On the relationship of associative compensatory operators to triangular norms and conorms

Abstract

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 one category, so-called compensatory operators have been proposed in the literature [H.-J. Zimmermann and P. Zysno, Fuzzy Sets and Systems 4 (1980) 37-51] 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 [J. Dombi, Europ. J. Oper. Res. 10 (1982) 282-293]) 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.