On the union and intersection operations of rough sets based on various approximation spaces

Algebraic structures and lattice structures of rough sets are basic and important topics in rough sets theory. In this paper we pointed out that a basic problem had been overlooked, that is the closeness of union and intersection operations of rough approximation pairs, i.e. (lower approximation, upper approximation). We present that the union and intersection operations of rough approximation pairs are closed for classical rough sets and two kinds of covering based rough sets, but not for twenty kinds of covering based rough sets and the generalized rough sets based on fuzzy approximation space. Moreover, we proved that the union and intersection operations of rough fuzzy approximation pairs are closed and a bounded distributive lattice can be constructed.