Goguen Categories: A Categorical Approach to L-Fuzzy Relations

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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

viii

GOGUEN CATEGORIES

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

x

GOGUEN CATEGORIES

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

GOGUEN CATEGORIES

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 === 4 3= == === 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
Since the class of ordinals is not a set but L is, it can be shown that there is a least ordinal u with f u (a) = f u+1 (a), which implies that f u (a) is the least ﬁxed point of f greater or equal to a. Now, the principle of ﬁxed point induction states that P holds for f x (a) for all ordinals x, and, hence, for that least ﬁxed point. The base case and the induction step of the ﬁxed point induction are exactly the base case and the induction step for successor ordinals of the transﬁnite induction. The property of being admissible implies the induction step for limiting ordinals. 2.7

THE COMPLETE BROUWERIAN LATTICE OF ANTIMORPHISMS

Later on, we will see that antimorphisms play an important role within our approach to fuzzy theory. In this chapter, we want to study the structure of anti the set of antimorphisms L1 → L2 between two complete lattices. Given an element u ∈ L2 we will identify u with the constant function u : L1 → L2 deﬁned by u(x) := u for all x ∈ L1 . This function is antitone, but in general not an antimorphism. Therefore, the quasi constant antimorphism induced by u is deﬁned by

1 iﬀ x = 0, u(x) ˙ = u iﬀ x = 0.

26

GOGUEN CATEGORIES anti

The set of all antimorphisms L1 → L2 need not be a (complete) sublattice ≥ ˙ of L1 → L2 or L1 → L2 . Nevertheless, it contains the greatest function 1. Lemma 2.39 The meet of a set of antimorphisms is again an antimorphism. Proof. The assertion follows immediately from the following computation: M = M deﬁnition of ∧ for functions fi fi i∈I

i∈I

=

fi (x)

fi antimorphism

i∈I x∈M

=

fi (x)

x∈M i∈I

=

x∈M

fi (x).

deﬁnition of ∧ for functions

i∈I

The previous lemma induces the following closure operation: τ (f ) := {h | f ≤ h and h antimorphism}. Notice that the antimorphism closure τ (u) ofthe constant function u is equal to u. ˙ This follows from u( ˙ ∅) = u(0) ˙ = 1 = ∅ and M \ {0} = ∅ implies u(x) ˙ = u(x) ˙ = u(x) = u = u M \ {0} = u˙ M . x∈M

x∈M \{0}

x∈M \{0}

Lemma 2.40 Let L1 and L2 be complete lattices. Then τ is a closure operation anti on L1 → L2 so that the image of τ is the set of antimorphisms L1 → L2 . Proof. First of all, by Lemma 2.39 τ (f ) is an antimorphism for every function f . On the other hand, any antimorphism g is greater or equal to itself so that τ (g) = g follows. Now, we want to show the three properties of a closure operation. (1) Suppose f ≤ g. Then the set {h | g ≤ h and h antimorphism} is a subset of {h | f ≤ h and h antimorphism}, which implies τ (f ) ≤ τ (g). (2) τ is extensive since any element of {h | f ≤ h and h antimorphism}, and, hence, its meet is greater or equal to f . (3) We have already shown that τ (g) = g for all antimorphisms g so that τ (τ (f )) = τ (f ) follows. anti

Now, by Theorem 2.10, we may establish the fact that L1 → L2 is indeed a complete lattice. In the rest of this chapter, we want to show that it is a complete Brouwerian lattice if L1 and L2 are.

LATTICES

27

anti

Corollary 2.41 Let L1 and L2 be complete lattices. Then the set L1 → L2 together with meet and join f ∨· g := τ (f ∨ g) is again a complete lattice. Unfortunately, the deﬁnition of τ gives us no information about the image of τ (f ) for a given value x ∈ L1 . Therefore, consider the following operation: ϕ(f )(x) := f (y). y∈M M ⊆L 1 M =x

We will show that τ (f ) is indeed the least ﬁxed point µϕ (f ) of ϕ greater or equal to f in the complete lattice of antitone function from L1 to L2 . Then we may prove properties of τ by using ϕ and the principle of ﬁxed point induction. Later on, we will use a slightly modiﬁed deﬁnition of ϕ. Lemma 2.42 Let L1 be a complete Brouwerian lattice, L2 a complete lattice and f : L1 → L2 be antitone. Then we have f (y). ϕ(f )(x) = y∈M M ⊆L 1 M ≥x

Proof. The inclusion ≤ is trivial. Suppose M is a subset of L1 so that M ≥ x. Then deﬁne Mx := {x ∧ y | y ∈ M } and conclude that Mx = (x ∧ y) = x ∧ M = x y∈M

x ∧ y ≤ y, and, hence, since L1 is a complete Brouwerian lattice. Furthermore, f (y) ≤ f (z), and, hence, f (y) ≤ f (x ∧ y). This implies y∈M

f (y) ≤

M ≥x y∈M

z∈Mx

f (y) = ϕ(f )(x).

M =x y∈M

In the next lemma we have summarized some basic properties of ϕ. Lemma 2.43 (1) ϕ is monotone on L1 → L2 . (2) f ≤ ϕ(f ) for all f ∈ L1 → L2 . (3) If f is an antimorphism, then ϕ(f ) = f . Proof. (1) Suppose f ≤g. Then we have f (y) ≤ g(y) for all y ∈ L1 , which implies f (y) ≤ g(y) for all subsets M of L1 . We obtain f (y) ≤ y∈M

M =x y∈M

y∈M

g(y), and, hence, ϕ(f ) ≤ ϕ(g).

M =x y∈M

28

GOGUEN CATEGORIES

(2) We immediately conclude that

f (x) =

f (y) ≤

f (y) = ϕ(f )(x).

M =x y∈M

y∈{x}

(3) The assertion follows from ϕ(f )(x) =

f (y)

M =x y∈M

=

f

M

f is an antimorphism

M =x

=

f (x)

M =x

M =x

= f (x).

The previous lemma implies that for every f there is a least ﬁxed point µϕ (f ) of ϕ greater or equal to f . Lemma 2.44 Let L1 be a complete Brouwerian lattice and L2 a complete lattice. Then f is antitone implies ϕ(f ) is antitone and µϕ (f ) is an antimorphism. Proof. Suppose x ≤ y and M is a subset of L1 so that the set Mx := {x ∧ u | u ∈ M }. Then we have

Mx =

(x ∧ u) = x ∧

M = y. Consider

M =x∧y =x

u∈M

since L1 is a Brouwerian lattice. Furthermore, for every u∈ M we have f (u) ≤ f (u) ≤ f (v). Now, we get f (u ∧ x) since f is antitone, and, hence, u∈M

v∈Mx

the ﬁrst assertion from ϕ(f )(y) =

f (u)

M =y u∈M

≤

f (v)

M =y v∈Mx

≤

N =x v∈N

= ϕ(f )(x).

f (v)

computation above

LATTICES

29

Now, let P be the predicate of being antitone, i.e., ∀x, y ∈ L1 : x ≤ y ⇒ g(y) ≤ g(x). This predicate is admissible since P(gi ) for all i ∈ I implies gi (y) ≤ gi (x) i∈I i∈I for all x, y ∈ L1 with x ≤ y, and, hence, P( gi ). P(g)

:⇔

i∈I

The base case P(f ) is trivial since f is antitone. Suppose P(g), i.e., g is antitone. As shown above, we conclude that ϕ(g) is antitone, and, hence, P(ϕ(g)). The principle of ﬁxed point induction shows that µϕ (f ) is antitone. It remains to show that µϕ (f )is an antimorphism. Since µϕ (f ) is antitone µϕ (f )(y). The other inclusion follows from we conclude that µϕ (f )( N ) ≤ y∈N

N = ϕ(µϕ (f )) N µϕ (f ) = µϕ (f )(y) y∈M

µϕ (f ) is a ﬁxed point

M= N

≥

µϕ (f )(y).

y∈N

Now, we are ready to establish the connection between ϕ and τ . Theorem 2.45 Let L1 be a complete Brouwerian lattice, L2 a complete lattice, and f : L1 → L2 be antitone. Then we have τ (f ) = µϕ (f ). Proof. Since τ (f ) is an antimorphism it is also a ﬁxed point of ϕ by Lemma 2.43 (3). Furthermore, τ (f ) is greater or equal to f . This implies µϕ (f ) ≤ τ (f ) since µϕ (f ) is the least ﬁxed point of ϕ greater or equal to f . The other inclusion follows immediately from the fact that µϕ (f ) is an antimorphism by Lemma 2.44 and f ≤ µϕ (f ) since τ (f ) is the least antimorphism greater or equal to f . As mentioned above, by Theorem 2.45 we may use the principle of ﬁxed point induction in order to prove properties of τ on the subset of antitone functions. As a ﬁrst example we obtain the next lemma. ≥

Lemma 2.46 Let L1 and L2 be complete Brouwerian lattices, and f, g : L1 → L2 . Then we have the following: (1) τ (f ) ∧ g ≤ τ (f ∧ g), (2) τ (f ) ∧ τ (g) = τ (f ∧ g). Proof. (1) We prove the assertion by ﬁxed point induction. Therefore, we deﬁne the following predicate: P(h, k) :⇔ h ∧ g ≤ k.

30

GOGUEN CATEGORIES

Suppose {(hi , ki ) | i ∈ I} is a set so that P(h i , ki ), and, hence, hi ∧ g ≤ ki holds for all i ∈ I. Then we conclude that (hi ∧g) ≤ ki . Since L1 → L2 i∈I i∈I is a complete Brouwerian lattice by Theorem 2.23 we have ( hi ) ∧ g ≤ i∈I ki , and, hence, P( hi , ki ). i∈I

i∈I

i∈I

The base case of the ﬁxed point induction is trivial since P(f, f ∧ g) holds. Now, suppose P(h, k). Then we conclude that (ϕ(h) ∧ g)(x) = ϕ(h)(x) ∧ g(x) ⎞ ⎛ ⎟ ⎜ h(y)⎠ ∧ g(x) =⎝ y∈M

deﬁnition of ∧

M =x

=

(h(y) ∧ g(x))

L1 → L2 is a complete Brouwerian lattice

(h(y) ∧ g(y))

g is antitone and y ≤ x

M =x y∈M

≤

M =x y∈M

=

(h ∧ g)(y)

M =x y∈M

≤

k(y)

induction hypothesis

M =x y∈M

= ϕ(k)(x), and, hence, P(ϕ(h), ϕ(k)). By the principle of ﬁxed point induction we get P(µϕ (f ), µϕ (f ∧ g))

⇔

τ (f ) ∧ g ≤ τ (f ∧ g).

(2) The inclusion τ (f ∧ g) ≤ τ (f ) ∧ τ (g) is trivial since τ is monotone. The other inclusion follows from τ (f ) ∧ τ (g) ≤ τ (f ∧ τ (g)) ≤ τ (f ∧ g) = τ (f ∧ g). 2

by (1) by (1)

We aim at the following theorem: anti

Theorem 2.47 Let L1 and L2 be complete Brouwerian lattices. Then L1 → L2 is again a complete Brouwerian lattice.

Proof. By Theorem 2.25 (2) and Corollary 2.41 it is suﬃcient to show that (1) (f ∧ g) ∨· h = (f ∨· h) ∧ (g ∨· h) for all antimorphisms f, g, h,

LATTICES

31

anti (2) f ∧ · gi = · (f ∧ gi ) for all i ∈ I and f, gi : L1 → L2 . i∈I

i∈I

The ﬁrst assertion follows from (f ∧ g) ∨· h = τ ((f ∧ g) ∨ h) = τ ((f ∨ h) ∧ (g ∨ h)) = τ (f ∨ h) ∧ τ (g ∨ h) = (f ∨· h) ∧ (g ∨· h)

L1 → L2 is distributive Lemma 2.46 (2)

and the second from f ∧ · gi = f ∧ τ gi i∈I

i∈I

=τ f∧ gi =τ

i∈I

(f ∧ gi )

Lemma 2.46(2) since τ (f ) = f L1 → L2 is a complete Brouwerian lattice

i∈I

= · (f ∧ gi ).

i∈I

2.8

FILTERS

We may use the ϕ operation and the principle of ﬁxed point induction in order to prove properties of τ . But, in general, we are not able to compute τ (f )(x) for a given x explicitly. Unfortunately, later on we will need this value in connection with validity of equations within crisp and noncrisp relations. If x is completely irreducible, we obtain f (y) = f (x) ϕ(f )(x) = y∈M M =x

for all antitone f since x ∈ M for all sets M with M = x. A simple veriﬁcation shows2 that this implies τ (f )(x) = µϕ (f )(x) = f (x), and, hence, that the problem mentioned above is trivial for completely irreducible elements. In other words, if x is completely irreducible, we are able to compute τ (f )(x) explicitly since the result is f (x). Unfortunately, there are complete Brouwerian lattices without such elements. In this chapter, we focus on the problem to ﬁnd for some given complete Brouwerian lattice L a suitable complete Brouwerian lattice L with at least one completely irreducible element so that for all complete lattices L the τ ≥ ≥ operation on L → L is a canonical extension of the τ operation on L → L . 2 This

property is shown by ﬁxed point induction using the predicate P(g) :⇔ g(x) = f (x).

32

GOGUEN CATEGORIES

The question arises what is a canonical extension, i.e., which property should be ≥ ≥ satisﬁed. Let ϑ be the mapping from L → L to L → L . Then the following property: (ext) τ (ϑ(τ (f ))) = τ (ϑ(f )) will be suitable. It allows us to compute the value of the image of the anti≥ morphism closure of f within L → L by using just the image of f and the corresponding τ operation. Then we may conclude that τ (ϑ(τ (f )))(x) = τ (ϑ(f ))(x) = ϑ(f )(x) for all completely irreducible elements in L . Notice that τ on the left-hand side of (ext) refers to two diﬀerent τ operations, namely the ≥ ≥ outer one to the operation on L → L and the inner one to that on L → L . Unfortunately, the usual embedding of L into the power set of all prime ﬁlters does not work. This embedding is not necessarily (upward) continuous so that the least upper bound of a subset M ⊆ L and the union of the image of ≥ M need not coincide. As a consequence, ϕ resp. τ on L → L is not a canonical ≥ extension of ϕ resp. τ for L → L in the sense mentioned above. We have to switch to the special class of complete prime ﬁlters. Deﬁnition 2.48 A subset F of a lattice L is called a ﬁlter of L iﬀ the following properties hold: (1) If x ∈ F and y ∈ F , then x ∧ y ∈ F , (2) If x ∈ F and y ∈ L, then x ∨ y ∈ F . In the next lemma we give an alternative deﬁnition of a ﬁlter, which is useful to compare the notion of a ﬁlter with the notions given later. Lemma 2.49 A subset F of a lattice L is a ﬁlter iﬀ (1) x ∈ F and y ∈ F iﬀ x ∧ y ∈ F , (2) If x ∈ F or y ∈ F , then x ∨ y ∈ F . Proof. Obviously, every set fulﬁlling (1) and (2) is a ﬁlter. Now, suppose F is a ﬁlter and x ∧ y ∈ F . By property (2) of a ﬁlter we have x = x ∨ (x ∧ y) ∈ F and y = y ∨ (x ∧ y) ∈ F . The second property is trivial. Filters that are generated by a single element are of special interest. Deﬁnition 2.50 For all elements x ∈ L the set [x] := {y ∈ L | x ≤ y} is called the principal ﬁlter induced by x. The deﬁnition above naturally leads to the following lemma: Lemma 2.51 A principal ﬁlter is a ﬁlter. Proof. Suppose y, z ∈ [x]. Then we have x ≤ y and x ≤ z, which implies x ≤ y ∧ z, and, hence, y ∧ z ∈ [x]. Now, suppose y ∈ F and z ∈ L. Then we get x ≤ y ≤ y ∨ z, and, hence, y ∨ z ∈ [x].

LATTICES

33

As mentioned above, a special class of ﬁlters is used to show that a distributive lattice may be embedded into a power set. Deﬁnition 2.52 A subset F of a lattice L with least element 0 is called a prime ﬁlter of L iﬀ for all x, y ∈ L the following properties hold: (1) 0 ∈ F , (2) x ∧ y ∈ F iﬀ x ∈ F and y ∈ F , (3) x ∨ y ∈ F iﬀ x ∈ F or y ∈ F . Using Lemma 2.49 it is obvious that any prime ﬁlter is a ﬁlter. That the other implication is not true is shown by the next lemma. Lemma 2.53 Let L be a distributive lattice and x ∈ L. Then the principal ﬁlter [x] is a prime ﬁlter iﬀ x is irreducible. Proof. ⇒: Suppose y ∨ z = x. Since x ∈ [x] we may assume without loss of generality y ∈ [x]. We conclude that y ≤ x since y ≤ y ∨ z = x and x ≤ y from y ∈ [x]. ⇐: First of all, x = 0 by deﬁnition, and, hence, 0 ∈ [x]. Furthermore, suppose y ∨ z ∈ [x] and deﬁne y := x ∧ y and z := x ∧ z. Then we have y ∨ z = (x ∧ y) ∨ (x ∧ z) = x ∧ (y ∨ z) = x since x ≤ y ∨ z. Without loss of generality we get x = y = x ∧ y, which implies x ≤ y, and, hence, y ∈ [x]. On the other hand, the maximal elements within the set of ﬁlters are of special interest. Deﬁnition 2.54 A ﬁlter F ⊆ L of lattice L with least element 0 is called a maximal ﬁlter or an ultraﬁlter iﬀ it is maximal with respect to the property 0 ∈ F . As above, there is a connection between ultraﬁlters and a set of special elements. Lemma 2.55 Let L be a lattice and x ∈ L. Then the principal ﬁlter [x] is an ultraﬁlter iﬀ x is an atom. Proof. ⇒: Suppose y ≤ x. If y ∈ [x], we conclude that x = y. On the other hand, if y ∈ [x], the principal ﬁlter [y] is a proper superset of [x], which implies 0 ∈ [y] since [x] is maximal. We get y = 0. ⇐: Suppose F is a ﬁlter so that 0 ∈ F and [x] ⊆ F . For y ∈ F we have x∧y ∈ F . Since x is an atom and 0 ∈ F we obtain x∧y = x, which immediately implies x ≤ y, and, hence, y ∈ [x]. The maximality of ultraﬁlters implies a nice property for pairs of elements x, x of a Boolean algebra.

34

GOGUEN CATEGORIES

Lemma 2.56 Let L be a Boolean algebra and F an ultraﬁlter. Then for every element x ∈ L exactly one of x, x belongs to F . Proof. Suppose, neither x nor x belongs to F . Deﬁne F := {z | ∃y ∈ F : z ≥ x ∧ y}. We show that (1) F is a ﬁlter, (2) x ∈ F , (3) 0 ∈ F , (4) F ⊆ F . The existence of such an F contradicts the maximality of F , and, hence, either x ∈ F or x ∈ F . (1) Let u, v ∈ F and compute u ∧ v ≥ (x ∧ y1 ) ∧ (x ∧ y2 ) = x ∧ (y1 ∧ y2 ) for suitable elements y1 , y2 ∈ F . This shows that u ∧ v ∈ F since y1 ∧ y2 ∈ F . Now, suppose u ∈ F and v ∈ L. Then u ≥ x ∧ y for a y ∈ F . We get u ∨ v ≥ (x ∧ y) ∨ v = (x ∨ v) ∧ (y ∨ v) ≥ x ∧ (y ∨ v), which implies u ∨ v ∈ F since y ∨ v ∈ F . (2) This follows immediately from x = x ∧ 1 and 1 ∈ F . (3) Suppose 0 ∈ F . Then there is an element y ∈ F so that x ∧ y = 0. This implies y ≤ x. Since F is a ﬁlter we conclude that x = y ∨ x ∈ F , a contradiction. (4) This follows immediately from y ≥ x ∧ y for all y ∈ F . Suppose x ∈ F and x ∈ F then we obtain x ∧ x = 0 ∈ F , a contradiction.

Using the previous lemma, we are able to show that ultraﬁlters within a Boolean algebra may be deﬁned just by properties of their elements. A quantiﬁcation over all subsets of the Boolean algebra, or at least over all ﬁlters, is not needed. Lemma 2.57 Let L be a Boolean algebra. A ﬁlter F is a prime ﬁlter iﬀ F is an ultraﬁlter. Proof. ⇒: Suppose F is a ﬁlter so that 0 ∈ F and F ⊆ F . For x ∈ F we have x ∨ x = 1 ∈ F . This implies either x ∈ F or x ∈ F . In the second case we conclude that x ∧ x = 0 ∈ F , a contradiction, since x ∈ F and x ∈ F ⊆ F . ⇐: It is suﬃcient to show that x ∨ y ∈ F implies x ∈ F or y ∈ F . Consider (x ∨ y) ∧ (x ∧ y) = 0 ∈ F . Since F is a ﬁlter we get x ∧ y ∈ F , and, hence, x ∈ F or y ∈ F . Using Lemma 2.56 we obtain x ∈ F or y ∈ F . The next lemma is in some sense a generalization of Lemma 2.55. Lemma 2.58 Let L be a complete Boolean algebra and F an ultraﬁlter. Then F equals 0 or is an atom.

LATTICES

35

Proof. Suppose x ≤ F and consider the principal ﬁlter [x]. Then we have F ⊆ [x], which impliesF = [x] or 0 ∈ [x] by the maximality of F . In the ﬁrst case we conclude x = F and in the second case x = 0. As mentioned at the beginning of this section, we will need a special class of prime ﬁlters. We may motivate this new notion also by the following fact: As shown in the previous lemmata properties of principal ﬁlters are related to properties of the generating element. So far we have encountered the following relationships: class of ﬁlters principal ﬁlter principal prime ﬁlter principal ultraﬁlter

= ˆ = ˆ = ˆ

class of elements arbitrary element irreducible element atom

A class of ﬁlters corresponding to completely irreducible elements is still missing. Since those elements are deﬁned using arbitrary unions it is not surprising that the corresponding class of ﬁlters requires such a property as well. Deﬁnition 2.59 A subset F ⊆ L of a complete Brouwerian lattice L is called a complete prime ﬁlter iﬀ for all x, y ∈ L and all subsets M ⊆ L the following properties hold: (1) 0 ∈ F , (2) x ∧ y ∈ F iﬀ x ∈ F and y ∈ F , (3) M ∈ F iﬀ ∃y ∈ M : y ∈ F . We will denote the set of all complete prime ﬁlters of L by FL . If FL = ∅, we call L proper. Again, there is a connection between complete prime ﬁlters and a set of special elements. Lemma 2.60 Let L be a complete Brouwerian lattice and x ∈ L. Then the principal ﬁlter [x] is a complete prime ﬁlter iﬀ x is completely irreducible. Proof. ⇒: Suppose M = x ∈ [x]. Then there is a y ∈ M ∩ [x] since [x] is a complete prime ﬁlter. We conclude y ≤ x from y ∈ M and x ≤ y from y ∈ [x]. This implies x = y ∈ M . ⇐: It is suﬃcient to show that M ∈ [x] implies thatthere is ay ∈ M with (x ∧ y) = x ≤ y. Deﬁne Mx := {x ∧ y | y ∈ M }. Then we have Mx = y∈M x ∧ M = x since L is a complete Brouwerian lattice. This implies x ∈ Mx since x is completely irreducible. We conclude that x = x ∧ y ≤ y for a suitable y ∈ M. We will prove property (ext) for proper lattices. But ﬁrst, we want to study this class of lattices.

36

GOGUEN CATEGORIES

Theorem 2.61 (1) Every linear ordering is proper. (2) If a lattice has at least one completely irreducible element, then it is proper. (3) The class of proper lattices is closed under arbitrary products. (4) A complete atomless Boolean algebra is not proper. Proof. (1) The set F := L \ {0} for a linear ordering L is obviously a complete prime ﬁlter. (2) If x is completely irreducible, then the principal ﬁlter [x] is a complete prime ﬁlter by Lemma 2.60. (3) Let Li for i ∈ I be proper lattices. Suppose forall i ∈ I the set Fi is a Fi . It is easy to verify complete prime ﬁlter of Li . Then deﬁne F := i∈I Li . that F is a complete prime ﬁlter of i∈I

(4) Suppose F is a complete prime ﬁlter of a complete atomless Boolean algebra L. Then F is a prime ﬁlter, and, hence, an ultraﬁlter of L by Lemma 2.57. Furthermore, by Lemma 2.58 we have F equals 0 or is an atom. Since L is atomless F = 0 follows. Deﬁne F := {x | x ∈ F }. Then F ∩ F = ∅ x = F = 1 ∈ F, by Lemma 2.56. Last but not least, we have F = x∈F

which shows that F is not a complete prime ﬁlter.

Notice that property (2) of the previous theorem implies that every ﬁnite lattice and every power set is proper. Furthermore, from (1) we conclude that the unit interval [0, 1] of the real numbers is also proper. Now, we deﬁne a function ψ : L → P(FL ) by ψ(x) := {F ∈ FL | x ∈ F }. Notice that ψ is deﬁned similar to the function used in the representation theorem of Boolean algebras by Stone. However, ψ is not necessarily injective. But we have the following: Lemma 2.62 (1) ψ(x ∧ y) = ψ(x) ∩ ψ(y), ψ(y). (2) ψ( M ) = y∈M

Proof. (1) Consider the following computation: F ∈ ψ(x ∧ y) ⇔ x ∧ y ∈ F ⇔ x ∈ F and y ∈ F ⇔ F ∈ ψ(x) and F ∈ ψ(y) ⇔ F ∈ ψ(x) ∩ ψ(y).

F is a prime ﬁlter

LATTICES

37

(2) The assertion follows immediately from F ∈ ψ( M ) ⇔ M ∈F ⇔ ∃y ∈ M : y ∈ F ⇔ ∃y ∈ M : F ∈ ψ(y) ψ(y). ⇔F ∈

F is a complete prime ﬁlter

y∈M

Using ψ we may extend a function f : L1 → L2 to a function ϑ(f ) : P(FL1 ) → L2 by ϑ(f )(M) := f (x). x∈L1 M⊆ψ(x)

In the next lemma we have summarized some basic properties of ϑ. Lemma 2.63 Let L1 and L2 be complete Brouwerian lattices. For all antitone ≥ functions f, g, fi : L1 → L2 for all i ∈ I we have (1) ϑ(f ) is antitone, ϑ(fi ) = ϑ( fi ), (2) i∈I

i∈I

(3) ϑ(f ∧ g) = ϑ(f ) ∧ ϑ(g), f (x). (4) ϑ(f )({F }) = x∈F

Proof. (1) If M ⊆ N, we obtain {x | N ⊆ ψ(x)} ⊆ {x | M ⊆ ψ(x)}, and, hence, f (x) ≤ f (x) = ϑ(f )(M). ϑ(f )(N) = N⊆ψ(x)

M⊆ψ(x)

(2) Consider the following computation: ϑ(fi ) (M) = ϑ(fi )(M) i∈I

i∈I

=

deﬁnition of ∨

fi (x)

deﬁnition of ϑ

i∈I M⊆ψ(x)

=

fi (x)

M⊆ψ(x) i∈I

=

M⊆ψ(x)

=

fi (x)

(M). ϑ fi i∈I

deﬁnition of ∨

i∈I

deﬁnition of ϑ

38

GOGUEN CATEGORIES

(3) First, we show that

(f (x) ∧ g(x)) = (

M⊆ψ(x)

f (x)) ∧ (

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

=

(f (x) ∧

M⊆ψ(x)

=

M⊆ψ(x)

g(x)

L1 → L2 is a complete Brouwerian lattice

M⊆ψ(x)

(f (x) ∧ g(y))

L1 → L2 is a complete Brouwerian lattice

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 of ϑ

M⊆ψ(x)

=

(f (x) ∧ g(x))

M⊆ψ(x)

=

deﬁnition of ∧

f (x) ∧

M⊆ψ(x)

g(x)

computation above

M⊆ψ(x)

= ϑ(f )(M) ∧ ϑ(g)(M)

deﬁnition of ϑ

= (ϑ(f ) ∧ ϑ(g))(M).

deﬁnition of ∧

(4) We immediately conclude that ϑ(f )({F }) = f (x) = {F }⊆ψ(x)

f (x) =

F ∈ψ(x)

f (x).

x∈F

Notice that (2) as well as (3) of the previous lemma implies that ϑ is monotone. The next lemma will show the key property (ext) introduced above. Notice that in the proof of this lemma it is essential that ψ is continuous. Lemma 2.64 If f : L1 → L2 is antitone, then we have τ (ϑ(τ (f ))) = τ (ϑ(f )). Proof. ≤: Since f ≤ τ (f ) we get ϑ(f ) ≤ ϑ(τ (f )) by Lemma 2.63 (2), and, hence, τ (ϑ(f )) ≤ τ (ϑ(τ (f ))). ≥: Consider the property (∗)

ϑ(τ (f )) ≤ τ (ϑ(f )).

