the input resp. output are given by OT = ˆ optimal temperature SW = ˆ slightly too warm

NC = ˆ no change PS = ˆ positive small

SC LW LC TW TC

NS PM NM PB NB

= ˆ = ˆ = ˆ = ˆ = ˆ

slightly too cold little bit too warm little bit too cold too warm too cold

MW = ˆ much too warm MC = ˆ much too cold

= ˆ = ˆ = ˆ = ˆ = ˆ

negative small positive medium negative medium positive big negative big

AL = ˆ alert

Since a temperature may be optimal with a certain degree in two senses, namely for the first and the second colony, we choose the lattice L to be [0, 1] × [0, 1]. We will denote the L-fuzzy relations corresponding to the linguistic entities as shown in the following table: OT SW LW TW MW

Q0 Q1 Q2 Q3 Q4

SC LC TC MC

Q−1 Q−2 Q−3 Q−4

NC PS PM PB

S0 S1 S2 S3

AL NS NM NB

S4 S−1 S−2 S−3

We start our interpretation of the linguistic entities with two L-fuzzy relations e24 , e25.5 : I → A defined by



(1, 0) iff x = 24, (0, 1) iff x = 25.5, and e25.5 (1, x) := e24 (1, x) := (0, 0) iff x = 24 (0, 0) iff x = 25.5, denoting the optimal temperature for the first and second colony. Furthermore, we define an approximate equality ΞA by ΞA (x, y) := (min(1, max(0, a1 − b1 |x − y|)), min(1, max(0, a2 − b2 |x − y|))) for suitable a1 , a2 , b1 and b2 . ΞA expresses, which temperatures are considered as equal in respect to the first (coefficients a1 and b1 ) and to the second (coefficients a2 and b2 ) colony. By weakening e24 resp. e25.5 and combining the results we obtain the L-fuzzy set Q0 := more or less(ΞA , e24 )  more or less(ΞA , e25.5 ).

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

175

Notice that we use  for ∗ in the definition above. As an example, we want to compute Q0 (1, 24.8) with a1 = a2 = 1.2, b1 = 0.35 and b2 = 0.25 Q0 (1, 24.8) = (e24 ; ΞA )(1, 24.8) ∨ (e25.5 ; ΞA )(1, 24.8) = ((1, 0) ∧ ΞA (24, 24.8)) ∨ ((0, 1) ∧ ΞA (25.5, 24.8)) = ((1, 0) ∧ (0.92, 1)) ∨ ((0, 1) ∧ (0.965, 1))

definition Q0 def. e24 , e25.5 and ; definition ΞA

= (0.92, 1). The relations for SW, . . . , TC are computed by shifting Q0 to the left resp. to the right. This may be achieved using a suitable crisp bijection sA : A → A. For example, for a suitable u let sA be defined by sA (x, y) = (1, 1) if y = (x + u) mod 50.0 and sA (x, y) = (0, 0) otherwise. It seems to be unnatural to require that sA is a bijection. But this property will make the proofs in Section 6.7 much more easier. Furthermore, sA fulfills a special kind of a monotonicity property. Suppose EA is the linear ordering on A. An element is in the domain of the univalent and injective relation sA  EA iff it is expanded by sA . Analogously, an element is in the range of that relation iff it is contracted by sA . Between its domain and range, sA  E is an order isomorphism such that we have EA ; (sA  EA ) = (sA  EA ); EA . Now, we define Qi+1 := Qi ; sA

and

Q−(i+1) := Q−i ; s
A

for 0 ≤ i ≤ 2. Finally, Q4 and Q−4 are defined by the order-based modifiers.  ukasiewicz t-norm extended to Suppose EA is the ordering on A and ∗ is the L pairs, i.e., (x1 , x2 ) ∗ (y1 , y2 ) := (tL (x1 , y1 ), tL (x2 , y2 )). Then we obtain Q4 := greater than∗ (EA , Q3 )

and

Q−4 := less than∗ (EA , Q−3 ).

The interpretation of the output linguistic entities is done analogously. We start with the crisp relation e0 : I → B defined by e0 (1, 0) = (1, 1) and e0 (1, x) = (0, 0) otherwise. Furthermore, we define an approximate equality ΞB on B by ΞB (x, y) := (min(1, max(0, a − b|x − y|)), min(1, max(0, a − b|x − y|))) for suitable a and b. Together with a suitable crisp bijection sB : B → B for shifting we define S0 := more or less(ΞB , e0 ),

 Si+1 := Si ; sB

and

  S−(i+1) := S−i ; s
B

for 0 ≤ i ≤ 2. Finally, using the crisp injections ι and κ from B resp. I to B + I we get Si := Si ; ι for −3 ≤ i ≤ 3, S4 := κ, and, hence, 

ιi ; Qi and Lout := ιi ; Si .  Lin := −4≤i≤4

−3≤i≤4

176 6.3

GOGUEN CATEGORIES

FUZZIFICATION

The fuzzification part of a controller consists of an operation F (x) mapping each input value x to an L-fuzzy set. In our approach the input values from A are interpreted by relational elements or points, i.e., by crisp functions x : I → A. Therefore, the easiest fuzzification operation is the identity since points are special L-fuzzy sets. Naturally, other operations are possible. For example, one may map such a point to another L-fuzzy sets using the weakening operators from Section 6.2. A suitable choice hardly depends on the application. Usually, the fuzzification part is used to validate the controller. Therefore, some arbitrary values are chosen for the possible parameters in a first attempt. By testing the controller for suitable inputs those parameters are modified until a desired behavior can be observed. Example 6.5 In our example we will use the identity as fuzzification. 6.4



THE RULE BASE

A control rule is usually formulated as a conditional expression using the linguistic variables, i.e., it is of the form if x is M, then y = N, where x and y are linguistic variables considered as the input resp. as the output and M and N are fuzzy sets. These sets are built up from basic linguistic entities and some ∗-based operations. We may require without loss of generality that M and N are indeed basic linguistic entities. If not, we introduce a new entity for the corresponding expression. With this convention a control rule may be seen as pair of such linguistic entities. A rule base, i.e., a finite list of control rules, may be described by a crisp relation R between the given sets of linguistic entities. As mentioned above, in an arbitrary Goguen category these sets are modelled by a suitable disjoint union of several copies of a unit I. R is given by the corresponding relation between them. For example the rule base if x is M1 , then y = N1 , if x is M2 , then y = N1 , if x is M3 , then y = N2 is modelled by a relation R : I + I + I → I + I defined by

R := ι
1 ; ι1  ι2 ; ι1  ι3 ; ι2 ,

where ι1 , ι2 , ι3 resp. ι1 , ι2 are the crisp injections from I to I + I + I resp. I + I. R may be also represented by the following matrix:   1 0 1 0 R= 0

1

In general, if R(i) denotes the set of indices of output linguistic entities related by the rule base to the input linguistic entity i, the relation R is of the from

FUZZY CONTROLLERS IN GOGUEN CATEGORIES



R=

177

(ι
i ; ιj ).

i∈I j∈R(i)

If we combine the interpretation of the rule base and of the linguistic entities for input and output, we obtain the following relational terms T : A → B for the core of the controller:


T := (Lin ;∗1 R);∗2 Lout

or




T := Lin ;∗1 (R;∗2 Lout ),

where (Sc[G], ∗1 , I, ζ) and (Sc[G], ∗2 , I, ζ) are closgs. The following and a similar computation for the second term show that just one closg is really needed:


