Free matroidal structure of covering-based rough sets

Matroids were defined as abstract generation of graphs and matrices to try to capture abstractly essence of (in)dependence, it has wide application in electrical and computer science, mechanical engineering. Rough set theory is a new tool to deal with various types of data, covering-based rough set is a generalization of traditional rough set theory. This paper concerns the relationships between covering-based rough sets and matroid. Firstly, we establish the free matroidal structure of covering-based rough sets. Any subset of a universe can generate a free matroid, therefore any covering of a universe can be characterized by a family of free matroids. On the other hand, a family of free matroids can be used to generate a covering. Secondly, reducible element is to remove the redundant element in covering-based rough sets, reducible matroid is to remove redundant matroid in a family of matroids. We mainly research the relationship between reducible element and the reducible matroid in the family of free matroids induced by covering. Actually they are equivalent, the reducible element of a covering is always consistent with the reducible matroid induced by the covering. These results enrich covering-based rough set theory and matroid theory.