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**Fuzzy equational logic.**
*(English)*
Zbl 1083.03030

Studies in Fuzziness and Soft Computing 186. Berlin: Springer (ISBN 3-540-26254-7/hbk). xii, 283 p. (2005).

This volume starts with an excellent introduction to the basics of classical set theory, relational structures and in particular order relations and equivalence relations, ending with the concept of a structure and in particular of an algebra. In a second section complete residuated lattices are advocated as suitable frameworks for the representation of graded or partial truth domains. Afterwards important concepts such as truth stresser (to represent the hedge ‘very’) and triangular norms are introduced and extensively illustrated. In the third section of chapter 1 the authors introduce the basic notions about fuzzy sets (in fact \(L\)-fuzzy sets) and fuzzy relations, in particular fuzzy equivalence (the well-known Zadeh fuzzy similarity relations) and as a special case \(L\)-equalities or fuzzy equalities. The first chapter ends with an introduction to Pavelka-style fuzzy logic.

The second chapter studies special algebras with fuzzy equalities and many examples of such algebras are provided. Then, fundamental algebraic constructions such as subalgebra, morphism, congruence, factor algebra, direct and indirect product, direct union, direct limit, reduced product are investigated in the framework of universal algebra.

Chapter 3 extends Birkhoff’s equational logic and theory of varieties to \(L\)-fuzzy set theory. After the introduction of the basic concepts of the syntax and semantics of fuzzy equational logic such as language, term, formula, structure and truth degree of a formula, the authors prove the completeness in the sense of Pavelka of a fuzzy equational logic determined by a complete residuated lattice. Furtheron, varieties and free algebras are treated in fuzzy equational logic and finally Birkhoff’s variety theorem is fuzzified.

In the final chapter 4 a generalization of an implication between identities is introduced into fuzzy logic. It is proved that the results of classical Horn logic can be extended to the fuzzy Horn logic developed in Pavelka’s style.

This volume is self-contained and is a must for lovers of fuzzy set theory and the beautiful but less known universal algebra. Special attention is paid in this volume to link the authors’ approach to other related approaches as well as to bibliographical remarks at the end of each chapter. I wonder if it would not be possible to use much simpler notations, what would certainly improve the readability for a broader audience.

The second chapter studies special algebras with fuzzy equalities and many examples of such algebras are provided. Then, fundamental algebraic constructions such as subalgebra, morphism, congruence, factor algebra, direct and indirect product, direct union, direct limit, reduced product are investigated in the framework of universal algebra.

Chapter 3 extends Birkhoff’s equational logic and theory of varieties to \(L\)-fuzzy set theory. After the introduction of the basic concepts of the syntax and semantics of fuzzy equational logic such as language, term, formula, structure and truth degree of a formula, the authors prove the completeness in the sense of Pavelka of a fuzzy equational logic determined by a complete residuated lattice. Furtheron, varieties and free algebras are treated in fuzzy equational logic and finally Birkhoff’s variety theorem is fuzzified.

In the final chapter 4 a generalization of an implication between identities is introduced into fuzzy logic. It is proved that the results of classical Horn logic can be extended to the fuzzy Horn logic developed in Pavelka’s style.

This volume is self-contained and is a must for lovers of fuzzy set theory and the beautiful but less known universal algebra. Special attention is paid in this volume to link the authors’ approach to other related approaches as well as to bibliographical remarks at the end of each chapter. I wonder if it would not be possible to use much simpler notations, what would certainly improve the readability for a broader audience.

Reviewer: E. Kerre (Gent)