Transitivity of comparison measures

A family of fuzzification schemes is proposed for transforming cardinality-based similarity and inclusion measures for ordinary sets into similarity and inclusion measures for fuzzy sets in a finite universe. The family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets. In particular, the fuzzy set intersections are pointwisely generated by Frank t-norms. The fuzzification schemes are applied to a variety of previously studied rational cardinality-based similarity and inclusion measures for ordinary sets and it is demonstrated that transitivity is preserved in the fuzzification process. Finally, we introduce a new parametrized family of similarity measures and a new parametrized family of inclusion measures and investigate the transitivity properties of both families and their fuzzification