Lin ;∗ (R;∗ Lout ) ⎛ ⎛ in


=L

;∗ ⎝R;∗ ⎝ ⎛

in


=L

⎛

;∗ ⎝R; ⎝



⎞⎞ ⎠⎠ (ι
j ; Sj )

j∈J



⎞⎞

⎠⎠ (ι
j ; Sj )

Lemma 5.66

j∈J

⎞ ⎛ ⎞⎞  
⎜⎜ ⎟ ⎝ ⎠⎟ = Lin ;∗ ⎝⎝ (ι
(ι
i ; ιj )⎠ ; j ; Sj ) ⎠ ⎛⎛

i∈I j∈R(i)

 =



⎞  ⎟ ⎜ (Q
(ι
i ; ιi ) ;∗ ⎝ i ; Sj )⎠ 

i∈I

=

  l∈I

= = = =

definition Lin

i∈I j∈R(i)


((Q
l ; ιl );∗ (ιi ; Sj ))

;∗ continuous


((Q
l ;∗ ιl );∗ (ιi ;∗ Sj ))

Lemma 5.66


(Q
l ;∗ (ιl ;∗ ιi );∗ Sj )

Theorem 5.73


(Q
l ;∗ (ιl ; ιi );∗ Sj )

Lemma 5.66

i∈I j∈R(i)

  l∈I

⎛

i∈I j∈R(i)

  l∈I

definition injections

i∈I j∈R(i)

  l∈I

interpretation R

j∈J

i∈I j∈R(i)

⎞ 
⎜ ⎟ (ι
= Lin ;∗ ⎝ i ; Sj )⎠ ⎛

definition Lout

i∈I j∈R(i)



(Q
i ;∗ Sj ),

i∈I j∈R(i)

where the last equality follows from the definition of the injections and Q;∗ IB = ⊥BC = Q; ⊥ ⊥BC = ⊥ ⊥AC for all Q : A → B since IB and Q; IB = Q and Q;∗ ⊥ ⊥ ⊥BC are crisp.

178

GOGUEN CATEGORIES

Linguistic entities (output) 1 S1 + @ @ S@ 1 HH2 @ @ H @ H @ @ + R H S j H 3 -B R crisp @ 1 *   +    S4 1 .. + . .. .

Linguistic entities (input) 1 Q1 + Q2 1   +   Q3    1 AH Y H HH + H H Q4 1 .. . + .. . Figure 6.4.

Core of a fuzzy controller.

The core T : A → B of a fuzzy controller is visualized in Figure 6.4. Notice again that we do not require that the underlying lattice L is the unit interval. Example 6.6 Now, we want to return to our running example. Consider the following rule base and its interpretation by a matrix: ⎛ 1 0 0 0 0 0 0 0 ⎞ if x is OT, then y = NC, 0 1 0 0 0 0 0 0 ⎜ 0 0 1 0 0 0 0 0 ⎟ if x is SW, then y = NS, ⎜ 0 0 0 1 0 0 0 0 ⎟ ⎟ R=⎜ ⎜ 00 00 00 00 10 01 00 00 ⎟ if x is SC, then y = PS, ⎝ 0 0 0 0 0 0 1 0 ⎠ 0 0 0 0 0 0 0 1 if x is LW, then y = NM, 0

if x is LC, then y = PM, if x is TW, then y = NB,

0

0

0

0

0

0

1

if x is TC, then y = PB, if x is MW, or x is MC, then y = AL. out for −3 ≤ i ≤ 4, If we denote the crisp injections by ιin i for −4 ≤ i ≤ 4 and ιi the relational term for R is given by ⎞ ⎛ 
out
out (ιin ; ιi )⎠  ιin R := ⎝ i −4 ; ι4 −3≤i≤4

We choose  for ∗ and get the following core of the controller T : A → B + I: ⎞ ⎛ ⎞ ⎛  
in ⎠ ⎝ (Q
(ιout ; Si )⎠ T := ⎝ i i ; ιi ) ; R; −4≤i≤4

=



−3≤i≤4

Q
i ; Si

−3≤i≤4



Q
−4 ; S4 .

computation above



FUZZY CONTROLLERS IN GOGUEN CATEGORIES

6.5

179

DECISION MODULE

The decision module describes to which degree a rule of the controller is activated for a given input. In our approach this corresponds to the question of how to combine the result of the fuzzification F (x) with the core of the controller T . Usually, a so-called optimistic view is taken (cf. [14]). Using the unit interval the degree of activation of a rule if x is M , then y = N for a fuzzy set P is computed in respect to a given t-norm by the expression sup t(P (x), M (x)), x∈A

i.e., as the supremum of the combined degrees of membership within P and M for all suitable x. This corresponds exactly to a ∗-based composition within Goguen categories such that we usually use U (x) := F (x);∗ T as the fuzzy part of the controller. Also, a more pessimistic view may be chosen. Such an approach corresponds to a relation U based on some residuals instead of ;∗ . For example, one may

choose U (x) := F (x) ∗ T or quite more pessimistic U (x) := syQ∗ (F (x) , T ). These expressions compute the degree of inclusion resp. the degree of equality of F (x) and the corresponding M . But, as mentioned in [14] none of those pessimistic views has been used for any fuzzy controller so far. Example 6.7 In our example we choose the optimistic view with ∗ =  such that ⎞ ⎛ 
⎠.  (Q
U (x) := F (x); T = x; T = x; ⎝ i ; Si )  Q−4 ; S4 −3≤i≤4

6.6

DEFUZZIFICATION

In the defuzzification part of controller the fuzzy output is transformed back into a crisp value. In our approach we need an operation D mapping the Lfuzzy relation U (x) : I → B to a crisp relation D(U (x)) : I → B. There are several proposals for D throughout the literature (e.g., cf. [15]). Most of them are closely related to the unit interval [0, 1] and the operations defined there. Therefore, the choice of D depends usually on the lattice L, which is used to model the degree of membership within the controller. But, at least one class of methods may be used for all lattices, methods arising from the cut operation. Given an operation ΘI , which maps any x to a scalar ΘI (x) on I we define ↓

D(U (x)) := (ΘI (x)\U (x)) . The most simple definition for ΘI would be a constant function mapping any x to a given scalar αI , i.e., ΘI (x) := αI . Another suitable choice for ΘI may

180

GOGUEN CATEGORIES

be the scalar of the maximal degree of membership. Within L-fuzzy relations BI ∩ II and compute consider the partial identity ΘI (x) := U (x) ◦  ΘI (x)(1, 1) = (U (x) ◦  BI ∩ II )(1, 1) = (U (x) ◦  BI )(1, 1) = (U (x)(1, v) ∧  BI (v, 1))

definition ΘI (x) definition ∩ and II definition ◦

v∈B

=



definition  BI

U (x)(1, v).

v∈B

With other words, this relation computes the supremum w of all degrees of membership of elements v ∈ B related to x within U (x). Under some circumstances the controller D(U (x)) may be expressed in quite a nice way. Suppose ΘI fulfills x
; ΘI (x) = SA ; x
for all x : I → A and a suitable partial identity SA : A → A. Furthermore, suppose F (x) is some weakening of x, i.e., F (x) = x;∗ Ξ for some closg and relation Ξ, and, hence, F (x) = x; Ξ since x is crisp. Then we have in the optimistic approach U (x) = F (x);∗ T = (x; Ξ);∗ T = (x;∗ Ξ);∗ T

definition of U (x) definition of F (x) Lemma 5.66 Theorem 5.73

= x;∗ (Ξ;∗ T ) = x; (Ξ;∗ T ).

