On three types of covering-based rough sets via definable sets

The study of definable sets in various generalized rough set models would provide better understanding to these models. Some algebraic structures of all definable sets have been investigated, and the relationships among the definable sets, the inner definable sets and the outer definable sets have been presented. In this paper, we further study the definable sets in three types of covering-based rough sets and present several necessary and sufficient conditions of definable sets. These three types of covering-based rough sets are based on three kinds of neighborhoods: the neighborhood, the complementary neighborhood and the indiscernible neighborhood, respectively. Some necessary and sufficient conditions of definable sets are presented through these three types of neighborhoods, and the relationships among the definable sets are investigated. Moreover, we study the relationships among these three types of neighborhoods, and present certain conditions that the union of the neighborhood and the complementary neighborhood is equal to the indiscernible neighborhood.