LATTICES

39

From (∗) we immediately conclude that τ (ϑ(τ (f ))) ≤ τ 2 (ϑ(f )) = τ (ϑ(f )). We prove (∗) by ﬁxed point induction. Therefore, we deﬁne the predicate ϑ(h) ≤ τ (ϑ(f ))

:⇔

P(h)

ϑ(hi ) ≤ τ (ϑ(f )), This predicate is admissible since P(hi ) for all i ∈ I implies i∈I and, hence, ϑ( hi ) ≤ τ (ϑ(f )) by Lemma 2.63 (2), which is equivalent to i∈I P( hi ). i∈I

The base case P(f ) is trivial. Let M be a subset of FL , x so that M ⊆ ψ(x) and M a set with M = x. Then we deﬁne PM := {M ∩ ψ(y) | y ∈ M }. Using Lemma 2.62 (2) we conclude that PM = (M ∩ ψ(y)) = M ∩ ψ(y) = M ∩ ψ( M ) = M. y∈M

y∈M

Furthermore, we have h(y) ≤ h(z) for all y ∈ M , which implies M∩ψ(y)⊆ψ(z) h(y) ≤ h(z). We obtain y∈M

N∈PM N⊆ψ(z)

(∗∗)

M⊆ψ(x)

h(y) ≤

M =x y∈M

Now, the induction step follows from ϑ(ϕ(h))(M) = ϕ(h)(x) M⊆ψ(x)

M⊆ψ(x)

≤ = ≤

=

h(z).

P =M N∈P N⊆ψ(z)

deﬁnition of ϑ

h(y)

deﬁnition of ϕ

h(z)

by (∗∗)

M =x y∈M

ϑ(h)(N)

deﬁnition of ϑ

τ (ϑ(f ))(N)

induction hypothesis

P =M N∈P

P =M N∈P N⊆ψ(z)

P =M N∈P

= ϕ(τ (ϑ(f )))(M) = τ (ϑ(f ))(M).

deﬁnition of ϕ τ (f ) ﬁxed point of ϕ

From the principle of ﬁxed point induction we obtain property (∗). We summarize Lemma 2.64 and 2.63 (4) and compute τ (ϑ(τ (f )))({F }) = τ (ϑ(f ))({F }) = ϑ(f )({F }) =

x∈F

f (x).

40

GOGUEN CATEGORIES

2.9

LATTICE-ORDERED SEMIGROUPS

In fuzzy theory t-norms and t-conorms are essential for deﬁning new operations for fuzzy sets and/or relations. The corresponding notion for L-fuzzy relations is given by complete lattice-ordered semigroups introduced in [12]. Deﬁnition 2.65 Let L be a distributive lattice with least element 0 and greatest element 1, ∗ a binary operation on L and e, z ∈ L. Then (L, ∗, e, z) is called a lattice-ordered operator set, abbreviated loos, iﬀ (1) ∗ is monotone in both arguments, (2) e is a left and right neutral element for ∗, i.e., x ∗ e = e ∗ x = x for all x ∈ L, (3) z is a left and right zero for ∗, i.e., x ∗ z = z ∗ x = z for all x ∈ L. If ∗ is associative, (L, ∗, e, z) is called a lattice-ordered semigroup (losg). Furthermore, if L is a complete Brouwerian lattice and ∗ is continuous (distributes over nonempty unions) in both arguments, i.e., yi = (x ∗ yi ) and yi ∗ x = (yi ∗ x) x∗ i∈I

i∈I

i∈I

i∈I

for all nonempty sets I, (L, ∗, e, z) is called a complete lattice-ordered operator set/semigroup (cloos/closg). Finally, the structures deﬁned above are called commutative if ∗ is. As usual, e and z are unique. Suppose e is another left and right neutral element and z is another left and right zero for ∗. Then we conclude that e = e ∗ e = e ,

e right neutral e left neutral

z = z ∗ z = z.

z left zero z right zero

Notice that for L = [0, 1], e = 1 and z = 0 we get the usual deﬁnition of t-norms and for e = 0 and z = 1 of t-conorms. For example, the product norm, i.e., (x, y) → xy, is a commutative closg. Further examples of commutative losgs are (L, ∧, 1, 0), (L, ∨, 0, 1). In addition, we may deﬁne the following operations: ⎧ ⎧ ⎨ x iﬀ y = 1, ⎨ x iﬀ y = 0, y iﬀ x = 1, y iﬀ x = 0, x y := x y := ⎩ ⎩ 0 otherwise. 1 otherwise. Again, (L, , 1, 0) and (L, , 0, 1) are commutative losgs. Lemma 2.66 Let (L, ∗, 1, z) be a loos. Then we have the following: (1) z = 0, i.e., x ∗ 0 = 0 ∗ x = 0 for all x ∈ L, (2) x y ≤ x ∗ y ≤ x ∧ y for all x, y ∈ L, (3) ∗ = ∧ iﬀ u ∗ u = u for all u ∈ L.

LATTICES

41

Proof. (1) x ∗ 0 ≤ 1 ∗ 0 = 0 and 0 ∗ x ≤ 0 ∗ 1 = 0, and, hence, z = 0. (2) The second inclusion follows immediately from x ∗ y ≤ x ∗ 1 = x and x ∗ y ≤ 1 ∗ y = y. Suppose x y = 0. Then x = 1 or y = 1, and, hence, x y = 1 y = y = 1 ∗ y = x ∗ y resp. x y = x 1 = x = x ∗ 1 = x ∗ y. (3) ⇒ is trivial, and ⇐ follows from 2. and x ∧ y = (x ∧ y) ∗ (x ∧ y) ≤ x ∗ y. If the identity 1 of (L, ∗, 1, z) in the previous lemma is replaced by 0, a dual version may be proved. Lemma 2.67 Let (L, ∗, 0, z) be a loos. Then we have the following: (1) z = 1, i.e., x ∗ 1 = 1 ∗ x = 1 for all x ∈ L, (2) x ∨ y ≤ x ∗ y ≤ x y for all x, y ∈ L, (3) ∗ = ∨ iﬀ u ∗ u = u for all u ∈ L. Proof. Similar to Lemma 2.66.

3 L-FUZZY RELATIONS

As mentioned in the introduction, for a complete Brouwerian lattice L an Lfuzzy relation R between two nonempty sets A and B is a function from A × B to L. Notice, if L = B, we get the set of regular binary relations between A and B. Therefore, we also use the denotation R : A → B to indicate that an L-fuzzy relation R has source A and target B. 3.1

BASIC OPERATIONS AND PROPERTIES

Let Q, R : A → B and S : B → C be L-fuzzy relations. Then we may introduce several operations as follows: (Q ∩ R)(x, y) (Q ∪ R)(x, y) QT (x, y) (Q ◦ S)(x, z)

:= := := :=

Q(x, y) ∧ R(x, y), Q(x, y) ∨ R(x, y), Q(y, x), (Q(x, y) ∧ S(y, z)). y∈B

Notice that these operations are generalizations of those deﬁned for regular (crisp) binary relations, i.e, for L = B they coincide with the corresponding set-theoretic operations deﬁned in Chapter 1. Furthermore, the inclusion ⊆, which is induced by the intersection or union of L-fuzzy relations, has to be read as follows: Q⊆R

⇐⇒

∀x ∈ A, y ∈ B : Q(x, y) ≤ R(x, y). 43

44

GOGUEN CATEGORIES

Again, this is a generalization of ⊆ deﬁned for regular relations. By Theorems 2.7, 2.23, and 2.25 (2) the set of all L-fuzzy relations between A and B is again a complete Brouwerian lattice with least and greatest element deﬁned by AB (x, y) := 1.

⊥ ⊥AB (x, y) := 0,

Furthermore, one may deﬁne the identity relation on the set A by

IA (x, y) :=

1 iﬀ x = y, 0 else.

Theorem 3.1 Let L be a complete Brouwerian lattice. Then for all L-fuzzy relations Q, Q , Qi : A → B, R, Ri : B → C, S : C → D for i ∈ I and T : A → C we have (1) Q ◦ IB = Q and IB ◦ R = R, (2) (Q ◦ R) ◦ S = Q ◦ (R ◦ S), (3) (Q ∩ Q ) = QT ∩ Q , T

T

(4) (Q ◦ R) = RT ◦ QT , T

T

(5) (QT ) = Q, (6) Q ◦ (

Ri ) ⊆

i∈I

(Q ◦ Ri ) and (

i∈I

Qi ) ◦ R ⊆

i∈I

(Qi ◦ R),

i∈I

(7) Q ◦ R ∩ T ⊆ Q ◦ (R ∩ QT ◦ T ), (8) Q ◦ ⊥ ⊥BC = ⊥ ⊥AC , (9) Q ◦ (

i∈I

Ri ) =

(Q ◦ Ri ) and (

i∈I

Qi ) ◦ R =

i∈I

(Qi ◦ R),

i∈I

Proof. Throughout this proof, (∗) refers to the fact that L is completely upward-distributive. (1) We just show the ﬁrst assertion. This follows immediately from (Q ◦ IB )(u, w) =

(Q(u, v) ∧ IB (v, w))

deﬁnition of ◦

v∈B

= Q(u, w) ∧ 1 = Q(u, w).

deﬁnition of IB

L-FUZZY RELATIONS

45

(2) The following computation shows the assertion: ((Q ◦ R)(u, w) ∧ S(w, x)) ((Q ◦ R) ◦ S)(u, x) = w∈C

=

w∈C

=

(Q(u, v) ∧ R(v, w)) ∧ S(w, x)

v∈B

(Q(u, v) ∧ R(v, w) ∧ S(w, x))

by (∗)

w∈C v∈B

=

(Q(u, v) ∧ R(v, w) ∧ S(w, x))

v∈B w∈C

=

Q(u, v) ∧

v∈B

=

(R(v, w) ∧ S(w, x)) by (∗)

w∈C

Q(u, v) ∧ (R ◦ S)(v, x)

v∈B

= (Q ◦ (R ◦ S)(u, x)). (3) Again, the following computation shows the assertion: (Q ∩ Q ) (u, v) = (Q ∩ Q )(v, u) = Q(v, u) ∧ Q (v, u) T

= QT (u, v) ∧ Q (u, v) T

= (QT ∩ Q )(u, v). T

(4) The assertion is shown as follows: (Q ◦ R) (u, w) = (Q ◦ R)(w, u) (Q(w, v) ∧ R(v, u)) = T

v∈B

=

(RT (u, v) ∧ QT (v, w))

v∈B T

= (R ◦ QT )(u, w). (5) follows immediately from T

(QT ) (u, v) = QT (v, u) = Q(u, v). (6) We just show the ﬁrst assertion. Therefore, consider the following computation: (u, w) = Q(u, v) ∧ Ri Ri (v, w) Q◦ i∈I

v∈B

=

i∈I

Q(u, v) ∧

v∈B

i∈I

Ri (v, w)

46

GOGUEN CATEGORIES

=

(Q(u, v) ∧ Ri (v, w))

v∈B i∈I

≤

(Q(u, v) ∧ Ri (v, w))

i∈I v∈B

=

(Q ◦ Ri )(u, w)

i∈I

=

(Q ◦ Ri ) (u, w)

i∈I

(7) Again, consider the following computation: (Q ◦ R ∩ T )(u, w) = (Q ◦ R)(u, w) ∧ T (u, w) = (Q(u, v) ∧ R(v, w)) ∧ T (u, w) v∈B

=

(Q(u, v) ∧ R(v, w) ∧ T (u, w))

by (∗)

v∈B

=

(Q(u, v) ∧ R(v, w) ∧ Q(u, v) ∧ T (u, w))

v∈B

≤

Q(u, v) ∧ R(v, w) ∧

=

u ∈A

v∈B

(Q(u , v) ∧ T (u , w))

(Q(u, v) ∧ R(v, w) ∧ (QT ◦ T )(v, w))

v∈B

=

(Q(u, v) ∧ (R ∩ QT ◦ T )(v, w))

v∈B

= (Q ◦ (R ∩ QT ◦ T ))(u, w). (8) The assertion follows immediately from (Q(u, v) ∧ ⊥ ⊥BC (v, w)) = 0 = ⊥ ⊥AC (u, w). (Q ◦ ⊥ ⊥BC )(u, w) = v∈B

(9) We just show the ﬁrst assertion. It follows from Q◦ (u, w) = Ri (Q(u, v) ∧ Ri (v, w)) i∈I

v∈B

=

i∈I

Q(u, v) ∧

v∈B

=

v∈B i∈I

Ri (v, w)

i∈I

(Q(u, v) ∧ Ri (v, w))

by (∗)

L-FUZZY RELATIONS

=

47

(Q(u, v) ∧ Ri (v, w))

i∈I v∈B

=

(Q ◦ Ri )(u, w)

i∈I

=

(Q ◦ Ri ) (u, w).

i∈I

Notice that (9) of the previous lemma implies that ◦ is a lower adjoint of a triple of residuated operations. The upper left adjoint is denoted by S ·· R and the upper right adjoint by Q ·· S. The next lemma shows the componentwise deﬁnition of the residuals. Lemma 3.2 Let L be a complete Brouwerian lattice. Then for all L-fuzzy relations Q : A → B, R : B → C, and S : A → C we have (S(v, w):Q(v, u)), (1) (Q ·· S)(u, w) = v∈A

(2) if Q ⊆ IA , then (Q ·· S)(v, w) = S(v, w):Q(v, v), (S(v, w):R(u, w)), (3) (S ·· R)(v, u) = w∈C

(4) if R ⊆ IB , then (S ·· R)(v, u) = S(v, w):R(u, u). Proof. (1) Deﬁne a relation i.e., we deﬁne U by the right-hand side of the assumption, U (u, w) := (S(v, w):Q(v, u)), and suppose (Q ·· S)(u, w) = x. First of v∈A

all, we have

Q(v, u) ∧ U (u, w) = Q(v, u) ∧

S(v , w):Q(v , u)

deﬁnition U

v ∈A

≤ Q(v, u) ∧ S(v, w):Q(v, u) ≤ S(v, w) such that

deﬁnition :

(Q(v, u)∧U (u, w)) ≤ S(v, w) follows. This implies Q◦U ⊆ S,

u∈B

and, hence, U ⊆Q ·· S by the deﬁnition of the residual. The previous (S(v, w):Q(v, u)) ≤ x. The property Q ◦ (Q ·· S) ⊆ S inclusion shows v∈A

implies Q(v, u) ∧ x ≤ S(v, w) for all v ∈ A. We obtain x ≤ S(v, w):Q(v, u) (S(v, w):Q(v, u)). for all v ∈ A, and, hence, x ≤ v∈A

(2) The assertion follows immediately from (1) since x:0 = 1. (3) and (4) are shown analogously.

48

GOGUEN CATEGORIES

3.2

CRISPNESS

As mentioned in the introduction, crispness is a fundamental notion within fuzzy theory. An L-fuzzy relation Q is called 0–1 crisp, iﬀ Q(x, y) = 0 or Q(x, y) = 1 for all x and y. If we identify f and 0 resp. t and 1 and regular binary relations with B-fuzzy relations, we may regard 0–1 crisp relations over an arbitrary complete Brouwerian lattice L as regular relations. Obviously, the set of 0–1 crisp relations is closed under all operations deﬁned above. Furthermore, under the identiﬁcation of 0–1 crisp and regular relations introduced above they coincide with the set-theoretic operations. Therefore, we will use the settheoretic notations and deﬁnitions also for 0–1 crisp relations, e.g., we write Q(x, y) instead of Q(x, y) = 1 and

instead of

(Q ◦ R)(u, w) ⇐⇒ ∃v : Q(u, v) and R(v, w) (Q ◦ R)(u, w) = 1 ⇐⇒ (Q(u, v) ∧ R(v, w)) = 1 v

for 0–1 crisp relations Q and R. There are several possibilities to identify a class of L-fuzzy relations with the lattice L itself, e.g., one could choose ideal elements. In our approach we u : A → A is called a scalar on will take scalar relations. An L-fuzzy relation αA A induced by u ∈ L iﬀ

u iﬀ x = y, u αA (x, y) = 0 else. Obviously, the set of scalars on A is closed under arbitrary intersections and unions and is isomorphic to L. The induced isomorphism is an isomorphism of complete Brouwerian lattices since it is surjective in respect to the set of scalars on A. A u-cut of an L-fuzzy relation is deﬁned as the following 0–1 crisp relation:

1 iﬀ R(x, y) ≥ u, Ru (x, y) := 0 else. The special cut with 1 will be denoted by R↓ . It is the greatest 0–1 crisp relation R contains. On the other hand, we may deﬁne

1 iﬀ R(x, y) = 0 ↑ R (x, y) := 0 else. R↑ is the least 0–1 crisp relation containing R. In fuzzy theory the relations R↑ and R↓ are called the support and the kernel of R, respectively. In the next lemma we have summarized some properties of the operations deﬁned above. Lemma 3.3 Let L be a complete Brouwerian lattice, and Q, R : A → B and S : B → C be L-fuzzy relations. Then we have

L-FUZZY RELATIONS

49

(1) Q is 0-1 crisp iﬀ Q↑ = Q iﬀ Q↓ = Q, (2) (↑ , ↓ ) is a Galois correspondence, ↑

T

(3) (RT ◦ S ↓ ) = R↑ ◦ S ↓ , ↑

(4) (Q ∩ R↓ ) = Q↑ ∩ R↓ , u↑ = IA , (5) if u = 0, then αA u · (6) Qu = (αA · Q) . ↓

Proof. (1) The assertion follows immediately from the deﬁnition of

↑

and ↓ .

(2) Suppose R↑ ⊆ S and R↑ (x, y) = 0. Then by the deﬁnition of ↑ we have 1 = R↑ (x, y) ≤ S(x, y), and, hence, R(x, y) ≤ 1 = S ↓ (x, y). The other implication is shown analogously. (3) Deﬁne two operations ↑ , ↓ : L → L on the lattice L by x↑ = 0 if x = 0 and x↑ = 1 otherwise, and by x↓ = 1 if x = 1 and x↓ = 0 otherwise, respectively. Then we have ↑

(a) R↑ (x, y) = (R(x, y)) , ↓

(b) R↓ (x, y) = (R(x, y)) . Furthermore, the operations satisfy ↑

(c) (x ∧ y ↓ ) = x↑ ∧ y ↓ for all x, y ∈ L,, ↑ ↑ x for all subset M of L. (d) ( M ) = x∈M

Finally, we obtain ↑

↑

(RT ◦ S ↓ ) (u, w) = ((RT ◦ S ↓ )(u, w)) ↑ (RT (u, v) ∧ S ↓ (u, w)) = v

= =

↑ ↓ (R(v, u) ∧ (S(u, w)) )

v

↓ ↑

((R(v, u) ∧ (S(u, w)) ) )

(a) deﬁnition of ◦ (b) and deﬁnition of (d)

v

=

↑

↓

((R(v, u)) ∧ (S(u, w)) )

(c)

(R↑ (v, u) ∧ S ↓ (u, w))

(a) and (b)

v

=

T (R↑ (u, v) ∧ S ↓ (v, w)) v

=

deﬁnition of

T

v T

= (R↑ ◦ S ↓ )(u, w).

deﬁnition of ◦

T

50

GOGUEN CATEGORIES

(4) Using the operations deﬁned in (3) the assertion follows from ↑

(Q ∩ R↓ ) (u, v) = ((Q ∩ R↓ )(u, v))

↑

(3a) ↑

↓

= (Q(u, v) ∧ R (u, v))

↓ ↑

= (Q(u, v) ∧ R(u, v) ) ↑

↓

= Q(u, v) ∧ R(u, v) ↑

↓

= Q (u, v) ∧ R (u, v) ↑

↓

= (Q ∩ R )(u, v).

deﬁnition of ∩ (3b) (3c) (3a) and (3b) deﬁnition of ∩

↑

we have for every u = 0

1 iﬀ x = y u↑ = IA (x, y). αA (x, y) = 0 iﬀ x = y

(5) By the deﬁnition of

(6) We will show the following property, which implies immediately the assertion: ↓ u · ⇐⇒ Q(x, y) ≥ u. (αA · Q) (x, y) = 1 Therefore, we deﬁne a crisp relation Ux,y : A → B by

1 iﬀ x = x and y = y Ux,y (x , y ) := 0 otherwise. Then we conclude ↓ ↓ u · u (αA · Q) (x, y) = 1 ⇔ Ux,y ⊆ (αA ·· Q) u · ⇔ Ux,y ⊆ αA ·Q u ⇔ αA ◦ Ux,y ⊆ Q ⇔ u ≤ Q(x, y),

by deﬁnition of Ux,y (2) and U is crisp

u u ◦ Ux,y )(x, y) = u and (αA ◦ where the last equivalence follows from (αA Ux,y )(x , y ) = 0 if x = x or y = y.

Notice that (5) of the previous lemma implies that the induced function fR from L to the set of 0–1 crisp relations deﬁned by fR (u) := Ru is an antimorphism. In the case of the unit interval fR is also called a tower resp. a chain of relations [9]. Furthermore, we have the following theorem: Theorem 3.4 Let L be a complete Brouwerian lattice and R : A → B a Lfuzzy relation. Then we have u (αA ◦ Ru ). R= u∈L

L-FUZZY RELATIONS

Proof. The assertion follows immediately from u u (αA ◦ Ru ) (w, y) = (αA ◦ Ru )(w, y) u∈L

51

deﬁnition ∪

u∈L

=

u (αA (w, x) ∧ Ru (x, y))

deﬁnition ◦

u∈L x∈A

=

(u ∧ Ru (w, y))

u deﬁnition of αA

u∈L

=

u

deﬁnition of Ru

u∈L R(w,y)≥u

= R(w, y).

The theorem above is known as the α-cut Theorem in fuzzy theory. On the other hand, the collection of the Ru is the least collection fulﬁlling the equation above. Notice that in the next lemma we identify Rel with the 0–1 crisp relations from L-Rel. anti

Rel[A, B] be an antimorphism, R : A → B an Lemma 3.5 Let f : L → u (αA ◦ f (u)). Then we have Rv ⊆ f (v) for all L-fuzzy relation with R ⊆ u∈L

v ∈ L.

Proof. Suppose Rv (x, y) = 1. By deﬁnition we have v ≤ (

u∈L

u (αA ◦ f (u)))

(x, y). Now, let M be the set of all w ∈ L such that f (w)(x, y) = 1 holds. Then we have u v≤ (αA ◦ f (u)) (x, y) u∈L

=

u (αA ◦ f (u))(x, y)

deﬁnition of ∪

u∈L

=

u (αA (x, w) ∧ f (u)(w, y))

deﬁnition of ◦

u∈L w∈A

=

(u ∧ f (u)(x, y))

u deﬁnition of αA

u∈L

=

M.

deﬁnition of M and f (u) is crisp

Now, let M := {v∧u | u ∈ M }. Then we get

M =

u∈M

(v∧u) = v∧

M =v

since L is completely upward-distributive. Every element w ∈ M is less or equal to an element u of M . Since f is antitone we get f (u) ⊆ f (w), and, hence, f (w)(x, y) = 1 for all w ∈ M by the deﬁnition of M . Finally, we

52

GOGUEN CATEGORIES

conclude that

1=

w∈M

=f

f (w)(x, y)

M (x, y)

= f (v)(x, y).

3.3

f antimorphism computation above

OPERATIONS DERIVED FROM LATTICE-ORDERED SEMIGROUPS

Within applications of fuzzy theory, usually union, meet and composition operators derived from t-norms resp. t-conorms, or more general from commutative complete lattice-ordered semigroups (commutative closg) as introduced in section 2.9, are used. Deﬁnition 3.6 Let Q, R : A → B and S : B → C be L-fuzzy relations, and (L, ∗, e, z) be a lattice-ordered operator set (loos). Then we deﬁne (1) (Q ∩∗ R)(x, y) := Q(x, y) ∗ R(x, y), (2) (Q ◦∗ S)(x, z) :=

(Q(x, y) ∗ S(y, z)).

y∈B

We want to show that the operations above may be deﬁned in a componentfree manner. Later on, this gives us the possibility to compare these operations with abstract deﬁned ∗-based connectives. Theorem 3.7 Let Q, R : A → B and S : B → C be L-fuzzy relations, and (L, ∗, e, z) be a loos. Then we have (1) Q ∩∗ R =

x∗y y x · (αA ◦ ((αA · Q) ∩ (αA ·· R) )), ↓

↓

x,y∈L

(2) Q ◦∗ S =

x∗y y x · (αA ◦ (αA · Q) ◦ (αB ·· R) ). ↓

↓

x,y∈L

Proof. (1) First of all, we have ↓ ↓ y x · ((αA · Q) ∩ (αA ·· R) )(u, v) ↓ ↓ ⇔ (αu ·· Q) (u, v) and (αy ·· R) (u, v) A

A

⇔ Q(u, v) ≥ x and R(u, v) ≥ y.

deﬁnition ∩ for crisp relations Lemma 3.3 (5)

L-FUZZY RELATIONS

53

This immediately implies ↓ ↓ x∗y y x · (αA ◦ ((αA · Q) ∩ (αA ·· R) )) (u, v) x,y∈L

=

x∗y y x · (αA ◦ ((αA · Q) ∩ (αA ·· R) ))(u, v) ↓

↓

x,y∈L

=

x∗y αA (u, u)

deﬁnition ◦ with a scalar and a crisp argument

x,y∈L · Q)↓ ∩(αyA ·· R)↓ )(u,v) ((αx A ·

=

deﬁnition ∪

x∗y deﬁnition αA

x∗y

x,y∈L · Q)↓ ∩(αyA ·· R)↓ )(u,v) ((αx A ·

=

x∗y

see above

x,y∈L Q(u,v)≥x and R(u,v)≥y

= Q(x, y) ∗ R(x, y).

monotonicity of ∗

(2) Again, we obtain y x · ((αA · Q) ◦ (αA ·· R) )(u, w) ↓ ↓ y u · ⇔ ∃v ∈ B : (αA · Q) (u, v) and (αA ·· R) (v, w) ↓

↓

⇔ ∃v ∈ B : Q(u, v) ≥ x and R(v, w) ≥ y.

deﬁnition ◦ for crisp relations Lemma 3.3 (5)

This implies ↓ ↓ x∗y y · x · (αA ◦ (αA · Q) ◦ (αB · S) ) (u, w) x,y∈L

=

x∗y y x · (αA ◦ (αA · Q) ◦ (αB ·· S) )(u, w)

x,y∈L

=

↓

x∗y αA (u, u)

x,y∈L · Q)↓ ◦(αyB ·· S)↓ )(u,w) ((αx A ·

=

x∗y

↓

deﬁnition ∪ deﬁnition ◦ with a scalar and a crisp argument x∗y deﬁnition αA

x,y∈L · Q)↓ ◦(αyB ·· S)↓ )(u,w) ((αx A ·

=

x∗y

x,y∈L ∃v∈B: Q(u,v)≥x and S(v,w)≥y

=

v∈B

(Q(u, v) ∗ S(v, w)),

see above

54

GOGUEN CATEGORIES

where the last equality is shown as follows: Suppose Q(u, v) ≥ x and S(v, w) ≥ y. Then we immediately conclude that x∗y ≤ Q(u, v)∗S(v, w) ≤ (Q(u, v) ∗ S(v, w)), and, hence, v∈B

x∗y ≤

x,y∈L ∃v∈B: Q(u,v)≥x and S(v,w)≥y

(Q(u, v) ∗ S(v, w)).

v∈B

On the other hand, for all v ∈ B we have Q(u, v) ∗ S(v, w) ≤

x∗y

x,y∈L ∃v∈B: Q(u,v)≥x and S(v,w)≥y

since Q(u, v) ≤ Q(u, v) and S(v, w) ≤ S(v, w).

