Algebraic Method of the Theory of Rough Sets over Two Universes

Algebraic method is one of important ways in the study of rough sets theory. One may profoundly understand the algebraic structure of approximation operator through this method research. In this paper, the related concepts with respect to Pawlak approximation space are briefly recalled. Then, the concepts of generalized approximation space and generalized approximation operators are firstly proposed, and the binary relation, defined over two universes, is reviewed. Afterwards, the relations of the binary relation and the approximation operator are discussed. On one hand, one can see that a binary relation, which satisfied some specific conditions, may determine the special property of the approximation operator. On the other hand, one defines a pair of dual approximation operators over two universes and states the axioms that those must satisfy, these axioms may sufficiently guarantee the existence of certain types of binary relations over two universes.