An axiomatic approach of fuzzy rough sets based on residuated lattices

Rough set theory was developed by Pawlak as a formal tool for approximate reasoning about data. Various fuzzy generalizations of rough approximations have been proposed in the literature. As a further generalization of the notion of rough sets, LL-fuzzy rough sets were proposed by Radzikowska and Kerre. In this paper, we present an operator-oriented characterization of LL-fuzzy rough sets, that is, LL-fuzzy approximation operators are defined by axioms. The methods of axiomatization of LL-fuzzy upper and LL-fuzzy lower set-theoretic operators guarantee the existence of corresponding LL-fuzzy relations which produce the operators. Moreover, the relationship between LL-fuzzy rough sets and LL-topological spaces is obtained. The sufficient and necessary condition for the conjecture that an LL-fuzzy interior (closure) operator derived from an LL-fuzzy topological space can associate with an LL-fuzzy reflexive and transitive relation such that the corresponding LL-fuzzy lower (upper) approximation operator is the LL-fuzzy interior (closure) operator is examined.