Notice that the lemma above is true for arbitrary loos, i.e., whatever the neutral element e is. As mentioned in Chapter 2, a commutative losg (L, ×, 1, 0) is a generalized version of a t-norm, and analogously a commutative losg (L, +, 0, 1) a generalized version of a t-conorm. In this context we may deﬁne the following meet and union operations on L-fuzzy relations Q, R : A → B by Q ∧× R := Q ∩× R,

Q ∨+ R := Q ∩+ R.

In the case L = [0, 1], these deﬁnitions are exactly the usual deﬁnitions of t-norm based meet, resp. t-conorm, based union of fuzzy relations. For the moment, we dispense with a further investigation of the properties of the ∗-based operations deﬁned above. This will be done in section 5.10.

4 CATEGORIES OF RELATIONS

Usually a binary relation acts between two diﬀerent sets. Therefore, an algebraic theory for relations should reﬂect this kind of typing, i.e., the theory should have a suitable notion of source and target of its elements. A convenient framework for that is given by category theory. 4.1

CATEGORIES

In this chapter we will introduce some basic notions from category theory. For a comprehensive introduction to this theory, especially for computer scientists, we refer to [1]. Deﬁnition 4.1 A category C consists of (1) a class of objects ObjC , (2) for every pair of objects A and B a class of morphisms C[A, B], (3) an associative binary (partial) operation ; mapping each pair of morphisms f in C[A, B] and g in C[B, C] to a morphism f ; g in C[A, C], (4) for every object A a morphism IA such that for all f in C[A, B] and g in C[C, A] we have IA ; f = f and g; IA = g. If f is a morphism in C[A, B], we will denote it by f : A → B. 55

56

GOGUEN CATEGORIES

The following table lists some common categories by specifying their objects and morphisms: Category

Objects

Morphisms

Set Rel [0, 1]-Rel L-Rel PO VctF ZF

sets sets nonempty sets nonempty sets posets vector spaces over the ﬁeld F models of Zermelo-Fraenkel set theory

functions relations fuzzy relations L-fuzzy relations monotone functions linear functions ∈-preserving functions, i.e., x ∈ A implies f (x) ∈ f (A)

Theorem 3.1 (1) and (2) show that Rel, [0, 1]-Rel, and L-Rel are indeed categories. Notice that the class of objects as well as the class of morphisms need not to be a set. For example, in ZF every class of morphisms ZF[A, B] as well as the class of objects is indeed a class and not a set. Also, in Set or Rel the class of objects is not a set but all classes of morphisms Set[A, B] or Rel[A, B] are indeed sets. Such a category is called locally small . The natural notion of a homomorphism between categories is given by functors. Deﬁnition 4.2 A functor F between two categories C1 and C2 is a pair of functions (FObj , FMor ) such that (1) FObj maps the objects of C1 to the objects C2 , (2) FMor maps morphisms of C1 to morphisms of C2 such that for all morphisms f : A → B and objects A and B of C1 the image FMor (f ) is a morphism from FObj (A) to FObj (B) in C2 , (3) FMor (f ); FMor (g) = FMor (f ; g) for all morphisms f : A → B and g : B → C and objects A, B, and C in C1 , (4) FMor (IA ) = IFObj (A) for all objects A in C1 . A functor F is called faithful iﬀ FMor is injective. It is called full iﬀ for all objects A and B and morphisms g : FMor (A) → FMor (B) there is a morphism f : A → B such that FMor (f ) = g. A full functor may be seen as a functor such that FMor is surjective on the image of FObj . As usual, an isomorphism between categories is a functor, −1 −1 , FMor ) is again which is bijective on objects and morphisms and F −1 := (FObj a functor. We will omit the subscripts Obj and Mor as it is always clear from the context whether the functor is meant to operate on objects or morphisms.

CATEGORIES OF RELATIONS

57

For technical reasons we call a pair of functions (FObj , FMor ) fulﬁlling (1) and (2) of Deﬁnition 4.2 a pre-functor. Lemma 4.3 A pre-functor F : C1 → C2 , which is full, faithful, and bijective on objects is an isomorphism iﬀ either F or F −1 respects composition, i.e., fulﬁlls (3) of Deﬁnition 4.2. Proof. Without loss of generality suppose F respects composition. The computation F (IA ) = F (IA ); IF (A) = F (IA ); F (F = F (IA ; F = F (F

−1

−1

−1

F surjective on objects (IF (A) ))

(IF (A) ))

F is full and faithful property (3) of a functor

(IF (A) ))

= IF (A)

F is full and faithful

shows that F is a functor. It remains to show that F −1 preserves identities and composition. Suppose A is an object of C2 . Since F is bijective on objects we have F (F −1 (A)) = A. We obtain F −1 (IA ) = F −1 (IF (F −1 (A)) ) = F −1 (F (IF −1 (A) )) = IF −1 (A) since F is a functor. Now, suppose A, B, and C are objects of C2 and f : A → B and g : B → C. Again, we have F (F −1 (A)) = A, F (F −1 (B)) = B, F (F −1 (C)) = C, F (F −1 (f )) = f, and F (F −1 (g)) = g since F is full, faithful, and bijective on objects. This implies F −1 (f ; g) = F −1 (F (F −1 (f )); F (F −1 (g))) =F =F

4.2

−1 −1

(F (F

−1

(f ); F

(f ); F

−1

(g).

−1

(g)))

computation above F is a functor F is full and faithful

ALLEGORIES

Throughout this book, phrases like “the class R[A, B] is a lattice” should not imply that R[A, B] is a set. It just states that there are several operations fulﬁlling the corresponding algebraic laws. On the other hand, if we refer to R[A, B] as some complete structure, we implicitly mean that it is a set since the notion of a subset is essential for complete structures. Deﬁnition 4.4 An allegory R is a category satisfying the following: (1) For all objects A and B the class R[A, B] is a lower semilattice. Meet and the induced ordering are denoted by , , respectively. The elements in R[A, B] are called relations.

58

GOGUEN CATEGORIES

(2) There is a monotone operation (called the converse operation) such that for all relations Q, R : A → B and S : B → C the following holds:

(Q; S) = S ; Q

and

(Q ) = Q.

(3) For all relations Q : A → B, R, S : B → C we have Q; (R S) Q; R Q; S. (4) For all relations Q : A → B, R : B → C and S : A → C the modular law Q; R S Q; (R Q ; S) holds.

Notice that [10] requires (Q R) = Q R instead of the monotonicity of . . Obviously, the property above implies monotonicity and the other implication is shown as follows: X Q R ⇔ X Q

and

X R

⇔ X Q and

X R

⇔X

monotone and (2b)

monotone and (2b)

QR

⇔ X (Q R) .

A homomorphism F : R1 → R2 between allegories is a functor, which preserves the converse operation and, restricted to every R1 [A, B], is a lower semilattice homomorphism. An allegory R, as well as the structures deﬁned later, is called representable iﬀ there is an embedding into Rel, i.e., there is a faithful homomorphism F (of convenient type) from R to Rel. By Theorem 3.1 (1)–(7), the category L-Rel of L-fuzzy relations with meet ∩ and conversion T is an allegory. Lemma 4.5 Let R be an allegory, A, B, C objects of R and Q, R : A → B, S : B → C, T : A → C, and U, V : A → A. Then we have (1) I A = IA , (2) (Q R); S Q; S R; S, (3) ; is monotone in both arguments, (4) Q; S T (Q T ; S ); S, (5) Q; S T (Q T ; S ); (S Q ; T ), (6) Q Q; Q ; Q,

(7) IA (U V ); (U V ) = IA U ; V = IA V ; U , (8) Q = (IA Q; Q ); Q = Q; (IB Q ; Q).

CATEGORIES OF RELATIONS

59

Proof. (1) The assertion follows from

IA = (I A) = = = =

axiom (2b)

(IA ; I A) (I A ) ; IA IA ; I A IA

identity law axiom (2a) axiom (2b) identity law

(2) Consider the following computation:

(Q R); S = (((Q R); S) )

axiom (2b)

= (S ; (Q R )

axiom (2a)

(S ; Q S ; R )

(1) and axiom (3)

= ((Q; S R; S) ) = Q; S R; S.

axiom (2a) axiom (2b)

(3) The assertion follows immediately from (2) and axiom (3). (4) is shown similar to (2). (5) The assertion is shown as follows: Q; S T = (Q; S T ) T (Q T ; S ); S T

(4)

(Q T ; S ); (S (Q T ; S ) ; T )

axiom (4)

(Q T ; S ); (S Q ; T ).

monotone and (3)

(6) Again, consider the following computation: Q = Q; IB Q

identity law

Q; (IB Q ; Q)

axiom (4)

Q; Q ; Q.

(3)

(7) First of all, we have IA (U V ); (U V ) inclusion follows from

IA U ; V . The other

IA U ; V = IA (IA (IA U ; V )) IA (IA (IA ; V U ); V )

modular law

= IA (IA (U V ); V )

identity law

IA (U V ); ((U V ) ; IA V )

= IA (U V ); (U V ) .

modular law

60

GOGUEN CATEGORIES

The second assertion is shown analogously. (8) We just show the ﬁrst assertion, which follows immediately from (IA Q; Q ); Q Q = IA ; Q Q (IA Q; Q ); Q.

(4)

In the remainder of this book we will use the properties (1)–(6) of the previous lemma without mentioning. An important class of relations is given by mappings. Deﬁnition 4.6 Let R be an allegory and Q : A → B. Then we call (1) Q univalent iﬀ Q ; Q IB , (2) Q total iﬀ IA Q; Q , (3) Q a map iﬀ Q is univalent and total, (4) Q injective iﬀ Q is univalent, (5) Q surjective iﬀ Q is total, (6) Q bijective iﬀ Q is a map, (7) Q a bijection iﬀ Q is a bijective map. Notice that in Rel the deﬁnitions above correspond to the set-theoretic deﬁnitions. As usual, we will denote mappings by lowercase letters. Since the notions of univalent and injective relations as well as the notions of total and surjective relations are dual via conversion we just state properties of univalent and/or total relations. The class of univalent relations, the class of total relations, and, hence, the class of mappings is closed under composition. This may be seen as follows:

(Q; R) ; Q; R = R ; Q ; Q; R R ; R IC ,

Q univalent R univalent

Q; R; (Q; R) = Q; R; R ; Q Q; Q

R total

IA .

Q total

On the other hand, if Q; R is total, then so is Q. This follows from IA = IA Q; R; (Q; R)

Q; R total

= IA Q; R; R ; Q

(Q; Q; R; R ); Q Q; Q .

modular law

CATEGORIES OF RELATIONS

61

Some other interesting properties of univalent relations are summarized in the next lemma. Lemma 4.7 Let R be an allegory, Q : A → B be univalent and R, S : B → C, T : C → A, and U : C → B. Then we have (1) Q; (R S) = Q; R Q; S, (2) T ; Q U = (T U ; Q ); Q. Proof. (1) It remains to show that Q; R Q; S Q; (R S). Therefore, consider the following computation: Q; R Q; S Q; (R Q ; Q; S) Q; (R S).

modular law Q univalent

(2) The assertion is shown as follows: (T U ; Q ); Q T ; Q U ; Q ; Q T;Q U

(T U ; Q ); Q.

Q univalent modular law

Now, suppose f is a concrete map, i.e., a set-theoretic function. Then we may state the following property of functions: The image of a set A under f is included in a set B iﬀ A is included in the inverse image of B or, alternatively, the operations Q → Q; f and R → R; f form a Galois correspondence. The next lemma shows that this property is valid in all allegories. Lemma 4.8 Let R be an allegory, Q : A → B, R : A → C, S : D → B be relations, and f : B → C and g : A → D be mappings. Then we have (1) Q; f R iﬀ Q R; f , (2) g ; Q S iﬀ Q g; S. Proof. (1) Suppose Q; f R. Then we get Q Q; f ; f R; f since f is total. Now, suppose Q R; f and compute Q; f R; f ; f R since f is univalent. (2) follows from (1) using converse.

In the view of scalars, partial identities, i.e., relations R : A → A with R IA are of special interest.

62

GOGUEN CATEGORIES

Lemma 4.9 Let R be an allegory, S, T : B → B partial identities and Q, U : A → B and R, V : B → C arbitrary relations. Then we have (1) S = S, (2) S; S = S, (3) S; T = S T , (4) Q; (S T ) = Q; S Q; T and (S T ); R = S; R T ; R, (5) (Q U ); (S T ) = Q; S U ; T and (S T ); (R V ) = S; R T ; V . Proof. (1) Consider the following computation: S S; S ; S

IB ; S ; IB

Lemma 4.5 (6) S partial identity

=S . The other inclusion is shown analogously. (2) The following computation shows the assertion: S S; S ; S S; IB ; S = S; S S; IB

Lemma 4.5 (6) S partial identity S partial identity

= S. (3) First, we have S; T S; IB = S and S; T IB ; T = T , and, hence, S; T S T . The other inclusion is shown as follows: S T = (S T ); (S T ) S; T

(2) since S T is a partial identity

(4) We just show the ﬁrst assertion, where the inclusion is trivial. The other inclusion follows from Q; S Q; T Q; (S Q ; Q; T )

modular law

= Q; (S IB Q ; Q; T )

S IB

= Q; (S IB Q ; Q; T )

(1)

= Q; (S IB (Q ; Q T ); (Q ; Q T ) )

Lemma 4.5 (7)

Q; (S IB T ; T ) = Q; (S IB T ) = Q; (S T ).

(1) and (2) S IB

CATEGORIES OF RELATIONS

63

(5) Notice that the relations S and S ∩ T are univalent. Again, we just show the ﬁrst assertion, which follows from Q; S U ; T = (Q U ; T ; S ); S = (Q U ; (S T )); S = (Q U ; (S T ); (S T )); S

Lemma 4.7 (2) (1) and (3) (2)

= (Q; (S T ) U ; (S T )); (S T ); S = (Q; (S T ) U ; (S T )); (S T ) = (Q; U ); (S T ); (S T )

Lemma 4.7 (2) (1) and (3) Lemma 4.7 (1)

= (Q; U ); (S T ).

(2)

As for lattices, a convenient allegory of antimorphism is embedded into an allegory of antitone functions, which is deﬁned as follows: Theorem 4.10 Let R be an allegory and P a poset. Then the structure RP ≥ deﬁned by (1) the objects of RP ≥ are the objects of R, ≥

(2) RP ≥ [A, B] is the lower semilattice P → R[A, B], (3) all other operations and constants are deﬁned componentwise, i.e., IA (x) := IA ,

(f ; g)(x) := f (x); g(x),

f (x) := (f (x))

is again an allegory. Proof. We just show that f ; g and f are antitone. The rest of the proof follows immediately since all operations and constants are deﬁned componentwise and the axioms of an allegory are equational. Suppose x y. Then we have (f ; g)(y) = f (y); g(y) f (x); g(x) = (f ; g)(x) and f (y) = (f (y)) (f (x)) = f (x). 4.3

DISTRIBUTIVE ALLEGORIES

Consider the collection of binary relations on a ﬁxed set. This structure constitutes a distributive lattice with a least element. This is our motivation to switch from lower semilattices to distributive lattices as the basic order structure. Deﬁnition 4.11 A distributive allegory R is an allegory satisfying the following: (1) The classes R[A, B] are distributive lattices with a least element. Union and the least element are denoted by , ⊥ ⊥AB , respectively.

64

GOGUEN CATEGORIES

(2) For all relations Q : A → B we have Q; ⊥ ⊥BC = ⊥ ⊥AC . (3) For all relations Q : A → B, R, S : B → C we have Q; (R S) = Q; R Q; S. Obviously, a homomorphism between distributive allegories is a homomorphism, which is also an upper semilattice homomorphism for every pair of objects. Again, Theorem 3.1 (8) and (9) show that the allegory L-Rel of L-fuzzy relations with union ∪ is a distributive allegory. Lemma 4.12 Let R be a distributive allegory. Then for all Q, R : A → B and S : B → C we have ⊥BA , (1) ⊥ ⊥ AB = ⊥ ⊥CB , (2) ⊥ ⊥CA ; Q = ⊥

(3) (Q R) = Q R , (4) (Q R); S = Q; S R; S. Proof. ⊥ (1) Suppose X : B → A. Then we have ⊥ ⊥AB X , and, hence, ⊥ AB X. (2) We immediately conclude that ⊥ ⊥CA ; Q = (Q ; ⊥ ⊥ CA ) ⊥ ⊥BC = ⊥ ⊥CB by (1) and axiom (2). (3) The inclusion is trivial since from

= (Q ; ⊥ ⊥AC )

=

is monotone. The other inclusion follows

(Q R) Q R ⇔ Q R (Q R ) ⇔ Q (Q R )

⇔ Q Q R

and R (Q R )

and R Q R .

(4) Consider the following computation:

(Q R); S = (S ; (Q R) )

= (S ; (Q R ))

= (S ; Q S ; R )

by (3) axiom (4)

= (Q; S R; S ) = Q; S R; S. ≥

by (3)

By Theorem 2.23 the structure P → R[A, B] is a distributive lattice for all A and B. This motivates the following theorem:

CATEGORIES OF RELATIONS

65

Theorem 4.13 Let R be a distributive allegory and P a poset. Then the allegory RP ≥ is again a distributive allegory. The axioms of a distributive allegory are equational and all new operations and constants are deﬁned componentwise. Therefore, we omit the proof of the previous theorem. 4.4

DIVISION ALLEGORIES

The next step in the hierarchy of allegories are division allegories. They are characterized by the fact that ; is a lower adjoint. Deﬁnition 4.14 A division allegory R is a distributive allegory such that ; has an upper left adjoint, i.e., for all relations R : B → C and S : A → C there is a relation S/R : A → B (called the left residual of S and R) such that for all Q : A → B the following holds: Q; R S

⇐⇒

Q S/R.

As before, a homomorphism F between division allegories is a homomorphism between distributive allegories, which reﬂects the residual operation, i.e., F (S/R) = F (S)/F (R) for all R : B → C and S : A → C. Again, as mentioned after Theorem 3.1 L-Rel is a division allegory with S ·· R as residual. The computation Q; R S ⇔ R ; Q S ⇔ R S /Q ⇔ R (S /Q )

shows that in a division allegory there is also an upper right adjoint (S /Q ) for ;, which will be denoted by Q\S and called the right residual of S and Q. A symmetric version, called the symmetric quotient, of the residuals may be deﬁned as syQ(Q, R) := (Q\R) (Q /R ). By deﬁnition, this relation is the greatest solution X of the inclusions Q; X R

and

X; R Q .

From the Corollaries 2.17 and 2.18 on triples of residuated operations we get the following: Corollary 4.15 Let R be a division allegory and Q, Q1 , Q2 : A → B, R, R1 , R2 : B → C, and S, S1 , S2 : A → C. Then we have (1) Q (Q; R)/R and R Q\(Q; R), (2) (S/R); R S and Q; (Q\S) S,

66

GOGUEN CATEGORIES

(3) S/(Q\S) Q and (S/R)\S R, (4) Q2 Q1 , R2 R1 and S1 S2 implies S1 /R1 S2 /R2 and Q1 \S1 Q2 \S2 , (5) (S1 S2 )/R = (S1 /R) (S2 /R) and Q\(S1 S2 ) = (Q\S1 ) (Q\S2 ), (6) S/(R1 R2 ) = (S/R1 ) (S/R2 ) and (Q1 Q2 )\S = (Q1 \S) (Q2 \S), Notice that the properties (1), (2), and (5) may also be used as axioms for the residuals. This shows that the notion of division allegories may also be deﬁned equationally. In the next lemma we have summarized further properties of the residuals. Lemma 4.16 Let R be a division allegory and Q : A → B, R : B → C, S : A → C, F : D → A, and G : C → E. Then we have (1) S/IC = S and IA \S = S, (2) F ; (S/R) (F ; S)/R and (Q\S); G Q\(S; G), (3) if F and G are mappings, then the inclusions in (2) are equalities, (4) S/R (S; G)/(R; G) and Q\S (F ; Q)\(F ; S), (5) if F and G are total and injective, then the inclusions in (4) are equalities. Proof. In all cases we just show the ﬁrst assertion. (1) Corollary 4.15 (1) implies S (S; IC )/IC = S/IC and we conclude S/IC = (S/IC ); IC S using Corollary 4.15 (2). (2) Again, Corollary 4.15 (2) implies F ; (S/R); R F ; S, which is equivalent to F ; (S/R) (F ; S)/R. (3) We immediately conclude that (F ; S)/R F ; F ; ((F ; S)/R)

F ; ((F ; F ; S)/R) F ; (S/R).

F total (2) F univalent and Corollary 4.15 (4)

(4) Consider the following computation: X S/R ⇔ X; R S ⇒ X; R; G S; G ⇔ X (S; G)/(R; G), which implies the assertion.

deﬁnition of / deﬁnition of /

CATEGORIES OF RELATIONS

67

(5) The assertion follows from X (S; G)/(R; G) ⇔ X; R; G S; G

deﬁnition of /

⇒ X; R; G; G S; G; G

G; G = IC by the assumption

⇔ X; R S ⇔ X S/R.

In the next lemma we have summarized some basic properties of symmetric quotients. Lemma 4.17 Let R be a division allegory, Q : A → B, R : A → C, S : A → D be relations, and f : D → A be a mapping. Then we have (1) f ; syQ(Q, R) = syQ(Q; f , R),

(2) syQ(Q, R) = syQ(R, Q), (3) syQ(Q, R); syQ(R, S) syQ(Q, S). Proof. (1) The following computation shows the assertion: X f ; syQ(Q, R) ⇔ f ; X syQ(Q, R)

Lemma 4.8 (2)

⇔ Q; f ; X R and f ; X; R Q

deﬁnition syQ

⇔ Q; f ; X R and X; R f ; Q

Lemma 4.8 (2)

⇔ X syQ(Q; f , R).

deﬁnition syQ

(2) We immediately conclude that X syQ(Q, R)

⇔ X syQ(Q, R) ⇔ Q; X R and X ; R Q

deﬁnition syQ

⇔ X; Q R and R; X Q ⇔ X syQ(R, Q).

deﬁnition syQ

(3) The assertion follows from syQ(Q, R); syQ(R, S) = ((Q\R) (Q /R )); ((R\S) (R /S ))

deﬁnition syQ

(Q\R); (R\S) (Q /R ); (R /S ) (Q\S) (Q /S ) = syQ(Q, S),

deﬁnition syQ

68

GOGUEN CATEGORIES

where the second inclusion is shown as follows: We have (Q\R); (R\S); S (Q\R); R Q by Corollary 4.15 (2), and, hence, (Q\R); (R\S) Q\S. (Q /R ); (R /S ) Q /S is shown analogously. Unfortunately, for antitone functions f and g, the function (f /g)(x) := f (x)/g(x) is not antitone since / itself is antitone in the second argument. Therefore, RP ≥ is not necessarily a division allegory. We will get a convenient theorem in the case of completeness, i.e., for Dedekind categories in the next chapter. The axioms of a division allegory are not independent. This is shown by the next lemma. Theorem 4.18 A category R is a division allegory iﬀ the following holds: (1) For all objects A and B the collection R[A, B] is a distributive lattice with ⊥AC for all relations Q : A → B, Q; ⊥ ⊥BC = ⊥ (2)

is a monotone operation with (Q; R) = R ; Q and (Q ) = Q for all relations Q : A → B and R : B → C.

(3) The modular law is valid. (4) The left residual exists. Proof. The implication ⇒ is trivial. For the other implication we have to show (1) Q; (R S) Q; R Q; S, (2) Q; (R S) = Q; R Q; S. The second assertion follows from Corollary 2.18 since ; is a lower adjoint of a triple of residuated operations. Therefore, ; is monotone in the second argument, which implies the ﬁrst assertion. 4.5

DEDEKIND CATEGORIES

Now, we will switch to complete structures. Remember, that this will imply that the corresponding categories are locally small. Deﬁnition 4.19 A Dedekind category R is a division allegory so that every R[A, B] is a complete Brouwerian lattice. The greatest element in R[A, B] is denoted by AB . A homomorphism between Dedekind categories is a homomorphism between division allegories, which is a complete Brouwerian lattice homomorphism for any pair of objects.

CATEGORIES OF RELATIONS

69

Lemma 4.20 Let F : R1 → R2 be a pre-functor between Dedekind categories, which is full, faithful, and bijective on objects. Furthermore, suppose (1) either F or F −1 respects composition, (2) either F or F −1 respects the converse operation, (3) either F or F −1 is a complete lower semilattice homomorphism for every pair of objects, and (4) either F or F −1 is an upper semilattice homomorphism for every pair of objects. Then F is an isomorphism. Proof. By Lemma 4.3 F is an isomorphism of categories. Suppose without loss of generality that F respects the converse operation. Then we have

F −1 (Q ) = F −1 (F (F −1 (Q)) ) =F

−1

(F (F

−1

(Q) ))

= F −1 (Q) .

F bijective on relations F respects

F bijective on relations

By Lemma 2.11 and the observation after Theorem 2.25 it is suﬃcient to show that F and F −1 preserve residuals. This follows immediately from the following computations: {X | X; R Q}

F (Q/R) = F = = =

{F (X) | X; R Q}

F complete lattice homo.

{F (X) | F (X); F (R) F (Q)}

F homomorphism and bijective

{Y | Y ; F (R) F (Q)}

F bijective

= F (Q)/F (R), F

−1

(S/T ) = F −1 (F (F −1 (S))/F (F −1 (T ))) =F =F

−1 −1

(F (F

−1

(S)/F

(S)/F

−1

(T ).

−1

(T )))

F bijective on relations F homomorphism F bijective on relations

As mentioned in Chapter 3, every L-Rel[A, B] is a complete Brouwerian lattice, and, hence, L-Rel is a Dedekind category. Now, we want to state some properties of the greatest elements within Dedekind categories. Lemma 4.21 Let R be a Dedekind category and A and B objects of R. Then we have BA , (1) AB =

70

GOGUEN CATEGORIES

AAE EE yy EE y yy R EE yS EE yy E y y ⊥ ⊥AA Q Q ; ⊥ ⊥AA R S AA ⊥ ⊥AB AB ⊥ ⊥BA BA ⊥ ⊥BB BB

⊥ ⊥AA R S AA ⊥ ⊥AA R S AA

AB

BA

BB

⊥ ⊥AB

⊥ ⊥BA

⊥ ⊥BB

⊥ ⊥AB AB ⊥ ⊥AB AB

⊥ ⊥BA BA ⊥ ⊥BA BA

⊥ ⊥AA R S AA ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA R ⊥ ⊥AA R ⊥ ⊥AA ⊥ ⊥AA S S ⊥ ⊥AA R S AA

⊥ ⊥AB ⊥ ⊥AB ⊥ ⊥AB ⊥ ⊥AB ⊥ ⊥AB

⊥ ⊥BA ⊥ ⊥BA ⊥ ⊥BA ⊥ ⊥BA ⊥ ⊥BA BA ⊥ ⊥BA BA

⊥ ⊥BB ⊥ ⊥BB ⊥ ⊥BB BB

Figure 4.1.

