The relationship among three types of rough approximation pairs

The aim of the paper is to clarify mathematical relationship among three types of rough approximation pairs. We propose subsystem-based definition of generalized rough sets and consider three different types of generalized rough approximation pairs from the viewpoint of three definitions, namely, the element, granule, and subsystem-based definitions. This paper investigates these three types of generalized rough sets with respect to arbitrary binary relations. The topologies induced by granule-based generalized rough sets are introduced. Finally, we give a characterization of granule-based generalized rough sets.