Fuzzy sets with triangular norms and their cardinality theory

This paper shows that the original forms of FGCounts, FLCounts, and FECounts are suitable only for fuzzy sets with the classical min and max operations, and proposes an appropriate generalization to fuzzy set with triangular norms and conorms.

Abstract

The most advanced and adequate approach to the question of cardinality of a fuzzy set seems to be that offering a fuzzy perception of cardinality. The resulting convex fuzzy sets of usual cardinals (of nonnegative integers, in the finite case) are then called generalized cardinal numbers. Three types of them are of special interest and importance, namely FGCounts, FLCounts, and FECounts. In this paper, first, we show that their original forms are suitable only for fuzzy sets with the classical min and max operations. Second, we propose an appropriate generalization to fuzzy sets with triangular norms and conorms. Further, we investigate the resulting generalized FGCounts, FLCounts, and FECounts from the viewpoint of the corresponding equipotency relations, inequalities, and arithmetical operations.