Lemma 5.66

If we use the abbreviation T  := Ξ;∗ T , the computation above implies ↓

Y  D(U (x)) ⇔ Y  (ΘI (x)\U (x)) 

⇔ Y  (ΘI (x)\(x; T )) ↑



⇔ Y  ΘI (x)\(x; T ) ↑

⇔ ΘI (x); Y  x; T



⇔ x
; ΘI (x); Y ↑  T 


↑

⇔ SA ; x ; Y  T


↑

⇔ x ; Y  SA \T ⇔x


↓

⇔x


↓


Galois correspondence definition residual Lemma 4.8 (2) definition residual

↑

; Y )  SA \T   ↓

; Y  (SA \T )  ↓

⇔ x ; Y  (SA \T )  ↓

computation above

property of ΘI



; Y ↑  SA \T 


↓

⇔ (x



definition D(U (x)) ↓

⇔ Y  x; (SA \T ) ,

x crisp Lemma 5.6 (9) Galois correspondence x crisp Lemma 4.8 (2)

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

181

↓

and, hence, D(U (x)) = x; (SA \T  ) . Consequently, the controller is given by a ↓ simple relational term C := (SA \T  ) . Its action on arguments D(U (x)) is just the usual relational application x; C of C on x. The constant function as well as the operation of maximal degree fulfills the property above. First, suppose ΘI (x) = αI for a given scalar αI . Then we have x
; ΘI (x) = x
; αI

definition ΘI




= αA ; x .

Lemma 5.14

Second, suppose ΘI (x) = U (x);  BI  II . Then we have x
; ΘI (x) = x
; (U (x);  BI  II )




= x ; (x; T ;  BI  II ) 




BI  x = T ; 




definition ΘI computation above Lemma 4.7 (2)




BA ; x  x = T ;

x total

BA  IA ); x
. = (T  ; 

Lemma 4.7 (1)

Usually, a controller defined using a cut as defuzzification is not a function. For example, there may be none or more than one possible values with maximal degree of membership for a given input x. But, in applications it is usually essential that the controller codes a function. Otherwise, it would not be clear which action, if any, should be done. Therefore, the cut approach has to be modified. Within the defuzzification operation one has to select a specific element as the result. This may be done by several operations. For example, if there is an ordering on the output domain, one could choose the greatest or least element among the set of results. Also, some weighted mean could be taken if the output domain offers such an operation. A general approach for selecting a specific element may be as follows: Suppose f : P∅ (B) → B is a crisp function mapping each (crisp) nonempty subset of B to an element of B. For example, any mean function and the order-based operations may be represented in that way. The class of mean functions shows that the empty set has to be excluded. A controller derived from an extension of a mean function to the empty set usually does not grasp the intended behavior of the system. The abstract description of the object P∅ (B) within an arbitrary Goguen category G is given by a total relational power, introduced in [39], in the Dedekind category G ↑ of crisp relations. This construction is defined by a crisp relation ε∅B : B → P∅ (B) such that syQ(ε∅B , ε∅B )  IP∅ (B) , syQ(X
, ε∅B );  P∅ (B)C = X;  BC

for all crisp relations X : A → B.

For a total relation X : A → B the relation syQ(X
, ε∅B ) corresponds to the map, which maps every x ∈ A to the image under X, i.e., the set of all y ∈ B, which are related to x. Now, we define a defuzzification by ↓


D(U (x)) := syQ((ΘI (x)\U (x))

, ε∅B ); f.

182

GOGUEN CATEGORIES

Again, if ΘI and F fulfill the properties above, the controller may be represented by a simple relational term, i.e., we have D(U (x)) = x; C where C : A → B is given by ↓


D(U (x)) = syQ((x; (SA \T  ) ) , ε∅B ); f = syQ((SA \T  )

↓


; x
, ε∅B ); f

↓


= x; syQ((SA \T  )

definition D and U

, ε∅B ); f.

Lemma 4.17 (1)

Example 6.8 Now, we want to return to our running example. We use the cut approach with a constant operation ΘI and an arbitrary mean function for selecting a result. It seems not to be ingenious to use the maximal degree of membership operation since our underlying lattice L is not linear. In fact, the interpretation of OT, the relation Q0 , shows that there is no temperature, which is optimal for both colonies. This implies that, in general, we will not find an element with maximal degree of membership within the set of possible results. Suppose f : P∅ (B + I) → B + I is a function. We require that f maps a set containing the “AL” element to that element. This property may be expressed by the following relational expression:





IB+I  IB+I ); f = (ε∅ B+I ; κ
;  IB+I  IB+I );  B+I I ; κ. (ε∅ B+I ; κ
; 


The partial identity ε∅ B+I ; κ
;  IB+I  IB+I describes those sets containing “AL.” Applied to this subset f should be the constant function  B+I I ; κ mapping each set to “AL.” In Section 6.7 we will investigate properties of the scalar αI (definition of the defuzzification), the parameter uA , uB (definition of the shifting functions sA resp. sB ), and a, b (definition of the approximate equality ΞB on B) in respect to given values a1 , a2 , b1 , b2 (definition of the approximate equality ΞA on A) such that our controller C = syQ((αA \T ) 6.7

↓


, ε∅B+I ); f is indeed a mapping.



PROVING PROPERTIES OF A CONTROLLER

Since our interpretation of a controller is an operation within a suitable Goguen category we are able to prove a lot of theorems about it by reasoning in the abstract theory of Goguen categories. Since this theory is basically equational and element-free it seems to be advantageous not to involve special properties of the underlying lattice L as far as possible. As indicated in the previous section, we want to find some properties of the parameters within our running example such that C is indeed a function. Suppose C is a controller defined by ⎞ ⎛ ↓
   Ti , ε∅B ⎠ ; f with Ti := (Q
C := syQ ⎝ αA \ i ;∗ Sj ). i∈I

j∈R(i)

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

183

↓

We will use the abbreviation Vi := (αA \Ti ) .  Lemma 6.9 If (Vi ;  BI ) =  AI , then C is a mapping. i∈I

Proof. First of all, we want to prove that every relation of the specific form syQ(X
, ε∅B ); f is univalent. This follows from


(syQ(X
, ε∅B ) ; f ); syQ(X
, ε∅B ); f = f
; syQ(ε∅B , X
); syQ(X
, ε∅B ); f f




; syQ(ε∅B , ε∅B ); f

Lemma 4.17 (2) Lemma 4.17 (3)




 f ;f

definition ε∅B

 IB .

f univalent

↓ Since the  the ↓left residual in the second argument are monotonic operation and Vi  (αA \( Ti )) , and, hence, for all objects D we get i∈I

i∈I

AI ;  ID  AD =   (Vi ;  BI );  ID = i∈I

 =



=



; BI ;  ID

Vi  Vi

i∈I

 

assumption



i∈I



Lemma 5.12 (5)

αA \



; BD 

Lemma 5.12 (5)

↓ Ti

; BD .

see above

i∈I

This immediately implies  ↓   αA \ Ti ; BD =  AD i∈I

⎞ ⎛ ↓
  P ⇔ syQ ⎝ αA \ Ti , ε∅B ⎠ ; 

∅ (B)D

= AD

definition ε∅B

i∈I

⎞ ⎛ ↓
  BD =  ⇔ syQ ⎝ αA \ Ti , ε∅B ⎠ ; f ;  AD

f total

i∈I

AD . ⇔ C;  BD = 

definition C




In the next lemma we want to investigate the relation between Vi+1 and Vi resp. V−(i+1) and V−i for 0 ≤ i ≤ 2 within our controller. Notice that it is essential that the shifting function sA and sB are chosen as bijections.

