M-fuzzifying submodular functions

Abstract

In this paper, the concept of M-fuzzifying submodular functions is introduced, which is a generalization of submodular functions in matroid theory. It is shown that M-fuzzifying matroids can be generated from an M-fuzzifying submodular function in different ways. A circuit-map, an M-fuzzifying family of independent sets and an M-fuzzifying family of dependent sets are obtained from an M-fuzzifying submodular function. We also present an M-fuzzifying submodular function on a lattice $\mathcal{L}_{E}$ and obtain an M-fuzzifying matroid from it. Moreover, new characterizations of base-maps and circuit-maps are obtained by means of an M-fuzzifying rank function. At last, the notion of the union of M-fuzzifying matriods is introduced and a result of the union of M-fuzzifying matriods associated with M-fuzzifying submodular functions is obtained. We also establish generalizations of Edmonds' Intersection Theorem and Edmonds' Covering Theorem in the framework of M-fuzzifying matriod.