Trends in Logic

A Categorical Approach to L-fuzzy Relations

By: Michael Winter

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Hardcover | 24 July 2007

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Hardcover
224 Pages

Dimensions(cm)
23.5 x 15.88 x 1.91

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Goguen categories extend the relational calculus and its categorical formalization to the fuzzy world. Starting from the fundamental concepts of sets, binary relations and lattices this book introduces several categorical formulations of an abstract theory of relations such as allegories, Dedekind categories and related structures. It is shown that neither theory is sufficiently rich to describe basic operations on fuzzy relations. The book then introduces Goguen categories and provides a comprehensive study of these structures including their representation theory, and the definability of norm-based operations.

The power of the theory is demonstrated by a comprehensive example. A certain Goguen category is used to specify and to develop a fuzzy controller. Based on its abstract description as well as certain desirable properties and their formal proofs, a verified controller is derived without compromising the - sometimes - intuitive choice of norm-based operations by fuzzy engineers.

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Introduction
Sets, Relations, and Functions	p. 1
Lattices	p. 5
Galois correspondences and residuated operations	p. 13
Distributive lattices	p. 17
Brouwerian lattices	p. 18
Boolean algebras	p. 20
Special elements	p. 22
Fixed points	p. 23
The complete Brouwerian lattice of antimorphisms	p. 25
Filters	p. 31
Lattice-ordered semigroups	p. 40
L-Fuzzy Relations	p. 43
Basic operations and properties	p. 43
Crispness	p. 48
Operations derived from lattice-ordered semigroups	p. 52
Categories of Relations	p. 55
Categories	p. 55
Allegories	p. 57
Distributive allegories	p. 63
Division allegories	p. 65
Dedekind categories	p. 68
Relational constructions in Dedekind categories	p. 74
The Dedekind category of antimorphisms	p. 75
Scalars and crispness in Dedekind categories	p. 79
Schroder categories	p. 85
Formal languages of relational categories	p. 86
Categories of L-Fuzzy Relations	p. 93
Arrow categories	p. 94
The arrow category of antimorphisms	p. 112
Arrow categories with cuts	p. 122
The arrow category with cuts of antimorphisms	p. 126
Goguen categories	p. 127
The Goguen category of antimorphisms	p. 131
Representation of Goguen categories	p. 134
Boolean-based Goguen categories	p. 139
Equations in Goguen categories	p. 142
Operations derived from lattice-ordered semigroups	p. 150
Fuzzy Controllers in Goguen Categories	p. 169
The Mamdani approach to fuzzy controllers	p. 169
Linguistic entities and variables	p. 170
Fuzzification	p. 176
The rule base	p. 176
Decision module	p. 179
Defuzzification	p. 179
Proving properties of a controller	p. 182
Discussion of the approach	p. 194
Index	p. 197
Symbols	p. 201
References	p. 203
Table of Contents provided by Blackwell. All Rights Reserved.

Trends in Logic

A Categorical Approach to L-fuzzy Relations

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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