On the representation of cardinalities of interval-valued fuzzy sets: The valuation property

In our earlier work we have developed the axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak’s axiomatic theory of scalar cardinalities of fuzzy sets. The cardinality has been defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞). We have shown that each scalar cardinality of interval-valued fuzzy set can be characterized using the appropriate mapping called cardinality pattern. Moreover, we have found some basic conditions under which the valuation property, the subadditivity property, the complementarity rule and the cartesian product rule are satisfied using different cardinality patterns tnorms, t-conorms and negations on the lattice L (the underlying lattice of interval-valued fuzzy set theory). The presented paper is the first from the collection of papers consisting of the further investigation of the proposed theory providing the description of cardinality patterns, t-norms, t-conorms and negations satisfying the above mentioned properties. We restrict ourselves to the valuation property here.