Relationship between the second type of covering-based rough set and matroid via closure operator

Recently, in order to broad the application and theoretical areas of rough sets and matroids, some authors have combined them from many different viewpoints, such as circuits, rank function, spanning sets and so on. In this paper, we connect the second type of covering-based rough sets and matroids from the view of closure operators. On one hand, we establish a closure system through the fixed point family of the second type of covering lower approximation operator, and then construct a closure operator. For a covering of a universe, the closure operator is a closure one of a matroid if and only if the reduct of the covering is a partition of the universe. On the other hand, we investigate the sufficient and necessary condition that the second type of covering upper approximation operation is a closure one of a matroid.