Rough approximation operators on two universes of discourse and their fuzzy extensions

Based on a crisp binary relation between two universes of discourse, one crisp covering and three crisp binary relations on the single universe of discourse are induced in this paper. Based on these induced notions, four pairs of rough approximation operators are formulated so that the approximating sets and the approximated sets are on the same universe of discourse. Furthermore, basic properties of the new approximation operators are investigated, and the relationships among the operators are also examined. Subsequently, conditions under which some or all of these approximation operators are equivalent are obtained. On the other hand, the generalizations of these rough approximation operators are also made in a fuzzy approximation space via a fuzzy implicator I and a triangular norm T. Consequently, four pairs of fuzzy rough approximation operators are constructed. Basic properties and comparison of the fuzzy rough approximation operators are discussed. If I is an R-implicator with respect to a t-norm T, then it is shown that three pairs of the fuzzy rough approximation operators are identical iff the induced fuzzy covering is a fuzzy T-partition.