The structures and the connections on four types of covering rough sets

Covering rough set model is an important extension of Pawlak rough set model, and its structure is the foundation of covering rough set theory. This paper considers four covering approximations and studies the structures of the families of their covering upper (or lower) definable sets by means of lattice theory. We provide some conditions under which the families of covering upper (or lower) definable sets with respect to these covering approximations are lattices of sets, or distributive lattices, or geometric lattices, or Boolean lattices. Furthermore, based on these results, we give the relationship among the four covering approximations and establish the connection between matroids and covering rough sets from the viewpoint of lattice theory.