On the cardinalities of interval-valued fuzzy sets

Wygralak has developed an axiomatic theory of scalar cardinalities of fuzzy sets with finite support which covers as particular cases all standard (historical) concepts of the scalar cardinality. In this paper we present a possible extension of this theory to interval-valued fuzzy set theory. We introduce the cardinality of interval-valued fuzzy sets as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞[. We study some properties of these cardinalities using t-norms, t-conorms and negations on the lattice LI (the underlying lattice of interval-valued fuzzy set theory). In particular, the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule will be discussed.