AB ⊥ ⊥AB AB ⊥ ⊥AB ⊥ ⊥AB

⊥ ⊥BB BB ⊥ ⊥BB BB

⊥ ⊥BA BA

⊥ ⊥BB BB

⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA R

⊥ ⊥AB ⊥ ⊥AB ⊥ ⊥AB AB

⊥ ⊥BA ⊥ ⊥BA ⊥ ⊥BA BA

⊥ ⊥BB ⊥ ⊥BB ⊥ ⊥BB BB

A Dedekind category with AB ; BA = AA .

(2) AA ; AB = AB ; BB = AB , AB ; BA ; AB . (3) AB = Proof. (1) Suppose X : B → A. Then we have X AB , and, hence, X AB . AB ; IBB AB ; BB . Analogously, we obtain AB = (2) We have AB = AB AA ; AB . IAA ; (3) The assertion follows immediately from (1) and Lemma 4.5 (6).

BC = AC is not valid. Consider Notice that the general property AB ; the four (ﬁnite) lattices with converse and composition deﬁned in Figure 4.1. BA = R = AA . This counterexIn this Dedekind category we have AB ; ample can also be found in [39]. BC = AC We will call a Dedekind category uniform iﬀ the equation AB ; is true for all objects A, B and C with IB = ⊥ ⊥BB . The so-called Tarski-rule R = ⊥ ⊥BC

=⇒

AB ; R; CD = AD

CATEGORIES OF RELATIONS

71

implies that R is uniform. The Dedekind category of L-fuzzy relations is uniform. But, if the lattice L has at least three elements, the converse implication does not hold. Let u ∈ L be an element with 0 = u = 1. Then we have u u ⊥AA , αA = ⊥ ⊥AA and ( AA ; αA ; AA )(x) = u for all x ∈ A, which shows IA = ⊥ u AA = AA . that AA ; αA ; According to the equivalence of the Tarski-rule to a generalized version of the notion of simplicity known from universal algebra, we call a Dedekind category simple iﬀ the Tarski-rule is valid. Corollary 2.19 for residuated triples gives us the following: Corollary 4.22 Let R be a Dedekind category Q, Qi : A → B, R, Ri : B → C and S, Si : A → C for i ∈ I. Then we have (Q; Ri ) and ( Qi ); R = (Qi ; R) (1) Q; ( Ri ) = i∈I

(2) Q; ( i∈I

i∈I

i∈I

Ri )

i∈I

Si )/R and

i∈I

(S/Ri ) = S/(

(4) i∈I

(Qi ; R),

i∈I

(Si /R) = (

(3)

Qi ); R

(Q; Ri ) and ( i∈I

i∈I

i∈I

(Q\Si ) = Q\( i∈I

i∈I

(Qi \S) = (

Ri ) and

Si ),

i∈I

i∈I

Qi )\S.

i∈I

In the next lemma we have summarized some other properties valid in Dedekind categories. Lemma 4.23 Let R be a Dedekind category Q : A → B, R : B → C, S : A → D, and T : D → C. Then we have DC and Q; (R BD ; T ) = Q; R AD ; T , (1) (Q S; DB ); R = Q; R S; (2) IA Q; Q = IA Q; BA = IA AB ; Q , (3) if Q and R are univalent, then the inclusions in Corollary 4.22 (2) are equalities. AC for all objects C. (4) Q is total iﬀ Q; BC = Proof. In all cases we just show the ﬁrst assertion. (1) Consider the following computation: (Q S; DB ); R Q; R S; DB ; R Q; R S; DC (Q S; DC ; R ); R (Q S; DB ); R.

modular law

72

GOGUEN CATEGORIES

(2) Again, consider the following computation: BA = IA Q; IA Q; AB

= IA (Q AB ); (Q AB )

Lemma 4.5 (7)

= IA Q; Q . (Q; Ri ) Q; Ri

(3) We just show the ﬁrst assertion. Since Q is univalent, i∈I

implies Q ; (

(Q; Ri )) Q ; Q; Ri Ri for all i ∈ I. We immediately

i∈I

conclude that Q ; (

(Q; Ri ))

i∈I

Ri , and, hence, i∈I

Q; Q ;

(∗)

Q;

(Q; Ri )

Ri

i∈I

.

i∈I

We obtain (Q; Ri Q; BC )

(Q; Ri ) = i∈I

i∈I

Q; BC

(Q; Ri )

= i∈I

= (IA Q; BA );

(Q; Ri ) i∈I

= (IA Q; Q );

Q; Q ;

(2)

(Q; Ri ) i∈I

Q;

(Q; Ri ) i∈I

(1)

Ri

.

(∗)

i∈I

AC Q; Q ; AC (4) Suppose Q is total. Then we have AC = IA ; BC = AC . Then IA Q; Q = IA Q; BA = Q; BC . Now, suppose Q; AA = IA follows. IA Within Dedekind categories there are several further properties of partial identities. These are collected in the next lemma. Lemma 4.24 Let R be a Dedekind category, S, T, Si : A → A for all i ∈ I partial identities and R : C → A, U : A → B. Then we have AA = IA AA ; S, (1) S = IA S; AB , (2) R; S = R CA ; S and S; U = U S; ( CA ; Si ) = CA ; (

(3) i∈I

i∈I

(Si ; AB ) = (

Si ), and i∈I

i∈I

Si ); AB ,

CATEGORIES OF RELATIONS

(R; Si ) = R; (

(4) i∈I

Si ), and

(Si ; U ) = (

i∈I

i∈I

73

Si ); U .

i∈I

Proof. In all cases we just show the ﬁrst assertion. (1) By Lemma 4.9 (1) and (2) we have S = IA S; S . From Lemma 4.23 (2) we conclude the assertion. (2) Consider the following computation: AA ; S) R; S = R; (IA = R CA ; S

(1) Lemma 4.23 (1)

(3) It is suﬃcient to show . This follows from ( CA ; Si ) = CA i∈I

( CA ; Si ) i∈I

CC ; ( CA ; Si ) CA i∈I = CA ; IA AC ; ( CA ; Si ) i∈I

IC CC Lemma 4.23 (1)

CA ; IA ( AC ; CA ; Si ) i∈I AA ; Si ) CA ; IA ( i∈I = CA ;

i∈I

(IA AA ; Si )

= CA ;

Si

.

(1)

i∈I

(4) We immediately conclude that (R CA ; Si )

(R; Si ) = i∈I

(2)

i∈I

=R

( CA ; Si ) i∈I

= R CA ; = R;

Si

Si

(3)

i∈I

.

(2)

i∈I

In contrast to division allegories, which are not necessarily complete, the residuals in RP ≥ can be constructed if R is complete. Theorem 4.25 Let R be a Dedekind category and P a poset. Then the allegory RP ≥ is again a Dedekind category.

74

GOGUEN CATEGORIES

Proof. First of all, RP ≥ [A, B] is a complete Brouwerian lattice by Theorem 2.23 and Theorem 2.25 (2). The residuals exist by Corollary 2.19 since ; is deﬁned componentwise and, therefore, an upper semilattice homomorphism in both arguments. Last but not least, we want to introduce a special endo-isomorphism of Dedekind categories. We will need this isomorphism for the motivation of Goguen categories. Lemma 4.26 Let R be a Dedekind category, and {fA | A object of R} be a class of bijections fA : A → A. Then F deﬁned by F (A) := A and F (g) := fA ; R; fB for R : A → B is an isomorphism. Proof. Suppose F (R) = F (S). Then we conclude R = fA ; fA ; R; fB ; fB = fA ; F (R); fB = fA ; F (S); fB = fA ; fA ; S; fB ; fB = S since fA and fB are bijections. Obviously, fA ; Q; fB is mapped to Q by F such that F is full, faithful, and bijective on objects. Using Lemma 4.20 the assertion follows from Corollary 4.22 (1) and Lemma 4.23 (3) since F −1 (Q) = fA ; Q; fB . 4.6

RELATIONAL CONSTRUCTIONS IN DEDEKIND CATEGORIES

In applications of Dedekind categories, especially in computer science, one often wants to use abstract counterparts of set-theoretic constructions as disjoint union of sets, Cartesian product of sets or subsets. Later on, we will give some examples in connection with the interpretation of fuzzy controllers within a Goguen category. In this chapter, we want to introduce those constructions. The abstract counterpart of a disjoint union of sets is called a relational sum. It is characterized by the injections in the following way: I} be a set of objects of a Dedekind category Deﬁnition 4.27 Let {Ai | i ∈ Ai , together with relations ιj ∈ R[Aj , Ai ] indexed by some set I. An object i∈I

i∈I

for all j ∈ I, is called a relational sum of {Ai | i ∈ I} iﬀ for all i, j ∈ I with i = j the following holds: = I , ι ; ι = ⊥ ⊥ , (ιi ; ιi ) = I Ai . ιi ; ι A i A A i i j i j i∈I

i∈I

R has relational sums iﬀ for every set of objects the relational sum does exist. A relational sum is unique up to isomorphism. We do not need that property so we omit the proof. L-Rel has relational sums, which are given by the disjoint union and the corresponding (crisp) set-theoretic injections. This is easy to verify and, therefore, we omit the proof. The dual notion of a sum is the notion of a product. Deﬁnition 4.28 Let A and B be objects of a Dedekind category. An object A × B, together with two relations π : A × B → A and ρ : A × B → B, is called

75

CATEGORIES OF RELATIONS

a relational product of A and B iﬀ the following holds: π ; π IA ,

ρ ; ρ IB ,

π ; ρ = AB ,

π; π ρ; ρ = IA×B .

R has relational products iﬀ for every pair of objects a relational product does exist. Again, this construction is unique up to isomorphism. Furthermore, L-Rel has relational products, which are given by the Cartesian products of sets and the corresponding set-theoretic projections. As above, we omit the proofs. There is also an abstract notion of singleton sets, called a unit. II and AI is total for Deﬁnition 4.29 An object I is called a unit iﬀ II = all objects A. Again, units are unique up to isomorphism. In L-Rel every singleton set is a unit. A subset M of a set N may be described by the canonical injection f : M → N . Furthermore, the set of equivalence classes of an equivalence relation is fully determined by the function mapping each element to its equivalence class. Combining both concept we aim at the notion of a splitting. Deﬁnition 4.30 Let Q : A → A be a symmetric idempotent relation, i.e., Q = Q and Q; Q = Q. An object B together with a relation R : B → A is called a splitting of Q (or R splits Q) iﬀ R; R = IB and R ; R = Q. As before, a splitting is unique up to isomorphism. If Q is a partial identity, the object B of the splitting corresponds to the subset given by Q. Analogously, if Q is an equivalence relation, B corresponds to the set of equivalence classes. This shows that in L-Rel every symmetric idempotent splits. We have seen that L-Rel oﬀers all relational constructions introduced in this chapter. Furthermore, the corresponding relations, i.e., the projections, injections, and splittings of crisp relations, are crisp. Later on, we will see that there may also be noncrisp versions of these constructions. This causes a problem within the application of the theory since it is usually supposed that they are crisp. In Section 5.9 we will focus on that problem. 4.7

THE DEDEKIND CATEGORY OF ANTIMORPHISMS anti

In Theorem 2.47 we have shown that the collection of antimorphisms L1 → L2 is a complete Brouwerian lattice if L1 and L2 are. Now, we want to extend this theorem to Dedekind categories. Throughout this chapter let R be a Dedekind category and L a complete Brouwerian lattice such that RL ≥ is again a Dedekind category. ≥

≥

Lemma 4.31 For all antitone functions f, fi : L → R[A, B], g, gi : L → ≥ R[B, C] for all i ∈ i and h : L → R[C, D] we have (1) τ (f ); g τ (f ; g) and f ; τ (g) τ (f ; g),

76

GOGUEN CATEGORIES

(2) τ (τ (f ); g) = τ (f ; g) and τ (f ; τ (g)) = τ (f ; g), (3) τ (τ ( fi ); g) = τ ( τ (fi ; g)) and τ (f ; τ ( gi )) = τ ( τ (f ; gi )). i∈I

i∈I

i∈I

i∈I

Proof. In all cases we just prove the ﬁrst assertion. (1) We prove the assertion using ﬁxed point induction. Therefore, we deﬁne the following predicate: : ⇐⇒

P(k, l)

k; g l.

This predicate is admissible since P(k i , li ), which is equivalent toki ; g li for all i ∈ I, implies (ki ; g) li , which is equivalent to ( ki ); g i∈I i∈I i∈I li and P( ki , li ). i∈I

i∈I

i∈I

The base case P(f, f ; g) is trivial. Now, suppose P(k, l). Then we have (ϕ(k); g)(x) = ϕ(k)(x); g(x) ⎞ ⎛ ⎟ ⎜ k(y)⎠ ; g(x) =⎝ y∈M M =x

=

deﬁnition of ϕ

k(y) ; g(x)

y∈M

M =x

deﬁnition of ;

(k(y); g(x))

M =x y∈M

(k(y); g(y))

k and g antitone

(k; g)(y)

deﬁnition of ;

l(y)

induction hypothesis

M =x y∈M

=

M =x y∈M

M =x y∈M

= ϕ(l)(x).

deﬁnition of ϕ

The principle of ﬁxed point induction gives us P(µϕ (f ), µϕ (f ; g)), and, hence, τ (f ); g τ (f ; g). (2) Using (1) we conclude that τ (τ (f ); g) τ 2 (f ; g) = τ (f ; g) τ (τ (f ); g). (3) Since τ (fi ; g) τ (τ ( fi ); g) for all i ∈ I we get i∈I

τ

i∈I

τ (fi ; g)

τ (τ 2

i∈I

fi ; g) = τ (τ

i∈I

fi

; g).

CATEGORIES OF RELATIONS

The other inclusion follows from fi ; g = τ fi ; g τ τ i∈I

=τ τ

i∈I

(2)

(fi ; g)

i∈I

77

; deﬁned componentwise

τ (fi ; g) .

i∈I

In the next lemma we have summarized some properties of the converse operation in RL ≥. anti

anti

Lemma 4.32 For all f : L → R[A, B] and g : L → R[B, C] we have (1) f is an antimorphism from L to R[B, A],

(2) τ (f ; g) = τ (g ; f ). Proof. (1) The assertion follows from f M = f M =

f (x)

deﬁnition of

f antimorphism

x∈M

=

(f (x)) x∈M

f (x).

=

deﬁnition of

x∈M

(2) The assertion is shown by ﬁxed point induction. Deﬁne the following predicate: P(h, k) : ⇐⇒ h = k. = ki This predicate is admissible since P(hi , ki ), which is equivalent to h i for all i ∈ I, implies hi = ki , and, hence, ( hi ) = ki and i∈I i∈I i∈I i∈I ki ). P( hi , i∈I

i∈I

The base case P(f ; g, g ; f ) is trivial. Now, suppose P(h, k). Then we have

ϕ(h) (x) = ϕ(h)(x) ⎛ ⎞ ⎜ ⎟ =⎝ h(y)⎠ y∈M M =x

=

M =x y∈M

(h(y))

deﬁnition of ϕ

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GOGUEN CATEGORIES

=

h (y)

deﬁnition of

k(y)

induction hypothesis

M =x y∈M

=

M =x y∈M

= ϕ(k)(x).

deﬁnition of ϕ

From the principle of ﬁxed point induction we get P(µϕ (f ; g), µϕ (g ; f )), and, hence, τ (f ; g) = τ (g ; f ). Now, we are able to prove the main theorem of this section. Theorem 4.33 Let R be a Dedekind category and L a complete Brouwerian lattice. Then the structure RL deﬁned by (1) the objects of RL are the objects of R, anti

(2) RL [A, B] is the complete Brouwerian lattice L → R[A, B] of antimorphisms, (3) the converse operation is deﬁned componentwise, (4) composition is deﬁned by f ··, g := τ (f ; g) with identity I˙ is again a Dedekind category. Proof. First, we want to show that RL is a category. Associativity follows from f ··, (g ··, h) = τ (f ; τ (g; h)) = τ (f ; (g; h)) = τ ((f ; g); h) = τ (τ (f ; g); h) = (f ··, g) ··, h.

deﬁnition of ··, Lemma 4.31 (2) ; deﬁned componentwise Lemma 4.31 (2) deﬁnition of ··,

Furthermore, we have f ··, I˙ B = τ (f ; τ (IB )) = τ (f ; IB ) = τ (f ) = f by Lemma 4.31 (2). I˙ A ··, f = f follows analogously. By Theorem 2.47 every RL [A, B] is a complete Brouwerian lattice, and Lemma 4.32 shows the properties of the converse operation. The modular law follows from (f ··, g) h = τ (f ; g) h = τ (f ; g) τ (h) = τ (f ; g h) τ (f ; (g f ; h))

deﬁnition of ··, h is an antimorphism Lemma 2.46 (2) ; and deﬁned componentwise

τ (f ; (g τ (f ; h)))

τ closure operation

= f ··, (g f ··, h).

deﬁnition of ··,

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79

Since ··, is a complete upper semilattice homomorphism by Lemma 4.31 (3) the residuals may be deﬁned using Theorem 2.19. Finally, from Theorem 4.18 we conclude the assertion. 4.8

SCALARS AND CRISPNESS IN DEDEKIND CATEGORIES

In some sense a relation of a Dedekind category may be seen as an L-fuzzy relation. The lattice L may equivalently be characterized by the ideal relations, AA ; JAB ; BB = JAB , or by the scalar i.e., a relation JAB : A → B satisfying relations. Deﬁnition 4.34 A relation αA : A → A is called a scalar on A iﬀ αA IA and AA ; αA = αA ; AA . The set of all scalars on A is denoted by ScR (A). The notion of ideals was introduced by J´ onsson and Tarski [17] and the notion of scalars by Furusawa and Kawahara [20]. Lemma 4.35 ScR (A) is a complete Brouwerian lattice. The pseudo-complement of αA in ScR (A) is denoted by ¬:αA . Proof. We have to show that ScR (A) is closed under arbitrary meet and union. This follows immediately from Lemma 4.24 (3). Consequently, we call a scalar linear iﬀ it is linear in ScR (A). As mentioned ⊥AA . Notice that IA = ⊥ ⊥AA implies in Section 2.3 this is equivalent to ¬:αA = ⊥ that IA is linear since IA is the greatest element in ScR (A). Lemma 4.36 Let Q : A → B be a relation. Then we have (1) AA ; Q; BA is an ideal element, AA ; Q; BA is a scalar on A, (2) IA (3) Q = ⊥ ⊥AB iﬀ the relation from (1) equals zero iﬀ the relation from (2) equals zero. Proof. CA ; Q; BD ); DD = (1) Using Lemma 4.21 (2) we immediately get CC ; ( BD . CA ; Q; (2) First of all, this relation is a partial identity. The following computation shows that it is indeed a scalar: AA ; Q; BA ) AA ; (IA = AA AA ; Q; BA AA ; Q; BA ); AA . = (IA

Lemma 4.23 (1) Lemma 4.23 (1)

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GOGUEN CATEGORIES

(3) Obviously, Q = ⊥ ⊥AB implies AA ; Q; BA = ⊥ ⊥AB and the latter implies AA ; Q; BA = ⊥ ⊥AA . Now, suppose IA AA ; Q; BA = ⊥ ⊥AA . Then IA we have BB Q AA ; Q; = AB AA ; Q; BB = (IA AA ; Q; BA ); AB AB =⊥ ⊥AA ; =⊥ ⊥AB ,

Lemma 4.23 (1)

which completes the proof.

The next lemma shows that we may use ideal elements instead of scalars. We have chosen scalars since they provide a nice algebraic term for cuts (cf. Lemma 3.3 (5)). AA constitute Lemma 4.37 The mappings φ(J) := IA J and φ−1 (α) := α; a bijection between the set of ideal elements on A and ScR (A). Proof. The assertion follows immediately from φ−1 (φ(J)) = (IA J); AA = (IA J; AA ); AA AA ) = ( AA J; = AA J

J ideal element Lemma 4.23 (1) J ideal element

= J, φ(φ

−1

(α)) = IA α; AA = IA ; α = α.

Lemma 4.24 (2)

Within L-fuzzy relations the sets ScL-Rel (A) and ScL-Rel (B) are isomorphic for arbitrary sets A and B. Unfortunately, that is not true for all Dedekind categories. Consider the following computation: BA ; αA ; AB IB ); BA IA AB ; ( = AA ; αA ; AA AB ; BA IA = αA AB ; BA .

Lemma 4.23 (1) Lemma 4.24 (1)

For αA = IA this shows that the sets of scalars are isomorphic iﬀ R is uniform. Consequently, a suitable abstract categorical description of L-fuzzy relations should be a uniform Dedekind category. Using the scalars, there are two notions of crispness in an arbitrary Dedekind category.

CATEGORIES OF RELATIONS

81

Deﬁnition 4.38 A relation R : A → B is called l-crisp/s-crisp iﬀ αA ; Q R implies Q R for all linear/nonzero scalars αA and all relations Q : A → B. Obviously, AB is s-crisp as well as l-crisp. Furthermore, we have the following lemma: Lemma 4.39 Let Ri : A → B for i ∈ I be (l-crisp) s-crisp relations. Then Ri is (l-crisp) s-crisp. i∈I

Proof. Suppose αA ; Q

Ri for a relation Q : A → B and a (linear) i∈I

nonzero scalar αA . Then we obtain αA ; Q Ri , and, hence, Q Ri for all Ri . i ∈ I, which implies Q i∈I

Again, this gives us the possibility to deﬁne closure operations mapping every relation R : A → B to the least s-crisp resp. l-crisp relation it is contained Rs := l

R :=

{Q : A → B | R Q and Q is s-crisp}, {Q : A → B | R Q and Q is l-crisp}.

Obviously, s-crispness implies l-crispness, and, hence, Rl Rs . As for τ there is another possibility to characterize Rs resp. Rl . Consider the functions (αA \R), Φs (R) := αA ∈ScG (A) ⊥AA αA =⊥

Φl (R) :=

(αA \R).

αA ∈ScG (A) αA linear

Notice that the unions on the right-hand side of the deﬁnitions above are ⊥AA since the IA is linear. nonempty if IA = ⊥ Lemma 4.40 (1) Φs and Φl are monotone, (2) R Φs (R) and R Φl (R) for all R : A → B, Proof. (1) The assertion follows immediately from the fact that \ is monotone in the second argument by Corollary 4.15 (4). (2) If there is no nonzero scalar, we conclude ⊥ ⊥AA = IA , which implies Q = ⊥AB = R = Φs (R) = ⊥ ⊥AB for all relations Q : A → B, and, hence, ⊥ ⊥AA and consider the following Φl (R) = Rs = Rl . Now, suppose IA = ⊥ computation: Lemma 4.16 (1) R = IA \R Corollary 4.15 (4) αA \R, which implies the assertion.

82

GOGUEN CATEGORIES

The previous lemma and Theorem 2.36 shows that for every R there exists a least ﬁxed point µΦs (R) of Φs and a least ﬁxed point µΦl (R) of Φl greater than R. Theorem 4.41 Let R be a Dedekind category. Then we have Rs = µΦs (R) and Rl = µΦl (R) for all relations R. Proof. : It is suﬃcient to show that µΦs (R) is s-crisp and µΦl (R) is l-crisp. Suppose there is a nonzero resp. linear scalar αA and a relation Q such that αA ; Q µΦs (R) resp. αA ; Q µΦl (R). Then we have Q αA \µΦs (R) (αA \µΦs (R))

deﬁnition \

αA =⊥ ⊥AA

= Φs (µΦs (R)) = µΦs (R)

deﬁnition Φs ﬁxed point property

and Q µΦl (R) analogously. : It is suﬃcient to show that Φs (Rs ) Rs resp. Φl (Rl ) Rl since then Rs resp. Rl is a ﬁxed point of Φs resp. Φl by Lemma 4.40 (2). For every nonzero/linear scalar αA we have αA \Rs Rs and αA \Rl Rl , respecs-crisp resp. αA ; (αA \Rl ) Rl and tively, since αA ; (αA \Rs ) Rs and Rs is l s (αA \Rs ) Rs and Φl (Rl ) = R is l-crisp. This implies Φs (R ) = αA =⊥ ⊥AA (αA \Rl ) Rl . αA linear

First, we want to concentrate on s-crispness. Unfortunately, ⊥ ⊥AB may be not s-crisp. Dedekind categories in which ⊥ ⊥AB is s-crisp are characterized by the following lemma: Lemma 4.42 In a Dedekind category R the following statements are equivalent: (1) All nonzero scalars are linear. (2) ⊥ ⊥AB is s-crisp. (3) For every R : A → B, its pseudo-complement ¬R is s-crisp. ⊥AB for a relation Q : A → B and a Proof. (1)⇒(2): Suppose αA ; Q ⊥ nonzero scalar αA . Then we get AA ; Q; BA IA ) = αA ; ( AA ; Q; BA IA ) αA ( αA ; AA ; Q; BA = AA ; αA ; Q; BA AA ; ⊥ ⊥AB ; BA =⊥ ⊥AA .

Lemma 4.24 (2) αA scalar assumption

CATEGORIES OF RELATIONS

83

BA IA is a scalar by Lemma 4.36 (2) and αA is linear by the Since AA ; Q; BA IA ¬:αA = ⊥ ⊥AA . Lemma 4.36 assumption we conclude that AA ; Q; (3) shows Q = ⊥ ⊥AB . (2)⇒(3): Suppose αA ; Q ¬R for a relation Q : A → B and a nonzero scalar ⊥AB , and, hence, αA . Then we obtain αA ; Q R ⊥ αA ; (Q R) = αA ; Q αA ; R αA ; Q R

Lemma 4.9 (4) αA IA

⊥ ⊥AB . Since ⊥ ⊥AB is s-crisp we have Q R ⊥ ⊥AB , and, hence, Q ¬R. AB = ⊥ ⊥AB . Now, suppose (3)⇒(1): First of all, ⊥ ⊥AA is s-crisp by (3) since ¬ ⊥AA and βA is a scalar such that αA βA = ⊥ ⊥AA . This implies αA ; βA = αA = ⊥ ⊥AA by Lemma 4.9 (3), and, hence, βA ⊥ ⊥AA since ⊥ ⊥AA is s-crisp. αA βA ⊥ The next lemma shows that in the situation not described by the previous lemma the notion of s-crispness is trivial. Lemma 4.43 Let A be an object of a Dedekind category. If there is a nonzero and nonlinear scalar αA on A, then there is no s-crisp relation R : A → B except AB for all objects B. Proof. Suppose R : A → B is s-crisp and αA , βA are nonzero scalars such ⊥AA holds. Then αA ; (βA ; AB ) = (αA βA ); AB = ⊥ ⊥AB R that αA βA = ⊥ AB R and further AB R since by Lemma 4.9 (3). We conclude that βA ; R is s-crisp. Considering L-fuzzy relations we have the following connection between 0–1 and s-crispness: Lemma 4.44 In L-Rel all s-crisp relations are 0–1 crisp. Proof. Suppose R : A → B is s-crisp and assume that R(x0 , y0 ) = u = 0 u : A → A on A induced by for a pair (x0 , y0 ) ∈ A × B. Consider the scalar αA u and the relation Q : A → B deﬁned by Q(x, y) := R(x, y):u. Then we have u u ; Q)(x, y) = (αA ; AB Q)(x, y) = u ∧ (R(x, y):u) ≤ R(x, y). Since R is (αA s-crisp we get Q R, and, hence, 1 = u:u = R(x0 , y0 ):u = Q(x0 , y0 ) ≤ R(x0 , y0 ), which shows that R is 0–1 crisp. The converse is, in general, not true. A property of the underlying lattice L equivalent for L-fuzzy relations to Lemma 4.42 (1) is required. Lemma 4.45 In L-Rel the following properties are equivalent: (1) All 0–1 crisp relations are s-crisp. (2) All nonzero elements of L are linear.

