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, L-fuzzy rough sets were proposed by Radzikowska and Kerre. In this paper, we present an operator-oriented characterization of L-fuzzy rough sets, that is, L-fuzzy approximation operators are defined by axioms. The methods of axiomatization of L-fuzzy upper and L-fuzzy lower set-theoretic operators guarantee the existence of corresponding L-fuzzy relations which produce the operators. Moreover, the relationship between L-fuzzy rough sets and L-topological spaces is obtained. The sufficient and necessary condition for the conjecture that an L-fuzzy interior (closure) operator derived from an L-fuzzy topological space can associate with an L-fuzzy reflexive and transitive relation such that the corresponding L-fuzzy lower (upper) approximation operator is the L-fuzzy interior (closure) operator is examined. (C) 2009 Elsevier Ltd. All rights reserved.