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**Fuzzy relational systems. Foundations and principles.**
*(English)*
Zbl 1067.03059

IFSR International Series on Systems Science and Engineering 20. New York, NY: Kluwer Academic Publishers (ISBN 0-306-46777-1/hbk). xii, 369 p. (2002).

Publisher’s description: This book deals with fuzzy relational systems, i.e. with systems of fuzzy relations defined on a set. Fuzzy relational systems represent a mathematical framework for fuzzy relational modeling 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 the 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 the classical “non-fuzzy” case. Chapter 4 deals with binary fuzzy relations and particularly with similarity and order, two 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 gives a 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 a bibliography and an index of key terms. The book provides a general framework for dealing with fuzzy relational systems and brings several new results.

### MSC:

03E72 | Theory of fuzzy sets, etc. |

03-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematical logic and foundations |

93A30 | Mathematical modelling of systems (MSC2010) |

03B52 | Fuzzy logic; logic of vagueness |

06A15 | Galois correspondences, closure operators (in relation to ordered sets) |