84

GOGUEN CATEGORIES

Proof. (1)⇒(2): Let be 0 = u ∈ L and v ∈ L with u ∧ v = 0. Then the scalar u u v u v on A induced by u is nonzero and we have αA ; αA = αA αA =⊥ ⊥AA . Since αA v ⊥AA , and, hence, ⊥ ⊥AA is s-crisp by the assumption we conclude that αA = ⊥ v = 0. u ; Q R for 0 = u ∈ L, Q : A → B is a relation and (2)⇒(1): Suppose that αA R : A → B is a 0–1 crisp relation. To prove that R is s-crisp it is suﬃcient to show that R(x, y) = 0 implies Q(x, y) = 0 since R is 0–1 crisp. Suppose u ; Q)(x, y) = u ∧ Q(x, y). The R(x, y) = 0. Then we have 0 = R(x, y) (αA assumption gives us Q(x, y) = 0 since u is nonzero and linear. We have seen that the notion of s-crispness coincides with 0–1 crispness under an assumption on L. In the rest of this chapter, we want to study the notion of l-crispness. Considering L-fuzzy relations we have the following connection between l-crispness and 0–1 crispness: Lemma 4.46 In L-Rel all 0–1 crisp relations are l-crisp. u u ; Q R for a linear scalar αA , Q : A → B is Proof. Suppose that αA a relation and R : A → B is a 0–1 crisp relation. Then u is also linear. In order to prove that R is l-crisp it is suﬃcient to show that R(x, y) = 0 implies Q(x, y) = 0 since R is 0–1 crisp. Suppose R(x, y) = 0. Then we have 0 = u ; Q)(x, y) = u ∧ Q(x, y). This implies Q(x, y) = 0 since u is R(x, y) (αA linear.

Notice that the previous lemma shows that ⊥ ⊥AB is l-crisp. The converse of the previous lemma is, in general, not true. The same property as in Lemma 4.45 is needed. Lemma 4.47 In L-Rel following properties are equivalent: (1) All l-crisp relations are 0–1 crisp. (2) All nonzero elements of L are linear. Proof. (1)⇒(2): Suppose there is a nonlinear element u ∈ L, i.e., ¬u = 0. We have to show that there is an l-crisp relation R, which is not 0–1 crisp. v ; Q R for linear scalar Deﬁne R : A → B by R(x, y) := ¬u and suppose αA v αA and a relation Q : A → B. First of all, v is linear since ScL-Rel (A) and L are isomorphic as complete Brouwerian lattices. Then we conclude v ∧ Q(x, y) = v ; Q)(x, y) ≤ R(x, y) = ¬u. This implies v ∧ (Q(x, y) ∧ u) = 0, and, hence, (αA Q(x, y) ∧ u = 0 since v is linear. The previous property gives us Q(x, y) ≤ ¬u = R(x, y), which shows that R is l-crisp. But R is not 0–1 crisp since ¬u = 0 by the assumption and ¬u = 1 since otherwise we have 0 = u ∧ ¬u = u ∧ 1 = u, a contradiction to u = 0. (2)⇒(1): If all nonzero elements of L are linear, then all nonzero scalars are linear, and, hence, the notions of s-crispness and l-crispness coincide. By Lemma 4.44 we conclude (1).

CATEGORIES OF RELATIONS

85

The previous lemma shows that the notion of l-crispness as well as the notion of s-crispness coincides with 0-1 crispness just in the case that all nonzero elements of L are linear. Furthermore, if L is a Boolean algebra, no scalar is linear, and, therefore, the notion of s-crispness by Lemma 4.43 as well as the notion of l-crispness is trivial. 4.9

¨ SCHRODER CATEGORIES

Last but not least, we want to switch from complete Brouwerian lattices to complete Boolean algebras as the underlying lattice structure of relations. Deﬁnition 4.48 A Schr¨ oder category is a Dedekind category in which every R[A, B] is a (complete) Boolean algebra. The Dedekind category L-Rel is, in general, not a Schr¨ oder category. It is iﬀ the lattice L is a Boolean algebra. On the other hand, the substructure of crisp relations constitutes a Schr¨ oder category since 0 = 1 and 1 = 0 is valid in any Brouwerian lattice. The next theorem states a version of the so-called Schr¨oder equivalences. Notice that these equivalences and the modular law are equivalent in the presence of the other axioms of a Schr¨ oder category. Theorem 4.49 (Schr¨ oder equivalences) Let R be a Schr¨ oder category, Q : A → B, R : B → C, and S : A → C. Then we have Q; R S ⇐⇒ Q ; S R ⇐⇒ S; R Q Proof. We just show “⇒” of the ﬁrst equivalence. All other implication ⊥AC . follow similarly. Assume Q; R S, which is equivalent to Q; R S = ⊥ Then we conclude Q ; S R Q ; (S Q; R)

⊥AC = Q ;⊥ =⊥ ⊥BC , and, hence, Q ; S R.

modular law see above

As an application of the previous theorem we want to show that the residuals are given by expression involving complements. Lemma 4.50 Let R be a Schr¨ oder category, Q : A → B, R : B → C, and S : A → C. Then we have (1) Q\S = Q ; S, (2) S/R = S; R .

86

GOGUEN CATEGORIES

Proof. The ﬁrst assertion follows immediately from X Q\S ⇔ Q; X S ⇔ Q ; S R ⇔X

residual Theorem 4.49

Q ; S.

The second is shown analogously. 4.10

FORMAL LANGUAGES OF RELATIONAL CATEGORIES

Later on, we will compare the validity of several formulae in the relational categories introduced so far. Therefore, we have to deﬁne formal languages over those categories. We require a set of object variables and a set of typed relation variables, i.e., every relation variable is of the form r : a → b where a and b are object variables. Now, general terms and formulae are deﬁned as follows: Deﬁnition 4.51 The set of terms of type a → b and the set of formulae are deﬁned inductively as follows: (1) Every relation variable r : a → b is a term of type a → b. (2) If a is an object variable, then Ia is a term of type a → a. (3) If t is a term of type a → b, then t is a term of type b → a. (4) If t1 and t2 are terms of type a → b, then t1 t2 is a term of type a → b. (5) If t1 and t2 are terms of type a → b resp. b → c, then t1 ; t2 is a term of type a → c. (6) If a and b are object variables, then ⊥ ⊥ab is a term of type a → b. (7) If t1 and t2 are terms of type a → b, then t1 t2 is a term of type a → b. (8) If a and b are object variables, then ab is a term of type a → b. (9) If t1 and t2 are terms of type a → c resp. b → c, then t1 /t2 is a term of type a → b. (10) If Θ is a formula and r : a → b a relation variable, then {r : a → b | Θ} and {r : a → b | Θ} are terms of type a → b. (11) If t1 and t2 are terms of type a → b, then t1 = t2 is a formula. (12) If Θ1 and Θ2 are formulas, then Θ1 ∧ Θ2 is a formula. (13) If Θ is a formula, then ¬Θ is a formula. (14) If Θ is a formula and r : a → b is a relation variable, then (∀r : a → b)Θ is a formula. (15) If Θ is a formula and a is an object variable, then (∀a)Θ is a formula.

CATEGORIES OF RELATIONS

87

The terms in the language of allegories are given by those terms that are built up by (1)–(5). Analogously, the terms of the language of distributive allegories are given by (1)–(7), of distributive allegories with greatest elements by (1)–(8), of division allegories by (1)–(9), and of Dedekind categories by (1)–(10). Notice that within the language of Dedekind categories (4)–(8) may be dropped since they are special cases of (10) in the following way: ˆ t 1 t2 =

{r : a → b | r = t1 ∨ r = t2 }, ˆ {r : a → b | r = t1 ∨ r = t2 }, t1 t2 = {r : a → b | r = r}, ˆ {r : a → b | r = r}, ab = t1 /t2 = ˆ {r : a → b | t2 ; r t1 = t1 }. ˆ ⊥ ⊥ab =

A formula in the language of a speciﬁc relational category is a formula such that all terms are terms in the language of this kind of a relational category. Finally, a set of equations is a set of formulae of the form (11). In the rest of this chapter we will prove several properties of terms and formulae within speciﬁc classes of relational categories. This is usually done by structural induction. In all cases, we will be careful that we just use properties of the most general kind of relational categories, which may be concerned. For example, if we prove a property of terms t by structural induction and we are considering the case t = t1 t2 , we will just use theorems valid in all allegories. This implies that the corresponding theorem is valid for all relational categories and their corresponding languages. Given a relational category R, an environment σ over R is a function mapping each object variable a to an object A of R and each relation variable r : a → b to a relation R : σ(a) → σ(b). The update σ[A/a] resp. σ[R/r : a → b] of σ at the object variable a resp. at the relation variable r : a → b with the object A resp. with the relation R : σ(a) → σ(b) is deﬁned by

σ[A/a](b) :=

σ(b) iﬀ a = b, A iﬀ a = b,

⎧ iﬀ a = c, ⊥Aσ(d) ⎨ ⊥ iﬀ a = d, ⊥ ⊥σ(c)A σ[A/a](r : c → d) := ⎩ σ(r : c → d) iﬀ a = c ∧ a = d, σ[R/r : a → b](c) := σ(c),

σ(s : c → d) iﬀ r : a → b = s : c → d, σ[R/r : a → b](s : c → d) := R iﬀ r : a → b = s : c → d. Notice that the update of an environment is again an environment. As usual we denote a sequence of updates σ[A/a][B/b][R/r : a → b] of an environment σ by σ[A/a, B/b, R/r : a → b].

88

GOGUEN CATEGORIES

The value of a term and the validity of a formula are deﬁned as usual. In order to avoid confusion we give here the formal deﬁnition. Deﬁnition 4.52 The value VR (t)(σ) of a term t of type a → b and the validity R |=σ Θ of a formula Θ in a suitable relational category R under an environment σ is deﬁned inductively as follows: (1) VR (r : a → b)(σ) := σ(r : a → b), (2) VR (Ia )(σ) := Iσ(a) ,

(3) VR (t )(σ) := (VR (t)(σ)) , (4) VR (t1 t2 )(σ) := VR (t1 ) VR (t2 ), (5) VR (t1 ; t2 )(σ) := VR (t1 )(σ); VR (t2 )(σ), ⊥ab )(σ) := ⊥ ⊥σ(a)σ(b) , (6) VR (⊥ (7) VR (t1 t2 )(σ) := VR (t1 ) VR (t2 ), ab )(σ) := σ(a)σ(b) , (8) VR ( (9) VR (t1 /t2 )(σ) := VR (t1 )(σ)/VR (t2 )(σ), (10) VR ( {r : a → b | ϕ})(σ) := {R | R |=σ[R/r:a→b] ϕ}, (11) VR ( {r : a → b | ϕ})(σ) :=

{R | R |=σ[R/r:a→b] ϕ},

(12) R |=σ t1 = t2 iﬀ VR (t1 )(σ) = VR (t2 )(σ), (13) R |=σ Θ1 ∧ Θ2 iﬀ R |=σ Θ1 and R |=σ Θ2 , (14) R |=σ ¬Θ iﬀ R |=σ Θ, (15) R |=σ (∀r : a → b)Θ iﬀ R |=σ[R/r:a→b] Θ for all R : σ(a) → σ(b), (16) R |=σ (∀a)Θ iﬀ R |=σ[A/a] Θ for all objects A. The set of relational variables RV(t) of a term t is deﬁned as the set of all relational variables occurring in t. Furthermore, the set OV(t) of object variables of t is deﬁned as the set of object variables a such that there is a relational variable in RV(t) typed with a, i.e., there is a variable in RV(t) of the form r : a → b or r : c → a. Obviously, the occurrences of an object variable within a term are completely determined by the occurrences of the relational variables. An occurrence of a relational variable r : a → b in a formula Θ is called bounded iﬀ it is within a subformula of Θ of the form (∀r : a → b)Θ and free otherwise. With RV(Θ) we denote the free relational variables of Θ, i.e., the relational variables, which have at least one free occurrence in Θ. Analogously, an occurrence of an object variable a in Θ is called bounded iﬀ it is within a subformula of the form (∀a)Θ and free otherwise. Again, OV(Θ) we denote the

CATEGORIES OF RELATIONS

89

set of free object variables of Θ, i.e., the object variables, which have at least one free occurrence in Θ. We will use the notation RV(S) resp. OV(S) also for a set of equations object within the S = {ti1 = ti2 | i ∈ I} denoting the relational variables resp. RV(ti1 ) ∪ RV(ti2 ) resp. OV(S) = terms of S. Formally, we have RV(S) = i∈I i∈I OV(ti1 ) ∪ OV(ti2 ). i∈I

i∈I

Theorem 4.53 Let t be a term and Θ a formula in the language of Dedekind categories. Furthermore, let R be a Dedekind category and σ1 and σ2 environments over R. Then we have the following: (1) if σ1 (r : a → b) = σ2 (r : a → b) for all r : a → b ∈ RV(t) and σ1 (a) = σ2 (a) for all a ∈ OV(t), then VR (t)(σ1 ) = VR (t)(σ2 ), (2) if σ1 (r : a → b) = σ2 (r : a → b) for all r : a → b ∈ RV(Θ) and σ1 (a) = σ2 (a) for all a ∈ OV(Θ), then R |=σ1 Θ iﬀ R |=σ2 Θ. The proof of the previous theorem is a straightforward structural induction and, therefore, omitted. As mentioned above, the previous lemma is also valid for allegories, distributive and division allegories. Given a functor F : R1 → R2 and an environment σ we denote with F (σ) the environment deﬁned by (1) F (σ)(a) = F (σ(a)) for all object variables a, (2) F (σ)(r : a → b) := F (σ(r : a → b)) for all relation variables r : a → b. Notice that F (σ) is indeed an environment. Since σ is an environment we have σ(r : a → b) : σ(a) → σ(b). This implies F (σ(r : a → b)) : F (σ(a)) → F (σ(b)) since F is a functor, and, hence, F (σ)(r : a → b) : F (σ)(a) → F (σ)(b). Consider the following computation with r : a → b = s : c → d: F (σ[R/r : a → b])(r : a → b) = F (σ[R/r : a → b](r : a → b)) = F (R) and

= F (σ)[F (R)/r : a → b](r : a → b), F (σ[R/r : a → b])(s : c → d) = F (σ[R/r : a → b](s : c → d)) = F (σ(s : c → d)) = F (σ)[F (R)/r : a → b](s : c → d),

which shows that F (σ[R/r : a → b]) = F (σ)[F (R)/r : a → b]. Analogously, we obtain F (σ[A/a]) = F (σ)[F (A)/a]. Furthermore, if F is an isomorphism, the function σ → F (σ) is a bijection on the class of environments. Lemma 4.54 Let t, t1 , t2 be a term of type a → b and Θ a formula in the language of division allegories. Furthermore, let R1 and R2 be division allegories,

90

GOGUEN CATEGORIES

σ an environment over R1 and F : R1 → R2 a homomorphism of division allegories. Then the following holds: (1) F (VR1 (t)(σ)) = VR2 (t)(F (σ)), (2) R1 |=σ t1 = t2 iﬀ R2 |=F (σ) t1 = t2 . Proof. (1) The assertion is shown by structural induction as follows: F (VR1 (r : a → b)(σ)) = F (σ(r : a → b)) = F (σ)(r : a → b) = VR2 (r : a → b)F (σ), F (VR1 (Ia )(σ)) = F (Iσ(a) )

def. of value

F (VR1 (t )(σ)) = F ((VR1 (t)(σ)) ) = F (VR1 (t)(σ))

def. of value def. of value F functor def. of F (σ)

= IF (σ(a)) = IF (σ)(a) = VR2 (Ia )F (σ),

def. of value def. of F (σ)

def. of value F homo.

= (VR2 (t)(F (σ)))

= VR2 (t )(F (σ)), F (VR1 (t1 t2 )(σ)) = F (VR1 (t1 )(σ) VR1 (t2 )(σ)) = F (VR1 (t1 )(σ)) F (VR1 (t2 )(σ)) = VR2 (t1 )(F (σ)) VR2 (t2 )(F (σ)) = VR2 (t1 t2 )(F (σ)) F (VR1 (t1 ; t2 )(σ)) = F (VR1 (t1 )(σ); VR1 (t2 )(σ)) = F (VR1 (t1 )(σ)); F (VR1 (t2 )(σ)) = VR2 (t1 )(F (σ)); VR2 (t2 )(F (σ)) = VR2 (t1 ; t2 )(F (σ)) F (VR1 (⊥ ⊥ab )(σ)) = F (⊥ ⊥σ(a)σ(b) ) =⊥ ⊥F (σ(a))F (σ(b)) =⊥ ⊥F (σ)(a)F (σ)(b) ⊥ab )(F (σ)), = VR2 (⊥ F (VR1 (t1 t2 )(σ)) = F (VR1 (t1 )(σ) VR1 (t2 )(σ)) = F (VR1 (t1 )(σ)) F (VR1 (t2 )(σ)) = VR2 (t1 )(F (σ)) VR2 (t2 )(F (σ)) = VR2 (t1 t2 )(F (σ))

ind. hyp. def. of value def. of value F homo. ind. hyp. def. of value def. of value F functor ind. hyp. def. of value def. of value F homo. def. of F (σ) def. of value def. of value F homo. ind. hyp. def. of value

CATEGORIES OF RELATIONS

F (VR1 (t1 /t2 )(σ)) = F (VR1 (t1 )(σ)/VR1 (t2 )(σ)) = F (VR1 (t1 )(σ))/F (VR1 (t2 )(σ)) = VR2 (t1 )(F (σ))/VR2 (t2 )(F (σ)) = VR2 (t1 /t2 )(F (σ))

91

def. of value F homo. ind. hyp. def. of value

(2) The assertion follows immediately from (1).

Again, the previous lemma is also valid for allegories and distributive and their formal languages. If F is an isomorphism, the previous lemma may be strengthened. Lemma 4.55 Let t be a term of type a → b and Θ a formula in the language of Dedekind categories. Furthermore, let R1 and R2 be Dedekind categories, σ an environment over R1 and F : R1 → R2 an isomorphism of Dedekind categories. Then the following holds: (1) F (VR1 (t)(σ)) = VR2 (t)(F (σ)), (2) R1 |=σ Θ iﬀ R2 |=F (σ) Θ. Proof. The assertions are shown simultaneously by structural induction. Using Lemma 4.54 (1) it remains to show that (1) holds for arbitrary unions and meets. This follows from the computation: F (VR1 ( {r : a → b | Θ})(σ)) def. of value =F {R | R1 |=σ[R/r:a→b] Θ} = {F (R) | R1 |=σ[R/r:a→b] Θ}) F homo. = {F (R) | R2 |=F (σ[R/r:a→b]) Θ}) ind. hyp. = {F (R) | R2 |=F (σ)[F (R)/r:a→b] Θ}) = {S | R2 |=F (σ)[S/r:a→b] Θ}) F iso. = VR2 {r : a → b | Θ} (σ) def. of value and a similar computation for meet. If Θ is an equality, (1) gives us the assertion. The remaining cases are shown as follows: R1 |=σ Θ1 ∧ Θ2 ⇐⇒ R1 |=σ Θ1 and R1 |=σ Θ2 ⇐⇒ R2 |=F (σ) Θ1 and R2 |=F (σ) Θ2 ⇐⇒ R2 |=F (σ) Θ1 ∧ Θ2 , R1 |=σ ¬Θ ⇐⇒ R1 |=σ Θ ⇐⇒ R2 |=F (σ) Θ ⇐⇒ R1 |=σ ¬Θ,

def. of validity ind. hyp. def. of validity def. of validity ind. hyp. def. of validity

92

GOGUEN CATEGORIES

R1 |=σ (∀r : a → b)Θ ⇐⇒ R1 |=σ[R/r:a→b] Θ for all R ⇐⇒ R2 |=F (σ[R/r:a→b]) Θ for all R ⇐⇒ R2 |=F (σ)[F (R)/r:a→b]) Θ for all R ⇐⇒ R2 |=F (σ)[S/r:a→b]) Θ for all S ⇐⇒ R2 |=F (σ) (∀r : a → b)Θ R1 |=σ (∀a)Θ ⇐⇒ R1 |=σ[A/a] Θ for all A ⇐⇒ R2 |=F (σ[A/a]) Θ for all A ⇐⇒ R2 |=F (σ)[F (A)/a]) Θ for all A ⇐⇒ R2 |=F (σ)[B/a]) Θ for all B ⇐⇒ R2 |=F (σ) (∀a)Θ.

def. of validity ind. hyp. F iso. def. of validity def. of validity ind. hyp. F iso. def. of validity

Notice that the last lemma is also valid for allegories, distributive and division allegories, and their formal languages.

5 CATEGORIES OF L-FUZZY RELATIONS

The notion of crispness is a basic property of L-fuzzy relations and sets such that a suitable algebraic theory should be able to express this property. We have shown that there are some notions of crispness within Dedekind categories, which grasp the notion of 0–1 crispness under an assumption on the underlying lattice. Unfortunately, a general notion, which coincides with 0–1 crispness has not yet been given. Theorem 5.1 There is no formula Θ in the language of Dedekind categories such that we have for all complete Brouwerian lattices L and L-fuzzy relations R:A→B L-Rel |=σ[A/a,B/b,R/r:a→b] Θ for all environments σ Proof. Consider the Boolean algebra of the set {a, b}. Let X = {x} and Y = relation f : Y → Y deﬁned by {a} f := {b}

R is 0–1 crisp.

B4 := P({a, b}), i.e., the power set {x, y} be sets. Consider the L-fuzzy {b} {a}

and the 0–1 crisp relation R : X → Y deﬁned by R := {a, b} ∅ . 93

⇐⇒

94

GOGUEN CATEGORIES

A simple veriﬁcation of the properties {a, b} f ◦f = f = f, ∅

∅ {a, b}

= IY .

shows that f is bijection in B4 -Rel[Y, Y ]. Obviously, the class consisting of f for Y and the identity on all other sets Z is a class of bijections. Now, take the isomorphism F induced by that class (cf. Lemma 4.26) and suppose such a formula Θ to exist. Then we have B4 -Rel |=σ[X/a,Y /b,R/r:a→b] Θ for all environments σ since R is 0–1 crisp. Lemma 4.55 implies that B4 -Rel |=F (σ[X/a,Y /b,R/r:a→b]) Θ, and, hence, B4 -Rel |=F (σ)[X/a,Y /b,F (R)/r:a→b]) Θ since F (X) = X and F (Y ) = Y . Since F is an isomorphism, we conclude B4 -Rel |=σ [X/a,Y /b,F (R)/r:a→b]) Θ for all environments σ , and, hence, F (R) = R; f is 0–1 crisp. But, this is a contradiction since {a} {b} F (R) = {a, b} ∅ ◦ = {a} {b} , {b} {a} which shows that F (R) is not 0–1 crisp.

The previous theorem shows that the theory of Dedekind categories is too weak to express crispness. This gives us the motivation to deﬁne extended algebraic structures for L-fuzziness, called arrow categories and Goguen categories. 5.1

ARROW CATEGORIES

In the previous theorem we have shown that we need an additional concept to deﬁne a suitable algebraic theory of L-fuzzy relations. Our approach introduces two operations mapping every relation to its support and kernel, respectively. In other terms, those operations map the relation to the greatest 0–1 crisp relation it contains and to the least 0–1 crisp relation it is included in. We now give an abstract deﬁnition. ⊥AB Deﬁnition 5.2 An arrow category A is a Dedekind category with AB = ⊥ for all objects A and B together with two operations ↑ and ↓ satisfying the following: (1) R↑ , R↓ : A → B for all R : A → B. (2) (↑ , ↓ ) is a Galois correspondence. ↑

(3) (R ; S ↓ ) = R↑ ; S ↓ for all R : B → A and S : B → C. ↑

(4) (Q R↓ ) = Q↑ R↓ for all Q, R : A → B. ↑ ⊥AA is a nonzero scalar, then αA = IA . (5) If αA = ⊥

Furthermore, an arrow category is called linear iﬀ all nonzero scalars are linear, and it is called Boolean iﬀ it is a Schr¨ oder category.

CATEGORIES OF L-FUZZY RELATIONS

95

A homomorphism F between arrow categories is a homomorphism between ↑ Dedekind categories preserving ↑ and ↓ , i.e, F (R↑ ) = F (R) and F (R↓ ) = ↓ F (R) for all R : A → B. Consequently, we call an arrow category A representable iﬀ there is an embedding of arrow categories into L-Rel for a suitable complete Brouwerian lattice L. The obvious deﬁnition of ↑ and ↓ introduced in Chapter 3 for L-fuzzy relations gives the standard model. Theorem 5.3 Let L be a complete Brouwerian lattice with 0 = 1. Then L-Rel together with ↑ and ↓ is an arrow category. ⊥AB for all nonempty sets since 0 = 1. Proof. First of all, we have AB = ⊥ Axiom (1) is trivial, and the axioms (2)–(5) were already shown in Lemma 3.3 (2)–(5). According to the standard model and Lemma 3.3 (1), we deﬁne crispness in an arbitrary arrow category as follows: Deﬁnition 5.4 A relation R : A → B of an arrow category A is called crisp iﬀ R↑ = R. The crisp fragment A↑ of A is deﬁned as the collection of all crisp relations of A. From Lemma 2.13 and Lemma 2.15 we get the following corollary: Corollary 5.5 Let A be an arrow category and R, Ri : A → B for i ∈ I. Then we have (1)

↑

and

↓

are monotone,

(2) R R↑↓ and R↓↑ R, (3) R↑ = R↑↓↑ and R↓ = R↓↑↓ , ↑ ↑ ↓ Ri and ( Ri ) = (4) ( Ri ) = i∈I

i∈I

i∈I

R↓ i . i∈I

In the next lemma we have summarized some basic properties of arrow categories. Lemma 5.6 Let A be an arrow category and Q, R : A → B, S : B → C, T : A → C. Then we have ⊥AA , (1) I↑A = IA = ⊥ (2) R↓↑ = R↓ , (3) R↑↓ = R↑ , (4)

↑

is a closure and

↓

a kernel operation,

(5) R = R↑ iﬀ R↓ = R↑ iﬀ R↓ = R,

96

GOGUEN CATEGORIES

(6) ⊥ ⊥↑AB = ⊥ ⊥AB and ↓ AB = AB ,

↑

(7) (R ; S ↑ ) = R↑ ; S ↑ , ↑

(8) R = R↑

↓

and R = R↓ ,

↑

↑

↑

↑

(9) (R; S ↓ ) = R↑ ; S ↓ and (R↓ ; S) = R↓ ; S ↑ , (10) (R; S ↑ ) = R↑ ; S ↑ and (R↑ ; S) = R↑ ; S ↑ , ↑

(11) (Q R↑ ) = Q↑ R↑ , (12) For all nonzero ideal relations J ↑ = AB holds, ↓

↓

↑

(13) R↓ :Q↑ = (R:Q↑ ) (R:Q) and (R:Q) R↑ :Q↓ , ↓

↓

↑

(14) Q↑ \T ↓ = (Q↑ \T ) (Q\T ) and (Q\T ) Q↓ \T ↑ , ↓

↓

↑

(15) T ↓ /S ↑ = (T /S ↑ ) (T /S) and (T /S) T ↑ /S ↓ . Proof. ⊥AA . Then we obtain AB = IA ; AB = ⊥ ⊥AA ; AB = (1) Suppose IA = ⊥ ⊥ ⊥AB , a contradiction. Axiom (5) implies I↑A = IA . (2) The following computation shows the assertion: ↑

↓ R↓↑ = (I A; R )

= I↑ A ; R↓ ↓

=R .