184

GOGUEN CATEGORIES

Lemma 6.10 For all 0 ≤ i ≤ 2 we have Vi+1 = s
ˆB and V−(i+1) = A ; Vi ; s

with s ˆ := ι ; s ; ι  κ ; κ. sA ; V−i ; sˆ
B B B Proof. We just show the first assertion. First of all, sˆB is a bijection, which follows from


ˆB = (ι
; s
sˆ
B; s B ; ι  κ ; κ); (ι ; sB ; ι  κ ; κ)

=

ι ; s
B ; sB ; ι







 κ ;κ

definition injections

= ι ;ι  κ ;κ = IB+I ,

sB bijection definition injections






sˆB ; sˆ
B = (ι ; sB ; ι  κ ; κ); (ι ; sB ; ι  κ ; κ)

=

ι ; sB ; s
B; ι




definition sˆB




 κ ;κ

definition sˆB definition injections

= ι ;ι  κ ;κ = IB+I .

sB bijection definition injections

Furthermore, we have ι; sˆB = ι; (ι
; sB ; ι  κ
; κ) = sB ; ι. Finally, the assertion follows immediately from X  Vi+1 ↓

⇔ X  (αA \Ti+1 )

definition Vi+1

↑

⇔ X  αA \Ti+1

Galois correspondence

↑

⇔ αA ; X  Ti+1 ↑

⇔ αA ; X  ⇔ αA ; X ↑  ↑

⇔ αA ; X  ↑

⇔ αA ; X  ↑

⇔ αA ; X  ↑

⇔ αA ; X  ↑

⇔ αA ; X  ↑

⇔ αA ; X  ⇔ ⇔ ⇔ ⇔ ⇔

definition residual

Q
i+1 ;∗ Si+1
 (s
A ; Qi );∗ (Si ; sB ; ι)
(s
A ; Qi );∗
(s
A ; Qi );∗

definition Ti+1 definition Qi+1 and Si+1

(Si ; ι; sˆB )

see above

(Si ; sˆB )

definition Si


(s
ˆB ) A ;∗ Qi );∗ (Si ;∗ s

sA ;∗ (Qi ;∗ Si );∗ sˆB
s
ˆB A ; (Qi ;∗ Si ); s
sA ; Ti ; sˆB

sA ; αA ; X ↑ ; sˆ
B  Ti ↑
αA ; sA ; X ; sˆB  Ti sA ; X ↑ ; sˆ
B  αA \Ti ↑ s↑A ; X ↑ ; sˆ
B  αA \Ti ↑ ↑ (s↑A ; X; sˆ
B )  αA \Ti

Lemma 5.66 Theorem 5.73 Lemma 5.66 definition Ti Lemma 4.8 (1)–(2) Lemma 5.14 definition residual sA and sB ; ι crisp Lemma 5.6 (10)

FUZZY CONTROLLERS IN GOGUEN CATEGORIES ↑

⇔ (sA ; X; sˆ
B )  αA \Ti ⇔ sA ; X; sˆ
B  (αA \Ti ) ⇔ ⇔

185

sA and sB ; ι crisp

↓

Galois correspondence

sA ; X; sˆ
B  Vi
X  sA ; Vi ; sˆB .

definition Vi


Lemma 4.8(1)–(2)

As a first property, we will require that the domain of V0 is an interval. An interval may be described within the language of relations as follows: Suppose x, y : I → A are crisp mappings such that y  x; EA , i.e., the element given by x is less or equal to the element given by y. Notice that the property above is
by Lemma 4.8. The corresponding interval is described equivalent to x  y; EA
by x; EA y; EA . For brevity, we will use the notation [x, y] for intervals. Notice that a term [x, y] should always imply the required order property for x and y. Therefore, our requirement for V0 may be formalized by (B+I)I = [x, y] V0 ; 




for suitable crisp mappings x, y : I → A. First, we want to show that the domain of V0 may be described by the L-fuzzy set Q0 . ↓

Lemma 6.11 Suppose V = (αA \(Q
;∗ S)) for relations Q : I → A and BI = (αI \Q) S : I → B such that S ↓ is total. Then we have V ; 

↓


.


IB ; S ↓  Proof. To prove the inclusion  notice that we have  II = 
↓
IB ; S = II since I is a unit. We  IB ; S since S is total, and, hence,  conclude IB ; V
=  IB ; αB ; V
αI ; 

definition scalar ↓


= IB ; αB ; (αA \(Q
;∗ S))





 IB ; αB ; (αA \(Q ;∗ S))




= IB ; αB ; (αB \(Q
;∗ S) )


 IB ; (Q
;∗ S)


= IB ; (S ;∗ Q)


= IB ;∗ (S ;∗ Q)


= ( IB ;∗ S );∗ Q


= ( IB ; S );∗ Q = II ;∗ Q = II ; Q = Q.

definition V Corollary 5.5 (2) Lemma 5.15 Corollary 4.15 (2) Theorem 5.75 Lemma 5.66 Theorem 5.73 Lemma 5.66 see above Lemma 5.66

186

GOGUEN CATEGORIES

By the definition of the residuals V ;  BI  (αI \Q) BI = II . We obtain inclusion notice again that S ↓ ;  (αI \Q)

↓


= (αI \Q)

↓


; S↓;  BI


↓

= (αA \Q ) ; S ;  BI

Lemma 5.15

↓

BI = ((αA \Q
) ; S ↓ ) ; 

Lemma 5.16 (3)

↓ ↓




follows. For the other

see above

↓

↓

↓


BI  ((αA \Q ); S ) ; 

Corollary 5.5 (2)

↓

BI  (αA \(Q
; S ↓ )) ; 

Lemma 4.16 (2)

↓

BI = (αA \(Q
;∗ S ↓ )) ; 

Lemma 5.66

↓

 (αA \(Q
;∗ S)) ;  BI = V ; BI .

;∗ and \ monotonic definition V




The previous lemma may be applied to our controller. ↓


Lemma 6.12 We have Vi ;  (B+I)I = (αI \Qi )

for all −4 ≤ i ≤ 4.

Proof. First of all, we have  ID = e0 ; ι;  (B+I)D ↓

 e0 ; ι; Ξ

e0 and ι are total

(B+I)D B; 

IB  ΞB

↓

= (e0 ; ι; Ξ↓ B ) ;  (B+I)D ↓

Lemma 5.16

(B+I)D  (e0 ; ι; ΞB ) ; 

Corollary 5.5 (2)

= S0 ↓ ;  (B+I)D ,

definition S0

such that the assertion is true for i = 0 by Lemma 6.11. The other cases are deduced by induction as follows: Suppose i ≥ 0. Then we conclude the assertion from (B+I)I = s
ˆB ;  (B+I)I Vi+1 ;  A ; Vi ; s = =

s
(B+I)I A ; Vi ;  ↓
s
A ; (αI \Qi )

Lemma 6.10 sˆB total induction hypothesis




↓

= ((αI \Qi ) ; sA ) = (αI \(Qi ; sA )) ↓


= (αI \Qi+1 )

↓


,

Lemma 4.16 (3) definition Qi+1

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

V−(i+1) ;  (B+I)I = sA ; V−i ; sˆ
(B+I)I B; 

Lemma 6.10 sˆ
B total

= sA ; V−i ;  (B+I)I ↓


= sA ; (αI \Q−i ) = = =

187

induction hypothesis


↓ ((αI \Q−i ) ; s
A) ↓
(αI \(Q−i ; s
A )) ↓
(αI \Q−(i+1) ) .

