Fuzzy Relational Systems: Foundations and Principles / Edition 1

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More About This Textbook

Overview

Fuzzy Relational Systems: Foundations and Principles presents a general theory of fuzzy relational systems and concentrates on selected general issues of fuzzy relational modeling in the framework of the developed theory. The text discusses phenomena hidden in the ordinary bivalent case, as well as new topics in fuzzy relational systems, such as object-attribute fuzzy relations and fuzzy concept latices, similarity, and fuzzy closure operators. Both mathematicians and engineers will find Fuzzy Relational Systems to be an invaluable teaching and reference resource in modeling and fuzzy logic.
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Product Details

Table of Contents

1 Preliminaries 1
1.1 Sets, relations, mappings 1
1.2 Ordered sets, lattices 3
1.3 Tolerances, equivalences 5
1.4 Algebras 6
2 Fuzzy Approach, Graded Truth, and Structures of Truth Values 11
2.1 Fuzzy approach: from resolving paradoxes to engineering applications 12
2.2 Graded truth and structures of truth values 25
2.3 Residuated lattices 27
2.4 Residuated lattices and t-norms 62
3 Fuzzy Structures 75
3.1 Fuzzy sets and fuzzy relations 76
3.2 Fuzzy logic: syntax and semantics 96
3.3 Similarity issues 110
3.4 Morphisms, products, substructures 125
3.5 Representation by crisp structures: cutlike semantics 154
3.6 Algebras with fuzzy equalities 159
3.7 Many-sorted case 173
4 Binary Fuzzy Relations 181
4.1 Operations and properties of binary fuzzy relations 182
4.2 Similarity: fuzzy equivalences 193
4.3 Hierarchy: fuzzy order 203
5 Object-Attribute Fuzzy Relations and Fuzzy Concept Lattices 215
5.1 Conceptual granulation: fuzzy context, fuzzy concepts, and fuzzy concept lattice 215
5.2 Similarity and logical precision 222
5.3 Fuzzy Galois connections 244
5.4 Main theorem of fuzzy concept lattices 261
6 Composition and Decomposition of Fuzzy Relations 273
6.1 Composition of fuzzy relations 273
6.2 Decomposition: fuzzy relational equations 279
7 Miscellanea 299
7.1 Fuzzy closure operators and similarity spaces 299
7.2 Applications of selected results to some problems and models 339
7.3 Fuzzy logic with truth degrees in residuated lattices 350
References 357
Index 367
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