Axiom (3) (1)

(3) R↑ = R↑↓↑ = R↑↓ by (2). (4) First of all, we have R R↑↓ = R↑ and R↑ = R↑↓↑ = R↑↑ using (2) and (3). Since ↑ is monotone, it is a closure operation. The second assertion is shown analogously. (5) Suppose R↑ = R. Then we have R↑↓ = R↓ and by (3) R↑ = R↓ . Analogously, from R↓ = R we get R↓↑ = R↑ and by (2) R↓ = R↑ . Now, suppose R↓ = R↑ . Then we conclude from (4) R R↑ = R↓ R, and, hence, R↓ = R and R↑ = R. (6) The assertion follows immediately from (4) and (5). ↑

↑

(7) We immediately conclude that (R ; S ↑ ) = (R ; S ↑↓ ) = R↑ ; S ↑↓ = R↑ ; S ↑ .

CATEGORIES OF L-FUZZY RELATIONS ↑

↑

↑

97

(8) Using (7) we obtain R = (R ; IA ) = (R ; I↑A ) = R↑ ; I↑A = R↑ ; IA = R↑ . The other assertion follows from X R

↓

⇐⇒ X ↑ R ↑

⇐⇒ X

R

↑

⇐⇒ X

Galois correspondence

R

see above

⇐⇒ X R↓ ⇐⇒ X R

↓

Galois correspondence .

(9) We immediately conclude that ↑

↑

(R; S ↓ ) = (R ; S ↓ ) = R = R↑

↑

↑

; S↓

Axiom (3)

; S↓

(8)

↓

= R ;S . The second assertion follows from the ﬁrst one by using conversion twice and (8). (10) This is shown analogously to (9) using (7) and (8). ↑

↑

(11) We immediately conclude (Q R↑ ) = (Q R↑↓ ) = Q↑ R↑↓ = Q↑ R↑ using Axiom (4) and (3) of this lemma. (12) Suppose J = ⊥ ⊥AB , and let αA := IA J; BA so that J = αA ; AB by ⊥AA by Lemma 4.36 (3) holds. We obtain Lemma 4.37 and αA = ⊥ ↑

J ↑ = (αA ; AB ) =

see above

↑ ↑AB ) (αA ; ↑ αA ; ↑AB

= AB = IA ;

(10) Axiom (5) since αA = ⊥ ⊥AA

= AB . (13) First of all, we have X R↓ :Q↑ ⇔ Q↑ X R↓ ↑

⇔ (Q↑ X) R ↑

↑

⇔Q X R ↑

↑

⇔ X R:Q

↑ ↓

⇔ X (R:Q )

deﬁnition : Galois correspondence (11) deﬁnition : Galois correspondence

98

GOGUEN CATEGORIES ↓

so that R↓ :Q↑ = (R:Q↑ ) follows. From Corollary 2.17 (5) we immediately ↓ ↓ conclude (R:Q↑ ) (R:Q) . The last assertion follows from ↑

Q↓ (R:Q) = (Q↓ (R:Q))

↑

Axiom (4)

↑

(Q (R:Q)) R↑ . (14) First of all, we have X Q↑ \T ↓ ⇔ Q↑ ; X T ↓

deﬁnition \

↑

↑

⇔ (Q ; X) T ↑

Galois correspondence

↑

⇔ Q ;X T ↑

(10)

↑

⇔ X Q \T ↑

⇔ X (Q \T )

deﬁnition \ ↓

Galois correspondence

↓

so that Q↑ \T ↓ = (Q↑ \T ) follows. From Corollary 4.15 (5) we immedi↓ ↓ ately conclude (Q↑ \T ) (Q\T ) . The last assertion follows from ↑

Q↓ ; (Q\T ) = (Q↓ ; (Q\T )) (Q; (Q\T ))

↑

(9)

↑

T ↑.

(15) This is shown analogously to (14).

Notice that the previous lemma shows that R is crisp iﬀ R↓ = R or R↓ = R↑ . In the rest of this book we will use the previous lemma without mentioning. For Boolean arrow categories we are able to show additional properties. Lemma 5.7 Let A be a Boolean arrow category and Q, R : A → B. Then we have ↑

↓

(1) Q↓ = Q and Q↑ = Q , ↓

(2) Q↑ R↓ = (Q↑ R) , ↓

↑

(3) R↓ :Q↑ = (R↓ :Q) and (R:Q) = R↑ :Q↓ . Proof. ↓

↑

(1) Lemma 5.6 (13) gives us Q↑ Q and Q Q↓ . From Q↓ = Q↓ ↑

Q

↓

Q

= Q↓

see above see above

CATEGORIES OF L-FUZZY RELATIONS ↑

99

↑

we conclude Q↓ = Q , and, hence, Q↓ = Q . ↓

↓

For Q instead of Q we get Q = Q↑ , and, hence, Q = Q↑ . (2) The assertion follows immediately from Q↑ R↓ = Q↑ R↓ ↓

=Q R

A[A, B] Boolean algebra

↑

(1) ↑

↓

= (Q R) ↓

↓

=Q R ↓ ↓ = Q R ↓

= (Q↑ R) .

(1)

(1)

(3) In a Boolean arrow category we have S:T = T S. We compute R↓ :Q↑ = Q↑ R↓ ↓

= Q R↓ ↓ ↓

= (Q R )

see above (1) (2)

↓

↓

= (R :Q) , R↑ :Q↓ = Q↓ R↑ ↑

=Q R

↑

see above (1)

↑

= (Q R) ↑

= (R:Q) .

As shown in the previous lemma in a Boolean arrow category we have ↓ ↓ ↑ ↓ R↑ :Q↓ = (R:Q) . On the other hand, R↓ :Q↑ = (R↓ :Q) (R:Q↑ ) (R:Q) by Lemma 5.6 (13) and the previous lemma. The corresponding equality is not valid as the following example shows: Example 5.8 Let B4 = {0, a, b, 1} be the Boolean algebra with 4 elements and a = b. Consider the 1 × 1 − B4 matrices, i.e., the B4 -fuzzy relations on a singleton set. Let Q and R be the matrices with coeﬃcient a. Then we have ↓ Q↑ R↓ = ⊥ ⊥ but (Q R) = . In Theorem 3.4 we have proved the so-called α-cut Theorem of fuzzy theory in L-Rel. The next lemma states a weak version valid in any arrow category.

100

GOGUEN CATEGORIES

Lemma 5.9 Let A be an arrow category and R : A → B be a relation. Then we have ↓ αA ; (αA \R) R, (1) αA ∈ScA (A)

(2)

↓

(αA \R) R↑ .

αA ∈ScA (A) αA =⊥ ⊥AA

Proof. (1) The assertion follows immediately from ↓ αA ; (αA \R) αA ; (αA \R) αA ∈ScA (A)

αA ∈ScA (A)

(2) We immediately conclude ↓ (αA \R) =

R = R.

αA ∈ScA (A)

↓

↑ αA ; (αA \R)

αA ∈ScA (A)

αA ∈ScA (A) αA =⊥ ⊥AA

=

αA ∈ScA (A)

⎛

=⎝

↓ ↑

(αA ; (αA \R) )

Lemma 5.6 (9)

⎞↑ ↓ αA ; (αA \R) ⎠

Corollary 5.5 (4)

αA ∈ScA (A)

R↑ .

(1)

Unfortunately, the general α-cut theorem is not necessarily valid in an arrow category. Let us consider the following examples: Example 5.10 Let L := {0, a, b, 1} be the complete (linear) Brouwerian lattice, and M1 be the substructure of L-fuzzy relations given in Figure 5.1. M1 is closed under all operations, including ↑ and ↓ , deﬁned on L-fuzzy relations, and, hence, an arrow category. But we have 1 a ↓ αA ; (αA \R) = = R a 1

αA ∈ScA (A)

1 b 1 a 1 b for R ∈ , , . The reason is that the scalar a 1 b 1 b 1 b 0 1 0 1 1 and the crisp relations and are not included 0 b 1 1 0 1 in M1 . The smallest set N1 of L-fuzzy relations containing M1 and the three relations above, which is again closed under all operations contains 30 relations. In this arrow category the α-cut theorem is valid. Notice that in this ﬁrst example the inclusion from Lemma 5.9 (2) is fact an equality.

CATEGORIES OF L-FUZZY RELATIONS

M1 L

1 a b 1

ss s KKK

1 b

1 0

0 1

ss s KKK

a 0 Figure 5.1.

1 1 1 1 1 b

b 1

1 a a 1

a 0 0 a 0 0 0 0

101

KKK

ss s KKK ss s

1 a

b 1

a a

a a

The lattice L and the structure M1 .

Example 5.11 For this example consider the substructure M2 of L-fuzzy relations given in Figure 5.2. As above M2 is closed under all operations deﬁned on L-fuzzy relations, and, hence, an arrow category. Again, we have ↓ αA ; (αA \R) = R αA ∈ScA (A)

b a 1 a a 0 , . This time the scalar is missing. a b a 1 0 a Furthermore, we have 1 0 1 1 ↓ (αA \R) = = = R↑ 0 1 1 1 for R ∈

αA ∈ScA (A) αA =⊥ ⊥AA

for those relations proving that Lemma an equality.

5.9 (2) is not necessarily a 0 a a , is again an arrow Notice that the set N2 := M2 ∪ 0 a a a category so that the α-cut theorem as well as Lemma 5.9 (2) is valid. In the next lemma we have summarized some further properties of arrow categories.

102

GOGUEN CATEGORIES

M2

b b

b b

ss s KKK

1 1

1 1

1 b

b 1

b a a b

KKK

ss s KKK

Figure 5.2.

1 a a 1

b 0

0 b

0 0 0 0

KKK

ss s

1 0

0 1

The structure M2 .

Lemma 5.12 Let A be an arrow category, Q, R : A → B relations and αA = ⊥ ⊥AA a scalar. Then we have ↓

(1) (αA \R↑ ) = R↑ , ↑

AA ; R; BA ) = IA AA ; R↑ ; BA , (2) (IA (3) AB ; BC = AC for all objects A, B and C, i.e., A is uniform, (4) if R = ⊥ ⊥AB , then CA ; R↑ ; BD = CD for all objects C and D. Proof. (1) The assertion follows from R↑ = R↑↓ = (IA \R↑ )

↓

↑ ↓

(αA \R ) ↓ (αA \R↑ )

Lemma 4.16 (1) Corollary 4.15 (4) αA = ⊥ ⊥AA

αA =⊥ ⊥AA

R↑↑ = R↑ .

Lemma 5.9 (2)

CATEGORIES OF L-FUZZY RELATIONS

103

(2) Consider the following computation: ↑

↑

AA ; R; BA ) = (I↑A AA ; R; BA ) (IA

↑

= I↑A ( AA ; R; BA )

↑

= IA ( ↑AA ; R; BA )

↑

= IA ↑AA ; (R; BA )

↑

= IA AA ; (R; ↑BA ) = IA AA ; R↑ ; ↑BA = IA AA ; R↑ ; BA .

BA = AA ; AB ; BA is a nonzero ideal element on (3) The relation AB ; ↑ BA ) = AA . A by 4.36 (1) and (3). Using Lemma 5.6 (12) we get ( AB ; On the other hand we have ↑

↑

BA ) = ( ↑AB ; BA ) = ↑AB ; ↑BA = AB ; BA . ( AB ; Together we have AC = AA ; AC = AB ; BA ; AC AB ; BC . AA ; R; BA is a nonzero ideal on A by (4) Since R = ⊥ ⊥AB the composite Lemma 4.36 (1) and (3). Using Lemma 5.6 (12) we have ↑

AA ; R; BA ) = AA ; R↑ ; BA AA = ( and conclude that CA ; R↑ ; BD = CA ; AA ; R↑ ; BA ; AD = CA ; AA ; AB = CD .

(3) see above (3)

In the next theorem we will show an important property of arrow categories. Notice that (3) of the previous lemma is essential. Theorem 5.13 Let A be an arrow category. For all objects A and B the complete Brouwerian lattices ScG (A) and ScG (B) are isomorphic. Proof. Deﬁne f : ScG (A) → ScG (B) and g : ScG (B) → ScG (A) by f (αA ) := BA ; αA ; AB and g(βB ) := IA AB ; βB ; BA . As pointed out in the IB previous chapter it is suﬃcient that f is a complete lattice isomorphism. First of all, f and g are inverse, which follows from AB ; (IB BA ; αA ; AB ); BA g(f (αA )) = IA AB ; BA AA ; αA ; AA ) = IA ( = IA ( AA AA ; αA ; AA )

deﬁnition f and g Lemma 4.23 (1) Lemma 5.12 (3)

AA ; αA ; AA = IA = IA αA ; AA = αA .

αA scalar Lemma 4.24 (1)

104

GOGUEN CATEGORIES

f (g(βB )) = βB is shown analogously. By Lemma 2.11 it is suﬃcient to show that f is a complete lower and g a complete upper semilattice homomorphism. This is shown for partial identities Si : A → A and relations Ri : B → B with i ∈ I as follows: AB S i = IB BA ; Si ; f i∈I

i∈I

= IB

( BA ; Si ; AB )

Lemma 4.24 (3)

i∈I

(IB BA ; Si ; AB )

= i∈I

= g

Ri

f (Si ), i∈I

= IA AB ;

i∈I

= IA =

BA Ri ;

i∈I

( AB ; Ri ; BA )

i∈I

(IA AB ; Ri ; BA )

i∈I

=

completely upwarddistributive

g(Ri ).

i∈I

We will identify all sets of scalars and denote them by Sc[A]. An abstract element from Sc[A] is denoted by α, i.e., without an index. The corresponding scalar on the object A is then denoted by αA . By this convention, we have the following connection αA = g(αB ) and αB = f (αA ) for an α ∈ Sc[A] and all objects A and B where f and g are the isomorphisms from the previous lemma: Lemma 5.14 Let A be an arrow category. Then we have αA ; R = R; αB for all scalars α ∈ Sc[A] and relations R : A → B. Proof. Using Lemma 4.23 (1) we have BA ; αA ; AB ) = AB AA ; αA ; AB = AA ; αA ; AB . AB ; (IB This implies AB R αA ; R = αA ; AA ; AB R = αA ; = AA ; αA ; AB R BA ; αA ; AB ) R = AB ; (IB = AB ; αB R = R; αB .

Lemma 4.24 (2) Lemma 4.21 (2) αA scalar see above convention Lemma 4.24 (2)

CATEGORIES OF L-FUZZY RELATIONS

105

As an application of the previous lemma, we obtain the following lemma about the relationship of the α-cuts of Q and Q : Lemma 5.15 Let A be an arrow category. Then we have (αA \Q)

↓

↓

= (αB \Q )

for all scalars α ∈ Sc[A] and relations Q : A → B. Proof. The assertion follows from X (αA \Q)

↓

⇔ X (αA \Q)

↓

↑

⇔ X αA \Q ⇔ αA ; X ⇔X ⇔X

↑ ↑

↑

Q

Galois correspondence deﬁnition residual

; αB Q

Lemma 5.14

; αB Q

Lemma 4.9(1) and 5.6(8)

↑

⇔ αB ; X Q

⇔ X ↑ αB \Q ↓

⇔ X (αB \Q ) .

deﬁnition residual Galois correspondence

In the next lemma we have collected some closure properties of the class of crisp relations. Lemma 5.16 Let A be an arrow category and Qi , Q, T : A → B for i ∈ I, R : A → C, and S : B → C crisp relations. Then the following holds: Qi and Qi are crisp. (1) i∈I

i∈I

(2) Q is crisp. (3) Q; S is crisp. (4) R/S and Q\R are crisp. (5) Q:T is crisp. Proof. For the ﬁrst three assertions consider the following computations: ↑ ↑ ↓ Qi = Qi and ( Qi ) = Q↓ i = Qi , (1) ( Qi ) = i∈I

i∈I

↑

(2) Q = Q↑ ↑

i∈I

i∈I

= Q , ↑

(3) (Q; S) = (Q; S ↑ ) = Q↑ ; S ↑ = Q; S.

i∈I

i∈I

106

GOGUEN CATEGORIES ↑

(4) From Lemma 5.6 (14) we conclude (Q\R) Q↓ \R↑ = Q\R. The other assertion follows analogously. ↑

(5) From Lemma 5.6 (13) we conclude (Q:T ) Q↑ :T ↓ = Q:T .

The previous lemma, Lemma 5.6 (1) and (6) and Lemma 5.12 (4) give us the following corollary: Corollary 5.17 If A is an arrow category, then A↑ is a simple Dedekind cat⊥AB . egory with AB = ⊥ The substructure of crisp relations may satisfy additional properties. Deﬁnition 5.18 An arrow category A such that A↑ is a Schr¨ oder category is called a Boolean-based arrow category. As already mentioned in Section 4.9 the crisp L-fuzzy relations constitute a Schr¨ oder category so that L-Rel is in fact a Boolean-based arrow category. The inclusion Q ⊆ R of L-fuzzy relations has two aspects. First of all, there is the usual relational aspect. The set of all pairs (x, y) such that Q(x, y) = ⊥ ⊥AB has to be a subset of the corresponding set of R. This may be expressed by the relational inclusion Q↑ ⊆ R↑ of crisp relations. Furthermore, there is an aspect induced by the lattice L. All values Q(x, y) have to be less or equal to R(x, y). The next lemma shows that these aspects of an inclusion can be found in all arrow categories. Lemma 5.19 Let A be an arrow category, X, Y : A → B be crisp relations and αA , βA ∈ ScA (A). Then we have αA ; X βA ; Y

⇐⇒

αA = ⊥ ⊥AA or X = ⊥ ⊥AB or (αA βA and X Y ).

Proof. The implication ⇐ is trivial. Now, suppose αA ; X βA ; Y , αA = ↑ ⊥AB . Then we have αA = IA and may conclude that ⊥ ⊥AA and X = ⊥ X = X↑ =

X crisp

↑ αA ; X↑

↑ αA = IA

= (αA ; X ↑ ) = (αA ; X)

↑

↑

Lemma 5.6 (10) X crisp

↑

(βA ; Y )

↑

= (βA ; Y ↑ ) =

↑ βA ;Y ↑ ↑

Y crisp Lemma 5.6 (10)

Y

= Y.

Y crisp

CATEGORIES OF L-FUZZY RELATIONS

107

From Lemma 5.12 (4) we obtain AB = αA ; AA ; X; BB αA ; = AA ; αA ; X; BB BB AA ; βA ; Y ; = βA ; AA ; Y ; BB βA ; AB ,

X = ⊥ ⊥AB αA scalar βA scalar

and, hence, AA αA = IA αA ; AB ; BA = IA αA ; IA βA ; AB ; BA

Lemma 4.24 (1) Lemma 5.12 (3)

= IA βA ; AA = βA .

Lemma 5.12 (3) Lemma 4.24 (1)

The previous lemma allows us to compute the α-cut of the composition from β with a crisp relation R. Lemma 5.20 Let A be an arrow category, R : A → B be a crisp relation and αA , βA ∈ ScA (A). Then we have ⎧ ⊥AA , AB iﬀ αA = ⊥ ⎨ ↓ R iﬀ α = ⊥ ⊥AA and αA βA , (αA \(βA ; R)) = ⎩ ⊥ ⊥AB else. Proof. Using the previous lemma the assertion follows immediately from ↓

X (αA \(βA ; R)) ⇔ X ↑ αA \(βA ; R) ⇔ αA ; X ↑ βA ; R ⇔ αA = ⊥ ⊥AA or X ↑ = ⊥ ⊥AB or (αA βA and X ↑ R) ⇔ αA = ⊥ ⊥AA or X = ⊥ ⊥AB or (αA βA and X R).

Lemma 5.19 ⊥ ⊥AB and R crisp

Let us consider the following operation in an arbitrary arrow category: (R) :=

↓

αA ; (αA \R) .

α∈Sc[A]

From Lemma 5.9 (1) we already know that (R) R. Furthermore, the examples have shown that the inclusion might be strict.

108

GOGUEN CATEGORIES

It is obvious that is monotone and the following computation shows that it is indeed a kernel operation: ↓

↓

βA ; (βA \R) (R) ⇐⇒ (βA \R) βA \ (R) ↓

↓

⇐⇒ (βA \R) (βA \ (R)) =⇒ (R) 2 (R).

Lemma 5.21 Let A be an arrow category and Q, R : A → B, S : B → C. Then we have

(1) (Q ) = (Q) , (2) (Q R) = (Q) (R), (3) (R; S) = (R); (S). Proof. (1) Consider the following computation: ↓ (Q ) = αB ; (αB \Q )

deﬁnition

α∈Sc[A]

=

↓

(αB \Q ) ; αA

Lemma 5.14

α∈Sc[A]

=

(αA \Q)

↓

; αA

Lemma 5.15

α∈Sc[A]

=

↓

(αA ; (αA \Q) )

α∈Sc[A]

=

αA ; (αA \Q)

↓

Lemma 4.9 (1)

α∈Sc[A]

= (Q) .

deﬁnition

(2) Again, consider the following computation: (Q) (R) ↓ αA ; (αA \Q) = α∈Sc[A]

=

↓

βA ; (βA \R)

deﬁnition

β∈Sc[A] ↓

↓

αA ; (αA \Q) ) βA ; (βA \R) )

α,β∈Sc[A]

=

↓

↓

(αA βA ); ((αA \Q) ) (βA \R) )

Lemma 4.9 (5)

α,β∈Sc[A]

↓

↓

(αA βA ); (((αA βA )\Q) ((αA βA )\R) )

α,β∈Sc[A]

CATEGORIES OF L-FUZZY RELATIONS

↓

109

↓

αA ; ((αA \Q) (αA \R) )

α∈Sc[A]

=

↓

αA ; ((αA \Q) (αA \R))

Corollary 5.5 (4)

α∈Sc[A]

=

↓

αA ; (αA \(Q R))

Corollary 4.15 (6)

α∈Sc[A]

= (Q R). The other inclusion is trivial since is a kernel operations. (3) First of all, we have ↓

↓

↓

αA ; (αA \R) ; (αB \S) = αA ; αA ; (αA \R) ; (αB \S) ↓

↓

= αA ; (αA \R) ; αB ; (αB \S) αA ; (αA \R); αB ; (αB \S) R; S. ↓

Lemma 4.9 (2)

↓

Lemma 5.14

↓

This implies (αA \R) ; (αB \S) αA \(R; S) and since the left-hand side ↓ ↓ ↓ is crisp (αA \R) ; (αB \S) (αA \(R; S)) . We conclude (R); (S) ↓ αA ; (αA \R) ; = α∈Sc[A]

=

↓

βB ; (βB \S)

deﬁnition

β∈Sc[A] ↓

αA ; (αA \R) ; βB ; (βB \S)

↓

α,β∈Sc[A]

=

↓

αA ; βA ; (αA \R) ; (βB \S)

↓

Lemma 5.14

α,β∈Sc[A]

=

↓

↓

(αA βA ); (αA \R) ; (βB \S)

Lemma 4.9 (3)

α,β∈Sc[A]

↓

↓

(αA βA ); ((αA βA )\R) ; ((αB βB )\S)

α,β∈Sc[A]

↓

αA ; (αA \R) ; (αB \S)

↓

α∈Sc[A]

αA ; (αA \(R; S))

↓

see above

α∈Sc[A]

= (R; S). Again, the other inclusion is trivial. As usual we are interested in the ﬁxed points of the kernel operation .

110

GOGUEN CATEGORIES

Deﬁnition 5.22 Let A be an arrow category. A relation R : A → B is called sliceable iﬀ (R) = R. Notice that if the α-cut theorem is valid, all relations are sliceable. However, we have the following closure properties of the class of sliceable relations: Theorem 5.23 Let A be an arrow category. Then we have (1) All scalar relations are sliceable. (2) All crisp relations are sliceable. (3) If Q : A → B is sliceable, then so is Q . (4) If Q, R : A → B are sliceable, then so is Q R. (5) If R : A → B and S : B → C are sliceable, then so is R; S. Proof. (1) Let βA be a scalar. Then we immediately conclude ↓ (βA ) = αA ; (αA \βA ) α∈Sc[A]

=

αA

Lemma 5.20

αA βA

= βA . (2) Let R be crisp and consider the following computation: ↓ (R) = αA ; (αA \R) α∈Sc[A]

=

αA ; (αA \R↑ )

↓

R crisp

α∈Sc[A]

=

αA ; R↑

α∈Sc[A]

=

Lemma 5.12 (1)

αA ; R

R crisp

α∈Sc[A]

= R. (3) –(5) follow from Lemma 5.21.

We want to close this section by providing an example of an arrow category where does not preserve arbitrary meets. Obviously, the α-cut theorem cannot be valid in that category.

CATEGORIES OF L-FUZZY RELATIONS

x0

x1

x 2 x∞

L

R

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

111

= =I =⊥ ⊥

0 Figure 5.3.

The lattice L and the simple Dedekind category R.

Example 5.24 Consider the complete Brouwerian lattice L and the simple Dedekind category R given in Figure 5.3. Notice that L is the reversed order on N∞ with an additional least element 0. Since the least upper bound of each subset of L is in fact a maximum all antitone functions are antimorphisms and all operations in RL are computed componentwise. Notice that all relations in R, and, hence, in RL are symmetric. The scalar gi in RL corresponding to the element xi (i ∈ N∞ ) is given by ⎧ iﬀ x = 0 ⎨ I iﬀ x xi gi (x) = ⎩ ⊥ ⊥ otherwise Now, let S be the subset of antimorphisms fulﬁlling the property ⊥, then there is an i ∈ N with f (xi ) = ⊥ ⊥. if f (x∞ ) = ⊥ It is easy to verify that the following properties are valid: (1) if f, g ∈ S, then so is f ; g and f g, (2) if fi ∈ S for all i ∈ I, then so is fi . i∈I

The conditions above show that S is an arrow category. But notice that the function g∞ , i.e, the scalar corresponding to x∞ , is not in S since g∞ (x∞ ) = ⊥ for all i ∈ N. Now, let fi for i ∈ N∞ be deﬁned by I = ⊥ ⊥ but g∞ (xi ) = ⊥

iﬀ x = 0 or x xi fi (x) = I otherwise Then fi ∈ S for all i ∈ N∞ and we have for i ∈ N and j ∈ N

fi = f∞ . It is easy to verify that i∈N

∞ ↓

(gi \fj ) =

˙ iﬀ j ≤ i ˙I otherwise

112

GOGUEN CATEGORIES

holds. Therefore, fi is sliceable in S iﬀ i ∈ N, which shows that the α-cut theorem is not valid and that the set of sliceable relations is not closed under arbitrary meets. 5.2

THE ARROW CATEGORY OF ANTIMORPHISMS

As shown in Theorem 4.33, the category RL is a Dedekind category. In order to establish that this structure is an arrow category, we ﬁrst want to characterize the set of scalar elements in RL [A, A]. Therefore, we need the following technical lemma: Lemma 5.25 Let L be a complete Brouwerian lattice, R be a Dedekind cateanti gory, f : L → R[A, A] be a partial identity in RL and R : C → A, S : A → B relations from R. Then we have ˙ f, (1) R˙ ··, f = R; ˙ (2) f ··, S˙ = f ; S. Proof. (1) First of all, we have ˙ f )(0) = R(0); ˙ ˙ f )(0) = (R˙ ··, f )(0) (R; f (0) = CA ; AA = CA = τ (R; ˙ f ) is an antimorphism. If x = 0 and M = x, then the set since τ (R; f (y) = f (y) since f (0) = M \ {0} is not empty, and, hence, y∈M

˙ f )(x) = ϕ(R;

0 =y∈M

˙ f )(y) = (R;

AA . Analogously, we have y∈M

˙ f )(y) and conclude (R; 0 =y∈M

˙ f )(y) (R;

deﬁnition ϕ

y∈M

M ⊆L M =x

=

M ⊆L

˙ f )(y) (R;

see above

(R; f (y))

deﬁnition ; and R˙

0 =y∈M

M =x

=

0 =y∈M

M ⊆L M =x

=

f (y)

R;

M =x

= R;

f (y)

M ⊆L

0 =y∈M

M =x

= R;

Lemma 4.24 (4)

0 =y∈M

M ⊆L

f (y)

M ⊆L M =x

y∈M

see above

CATEGORIES OF L-FUZZY RELATIONS

= R; ϕ(f )(x) = R; f (x) ˙ f )(x). = (R;

113

deﬁnition ϕ f antimorphism deﬁnition ; and R˙

˙ f ) = R; ˙ f , and, hence, τ (R; ˙ f ) R; ˙ f . The other Together we have ϕ(R; inclusion is trivial.

