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Zbl 1232.03041
Klement, Erich Peter; Mesiar, Radko; Pap, Endre
On the relationship of associative compensatory operators to triangular norms and conorms. (English)
[J] Int. J. Uncertain. Fuzziness Knowl.-Based Syst. 4, No. 2, 129-144 (1996). ISSN 0218-4885

Summary: When using a t-norm for combining fuzzy sets, no compensation between small and large degrees of membership takes place. On the other hand, a t-conorm provides full compensation. Since many real situations do not fall into either category, so-called compensatory operators have been proposed in the literature [{\it H.-J. Zimmermann} and {\it P. Zysno}, Fuzzy Sets Syst. 4, No. 1, 37--51 (1980; Zbl 0435.90009)] which are non-associative in nature. In this paper, associative compensatory operators (whose domain is the unit square with the exception of the two points $(0, 1)$ and $(1, 0)$ and whose only associative extensions to the whole unit square are the aggregative operators suggested in [{\it J. Dombi}, Eur. J. Oper. Res. 10, No. 3, 282--293 (1982; Zbl 0488.90003)]) are studied and their representation in terms of multiplicative generators is given. It is shown that these operators are constructed with the help of strict t-norms and t-conorms, in a way which is similar to ordinal sums. Finally, the duals of such operators are shown to be again associative compensatory operators, and a characterization of self-dual operators is given.

MSC 2000:: *03E72 Fuzzy sets (logic)
91B06 Decision theory

Keywords: compensatory operators; associative functions; t-norms; t-conorms; multiplicative generators

Citations: Zbl 0435.90009; Zbl 0488.90003

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