Lemma 4.16 (3) definition Q−(i+1)




Now, we want to prove two technical lemmata on elements and intervals within our application, i.e., EA and sA fulfil EA ; (sA  EA ) = (sA  EA ); EA . Lemma 6.13 Suppose y : I → A is a map such that y; sA  y; EA . Then we have
 IA  EA ; s
(1) IA  EA ; y
; y; EA A,
, then R; sA = R; (sA  EA ), (2) if IA  R
; R  IA  EA ; y
; y; EA

(3) if y  x; EA , then x; sA  x; EA , (4) if [x, y] is an interval, then so is [x; sA , y; sA ] and [x, y]; sA = [x; sA , y; sA ]. Proof. (1) First of all, we have y = y  y; sA ; s
A

sA bijection

y; EA ; s
A

assumption

y

= y; (IA 

EA ; s
A ).

y map

By Lemma 4.8 (2) we conclude y
; y  IA  EA ; s
A , and, hence,






y
; y = (y
; y)  (IA  EA ; s
A ) = IA  sA ; EA .

Now, the assertion follows from
IA  EA ; y
; y; EA

 IA  EA ; (IA  sA ; EA ); EA

 IA   IA  = IA 



EA ; (EA  sA ); EA ; EA
EA ; (EA  sA ); EA
(EA  sA ); EA ; EA

see above modular law EA transitive property of sA  EA

 IA  (EA  sA );  AA


= IA  (EA  sA ); (EA  sA )  IA 

EA ; s
A.

Lemma 4.23 (2)

188

GOGUEN CATEGORIES

(2) We immediately conclude that R; sA = R; (IA  R
; R); sA  R; (IA 

Lemma 4.5 (8)


EA ; y
; y; EA ); sA
EA ; sA ); sA

assumption

 R; (IA  = R; (sA  EA ).

by (1) Lemma 4.7 (2)

The other inclusion is trivial.
(3) First of all, we have IA  x
; x  IA  EA ; y
; y; EA , which implies x; sA = x; (sA  EA ) by (2). We get

x; sA = x; (sA  EA ) = x; sA  x; EA

by (2) x map,

and, hence, x; sA  x; EA . (4) First of all, we have to show y; sA  x; sA ; EA . Notice that we have y 
since [x, y] is an interval. We obtain x; EA and x  y; EA y; sA = y; sA  y; EA = y; (sA  EA )  x; EA ; (sA  EA ) = x; (sA  EA ); EA

assumption y map see above property of sA  EA

 x; sA ; EA .
Furthermore, we have IA  x
; x  IA  EA ; y
; y; EA and


IA  [x, y] ; [x, y]

= IA  (EA ; x
 EA ; y
); (x; EA  y; EA )

 IA  EA ; y




definition [x, y]


; y; EA ,

such that x; (sA  EA ) = x; sA and [x, y]; (sA  EA ) = [x, y]; sA follows from (2). We have


= (y  y; sA ; EA ); EA y; EA

= = = =

(y; sA ; s
A y; sA ; (s
A



 y; sA ; EA ); EA

 EA ); EA

y; sA ; (sA  EA ) ; EA

y; sA ; EA ; (sA  EA ) .

assumption on y sA bijection y; sA map property EA

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

189

Finally, the assertion follows from [x, y]; sA = [x, y]; (sA  EA ) = = = =

see above


(x; EA  y; EA ); (sA  EA )

(x; EA  y; sA ; EA ; (sA  EA ) ); (sA
x; EA ; (sA  EA )  y; sA ; EA
x; (sA  EA ); EA  y; sA ; EA
x; sA ; EA  y; sA ; EA

definition [x, y]  EA ) see above

= = [x; sA , y; sA ].

Lemma 4.7 (2) property EA see above definition [x; sA , y; sA ]




If we use s
A instead of sA , we get similar results. We have summarized these properties in the next corollary without a proof. Corollary 6.14 Suppose x : I → A is a map such that x  x; s
A ; EA . Then we have

; x
; x; EA  IA  EA ; sA , (1) IA  EA



(2) if IA  R
; R  IA  EA ; x
; x; EA , then R; s
A = R; (sA  EA ) ,

(3) if y  x; EA , then y  y; s
A ; EA ,



(4) if [x, y] is an interval, then so is [x; s
A , y; sA ] and [x, y]; sA = [x; sA , y; sA ].

Before we return to our example we state a lemma about the union of intervals. Obviously, the union of two intervals is again an interval if they overlap. Lemma 6.15 If v  [t, u] and u  [v, w], then [t, u]  [v, w] = [t, w]. Proof. First of all, u  t; EA and u  [v, w] implies w  u; EA  t; EA such that [t, w] is indeed an interval. Furthermore, v  [t, u]  t; EA and



implies t; EA  v; EA = t; EA and u; EA  w; EA = w; EA , u  [v, w]  w; EA respectively, and we have AA  IA = u;  = = 


u; (EA  EA )
u; EA  u; EA
v; EA  u; EA .

u total EA linear ordering u  [v, w]

190

GOGUEN CATEGORIES

The assertion follows from [t, u]  [v, w]

)  (v; EA  w; EA ) = (t; EA  u; EA

definition interval


= (t; EA  v; EA )  (t; EA  w; EA )


 (u; EA  v; EA )  (u; EA  w; EA )

= t; EA  w; EA  (t; EA  w; EA )
= t; EA  w; EA

see above

= [t, w].

definition interval


i+1 i 0 Now, let s0A := IA and si+1 A := sA ; sA resp. sA := IA and sA We may summarize our attempt so far as follows:

(B+I)I = [x, y] Lemma 6.16 If V0 ;  x; siA







 x; si+1 ; EA , A




 −3≤i≤3




:= siA ; s
A.

such that

i y; si+1 A  y; sA ; EA

for all 0 ≤ i ≤ 2, then we have







and

y; siA  x; si+1 A ; EA





(Vi ;  (B+I)I ) = [x; s3A , y; s3A ] .

Proof. First of all, we show (B+I)I = [x; siA , y; siA ] Vi ; 




and








V−i ;  (B+I)I = [x; siA , y; siA ]

for 0 ≤ i ≤ 3. The first assertion follows by induction from (B+I)I = sA
; Vi ; sˆB ;  (B+I)I Vi+1 ; 


BI = sA ; Vi ;  = =

Lemma 6.10 sˆB total


sA ; [x; siA , y; siA ] i+1
[x; si+1 A , y; sA ] ,


induction hypothesis Lemma 6.13

and the second is shown analogously. The required order properties give us the possibility to apply Lemma 6.15 such that the assertion follows.
Since EA has a least element 0 and greatest element 50.0 the intervals [0, x]
and x; EA , respectively. The reand [x, 50.0] are given by the relations x; EA
maining parts of the domain of C are exactly the intervals [0, x; s3A ] and [y; s3A , 50.0]. In the next lemma we want to investigate a property such that the domain of V−4 resp. of V4 fills this gap.


(B+I)I = [x, y] , Lemma 6.17 Suppose V0 ;  x; siA







 x; si+1 ; EA A

and

i y; si+1 A  y; sA ; EA

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

191





(B+I)I and for all 0 ≤ i ≤ 2. Then αI ;∗ Q0  [x, y] implies [0, x; s3A ]  V−4 ; 


[y; s3A , 50.0]  V4 ;  (B+I)I .


