"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 a 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.
Author(s): Winter M.
Series: Trends in Logic 25
Year: 2007
Language: English
Pages: 217
CONTENTS......Page 7
INTRODUCTION......Page 9
1. SETS, RELATIONS, AND FUNCTIONS......Page 13
2. LATTICES......Page 16
2.1 Galois correspondences and residuated operations......Page 24
2.2 Distributive lattices......Page 28
2.3 Brouwerian lattices......Page 29
2.4 Boolean algebras......Page 31
2.5 Special elements......Page 33
2.6 Fixed points......Page 34
2.7 The complete Brouwerian lattice of antimorphisms......Page 36
2.8 Filters......Page 42
2.9 Lattice-ordered semigroups......Page 51
3.1 Basic operations and properties......Page 53
3.2 Crispness......Page 58
3.3 Operations derived from lattice-ordered semigroups......Page 62
4.1 Categories......Page 65
4.2 Allegories......Page 67
4.3 Distributive allegories......Page 73
4.4 Division allegories......Page 75
4.5 Dedekind categories......Page 78
4.6 Relational constructions in Dedekind categories......Page 84
4.7 The Dedekind category of antimorphisms......Page 85
4.8 Scalars and crispness in Dedekind categories......Page 89
4.9 Schröder categories......Page 95
4.10 Formal languages of relational categories......Page 96
5. CATEGORIES OF L-FUZZY RELATIONS......Page 103
5.1 Arrow categories......Page 104
5.2 The arrow category of antimorphisms......Page 122
5.3 Arrow categories with cuts......Page 132
5.4 The arrow category with cuts of antimorphisms......Page 136
5.5 Goguen categories......Page 137
5.6 The Goguen category of antimorphisms......Page 141
5.7 Representation of Goguen categories......Page 144
5.8 Boolean-based Goguen categories......Page 149
5.9 Equations in Goguen categories......Page 152
5.10 Operations derived from lattice-ordered semigroups......Page 160
6.1 The Mamdani approach to fuzzy controllers......Page 179
6.2 Linguistic entities and variables......Page 180
6.4 The rule base......Page 186
6.6 Defuzzification......Page 189
6.7 Proving properties of a controller......Page 192
6.8 Discussion of the approach......Page 204
C......Page 206
L......Page 207
U......Page 208
Z......Page 209
SYMBOLS......Page 210
REFERENCES......Page 212
