On fuzzy rough sets based on tolerance relations

Dubois and Prade (1990) [1] introduced the notion of fuzzy rough sets as a fuzzy generalization of rough sets, which was originally proposed by Pawlak (1982) [8]. Later, Radzikowska and Kerre introduced the so-called (I,T)(I,T)-fuzzy rough sets, where I is an implication and T   is a triangular norm. In the present paper, by using a pair of implications (I,J)(I,J), we define the so-called (I,J)(I,J)-fuzzy rough sets, which generalize the concept of fuzzy rough sets in the sense of Radzikowska and Kerre, and that of Mi and Zhang. Basic properties of (I,J)(I,J)-fuzzy rough sets are investigated in detail.