M-fuzzifying submodular functions

Article type: Research Article

Authors: Xiu, Zhen-Yu | Shi, Fu-Gui

Affiliations: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, P.R. China

Note: [] Corresponding author. Zhen-Yu Xiu, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R. China. Emails: [email protected], [email protected] (Fu-Gui Shi).

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.

Keywords: M-fuzzifying submodular function, M-fuzzifying rank function, M-fuzzifying matriod, union of M-fuzzifying matriods

DOI: 10.3233/IFS-131089

Journal: Journal of Intelligent & Fuzzy Systems, vol. 27, no. 3, pp. 1243-1255, 2014