On characterizations of (I,T)-fuzzy rough approximation operators

In rough set theory, the lower and upper approximation operators defined by a fixed binary relation satisfy many interesting properties. Various fuzzy generalizations of rough approximations have been made in the literature. This paper proposes a general framework for the study of (I,T)-fuzzy rough approximation operators within which both constructive and axiomatic approaches are used. In the constructive approach, a pair of lower and upper generalized fuzzy rough approximation operators, determined by an implicator I and a triangular norm T, is first defined. Basic properties of (I,T)-fuzzy rough approximation operators are investigated. The connections between fuzzy relations and fuzzy rough approximation operators are further established. In the axiomatic approach, an operator-oriented characterization of rough sets is proposed, that is, (I,T)-fuzzy approximation operators are defined by axioms. Different axiom sets of T-upper and I-lower fuzzy set-theoretic operators guarantee the existence of different types of fuzzy relations which produce the same operators. Finally, an open problem proposed by Radzikowska and Kerre in (Fuzzy Sets and Systems 126 (2002) 137) is solved.