Covering-based rough sets on covering-circuit matroids

Rough set theory has been proposed by Pawlak as a useful and powerful tool for dealing with uncertainty, granularity, and incompleteness of knowledge in information systems. Matroid theory is a branch of combinatorial mathematics and widely used in optimization. Therefore, it is a good idea to integrate rough sets with matroids. In this paper, four types of covering approximation operators and their relationships are studied from the viewpoint of matroids. First, we define a new type of matroids named covering-circuit matroids whose all circuits form a covering. Second, for a covering-circuit matroid, we study the properties of the circuits of it from the perspective of coverings. Third, four types of covering approximation operators are represented by the circuits of the covering-circuit matroids. Moreover, we also investigate the relationships among four covering upper approximation operators. Finally, the conditions under which every type of covering upper approximation operator is the closure operator of the matroid are revealed. These results show many potential connections between covering-based rough sets and matroids.