Proof. For brevity, let u := x; s3A . Using the order properties on x we


may apply Corollary 6.14 (4) and conclude [x, y]; s3A = [x; s3A , y; s3A ]. This implies αI ; Q−3 = αI ; Q0 ; s3A  = 





[x, y]; s3A

[x; s3A , y; s3A ]
x; s3A ; EA .

definition Q−3 assumption Corollary 6.14 (4)

The following computation shows the first assertion: αI ;∗ Q−3  u; EA ⇔ u
; (αI ;∗ Q−3 )  EA


⇒ EA ; u ; (αI ;∗ Q−3 )  EA


⇔ (EA ; u );∗ (αI ;∗ Q−3 )  EA


⇔ ((EA ; u );∗ αI );∗ Q−3  EA


⇔ (EA ; u ; αI );∗ Q−3  EA


⇔ EA ; u ; αI  EA ∗ Q−3 ⇔ ⇔ ⇔

EA ; u ; αI  Q
−4
αA ; EA ; u  Q
−4
EA ; u  αA \Q
−4


↓

⇒ EA ; u
 ⇒ E A ; u



Theorem 5.73 Lemma 5.66 definition ∗

definition residual

⇔ EA ; u
 (αA \Q
II −4 ) ; 

⇔ EA ; u


Lemma 5.66

Lemma 5.14

⇔ EA ; u
 (αA \Q
−4 ) ⇔ EA ; u 

EA transitive

definition Q−4

↓




Lemma 4.8 (2)

↓ (αA \Q
(B+I)I −4 ) ; κ;  ↓ ↓ ((αA \Q
(B+I)I −4 ) ; κ) ;  ↓ ((αA \Q
(B+I)I −4 ); κ) ;  ↓ (αA \(Q
(B+I)I −4 ; κ)) ;  ↓

(B+I)I ⇔ EA ; u  (αA \T−4 ) ; 


⇔ [0, u]  V−4 ;  (B+I)I . The second assertion is shown analogously.

EA ; u
crisp  II = II κ total Lemma 5.16 R↓  R Lemma 4.16 (2) definition T−4 definition V−4 and [0, u]


We want to interpret the property above. Suppose α is the scalar given by the element (a0 , b0 ) ∈ [0, 1] × [0, 1]. Then the previous lemma requires that for

192

GOGUEN CATEGORIES

any element z, which is not in the interval [x, y] but in the L-fuzzy set Q0 with a degree (a, b), i.e., Q0 (1, z) = (a, b), we have (a0 , b0 ) ∗ (a, b) = (0, 0), which is equivalent to a0 + a ≤ 1 and b0 + b ≤ 1. Notice that (a, b) ≤ (a0 , b0 ) since z is not in the domain [x, y] of V0 and V0 is defined by an α-cut. We may choose a0 , b0 ≤ 12 and conclude a0 + a ≤ a0 + a0 = 2a0 ≤ 1

and

b0 + b ≤ b0 + b0 = 2b0 ≤ 1.

Now, we want to summarize the requirements on the controller on the level of elements. Again, suppose α is the scalar given by the element (a0 , b0 ) ∈ [0, 1] × [0, 1] with a0 > 0 and b0 > 0.


↓

(1) V0 ;  (B+I)I = [x, y] . By Lemma 6.12 this is equivalent to (αI \Q0 ) = [x, y] and on the level of elements to ↓

z ∈ [x, y] ⇔ (αI \Q0 ) (1, z) for all z ∈ A. The right side may be computed as follows: ↓

(αI \Q0 ) (1, z) ⇔ Q0 (1, z) ≥ (a0 , b0 ) ⇔ min(1, max(0, a1 − b1 |24 − z|)) ≥ a0

Lemma 3.3 (5) definition Q0

and min(1, max(0, a2 − b2 |25.5 − z|)) ≥ b0 ⇔ a1 − b1 |24 − z| ≥ a0 and a2 − b2 |25.5 − z| ≥ b0 a1 − a0 a 2 − b0 ⇔ ≥ |24 − z| and ≥ |25.5 − z| b1 b2 a1 − a0 a1 − a0 ⇔ z ∈ [24 − , 24 + ] b1 b1 a2 − b0 a2 − b0 and z ∈ [25.5 − , 25.5 + ] b2 b2

a0 , b0 ∈ [0, 1] b1 , b2 ≥ 0

Consequently, x resp. y is the least resp. greatest element of A such that a1 − a0 a2 − b0 , 25.5 − )≤x b1 b2 a1 − a0 a2 − b0 andy ≤ min(24 + , 25.5 + ). b1 b2 max(24 −







(2) x; siA  x; si+1 ; EA , y; si+1  y; siA ; EA and y; siA  x; si+1 A A A ; EA for all 0 ≤ i ≤ 2. If we choose the u in the definition of sA such that 3u ≤ min(50.0 − y, x), these properties are fulfilled. (3) αI ;∗ Q0  [x, y]. As indicated above, this property is fulfilled if a0 , b0 ≤ 12 . As a last requirement we will demand that our controller permits an ALsignal if the temperature is less than or equal to 5◦ or greater than or equal to 45◦ . In the language of relations this may be expressed by e5  u; EA or u  e45 ; EA implies u; C = AL.

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

193

In the next lemma we give a sufficient condition for that property.


Lemma 6.18 Suppose x; s3A  e5 ; EA and e45  y; s3A ; EA . Then e5  u; EA or u  e45 ; EA implies u; C = AL.
Proof. From u  e5 ; EA we conclude using Lemma 6.17 and Lemma 6.12





I(B+I) ) = αI \Q−4 . u  [0, x; s3A ]  (V−4 ;  Furthermore, we have


(Q
−4 ;∗ κ); κ = (Q−4 ;∗ κ);∗ κ

Lemma 5.66


= Q
−4 ;∗ (κ;∗ κ )

= = = =

Theorem 5.73


Q
−4 ;∗ (κ; κ ) Q
−4 ;∗ II

Lemma 5.66 definition κ

Q
−4 ; II Q
−4 .

Lemma 5.66

Together, we obtain u  αI \Q−4 ⇔ αI ; u  Q−4 ⇔ ⇔ ⇔ ⇔ ⇔

definition residual

u
; αI  Q
−4
αA ; u  Q
−4
αA ; u
 (Q
−4 ;∗ κ); κ αA ; u
; κ  Q
−4 ;∗ κ u
; κ  αA \(Q
−4 ;∗ κ)

⇔ u
; κ 

(αA \(Q
−4 ;∗

κ))

Lemma 5.14 see above Lemma 4.8 (1) definition residual ↓




u and κ crisp

⇔ u ; κ  V−4

definition V−4

⇔ κ  u; V−4 .

Lemma 4.8 (2)

This immediately implies  I(B+I) = κ; κ
;  I(B+I)

definition κ




I(B+I)  u; V−4 ; κ ;  ⎞⎞ ↓ ⎛ ⎛  Ti ⎠⎠ ; κ
;  I(B+I) .  u; ⎝αA \ ⎝ −4≤i≤4

⎛

For brevity, let R := u; syQ ⎝ αA \



 −4≤i≤4

↓
Ti

see above definition V−4 ⎞

, ε∅B+I ⎠. Then we have




I(B+I)  IB+I ) R; (ε∅B+I ; κ
; 


= R; ε∅B+I ; κ
;  I(B+I)  R

Lemma 4.7 (1)

194

GOGUEN CATEGORIES

⎛

⎛

= u; ⎝αA \ ⎝



⎞⎞↓ Ti ⎠⎠ ; κ
;  I(B+I)  R

property ε∅

−4≤i≤4

= R,

see above

and, hence, the first assertion from u; C = R; f = =


R; (ε∅B+I ; κ
;  I(B+I)
R; (ε∅B+I ; κ
;  I(B+I)

definition C  IB+I ); f

see above

 IB+I );  (B+I)I ; κ

definition f see above

= R;  (B+I)I ; κ = II ; κ =κ

R total I unit

= AL.

