fuzzy quantifier; fuzzy measure; residuated lattice operation; monotonicity of fuzzy quantifiers

In this contribution, we study a very important semantic property of generalized quantifiers called the monotonicity for fuzzy quantifiers of type <1, 1> defined using fuzzy measures and Sugeno type of fuzzy integrals. We show that fuzzy integrals can ensure under some natural conditions the monotonicity of fuzzy quantifiers. Finally, we propose the concept of concave fuzzy quantifiers and prove that each concave fuzzy quantifier is expressible as the conjunction of a non-increasing and a nondecreasing fuzzy quantifier.

In the classical set theory, there is the well known correspondence between the equipollence of finite sets and the equality of natural numbers expressing their cardinalities. In the fuzzy set theory, an equivalent correspondence between the equipollence of finite fuzzy sets and the equality of generalized cardinals has been proposed in [1]. This paper is devoted to the situation when we consider the mentioned relations to be graded, that means, the crisp equipollence of fuzzy sets is replaced now by the graded equipollence of fuzzy sets and the equality of fuzzy sets by their similarity.