(2) is shown analogously. The previous lemma gives us the following:

Lemma 5.26 Let L be a complete Brouwerian lattice, R be a Dedekind cateanti gory and αA : L → R[A, A] be a scalar in RL . Then αA (x) is a scalar for all 0 = x ∈ L. Proof. Suppose x = 0. Then αA (x) I˙ A (x) = IA shows that αA (x) is a partial identity for x = 0. Using the previous lemma we get ˙ AA (x) AA = αA (x); αA (x); ˙ AA )(x) = (αA ;

˙ AA deﬁnition deﬁnition ;

˙ AA )(x) = (αA ··, ˙ AA ··, αA )(x) = (

Lemma 5.25 (2)

˙ AA ; αA )(x) = (

Lemma 5.25 (1)

= AA ; αA (x).

˙ AA deﬁnition ; and

αA scalar

For simple Dedekind categories we obtain the following result as an immediate consequence: Corollary 5.27 Let L be a complete Brouwerian lattice, R be a simple anti Dedekind category and αA : L → R[A, A] be a scalar in RL . Then αA (x) ∈ {⊥ ⊥AA , IA } for all 0 = x ∈ L. AA we may conclude By the previous corollary and the fact that αA (0) = ⊥AA , IA , AA } for all x and all scalars αA in RL if R is simple. that αA (x) ∈ {⊥ We may deﬁne two functions γ : L → ScRL (A) and δ : ScRL (A) → L between the underlying lattice of scalars over A in RL and L by {βA ∈ ScRL (A) | βA (x) = ⊥ ⊥AA } ⊥AA }. and δ(αA ) := {y ∈ L | αA (y) = ⊥ γ(x) :=

Notice that the sets on the right-hand side of the deﬁnitions above are not empty if AB = ⊥ ⊥AB for all objects A and B holds in R. Furthermore, we ˙ AA ) = 0. ˙ AA and δ(⊥ ⊥ have γ(0) = ⊥ ⊥

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Lemma 5.28 Let L be a complete Brouwerian lattice and R be a simple ⊥AB for all objects A and B. Then for all Dedekind category with AB = ⊥ x, y ∈ L, and αA , βA ∈ ScRL (A) we have (1) γ(x)(y) = ⊥ ⊥AA ⇐⇒ y ≤ x, ⊥AA ⇐⇒ αA βA . (2) βA (δ(αA )) = ⊥ Proof. (1) The case x = 0 is obvious. For ⇐ suppose y ≤ x = 0. Since every scalar is antitone, we immediately compute βA (x) βA (y), and, hence, βA (y) ∈ AA } for all βA with βA (x) = ⊥ ⊥AA . It follows {IA , βA (y) = βA (y) = ⊥ ⊥AA . γ(x)(y) = βA (x) = ⊥ ⊥AA

βA (x) = ⊥ ⊥AA

For the other implication we deﬁne βˆA by ⎧ AA iﬀ z = 0, ⎨ IA iﬀ z = 0 and z ≤ x, βˆA (z) := ⎩ x. ⊥ ⊥AA iﬀ z = 0 and z ≤ βˆA is an antimorphism, which follows from ⎧ AA iﬀ M = ∅ ∨ M = {0}, ⎨ IA iﬀ ∀z ∈ M : z ≤ x and ∃z ∈ M : z = 0, βˆA (z) = ⎩ z∈M ⊥ ⊥AA iﬀ ∃z ∈ M : z = 0 and z ≤ x. ⎧ AA iﬀ M = 0, ⎨ IA iﬀ M = 0 and M ≤ x, = ⎩ ⊥ ⊥AA iﬀ M = 0 and M ≤ x. M . = βˆA Furthermore, βˆA is a scalar with βˆA (x) = ⊥ ⊥AA by deﬁnition. Now, suppose ⊥AA , and, hence, y ≤ x. γ(x)(y) = ⊥ ⊥AA . Then we have βˆA (y) = ⊥ (2) First, the computation βA (δ(αA )) = βA

αA (y) = ⊥ ⊥AA

y

=

βA (y) αA (y) = ⊥ ⊥AA

⊥AA is equivalent to the property shows that βA (δ(αA )) = ⊥ ⊥AA ⇒ βA (y) = ⊥ ⊥AA αA (y) = ⊥ for all y since βA (y) ∈ {⊥ ⊥AA , IA , AA }. The last property is equivalent to AA iﬀ z = 0 by Corollary 5.27. αA βA since αA (z) =

CATEGORIES OF L-FUZZY RELATIONS

115

Now, we may characterize the set of scalars in RL [A, A] by the following theorem: Theorem 5.29 Let L be a complete Brouwerian lattice and R be a simple ⊥AB for all objects A and B. Then L and Dedekind category with AB = ⊥ ScRL (A) are isomorphic. Proof. First, we show that γ and δ are inverse. Using the previous lemma this follows from y deﬁnition δ δ(γ(x)) = γ(x)(y) = ⊥ ⊥AA

=

y

Lemma 5.28 (1)

y≤x

=x and γ(δ(αA )) =

βA

deﬁnition γ

βA (δ(αA )) = ⊥ ⊥AA

=

βA

Lemma 5.28 (2)

αA βA

= αA . Since the notions of complete lattice isomorphisms and complete Brouwerian lattice isomorphisms are equivalent, it is suﬃcient to show that γ respects arbitrary intersections and unions. Let M be a subset of L. Then we have γ

M M (y) = ⊥ ⊥AA ⇔ y ≤

Lemma 5.28 (1)

⇔ ∀x ∈ M : y ≤ x ⇔ ∀x ∈ M : γ(x)(y) = ⊥ ⊥AA

Lemma 5.28 (1)

⇔

γ(x)(y) ∈ {IA , AA }

γ(x)(y) = ⊥ ⊥AA x∈M

⇔

γ(x) (y) = ⊥ ⊥AA .

x∈M

Since γ(x)(y) = AA iﬀ y = 0 we conclude γ( M )(y) = ( γ(x))(y), and, x∈M hence, γ( M ) = γ(x). x∈M It is suﬃcient to show that γ( M ) · γ(x) since γ is monotone as shown x∈M above. Suppose γ( M )(y) = ⊥ ⊥AA , which is equivalent toy ≤ M by Lemma 5.28 (1). Deﬁne My := {y ∧ x | x ∈ M }. Then we have My = y ∧ M = y since L is a complete Brouwerian lattice. Furthermore, for all z ∈ My there is

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an x ∈ M such that z ≤ x, which is equivalent to γ(x)(z) = ⊥ ⊥AA . This implies γ(x)(z) = ⊥ ⊥AA , z∈My x∈M

and, hence, ϕ γ(x) (y) = x∈M

γ(x)(z) = ⊥ ⊥AA .

N =y z∈N x∈M

Furthermore, ϕ( γ(x))(0) = ∅ γ(x)(0) = AA . Together we get x∈M x∈M γ( M ) ϕ( γ(x)), and, hence, γ( M ) ϕ( γ(x)) τ ( γ(x)) = x∈M x∈M x∈M · γ(x). x∈M

Now, we deﬁne an up- and down-operation in RL by ˙ f (y) and f ↓ := f (1) f ↑ := R˙ with R = y =0

or componentwise by AB f (y)

↑

f (x) :=

iﬀ x = 0 else,

↓

f (x) :=

y =0

AB f (1)

iﬀ x = 0 else.

Lemma 5.30 Let L be a complete Brouwerian lattice, and R be a simple ⊥AB for all objects A and B. Then for all Dedekind category with AB = ⊥ anti anti f, gi : L → R[A, B] with i ∈ I, g : L → R[A, B] and h : L → R[B, C] we have (1) f ↑ and f ↓ are antimorphisms, ↑ ↑ (2) gi = ( gi ) , i∈I

i∈I

↑

(3) (g; h↓ ) = g ↑ ; h↓ , ↑

(4) ϕ(f ) = f ↑ . Proof.

(1) Let be M ⊆ L. If M = 0, the assertion is trivial. Now, suppose M = 0. Then we immediately conclude that M = f (y) = f ↑ (x) f↑ and f ↓

y =0

M

x∈M

f ↓ (x).

= f (1) = x∈M

(2) If x = 0, we obtain ↑ ↑ gi↑ (0) = gi (0) = AB = gi (0). i∈I

i∈I

i∈I

CATEGORIES OF L-FUZZY RELATIONS

Now, let x = 0. Then we have ↑ gi↑ (x) = gi (x) i∈I

117

deﬁnition

i∈I

=

gi (y)

deﬁnition

↑

deﬁnition

↑

i∈I y =0

=

gi (y)

y =0 i∈I

=

↑

gi

(x).

i∈I

(3) First of all, IB = ⊥ ⊥BB since BB = ⊥ ⊥BB . For x = 0 we have ↑

(g; h↓ ) (0) = AC = AB ; BC

deﬁnition ↑ R simple and IB = ⊥ ⊥BB

= g ↑ (0); h↓ (0) = (g ↑ ; h↓ )(0). If x = 0, we conclude that ↑

(g; h↓ ) (x) =

(g; h↓ )(y)

deﬁnition

↑

y =0

g(y); h↓ (y)

deﬁnition ;

g(y); h(1)

deﬁnition

↓

= g (x); h↓ (x)

deﬁnition

↑

= (g ↑ ; h↓ )(x).

deﬁnition ;

=

y =0

=

y =0

=

g(y) ; h(1)

y =0 ↑

(4) If x = 0, we get

↑

AB = f ↑ (0). ϕ(f ) (0) = ↑

Now, let x = 0. Then we conclude that ϕ(f ) (x) f ↑ (x) since ϕ is expandf (z) f (y) = ing and ↑ monotone. For the other inclusion we have z∈M y =0 f (z) f ↑ (x), f ↑ (x) for all M such that M = 0. This implies M ⊆L M =y

↑

z∈M

and, hence, ϕ(f )(y) f (x) for all y = 0. We conclude that

ϕ(f )(y)

y =0 ↑

f ↑ (x), which is equivalent to ϕ(f ) (x) f ↑ (x). We are now ready to prove our main theorem in this section.

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Theorem 5.31 Let L be a complete Brouwerian lattice with 0 = 1 and R be ⊥AB for all objects A and B. Then a simple Dedekind category with AB = ⊥ RL is an arrow category. Proof. We have already shown in Theorem 4.33 that RL is a Dedekind ˙ AB and ˙ AB are not ⊥AB the relations ⊥ ⊥ category. Since 0 = 1 and AB = ⊥ equal. Axiom (1) is trivial and (2) follows from f ↑ g ⇔ ∀x = 0 : f ↑ (x) g(x) f (y) g(x) ⇔ ∀x = 0 : ⇔

since f ↑ (0) = AB = g(0) deﬁnition

↑

y =0

f (y)

since g(0) = AB

g(z) z∈L

y =0

⇔ ∀x = 0 : f (x)

g(z) z∈L

⇔ ∀x = 0 : f (x) g L

g antimorphism

⇔ ∀x = 0 : f (x) g(1) ⇔ ∀x = 0 : f (x) g ↓ (x) ⇔ f g↓ .

deﬁnition ↓ since f (0) = AB = g ↓ (0)

Consider the following computation for x = 0:

(f ↑ )(x) = (f ↑ (x)) = f (y)

deﬁnition

deﬁnition

↑

deﬁnition

deﬁnition

↑

y =0

=

f (y)

y =0

=

f (y)

y =0 ↑

= (f )(x).

↑

↑

Since f ↑ and f are antimorphisms f ↑ = f follows. In order to prove Axiom (3), we deﬁne the following predicate: P(h, k) :⇔ h↑ = k. This predi ↑ cate is admissible since P(hi , ki ) for all i ∈ I implies hi = ki , and, hence, i∈I i∈I ↑ ( hi ) = ki by Lemma 5.30 (2), which is equivalent to P( hi , ki ). The i∈I

i∈I

i∈I

i∈I

base case P(f ; g ↓ , f ↑ ; g ↓ ) follows from Lemma 5.30 (3). Now, suppose P(h, k). Then we conclude ↑

ϕ(h) = h↑

Lemma 5.30 (4) ↑

= ϕ(h ) = ϕ(k)

Lemma 5.30 (1) induction hypothesis

CATEGORIES OF L-FUZZY RELATIONS

119

The principle of ﬁxed point induction gives us P(µϕ (f ; g ↓ ), µϕ (f ↑ ; g ↓ )) and P(µϕ (f h↓ ), µϕ (f ↑ h↓ )). Together we conclude ↑

↑

(f ··, g ↓ ) = τ (f ; g ↓ ) ↑

= τ (f ; g ↓ )

= τ (f ↑ ; g ↓ ) =

f ↑ ··,

see above

↓

g ,

i.e, Axiom (3). Consider the following computation for x = 0: ↑

(f h↓ ) (x) =

(f h↓ )(y)

y =0

=

f (y) h(1)

y =0

=

f (y) h(1)

y =0

completely upwarddistributive

= f ↑ (x) h(1) = f ↑ (x) h↓ (x) = (f ↑ h↓ )(x). ↑

Again, (f h↓ ) and f ↑ h↓ are antimorphisms so that Axiom (4) follows. ˙ AA be a scalar. From Corollary 5.27 we conclude that αA (x) ∈ ⊥ Let αA = ⊥ {⊥ ⊥AA , IA } for all x = 0 and that there is at least one y = 0 such that αA (y) = ↑ ↑ IA . This implies αA (x) = αA (y) = IA for x = 0, and, hence, αA = I˙ A , i.e., y =0

Axiom (5).

Let F : R1 → R2 be a homomorphism between the Dedekind categories R1 and R2 . Then we may deﬁne an extension Fˆ of F on the antitone functions from L to R1 [A, B] by (1) Fˆ (A) := F (A) for all objects A, (2) Fˆ (f )(x) := F (f (x)) for all antitone functions f : L → R1 [A, B] and x ∈ L.

Fˆ (f ) is antitone since x ≤ y implies f (y) ≤ f (x), and, hence, F (f (y)) F (f (x)) by the monotonicity of F . Furthermore, we have the following lemma: Lemma 5.32 Let F : R1 → R2 be a homomorphism between the Dedekind ≥ categories R1 and R2 , L be a complete Brouwerian lattice, f : L → R1 [A, B], ≥ ≥ g : L → R1 [B, C] and fi : L → R1 [A, B] for all i ∈ I. Then we have

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GOGUEN CATEGORIES

(1) Fˆ (f ; g) = Fˆ (f ); Fˆ (g), (2) Fˆ (I˙ A ) = I˙ Fˆ (A) , ˆ (3) Fˆ ( fi ) = F (fi ), i∈I

(4) Fˆ ( i∈I

i∈I

Fˆ (fi ),

fi ) = i∈I

(5) Fˆ (f ) = Fˆ (f ) , ↓ (6) Fˆ (f ↓ ) = Fˆ (f ) , ↑ (7) Fˆ (f ↑ ) = Fˆ (f ) ,

(8) τ (Fˆ (f )) = Fˆ (τ (f )). Proof. (1) The assertion follows immediately from Fˆ (f ; g)(x) = F ((f ; g)(x)) = F (f (x); g(x)) = F (f (x)); F (g(x)) = Fˆ (f )(x); Fˆ (g)(x) = (Fˆ (f ); Fˆ (g))(x).

deﬁnition Fˆ deﬁnition ; F functor deﬁnition Fˆ deﬁnition ;

(2) –(5) similar to (1). (6) First, we have Fˆ (f ↓ )(0) = F (f ↓ (0)) = F( AB ) = F (A)F (B) ↓ = Fˆ (f ) (0).

deﬁnition Fˆ f ↓ antimorphism by Lemma 5.30 (1) F homomorphism ↓ Fˆ (f ) antimorphism by Lemma 5.30 (1)

Now, suppose x = 0 and compute Fˆ (f ↓ )(x) = F (f ↓ (x))

deﬁnition Fˆ deﬁnition

↓

= F (f (1)) = Fˆ (f )(1)

deﬁnition Fˆ

↓ = Fˆ (f ) (x).

deﬁnition

↓

(7) The case x = 0 follows similar to the corresponding case in (6). Now, suppose x = 0 and compute Fˆ (f ↑ )(x) = F (f ↑ (x)) f (y) =F y =0

deﬁnition Fˆ deﬁnition

↑

CATEGORIES OF L-FUZZY RELATIONS

=

F (f (y))

F homomorphism

Fˆ (f )(y)

deﬁnition F

121

y =0

=

y =0 ↑ = Fˆ (f ) (x).

deﬁnition

↑

(8) We prove the assertion by ﬁxed point induction. Therefore, we deﬁne a predicate P(g, h) :⇔ g = Fˆ (h). This predicate is admissible since P(gi , hi ) for all i ∈ I implies ˆ ˆ gi = hi F (hi ) = F i∈I

by (3), and, hence, P(

i∈I

gi ,

i∈I

i∈I

hi ). The base case is trivial since we obvi-

i∈I

ously have P(Fˆ (f ), f ). Now, suppose P(g, h). Then we conclude ϕ(g)(x) = g(y) deﬁnition ϕ y∈M M =x

=

Fˆ (h)(y)

induction hypothesis

(F (h(y))

deﬁnition Fˆ

M =x y∈M

=

M =x y∈M

=

F

M =x

=F

h(y)

F homomorphism

h(y)

F homomorphism

y∈M

M =x y∈M

= F (ϕ(h)(x)) = Fˆ (ϕ(h))(x),

deﬁnition ϕ deﬁnition Fˆ

and, hence, P(ϕ(g), ϕ(h)). From the principle of ﬁxed point induction we get τ (Fˆ (f )) = Fˆ (τ (f )) for all antitone functions f . The previous lemma gives us the following theorem: Theorem 5.33 Let R1 and R2 be Dedekind categories with AB = ⊥ ⊥AB for all objects A and B, L a complete Brouwerian lattice with 0 = 1 and F : R1 → R2 a homomorphism. Then the restriction of Fˆ to antimorphisms is L a homomorphism between the arrow categories RL 1 and R2 . Furthermore, if F ˆ is faithful, then so is F . Proof. First of all, Fˆ (f ) = Fˆ (τ (f )) = τ (Fˆ (f )) implies that Fˆ (f ) is an antimorphism if f is. To show the ﬁrst assertion we just prove that Fˆ preserves

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composition. All other properties are shown analogously using Lemma 5.32 (1)–(8) Fˆ (f ··, g) = Fˆ (τ (f ; g)) = τ (Fˆ (f ; g))

deﬁnition ··, Lemma 5.32(8)

= τ (Fˆ (f ); Fˆ (g)) = Fˆ (f ) ··, Fˆ (g).

Lemma 5.32(1) deﬁnition ··,

Now, suppose F is faithful and Fˆ (f ) = Fˆ (g). Then we conclude F (f (x)) = Fˆ (f )(x) = Fˆ (g)(x) = F (g(x)), and, hence, f (x) = g(x) for all x ∈ L, which shows that Fˆ is faithful. 5.3

ARROW CATEGORIES WITH CUTS

We have already seen that the α-cut theorem might not be valid in an arbitrary arrow category. In such a case, a relation, which is not sliceable cannot be represented by its cuts. The reason simply is, as shown in the examples of Section 5.1, that certain scalars or cuts are missing. But later on, we want to deﬁne new operations on relations based on their representation by cuts. Therefore, we consider the following structure: Deﬁnition 5.34 An arrow category with cuts A is an arrow category so that

R

↓

αA ; (αA \R)

α∈Sc[A]

for all relations R : A → B holds. In other words, an arrow category with cuts is an arrow category so that all relations are sliceable. First of all, in an arrow category with cuts a stronger version of Lemma 5.9 is valid. Theorem 5.35 (α-cut Theorem) Let A be an arrow category with cuts and R : A → B. Then we have ↓ (αA ; (αA \R) ), (1) R = α∈Sc[A]

(2) R↑ =

↓

(αA \R) .

α∈Sc[A] αA =⊥ ⊥AA

Proof. (1) follows immediately from Lemma 5.9(1).

CATEGORIES OF L-FUZZY RELATIONS

123

(2) By Lemma 5.9(2) it remains to show the inclusion . Therefore, consider the following computation: ↑ ↓ ↑ (αA ; (αA \R) ) (1) R = α∈Sc[A]

=

↓ ↑

(αA ; (αA \R) )

Corollary 5.5 (4)

α∈Sc[A]

=

↓

↑ αA ; (αA \R)

α∈Sc[A]

=

↓

(αA \R) .

α∈Sc[A] αA =⊥ ⊥AA

Notice that the axioms of an arrow category with cuts are not independent. Axiom (4) of an arrow category can be derived as follows: ↓ ↑ ↓ Q R = (αA \Q) R↓ Theorem 5.35 (2) αA =⊥ ⊥AA

=

↓

(αA \Q) R↓

αA =⊥ ⊥AA

=

↓

(αA \Q) R↓↑

αA =⊥ ⊥AA

=

↓

(αA \Q) (αA \R↓↑ )

↓

Lemma 5.12 (1)

αA =⊥ ⊥AA

=

((αA \Q) (αA \R↓ ))

↓

Corollary 5.5 (4)

αA =⊥ ⊥AA

=

(αA \(Q R↓ ))

↓

Corollary 4.15 (5)

αA =⊥ ⊥AA ↑

= (Q R↓ ) .

Theorem 5.35 (2)

In a Boolean arrow category with cuts Q↓ can be expressed in terms of .↑ in a similar way as Q↑ can be expressed in terms of .↓ shown in Theorem 5.35 (2). Theorem 5.36 Let A be a Boolean arrow category with cuts. Then we have ↑ (αA ; Q) . Q↓ = α∈Sc[A]

Proof. Consider the following computation: ↑

Q↓ = Q =

αA =⊥ ⊥AA

Lemma 5.7 (1) (αA \Q)

↓

Theorem 5.35 (2)

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GOGUEN CATEGORIES

=

(αA ; Q)

↓

Lemma 4.50 (1)

αA =⊥ ⊥AA

=

(αA ; Q)

↓

αA =⊥ ⊥AA

=

(αA ; Q)

↑

Lemma 5.7 (1)

α∈Sc[A] αA =⊥ ⊥AA

↑

=

(αA ; Q) . α∈Sc[A]

Again, the L-fuzzy relations can be considered as the standard model of our theory. Theorem 5.37 Let L be a complete Brouwerian lattice with 0 = 1. Then L-Rel is an arrow category with cuts. Proof. By Theorem 5.3 L-Rel is an arrow category. Lemma 3.3 (6) and Theorem 3.4 show the assertion. In the rest of this section we want to study the connection between crisp and s-crisp resp. l-crisp relations within an arbitrary arrow category with cuts. First of all, we need the following lemma: Lemma 5.38 Let A be an arrow category with cuts and R : A → B. Then we have ↑

(1) if αA is linear, then R↑ = (αA ; R) , (2) Rl R↑ Rs . Proof. ↑

(1) It is suﬃcient to show R↑ (αA ; R) . First, we have (αA βA ); (βA \R) = αA ; βA ; (βA \R) αA ; R, which proves βA \R (αA βA )\(αA ; R), and, ↓ ↓ ⊥AA iﬀ βA = hence, (βA \R) ((αA βA )\(αA ; R)) . Since αA βA = ⊥ ⊥AA = βA ∈ ScG (A)} is a subset of all ⊥ ⊥AA and the set M := {αA βA | ⊥ nonzero scalars we obtain ↓ (βA \R) Theorem 5.35 (2) R↑ = βA =⊥ ⊥AA

↓

((αA βA )\(αA ; R))

see above

βA =⊥ ⊥AA

=

(γA \(αA ; R))

↓

deﬁnition M

γA ∈M

↓

(γA \(αA ; R))

M subset of all nonzero scalars

γA =⊥ ⊥AA ↑

= (αA ; R) .

Theorem 5.35 (2)

CATEGORIES OF L-FUZZY RELATIONS

(2) From the monotonicity of Φs we get ↓ (αA \R) R↑ =

125

Theorem 5.35 (2)

αA =⊥ ⊥AA

(αA \R)

αA =⊥ ⊥AA

= Φs (R) Φs (µΦs (R)) = µΦs (R) = Rs .

R µΦs (R) µΦs (R) ﬁxed point Theorem 4.41 ↑

Now, suppose αA is linear. Then αA ; (αA \R) R implies (αA ; (αA \R)) ↑ R↑ . By (1) we conclude that (αA \R) R↑ . Using the Galois correspon(αA \R) dence we obtain αA \R R↑↓ = R↑ , and, hence, Φl (R) = αA linear

R↑ . An easy ﬁxed point induction shows µΦl (R) R↑ , which is by Theorem 4.41 equivalent to the assertion. The previous lemma leads to the following connection between the three notions of crispness: Theorem 5.39 Let A be an arrow category with cuts. Then we have the following: (1) All s-crisp relations are crisp. (2) All crisp relations are l-crisp. Proof. (1) Suppose R is s-crisp. Using Lemma 5.38 (2) we conclude R↑ Rs = R R↑ . (2) Suppose R is crisp. Again, using Lemma 5.38 (2) we conclude Rl R↑ = R Rl . The previous theorem is an abstract version of Lemma 4.44 and Lemma 4.46. Furthermore, we have a result similar to Lemma 4.45 and Lemma 4.47 concerning the characterization of those arrow categories with cuts where s-crispness resp. l-crispness and crispness coincide. Theorem 5.40 Let A be an arrow category with cuts. Then the following statements are equivalent: (1) A is linear. (2) All crisp relations are s-crisp, i.e., R↑ = Rs . (3) All l-crisp are crisp, i.e., R↑ = Rl .

