Reducible matroid and reducible element of covering-based rough sets

This paper connects covering-based rough sets with matroids. Firstly, we defined the r-restrict uniform matroid to establish the matroidal structure of covering-based rough sets. Any subset of a universe can generate an r-restrict uniform matroid, therefore any covering of a universe can be characterized by a family of r-restrict uniform matroids. On the other hand, a family of r-restrict uniform 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. In this paper we mainly research the relation between reducible element and the reducible matroid in the family of r-restrict uniform matroids induced by covering. Especially they are equivalent when the r-restrict uniform matroid degenerated to the 1-restrict uniform matroid. These results enrich covering-based rough set theory and matroid theory.