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Fuzzy set-theoretic operators and quantifiers. (English)
Dubois, Didier (ed.) et al., Fundamentals of fuzzy sets. Foreword by Lotfi A. Zadeh. Dordrecht: Kluwer Academic Publishers. Handb. Fuzzy Sets Ser. 7, 125-193 (2000).
This chapter in Kluwer’s handbook series devoted to fuzzy sets brings an overview of fuzzy set-theoretic operators and quantifiers. First of all, basic operations for fuzzy sets as well as connectives for fuzzy logics are discussed, such as negation (complementation), conjunction (intersection), disjunction (union), implication, coimplication, equivalence, etc. The authors present here basic characterization results for several unary and binary operations, including most important examples. Note that t-norms (models for fuzzy conjunction) and related operations are, in an exhaustive form, discussed also in a recent monograph of {\it E. P. Klement}, {\it R. Mesiar} and {\it E. Pap} [Triangular norms, Kluwer Academic Publishers, Dordrecht (2000; Zbl 0972.03002)]. Several other operators frequently applied in applications of fuzzy set theory (e.g., in fuzzy control) are also recalled, such as uninorms, OWA operators, root-power mean operators, symmetric sums, etc. Also, several types of quantifiers, especially linguistic quantifiers (“most”, “few”, “almost all”, etc.) are included. Finally, prioritized fuzzy operations are discussed. This contribution is a good state-of-the-art overview with more than 120 relevant references and it can be recommended to any user or person interested in fuzzy sets and their applications.
Reviewer: R.Mesiar (Bratislava)