A new approach to generalized neighborhood system-based rough sets via convex structures and convex matroids

Generalized neighborhood system-based rough sets (GNSs) play a key part in rough set theory. In this paper, we introduce a new approach to GNSs from the aspects of convex structures and convex matroids. Firstly, we propose the concepts of matroidal exterior operators and matroidal interior operators and investigate their relationship with convex matroids, respectively. We prove that there is a one-to-one correspondence among matroidal exterior operators, matroidal interior operators and convex matroids. Secondly, we introduce two new types of GNSs which are called co-directed intersection closed and exchangeable. Then we establish the compatible relationships between convex structures (resp. convex matroids) and co-directed intersection closed (resp. exchangeable) GNSs by using matroidal exterior operators and matroidal interior operators as the linkages. Finally, we present axiomatic characterizations of rough approximation operators corresponding to a co-directed intersection closed generalized neighborhood system operator (gns operator) as well as its combinations with serial, reflexive, weak-transitive and weak-unary gns operators.