On the transitivity of a parametric family of cardinality-based similarity measures

We introduce a parametric family of cardinality-based similarity measures for ordinary sets (on a finite universe) harbouring numerous well-known similarity measures. We characterize the Łukasiewicz-transitive and product-transitive members of this family. Their importance derives from their one-to-one correspondence with pseudo-metrics. Fuzzification schemes based on a commutative quasi-copula are then used to transform these similarity measures for ordinary sets into similarity measures for fuzzy sets, rendering them applicable on graded feature set representations of objects. The main result of this paper is that transitivity, and hence also the corresponding dual metric interpretation, is preserved along this fuzzification process.