Hybrid monotonic inclusion measure and its use in measuring similarity and distance between fuzzy sets

In fuzzy set theory, an inclusion measure indicates the degree to which a given fuzzy set is contained in another fuzzy set. In this paper, we introduce a new definition of inclusion measure: hybrid monotonic inclusion measure (S inclusion measure for short). Some inclusion measures defined by other authors and us are proved to be S inclusion measures. In addition, as the rational generalization of crisp inclusion relation, we prove most of the S inclusion measures possess some kind of T-transitivity and some properties postulated by Sinha and Dougherty for a reasonable inclusion measure. Similarity measure and distance between two fuzzy sets derived from S inclusion measure possess properties of normal similarity measure and distance. Furthermore, some similarity measures possess T-transitivity and distributivity.