126

GOGUEN CATEGORIES

Proof. (1) ⇒ (2): Suppose R is crisp. Then by Theorem 5.39 (2) R is l-crisp, and, hence, s-crisp since all nonzero scalars are linear. ⊥AA (2) ⇒ (1): Let αA be a nonzero scalar and βA a scalar such that αA βA = ⊥ holds. Then we have αA ; βA = αA βA = ⊥ ⊥AA and conclude that βA = ⊥ ⊥AA since ⊥ ⊥AA is crisp, and, hence, s-crisp. (1) ⇒ (3): Suppose R is l-crisp. Then R is s-crisp since all nonzero scalars are linear. Theorem 5.39 (1) shows that R is crisp. ⊥AA . Suppose (3) ⇒ (1): Suppose there is a nonlinear scalar αA , i.e., ¬:αA = ⊥ βA ; Q ¬:αA for a linear scalar βA and a partial identity Q : A → A. Then we ⊥AB . Since βA conclude βA Q = βA ; Q ¬:αA , and, hence, αA βA Q ⊥ is linear we get αA Q ⊥ ⊥AB , and, hence, Q ¬:αA , which shows that ¬:αA is l-crisp. On the other hand ¬:αA is not crisp. We have ¬:αA = ⊥ ⊥AA by the ⊥AA = αA ¬:αA = αA , assumption and ¬:αA = IA since otherwise we have ⊥ ⊥AA . a contradiction to αA = ⊥ 5.4

THE ARROW CATEGORY WITH CUTS OF ANTIMORPHISMS

In this section we want to establish that the arrow category of antimorphisms is indeed an arrow category with cuts. Theorem 5.41 Let L be a complete Brouwerian lattice with 0 = 1 and R be ⊥AB for all objects A and B. Then a simple Dedekind category with AB = ⊥ RL is an arrow category with cuts. Proof. By Lemma 5.31 RL is an arrow category. In order to prove the additional cut property, we deﬁne a function h : Sc[RL ] → RL [A, B] for a given f ∈ RL [A, B] by h(α)(x) := f (δ(α) ∧ x). AB = f (x). Suppose αA (x) = If x = 0, we conclude that (αA ; h(α))(x) ⊥AA ; h(α)(x) = ⊥ ⊥AB ⊥ ⊥AA . Then we get (αA ; h(α))(x) = αA (x); h(α)(x) = ⊥ ⊥AA and x = 0. Then we have γ(δ(α))(x) = f (x). Now, suppose αA (x) = ⊥ ⊥ ⊥AA by Theorem 5.29, and, hence, x ≤ δ(α) by Lemma 5.28 (1). This implies h(α)(x) = f (δ(α) ∧ x) = f (x), and, hence, (αA ; h(α))(x) = αA (x); h(α)(x) = IA ; h(α)(x) = f (x).

deﬁnition ; Corollary 5.27 see above

Together we conclude that αA ; h(α) f , and, hence, αA ··, τ (h(α)) = τ (αA ; τ (h(α))) = τ (αA ; h(α)) τ (f ) = f,

deﬁnition ··, Lemma 4.31 (2) see above f antimorphism

CATEGORIES OF L-FUZZY RELATIONS ↓

127

↓

which shows h(α) τ (h(α)) αA \f and h(αA ) (αA \f ) . Furthermore, for x = 0 we have ↓

↓

(γ(x); h(γ(x)) )(x) = γ(x)(x); h(γ(x)) (x) ↓ ˙ AA (x); h(⊥ ⊥AA ) (x) =⊥ ⊥

deﬁnition ; γ isomorphism ˙ AA (0) = AA ⊥ ⊥

= AA ; AB

↓

and h(⊥ ⊥AA ) antimorphism = AB = f (x)

f antimorphism

and for x = 0 ↓

↓

(γ(x); h(γ(x)) )(x) = γ(x)(x); h(γ(x)) (x)

deﬁnition ;

↓

= IA ; h(γ(x)) (x)

Lemma 5.28 (1) and Corollary 5.27

↓

= h(γ(x)) (x) deﬁnition ↓ deﬁnition h Theorem 5.29

= h(γ(x))(1) = f (δ(γ(x)) 1) = f (x). Together, this implies ↓

f (x) = (γ(x); h(γ(x)) )(x)

see above

↓

τ (γ(x); h(γ(x)) )(x) ↓

= (γ(x) ··, h(γ(x)) )(x)

deﬁnition ··,

↓

(γ(x) ··, (γ(x)\f ) )(x).

see above

Finally we conclude

f

↓

(αA ··, (αA \f ) )

αA ∈Sc[RL ]

τ

(αA

·· ,

(αA \f ) ) ↓

αA ∈Sc[RL ]

=

·

↓

(αA ··, (αA \f ) ).

αA ∈Sc[RL ]

5.5

GOGUEN CATEGORIES

In an arrow category with cuts all relations can be represented by their cuts. One might ask the question whether the converse is also valid. Does every antimorphism from the scalars to the crisp relations naturally give rise to a

128

GOGUEN CATEGORIES

AAE EE x x EE x xx IA G xR GG GG xx x x αA

⊥ ⊥AA αA IA R AA

⊥ ⊥AA ; ⊥ ⊥AA αA IA R AA

. .↑ ⊥ ⊥AA ⊥ ⊥AA αA IA IA IA R AA AA AA

⊥ ⊥AA αA IA R ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA ⊥ ⊥AA αA αA R ⊥ ⊥AA αA IA R ⊥ ⊥AA R R R ⊥ ⊥AA R AA R

.↓ ⊥ ⊥AA ⊥ ⊥AA IA ⊥ ⊥AA AA

AA ⊥ ⊥AA R AA R AA

The arrow category A.

Figure 5.4.

relation in the category? Of course, such an antimorphism f can be map to αA ; f (α) but the question remains whether the set of cuts the relation α∈Sc[G]

of this relation correspond to the image of the antimorphism f . Unfortunately, this is not the case as our example will show. Example 5.42 Consider the one-object arrow category A induced by the deﬁnitions in Figure 5.4. It is easy to verify that this structure is indeed an arrow category with cuts. ⊥AA , IA , AA } is the set Obviously, {⊥ ⊥AA , αA , IA } is the set of scalars, and {⊥ of crisp relations. The function f deﬁned by AA , f (⊥ ⊥AA ) :=

f (αA ) := AA ,

f (IA ) := IA

is an antimorphism and we have βA ; f (β) = ⊥ ⊥AA ; AA αA ; AA IA ; IA = ⊥ ⊥AA R IA = AA . β∈Sc[A]

This implies that the I-cut of

βA ; f (β) is . But, on the other hand, we

β∈Sc[A]

have AA IA = f (IA ).

The observation above give rise to the following deﬁnition: Deﬁnition 5.43 A Goguen category G is an arrow category with cuts so that αA ; f (α) =⇒ R f (β) βA ; R ↑ α∈Sc[G]

CATEGORIES OF L-FUZZY RELATIONS

129

for all relations R : A → B, scalars β ∈ Sc[G] and antimorphisms f : Sc[G] → G ↑ [A, B] holds. The two additional properties of a Goguen category (with respect to an arrow category) can be expressed by a single equivalence. Notice that this equivalence was originally used to deﬁne Goguen categories [40, 41, 42, 43]. Lemma 5.44 Let A be an arrow category. Then the following statements are equivalent: (1) A is a Goguen category. (2) For all relations R : A → B and antimorphisms f : Sc[A] → A↑ [A, B] we have ↓ αA ; f (α) ⇐⇒ (βA \R) f (β) for all β ∈ Sc[A]. R α∈Sc[A]

Proof. (1) ⇒ (2) : Suppose R

αA ; f (α). Then we have

α∈Sc[A] ↓

X (βA \R) ⇔ X ↑ βA \R

Galois correspondence

↑

⇔ βA ; X R ⇒ βA ; X ↑

αA ; f (α)

α∈Sc[A]

⇒ X f (β),

Goguen category

↓

and, hence, (βA \R) f (β). In order to prove the other implication suppose ↓ (αA \R) f (α) for all α ∈ Sc[A]. Since A is an arrow category with cuts we conclude that ↓ αA ; (αA \R) αA ; f (α). R α∈Sc[A]

α∈Sc[A] ↓

(2) ⇒ (1) : Let be R ∈ A[A, B]. Since (αA \R) is crisp and ↓ ↓ (αA \R) = (αA \R) Corollary 5.5 (4) α∈M

α∈M

=

M \R

↓ Corollary 4.22 (4) ↓

the function f : Sc[A] → A↑ [A, B] deﬁned by f (α) = (αA \R) is an antimorphism. We conclude thatA is an arrow category with cuts from 2. ⇐. αA ; f (α). If βA = ⊥ ⊥AA , the assertion is trivial. Suppose βA ; R↑ α∈Sc[A] ↓

⊥AA . Then 2. ⇒ implies (βA \(βA ; R↑ )) f (β). Lemma Now, suppose βA = ⊥ ↓ 5.20 gives us (β\(β; R↑ )) = R↑ , and, hence, R R↑ f (β).

130

GOGUEN CATEGORIES

We will use Property 2. of the previous lemma as an alternative deﬁnition of Goguen categories without mentioning. The inclusions on both sides of the equivalence can be replaced by an equality as the next lemma shows. Lemma 5.45 Let G be a Goguen category. Then we have ↓ αA ; f (α) ⇐⇒ (βA \R) = f (β) for all β ∈ Sc[G]. R= α∈Sc[G]

for all relations R : A → B and antimorphisms f : Sc[G] → G ↑ [A, B]. Proof. In order to prove the implication ⇒ it is suﬃcient to show that ↓ (αA ; f (α)) = R, f (β) (βA \R) for all scalars βA . We have βA ; f (β) α∈Sc[G] ↑

which implies f (β) βA \R. Since f (β) is crisp we obtain f (β) βA \R, and, ↓ hence, f (β) (βA \R) . (αA ; f (α)) R. This follows from ⇐: It is suﬃcient to show that α∈Sc[G] ↓

αA ; f (α) = αA ; (αA \R) αA ; (αA \R) R.

One might ask whether an arrow category with cuts can be embedded into a suitable Goguen category. The general answer to this question is no. Example 5.46 Consider again the arrow category with cuts A deﬁned in Example 5.42. Suppose A can be embedded into a Goguen category G. Then we identify A with its image in G and consider A to be a substructure of G. Now, consider the function fˆ : Sc[G] → G ↑ [A, A] deﬁned by

AA iﬀ βA αA ˆ f (β) = IA otherwise Notice that the restriction of fˆ to A is exactly the antimorphism f . It is easy to verify that fˆ is also an antimorphism. As before, we have AA IA ; IA βA ; fˆ(β) AA = R IA = αA ; β∈Sc[G]

↑AA so that IA ;

βA ; fˆ(β) follows. But we have AA IA = fˆ(IA ).

β∈Sc[G]

We want to prove a lemma corresponding to Lemma 4.20. A homomorphism F between Goguen categories is a homomorphism between Dedekind categories ↑ so that the operation .↑ and .↓ are preserved, i.e., F (R↑ ) = F (R) and F (R↓ ) = ↓ F (R) . Lemma 5.47 Let F : G1 → G2 be a pre-functor between Goguen categories, which is full, faithful, and bijective on objects. Furthermore, suppose

CATEGORIES OF L-FUZZY RELATIONS

131

(1) either F or F −1 respects composition, (2) either F or F −1 respects the converse operation, (3) either F or F −1 respects the

↓

operation,

(4) either F or F −1 is a complete lower semilattice homomorphism for every pair of objects, and (5) either F or F −1 is an upper semilattice homomorphism for every pair of objects. Then F is an isomorphism. Proof. By Lemma 4.20, F is an isomorphism between Dedekind categories. ↓ Now, suppose without loss of generality that F (R↓ ) = F (R) for all R in G1 . Then the computation ↓ Lemma 5.35 (2) (αA \R) F (R↑ ) = F αA =⊥ ⊥AA

=

↓

F ((αA \R) )

F homomorphism

αA =⊥ ⊥AA

=

↓

F (αA \R)

assumption

αA =⊥ ⊥AA

=

↓

(F (αA )\F (R))

F homomorphism

αA =⊥ ⊥AA

=

↓

(βA \F (R))

F bijective

βA =⊥ ⊥AA ↑

= F (R)

shows that F preserves ↑ . From ↑

F −1 (S ↑ ) = F −1 (F (F −1 (S)) ) ↑

= F −1 (F (F −1 (S) )) =F

−1

and a similar computation for 5.6

↓

↑

(S)

F bijective F see above F bijective

we conclude the assertion.

THE GOGUEN CATEGORY OF ANTIMORPHISMS

In this section we want to establish that the arrow category of antimorphisms is indeed a Goguen category. Theorem 5.48 Let L be a complete Brouwerian lattice with 0 = 1 and R be a simple Dedekind category with AB = ⊥ ⊥AB for all objects A and B. Then RL is a Goguen category.

132

GOGUEN CATEGORIES

Proof. By Theorem 5.41 RL is an arrow category with cuts. Now, suppose anti · (αA ··, h(αA )) for h : ScRL (A) → RL [A, B] such that βA ··, f ↑ αA ∈ScRL (A)

˙ AA , the assertion is trivial. ⊥ h(αA ) is crisp for all αA ∈ ScRL (A). If βA = ⊥ ˙ ⊥AA . Then we get Suppose βA = ⊥ (βA ··, f ↑ )(x) = (βA ; f ↑ )(x)

Lemma 5.25 (2)

= βA (x); f ↑ (x),

deﬁnition ;

⎧ ⎪ ⎨ AB ↑ f (y) ·· (βA , f )(x) = ⎪ ⎩ y =0 ⊥ ⊥AB

such that

iﬀ x = 0 iﬀ βA (x) = IA else.

⊥AA . Furtherfollows. Since βA βA Lemma 5.28 (2) gives us βA (δ(βA )) = ⊥ ˙ AA and δ is an isomorphism. By Corollary 5.27 ⊥ more, δ(βA ) = 0 since βA = ⊥ f (y). we conclude βA (δ(βA )) = IA , and, hence, (βA ··, f ↑ )(δ(βA )) = y =0 ↑ h(αA )(y) we conclude Since h(αA ) = h(αA ) = R˙ for R = y =0

αA ∈ScRL (A)

(αA

=

·· ,

h(αA )) (x)

(αA ··, h(αA ))(x)

deﬁnition

(αA ; h(αA ))(x),

Lemma 5.25 (2)

αA ∈ScRL (A)

=

αA ∈ScRL (A)

such that by Corollary 5.27

(αA

·· ,

h(αA )) (x) =

AB αA (x) = ⊥ ⊥AA

αA ∈ScRL (A)

h(αA )(x)

iﬀ x = 0 else

follows. Since h(αA ) iscrisp and 0 = 1 we have h(αA )(x) = h(αA )(1) for all (αA ··, h(αA )) = Q˙ with Q = h(αA )(1). x = 0, and, hence, αA ∈ScRL (A)

αA (x) = ⊥ ⊥AA

˙ Now, we want to show that Q(x) = h(γ(x))(1). Suppose x = 0. Then we have ˙ AB (1) = ˙ AA )(1) = ˙ AB = Q(0) since γ is an isomorphism h(γ(0))(1) = h(⊥ ⊥ ⊥AA , we conclude αA (δ(γ(x))) = and h an antimorphism. If x = 0 and αA (x) = ⊥ ⊥ ⊥AA since γ and δ are inverse, and, hence, γ(x) αA by Lemma 5.28 (2). This h(αA )(1) h(γ(x))(1). implies h(αA ) h(γ(x)), and, hence, Q = αA (x) = ⊥ ⊥AA

⊥AA . Together The other inclusion follows from γ(x)(x) = IA = ⊥ immediately (αA ··, h(αA )))(x) = h(γ(x))(1). we have shown that ( αA ∈ScRL (A)

CATEGORIES OF L-FUZZY RELATIONS

133

Now, consider the following computation: Since γ is an isomorphism of complete lattices we have

ϕ

αA ∈ScRL (A)

=

(αA ··, h(αA )) (x)

⎛

⎝

⎛

h⎝

M =x

=

(αA ··, h(αA ))⎠ (y)

h(γ(y))(1)

M =x y∈M

=

⎞

deﬁnition ϕ

αA ∈ScRL (A)

M =x y∈M

=

see above

⎞ γ(y)⎠ (1)

h antimorphism

y∈M

h γ M (1)

γ homomorphism

M =x

= h(γ(x))(1) ⎛ =⎝

⎞ (αA ··, h(αA ))⎠ (x),

see above

αA ∈ScRL (A)

·

and, hence,

αA ∈ScRL (A)

(αA ··, h(αA ))

αA ∈ScRL (A)

(αA ··, h(αA )).

Finally, we obtain the assertion as follows: The inclusion f (0) h(βA )(0) is trivial and for x = 0 we get f (x)

f (y)

y =0

= (βA ··, f ↑ )(δ(βA )) ⎞ ⎛ (αA ··, h(αA ))⎠ (δ(βA )) ⎝ · αA ∈ScRL (A)

⎛ ⎝

see above assumption

⎞ (αA ··, h(αA ))⎠ (δ(βA ))

see above

αA ∈ScRL (A)

= h(γ(δ(βA )))(1) = h(βA )(1) ↓

see above γ and δ are inverse ↓

= h(βA ) (x)

deﬁnition

= h(βA )(x).

h(βA ) crisp

134 5.7

GOGUEN CATEGORIES

REPRESENTATION OF GOGUEN CATEGORIES

Suppose G is a Goguen category. By Lemma 4.35 Sc[G] is a complete Brouwerian lattice with 0 = 1 since ⊥ ⊥AA = AA for all objects A. Furthermore, by AB = ⊥ ⊥AB for all Corollary 5.17 G ↑ is a simple Dedekind category with objects A and B. Theorem 5.48 shows that the category of antimorphisms Sc[G] is again a Goguen category. In this section, we want to prove that G↑ Sc[G] G and G ↑ are isomorphic, i.e., we prove a pseudo-representation theorem for Goguen categories. Afterwards, we show that the representation theory of Goguen categories is equivalent to the representation theory of the simple Dedekind categories. This result allows us to transfer known representation results to the theory of Goguen categories. Lemma 5.49 Let G be a Goguen category and f : ScG (A) → G ↑ [A, B]. Then we have ↓ (αA ; f (αA )) , (1) τ (f )(βA ) = βA \ (2)

αA ∈ScG (A)

αA ∈ScG (A)

(αA ; τ (f )(αA )) =

αA ∈ScG (A)

(αA ; f (αA )).

Proof. (1) Let f˜ be deﬁned by the right-hand side of the assertion, i.e., ⎛ ⎛ ⎞⎞↓ f˜(βA ) := ⎝βA \ ⎝ (αA ; f (αA ))⎠⎠ . αA ∈ScG (A)

Then f f˜ since βA ; f (βA )

αA ∈ScG (A)

(αA ; f (αA ))

↑

⇔ βA ; f (βA )

(αA ; f (αA ))

αA ∈ScG (A)

⎛

↑

⇔ f (βA ) βA \ ⎝ ⎛

⎞ (αA ; f (αA ))⎠

αA ∈ScG (A)

⎛

⇔ f (βA ) ⎝βA \ ⎝

f (βA ) is crisp

deﬁnition residual

⎞⎞↓ (αA ; f (αA ))⎠⎠ .

Galois correspondence

αA ∈ScG (A)

Furthermore, the computation ⎛ ⎛ ⎝βA \ ⎝

f˜(βA ) = βA ∈M

βA ∈M

αA ∈ScG (A)

⎞ ⎞↓ (αA ; f (αA ))⎠⎠

deﬁnition f˜

135

CATEGORIES OF L-FUZZY RELATIONS

⎛

⎛

=⎝

=⎝ = f˜

⎞⎞⎞↓

⎝βA \ ⎝ βA ∈M

⎛

⎛

(αA ; f (αA ))⎠⎠⎠

αA ∈ScG (A)

⎛

M \⎝

Corollary 5.5 (4)

⎞⎞↓

(αA ; f (αA ))⎠⎠

Corollary 4.22 (4)

αA ∈ScG (A)

deﬁnition f˜

M

shows that f˜ is an antimorphism. Together we have τ (f ) τ (f˜) = f˜. For the other inclusion consider the following computation: X f˜(βA ) ⎛

⎛

⇔ X ⎝βA \ ⎝ ⎛

⎞⎞↓

(αA ; f (αA ))⎠⎠

αA ∈ScG (A)

⇔ X ↑ βA \ ⎝

deﬁnition f˜

⎞

(αA ; f (αA ))⎠

Galois correspondence

αA ∈ScG (A)

⇔ βA ; X ↑

(αA ; f (αA ))

deﬁnition residual

(αA ; τ (f )(αA ))

f τ (f )

αA ∈ScG (A)

⇒ βA ; X ↑

αA ∈ScG (A)

⇒ X ↑ τ (f )(βA ) ⇔ X τ (f )(βA )

Property (5b) for τ (f )

↓

Galois correspondence

⇔ X τ (f )(βA ),

τ (f )(βA ) is crisp

which shows f˜ τ (f ). (2) The inclusion is trivial. The other inclusion is shown as follows: (αA ; τ (f )(αA )) αA ∈ScG (A)

=

αA ∈ScG (A)

αA ∈ScG (A)

αA ∈ScG (A)

=

αA ∈ScG (A)

⎛

⎛

⎛

⎝αA ; ⎝αA \ ⎝ ⎛

⎛

⎛

⎝αA ; ⎝αA \ ⎝ ⎛ ⎝

⎞⎞⎞↓

(βA ; f (βA ))⎠⎠⎠

βA ∈ScG (A)

(1)

⎞⎞⎞ (βA ; f (βA )⎠⎠⎠

βA ∈ScG (A)

⎞

(βA ; f (βA ))⎠

βA ∈ScG (A)

(αA ; f (αA )).

136

GOGUEN CATEGORIES

Now, we deﬁne the pseudo-representation functor PG : G → G ↑ PG (A) := A PG (R)(α) := (αA \R)

PG−1 (A) := A PG−1 (f )

:=

by

for all objects A, ↓

and its inverse PG−1 : G ↑

Sc[G]

Sc[G]

for all R : A → B and α ∈ Sc[G],

→ G by for all objects A,

anti

for all f : Sc[G] → G ↑ [A, B].

(αA ; f (α))

α∈Sc[G]

Using these deﬁnitions we are able to state our main theorem. Theorem 5.50 (Pseudo-Representation Theorem) Let G be a Goguen Sc[G] are isomorphic via PG . category. Then G and G ↑ Proof. Since ↓

(αA \R)

PG (R)(α) = αA ∈M

αA ∈M

deﬁnition PG ↓

(αA \R)

=

Corollary 5.5 (4)

αA ∈M

↓ M \R M = PG (R) =

Corollary 4.22 (4) deﬁnition PG

PG (R) is an antimorphism, and, hence, PG well-deﬁned. Suppose PG (R) = PG (S). Then we have R=

↓

(αA ; (αA \R) )

Theorem 5.35

α∈Sc[G]

=

(αA ; PG (R)(α))

deﬁnition PG

α∈Sc[G]

=

(αA ; PG (S)(α))

α∈Sc[G]

=

↓

(αA ; (αA \S) )

deﬁnition PG

α∈Sc[G]

= S,

Theorem 5.35

CATEGORIES OF L-FUZZY RELATIONS

137

anti

which shows that PG is faithful. Now, suppose f : Sc[G] → G ↑ [A, B] and compute ⎛ PG ⎝

⎞

⎛

⎛

(αA ; f (α))⎠ (β) = ⎝βA \ ⎝

α∈Sc[G]

⎞ ⎞↓ (αA ; f (α))⎠⎠

deﬁnition PG

α∈Sc[G]

= τ (f )(β)

Lemma 5.49 (1)

= f (β)

f antimorphism

for every scalar β ∈ Sc[G], which shows that PG is full and that the inverse of PG is indeed PG−1 . Now, we want to show ﬁve properties required by Lemma 5.47. (1) First, the following computation:

(αA ; f (α); g(α))

α∈Sc[G]

=

(αA ; f (α); αB ; g(α))

α∈Sc[G]

⎛

⎝

⎞ ⎛

(αA ; f (α))⎠ ; ⎝

α∈Sc[G]

=

⎛

=

⎝αA ; f (α); ⎝

⎞

(βB ; g(β))⎠

β∈Sc[G]

⎛

α∈Sc[G]

Lemma 5.14

⎞⎞

(βB ; g(β))⎠⎠

β∈Sc[G]

(αA ; f (α); βB ; g(β))

α,β∈Sc[G]

=

(αA ; βA ; f (α); g(β))

Lemma 5.14

α,β∈Sc[G]

=

((αA βA ); f (α); g(β))

α,β∈Sc[G]

((αA βA ); f (α β); g(α β))

f and g are antitone

α,β∈Sc[G]

=

(αA ; f (α); g(α))

α∈Sc[G]

implies

(αA ; f (α); g(α)) = (

α∈Sc[G]

(αA ; f (α))); (

α∈Sc[G]

(βB ; g(β))).

β∈Sc[G]

138

GOGUEN CATEGORIES

We conclude PG−1 (f ··, g) =

deﬁnition PG−1

(αA ; (f ··, g)(α))

α∈Sc[G]

=

deﬁnition ··,

(αA ; τ (f ; g)(α))

α∈Sc[G]

=

(αA ; (f ; g)(α))

Lemma 5.49 (2)

α∈Sc[G]

=

(αA ; f (α); g(α))

α∈Sc[G]

⎛ =⎝

deﬁnition ;

⎞ ⎛

(αA ; f (α))⎠ ; ⎝

α∈Sc[G]

⎞ (βB ; g(β))⎠

see above

β∈Sc[G]

= PG−1 (f ); PG−1 (g).

deﬁnition PG−1

(2) We immediately conclude that ↓

PG (Q )(α) = (αA \Q ) = (αB \Q)

deﬁnition PG

↓

= (PG (Q)(α))

Lemma 5.15

= PG (Q) (α).

deﬁnition PG deﬁnition

(3) First of all, we have ↓

⊥) = AB = PG (Q) (⊥ ⊥) PG (Q↓ )(⊥ since both functions are antimorphisms. Now, suppose α = ⊥ ⊥. Then we have PG (Q↓ )(α) = (αA \Q↓ )

↓

↓

=Q

deﬁnition PG Lemma 5.12 (1)

↓

= (IA \Q) = PG (Q)(I) ↓

= PG (Q) (α).

Lemma 4.16 (1) deﬁnition PG deﬁnition

↓

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139

(4) We immediately conclude that

i∈I

↓

αA \

(α) =

Ri

PG

Ri i∈I

deﬁnition PG

↓

(αA \Ri )

=

Corollary 4.22 (3)

i∈I

(αA \Ri )

=

↓

Corollary 5.5 (4)

i∈I

=

PG (Ri )(α).

deﬁnition PG

i∈I

(5) Again, we immediately conclude that −1 · fi = αA ; · fi (α) PG i∈I

α∈Sc[G]

=

αA ;

α∈Sc[G]

=

αA ; τ

α∈Sc[G]

=

i∈I

(α)

fi

i∈I

deﬁnition PG−1 deﬁnition ·

fi (α)

Lemma 5.49 (2)

i∈I

(αA ; fi (α))

α∈Sc[G] i∈I

=

(αA ; fi (α))

i∈I α∈Sc[G]

=

PG−1 (fi ).

deﬁnition PG−1

i∈I

The extension PˆG of the pseudo-representation functor gives us the following: Theorem 5.51 A Goguen category G is representable iﬀ G ↑ is representable. Proof. The implication ⇒ is trivial since crisp relations are represented by 0-1 crisp relations. Now, suppose there is an embedding F : G ↑ → Rel. Then Sc[G] and RelSc[G] . By Theorem 5.33 implies that Fˆ is an embedding between G ↑ Sc[G] ↑ Sc[G] Theorem 5.50 G and G as well as Rel and Sc[G]-Rel are isomorphic −1 is an embedding from G to Sc[G]-Rel. such that PG ◦ Fˆ ◦ PSc[G]-Rel 5.8

BOOLEAN-BASED GOGUEN CATEGORIES

The axioms of an arrow category, and, hence, of a Goguen category are somehow ↑ asymmetric. The property (↑ ) Q↑ R↓ = (Q R↓ ) is an axiom of this theory

140

GOGUEN CATEGORIES

|| || y1 C CC {{ C { {

zi := {(j, j) | 0 < j ≤ i} ∪ {(j, k) | j, k > 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

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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

2.

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

3.

H. Wansing: Displaying Modal Logic. 1998

ISBN 0-7923-5205-X

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

ISBN 0-7923-5238-6

5.

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

6.

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

7.

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

ISBN 0-7923-5720-5

8.

E.P. Klement, R. Mesiar and E. Pap: Triangular Norms. 2000 ISBN 0-7923-6416-3

9.

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

10.

J. Czelakowski: Protoalgebraic Logics. 2001

ISBN 0-7923-6940-8

11.

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

12.

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

ISBN 1-4020-0604-7

13.

F. Paoli: Substructural Logics: A Primer. 2002

ISBN 1-4020-0605-5

14.

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

15.

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

16.

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

17.

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

19.

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

20.

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

22.

M. Dalla Chiara, R. Giuntini and R Greechie: Reasoning in Quantum Theory. Sharp and Unsharp Quantum Logics. 2004 ISBN 1-4020-1978-5

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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