residuated-dually residuated lattice; graded equipollence; fuzzy c-measure

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.

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.