Inclusion-exclusion families with respect to fuzzy measures

Abstract

This paper generalizes the concept of inclusion-exclusion coveringto the inclusion-exclusion family of subsets, which is not necessarily a covering, and describes its mathematical properties. Especially, the class of inclusion-exclusion families of subsets of a finite set $X$ is isomorphic to the free upper-bounded distributive lattice generated by $X$, where an upper-bounded lattice is a lattice with the largest element. In addition, the paper proposes inclusion-exclusion family models, provide variable selection and the adjustment of qualitative strength of additivity.