Reducible matroid and its relationship with covering-based rough sets

Covering is an important form to organize data. Covering-based rough set theory provides a systematic approach to deal with this form of data. A matroid is a generalization of linear independence in vector space. This paper connects matroid theory with covering-based rough sets from the reducible point of view. Firstly, similar to the definition of reducible elements in covering-based rough sets, we define reducible matroids in matroid theory to remove redundant matroids. Secondly, we establish a matroidal structure of covering-based rough sets. Specifically, we propose an approach to induce a matroid by subset of a covering. Therefore a matroid family can be induced by all subsets of a covering. The subset having only one element induces a 1-rank matroid. Each matroid in this family can be expressed as a union of some 1-rank matroids induced by elements of covering. We show that the reduct of the matroid family is equal to the family of these 1-rank matroids. Finally, the reducible element, neighborhood, minimal description and approximations in covering-based rough sets can be represented by closure operators of these 1-rank matroids.