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An axiomatic approach of fuzzy rough sets based on residuated lattices. (English) Zbl 1189.03059

Summary: 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 A. M. Radzikowska and E. E. Kerre [Fuzzy Sets Syst. 126, 137–155 (2002; Zbl 1004.03043)]. 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.

MSC:

03E72 Theory of fuzzy sets, etc.

Citations:

Zbl 1004.03043
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References:

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