definition AL


The second assertion follows analogously.
x; s3A

as small Since we do not want an AL-signal too often we choose and y; s3A as great as possible. On the element level, we have to choose u as great as possible such that 5 ≤ x − 3u and y + 3u ≤ 45, or equivalently 3u ≤ min(x − 5, 45 − y), holds. Let us consider an example. Suppose a1 = a2 = 1.2, b1 = 0.35 and b2 = 0.25. As indicated in (3) above, we choose let a0 = b0 = 12 . Using (1) we get a1 − a0 a2 − b0 , 25.5 − ) = max(24 − 2, 25.5 − 2.8) = 22.7, b1 b2 a1 − a0 a2 − b0 min(24 + , 25.5 + ) = min(24 + 2, 25.5 + 2.8) = 26, b1 b2 max(24 −

and, hence, x = 22.7 and y = 26. Furthermore, the previous property above gives us min(x − 5, 45 − y) = min(17.7, 19) = 17.7 such that we choose u = 5.9. 6.8

DISCUSSION OF THE APPROACH

Our approach has several advantages. We want to discuss them in detail. Of course, it is possible to implement the controller by using the unit interval [0, 1] as underlying lattice instead of [0, 1] × [0, 1]. But, in that case, we have to fix a temperature t0 , or at least a set of temperatures, in advance, which is good with respect to both colonies. The controller would try to keep the actual temperature closely to t0 . Selecting a suitable t0 is a nontrivial problem. It either has to be computed once the parameters are fixed or be chosen by an expert. In our approach this is not necessary. The controller will find that temperature on its own depending on the chosen parameters. During the development process of the controller we have selected specific semigroup operations in several places in advance. We kept the controller parametric by not fixing the parameters of that operation. From an abstract proof

FUZZY CONTROLLERS IN GOGUEN CATEGORIES

195

we then derived several properties that have to be satisfied by the parameters of the controller to ensure totality. Finally, we haven chosen suitable values to aim at a concrete controller. Notice that the abstract proof does not use any specific property of the selected semigroup operations so that the whole approach is in fact parametric in almost all those operations. The bottom line is that the abstract approach allows to make decisions as late as possible, which may save a considerable amount of money and effort. It is well known that early mistakes detected late are most painful. Last but not least, the relational term derived in the previous section can be used as a prototype of the controller. This was actually done using the Rath system [19] and its extension for Goguen categories [38].

INDEX

α-cut Theorem, 51, 99, 122 I-indexed product, 3 l-crisp, 81 s-crisp, 81 t-conorm, 40, 52, 155 t-norm, 40, 52, 154, 171 0–1 crisp, 48 Adjoint, 13, 47, 65, 167 lower, 13, 15, 47, 65 upper, 13 upper left, 15, 47 upper right, 15, 47, 65 Allegory, 57 distributive, 63 division, 65 representable, 58, 95 Antimorphism, 11, 25, 75, 111–112, 126–127, 131, 134, 140 Antisymmetric, 5 Antitone, 6, 16, 25, 63, 68 Approximate equality, 171 Arrow category, 94 representable, 95 with cuts, 122 Associative, 2, 9, 40, 163 Atom, 22 Bijection, 3, 60 Bijective, 2, 60

197

Boolean algebra, 20, 85, 93, 99 Boolean values, 3 Brouwerian lattice, 18, 43, 68 complete, 43, 68 proper, 147 Cartesian product, 2 Category, 55 arrow, 94 representable, 95 with cuts, 122 Dedekind, 68, 94 Goguen, 128 representable, 139 locally small, 56, 68 Class, 55 Cloos, 40, 160 Closed set, 23 Closg, 40 Closure operation, 11, 14 Coclosure operation, 11 Commutative, 9, 40 Complement, 1, 20 Complete, 10 Completely irreducible, 22 Composition, 2, 52, 57, 164 Consistent, 9 Continuous, 10, 24, 32, 40 Contractive, 11

198

INDEX

Control rule, 176 Converse, 58 Converse operation, 58 Conversion, 2 Crisp, 48, 95 l-crisp, 81 s-crisp, 81 0–1 crisp, 48 Crispness, 48 Cut, 48 Decision module, 170, 179 Dedekind category, 68, 94 simple, 71, 106 uniform, 70, 102 Defuzzification, 170, 179 Difference, 1 Distributive, 17, 63 Domain, 2 Endofunction, 11, 24 Endorelation, 2 Environment, 87, 144 Equation, 87, 144 Extensive, 11 Faithful, 56 Field, 56 Filter, 32 maximal, 33 prime, 33 complete, 35 principal, 32 Fixed point, 23 induction, 25, 27, 29 least, 23, 27 theorem, 24 Formula, 86 Full, 56 Function, 2 antitone, 6, 16, 25, 63, 68 contractive, 11 extensive, 11 idempotent, 11 linear, 56 monotone, 6, 11, 16 Functor, 56 faithful, 56 full, 56 Fuzzification, 142, 170, 176 Fuzzy controller, 142, 169 Galois correspondence, 13, 49 Goguen category, 128 representable, 139 Greatest element, 7 Greatest lower bound, 7 Heyting algebra, 18 Heyting algebras, 18 Homogenous relation, 2

Homomorphism, 3, 9, 11, 19, 56, 58, 65, 68, 95, 130 Brouwerian lattice, 19 continuous, 10 lattice, 11 complete, 11 lower co-semilattice, 9 complete, 10 lower semilattice, 9 complete, 10 upper co-semilattice, 10 complete, 10 upper semilattice, 9 complete, 10 Ideal relation, 48, 79 Idempotent, 9, 11 Identity, 2 Image, 2 Injection, 74, 173 crisp, 173 Injective, 2, 60 Intensifying modifier, 170 Intersection, 1 Interval, 185 Irreducible, 22 completely, 22 Isomorphic, 3 Isomorphism, 3, 56 Join, 8 Kernel, 48, 94 Kernel operation, 11, 14, 108 Lattice-ordered operator set, 40 commutative, 40 complete, 40 Lattice-ordered semigroup, 40, 52 commutative, 40 complete, 40 Lattice, 10 atomic, 23 complete, 10 completely distributive, 17 completely downwards-distributive, 17 completely upwards-distributive, 17 distributive, 17, 63 proper, 35, 147 sublattice, 10 complete, 10 Least element, 7 Least upper bound, 7 Left residual, 65 Linear, 5 Linear element, 8, 79 Linear function, 56 Linguistic entity, 170 Linguistic modifier, 170 Linguistic variable, 170 Locally small, 56, 68

