(L, M)-fuzzy matroids

In this paper, an extension of matroids is introduced in lattice-valued fuzzy set theory. Such a generalized matroid is a pair made of a finite set and a fuzzy subfamily of its lattice-valued fuzzy subsets satisfying three axioms. It uses two lattices: one for membership grades of elements to fuzzy sets, and another for membership degrees of fuzzy sets in a fuzzy family of fuzzy sets. It is a logical generalization of the notions of matroids and fuzzifying matroids. In our definition of generalized fuzzy matroids, each L-fuzzy subset can be regarded to be independent of other ones to some degree. The relevance of generalized fuzzy matroids is shown in the setting of fuzzy vector spaces and fuzzy graphs.