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

In the previous work, we developed an axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak's axiomatic theory of the scalar cardinality of fuzzy sets. Cardinality was defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+∞). We showed that the scalar cardinality of each interval-valued fuzzy set can be characterized using an appropriate mapping called a cardinality pattern. Moreover, we 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, t-norms, t-conorms and negations on the lattice LI (the underlying lattice of interval-valued fuzzy set theory). This paper is the first in a series that further investigates the proposed theory, providing a description of cardinality patterns, t-norms, t-conorms and negations satisfying the properties mentioned above. This paper focuses on the valuation property.