INDEX Loos, 40, 52, 150 Losg, 40, 163 Lower adjoint, 65 Lower bound, 7, 170 greatest, 7 Lukasiewicz, 171 Map, 60 Meet, 7 Modifier, 170 intensifying, 170 ordering-based, 170 weakening, 170 Modular law, 58 Monotone, 6, 9, 16 Morphism, 55 Natural numbers, 3 Object, 55 Ordering-based modifier, 170 Ordering, 5, 170 Partial identity, 61, 72 Poset, 5 linear, 5 Power set, 2 Pre-functor, 57, 69, 130 Predicate, 1, 24 admissible, 24 continuous, 24 Projection, 75, 142 Pseudo-complement, 20 Pseudo-Representation Theorem, 136 Range, 2 Real numbers, 3 Reflexive, 5 Relation, 2, 43, 57 L-fuzzy, 43 antisymmetric, 5 approximate equality, 171 bijective, 2, 60 composition, 2 converse, 2 crisp, 48, 95 domain, 2 endorelation, 2 homogeneous, 2 ideal, 48, 79 identity, 2 image, 2 injective, 2, 60 ordering, 5 reversed, 6 partial identity, 61, 72 range, 2 reflexive, 5 scalar, 48, 79 linear, 79 sliceable, 110 source, 2, 43

surjective, 2, 60 symmetric idempotent, 75 target, 2, 43 total, 2, 60 transitive, 5 univalent, 2, 60 Relational product, 75, 142 Relational sum, 74, 142 Relative pseudo-complement, 18 Representable, 58, 95, 139 Residual left, 65 right, 65 Residuated operations, 15, 18, 47, 167 Reversed ordering, 6 Right residual, 65 Rule base, 170, 176 Scalar, 48, 79 linear, 79 Semilattice, 7 complete, 7 lower, 7 upper, 8 Set-theoretic injection, 74 Set-theoretic projection, 75 Set, 1 cartesian product, 2 closed, 23 complement, 1 comprehension, 1 difference, 1 intersection, 1 power set, 2 union, 1 Simple, 71, 106 Sliceable, 110 Source, 2, 43 Splitting, 75, 142 Sublattice, 10 Subsemilattice, 8 lower, 8 upper, 8 Support, 48, 94 Surjective, 2, 60 Symmetric idempotent relation, 75 Symmetric quotient, 65 Target, 2, 43 Tarski-rule, 70 Term, 86 Total, 2, 60 Transitive, 5 Triangular fuzzy set, 171 Ultrafilter, 33 Uniform, 70, 102 Union, 1 Unit, 75 Unit interval, 3

199

200

INDEX

Univalent, 2, 60 Upper bound, 7, 170 least, 7 Upper right adjoint, 65 Validity, 88

Value, 88 Vector space, 56 Weakening modifier, 170 Zermelo-Fraenkel, 1, 56 ZF, 1, 56 ZFC, 1

SYMBOLS

x∈A A⊆B P {x ∈ A | P(x)} A∪B A∩B A\B A, x A×B P(A) R:A→B RT R◦S R(x, y) IA ran(R) dom(R) f (x), f (A) A→B f−1 Ai

1 1 1 1 1, 1, 1 1, 2, 2 2 2, 2, 2 2, 2 2 2 2 3 3

B N N∞ R [0, 1] ≤  ≤ P1 → P2 ≥ P1 → P2 ∗ P1 → P2 L x∧y x∨y 0, 1 x ∨· y x:y ¬x At(L) µf (a) Xf,a anti L1 → L2

43 43 20 74

43 43 44, 55

i∈I

201

3 3 3, 23 3 3 5 6 6 6 6 7 7 8 8 12, 27 18 20 23 23 24 25

202

SYMBOLS

τ (f ) ϕ(f ) FL ψ ϑ(f ) (L, ∗, e, z) ⊥ ⊥AB  AB S ·· R Q ·· S u αA Ru R↓ R↑ Q ∩∗ R Q ◦∗ S C[A, B] f; g Set Rel [0, 1]-Rel L-Rel PO VctF ZF F, FObj , FM or R QR  Q
RP ≥ QR S/R Q\S syQ(Q, R)  Ai , A + B

26 27 35 36 37 40 44, 44, 47 47 48 48 48, 48, 52 52 55 55 56 56 56 56 56 56 56 56 57 57 57 58 63 63 65 65 65 74

i∈I

I

75

63 68

94 94

RL αA ScR (A) ¬:αA φ, φ−1 Rs Rl Φs Φl σ[A/a] σ[R/r : a → b] VR (t)(σ) R |=σ Θ A A↑ Sc[A] Fˆ G PG , PG−1 Q ⊗∗ R ˜ ∗ βA αA ⊗ Q ⊗ ∗ R R ⊗ ∗ S greater than∗ (E, M ) less than∗ (E, M ) tL extremely∗ (M ) very∗ (M ) more or less∗ (M ) roughly∗ (M ) veryi∗ (M ) roughlyi∗ (M ) P∅ (A) ε∅A [x.y]

78 79 79 79 80 81 81 81 81 87 87 88 88 94 95 104 107 119 128 136 151 153 168 168 171 171 171 172 172 172 172 172 172 181 181 185

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TRENDS IN LOGIC 1.

G. Schurz: The Is-Ought Problem. An Investigation in Philosophical Logic. 1997 ISBN 0-7923-4410-3

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E. Ejerhed and S. Lindstr¨om (eds.): Logic, Action and Cognition. Essays in Philosophical Logic. 1997 ISBN 0-7923-4560-6

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H. Wansing: Displaying Modal Logic. 1998

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P. H´ajek: Metamathematics of Fuzzy Logic. 1998

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H.J. Ohlbach and U. Reyle (eds.): Logic, Language and Reasoning. Essays in Honour of Dov Gabbay. 1999 ISBN 0-7923-5687-X

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K. Do˘sen: Cut Elimination in Categories. 2000

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R.L.O. Cignoli, I.M.L. D’Ottaviano and D. Mundici: Algebraic Foundations of manyvalued Reasoning. 2000 ISBN 0-7923-6009-5

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V.F. Hendricks: The Convergence of Scientific Knowledge. A View From the Limit. 2001 ISBN 0-7923-6929-7

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J. Czelakowski: Protoalgebraic Logics. 2001

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G. Gerla: Fuzzy Logic. Mathematical Tools for Approximate Reasoning. 2001 ISBN 0-7923-6941-6

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M. Fitting: Types, Tableaus, and G¨odel’s God. 2002

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F. Paoli: Substructural Logics: A Primer. 2002

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S. Ghilardi and M. Zawadowki: Sheaves, Games, and Model Completions. A Categorical Approach to Nonclassical Propositional Logics. 2002 ISBN 1-4020-0660-8

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G. Coletti and R. Scozzafava: Probabilistic Logic in a Coherent Setting. 2002 ISBN 1-4020-0917-8; Pb: 1-4020-0970-4

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P. Kawalec: Structural Reliabilism. Inductive Logic as a Theory of Justification. 2002 ISBN 1-4020-1013-3

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B. L¨owe, W. Malzkorn and T. R¨asch (eds.): Foundations of the Formal Sciences II. Applications of Mathematical Logic in Philosophy and Linguistics, Papers of a conference held in Bonn, November 10-13, 2000. 2003 ISBN 1-4020-1154-7

18.

R.J.G.B. de Queiroz (ed.): Logic for Concurrency and Synchronisation. 2003 ISBN 1-4020-1270-5

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A. Marcja and C. Toffalori: A Guide to Classical and Modern Model Theory. 2003 ISBN 1-4020-1330-2; Pb 1-4020-1331-0

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S.E. Rodabaugh and E.P. Klement (eds.): Topological and Algebraic Structures in Fuzzy Sets. A Handbook of Recent Developments in the Mathematics of Fuzzy Sets. 2003 ISBN 1-4020-1515-1; Pb 1-4020-1516-X

21.

V.F. Hendricks and J. Malinowski: Trends in Logic. 50 Years Studia Logica. 2003 ISBN 1-4020-1601-8

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23.

¨ ¨ B. Lowe, B. Piwinger and T. Rasch (eds.): Classical and New Paradigms of Computation and their Complexity Hierarchies. Papers of the conference "Foundations of the Formal Sciences III" held in Vienna, September 21–24, 2001 ISBN 1-4020-2775-3

24.

G. Jager: Anaphora and Type Logical Grammar. 2005 ¨

25.

M. Winter: Goguen Categories. A Categorical Approach to L-fuzzy Relations. 2007 ISBN 978-1-4020-6163-9

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