Overview

"This study deals with fuzzy relational systems, in other words, systems of fuzzy relations defined on a set. Fuzzy relational systems represent a mathematical framework for fuzzy relational modelling which is the most successful part of fuzzy logic. The book deals with foundational aspects of fuzzy relational systems. It starts (Chapter 2) with motivations and discussions about fuzzy approach. The result of this are some requirements for the structures of truth values for fuzzy logic. These structures are analyzed in subsequent sections. Chapter 3 is a key one and develops a general theory of fuzzy relational systems, paying special attention to issues which are degenerate in classical ""non-fuzzy"" case. Chapter 4 deals with binary fuzzy relations and particularly with similarity and order, two of the most frequently used types of binary relations. Chapter 5 deals with binary fuzzy relations (interpreted as fuzzy relations between a set of objects and a set of attributes) and formal analysis of such relations. Chapter 6 focuses on the problem of composition and decomposition of binary fuzzy relations. Chapter 7 contains miscellaneous topics: fuzzy closure operators; similarity spaces; selected applications; and a formal deductive system of fuzzy logic. Each Chapter is closed by bibliographical remarks. The book contains bibliography and an index of key terms. The book provides a general framework for dealing with fuzzy relational systems and brings several new results."

Full Product Details

Author:   Radim Belohlávek
Publisher:   Springer Science+Business Media
Imprint:   Kluwer Academic/Plenum Publishers
Edition:   2002 ed.
Volume:   20
Dimensions:   Width: 17.80cm , Height: 2.20cm , Length: 25.40cm
Weight:   2.090kg
ISBN:  

9780306467776

ISBN 10:   0306467771
Pages:   369
Publication Date:   30 June 2002
Audience:   College/higher education ,  Professional and scholarly ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

1. Preliminaries.- 1.1 Sets, relations, mappings.- 1.2 Ordered sets, lattices.- 1.3 Tolerances, equivalences.- 1.4 Algebras.- 2. Fuzzy Approach, Graded Truth, and Structures of Truth Values.- 2.1 Fuzzy approach: from resolving paradoxes to engineering applications.- 2.2 Graded truth and structures of truth values.- 2.3 Residuated lattices.- 2.4 Residuated lattices and t-norms.- 2.5 Bibliographical remarks.- 3. Fuzzy Structures.- 3.1 Fuzzy sets and fuzzy relations.- 3.2 Fuzzy logic: syntax and semantics.- 3.3 Similarity issues.- 3.4 Morphisms, products, substructures.- 3.5 Representation by crisp structures: cutlike semantics.- 3.6 Algebras with fuzzy equalities.- 3.7 Many-sorted case.- 3.8 Bibliographical remarks.- 4. Binary Fuzzy Relations.- 4.1 Operations and properties of binary fuzzy relations.- 4.2 Similarity: fuzzy equivalences.- 4.3 Hierarchy: fuzzy order.- 4.4 Bibliographical remarks.- 5. Object-Attribute Fuzzy Relations and Fuzzy Concept Lattices.- 5.1 Conceptual granulation: fuzzy context, fuzzy concepts, and fuzzy concept lattice.- 5.2 Similarity and logical precision.- 5.3 Fuzzy Galois connections.- 5.4 Main theorem of fuzzy concept lattices.- 5.5 Bibliographical remarks.- 6. Composition and Decomposition of Fuzzy Relations.- 6.1 Composition of fuzzy relations.- 6.2 Decomposition: fuzzy relational equations.- 6.3 Bibliographical remarks.- 7. Miscellanea.- 7.1 Fuzzy closure operators and similarity spaces.- 7.2 Applications of selected results to some problems and models.- 7.3 Fuzzy logic with truth degrees in residuated lattices.- 7.4 Bibliographical remarks.- References.

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