a b s t r a c tRough 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.

Smarandache initiated neutrosophic sets (NSs) which can be used as a mathematical tool for dealing with indeterminate and inconsistent information. In order to apply NSs conveniently, single valued neutrosophic sets (SVNSs) were proposed by Wang et al. In this paper, we propose single valued neutrosophic relations (SVNRs) and study their properties. The notions of anti-reflexive kernel, symmetric kernel, reflexive closure, and symmetric closure of a SVNR are introduced, respectively. Their accurate calculate formulas and some properties are explored. Finally, single valued neutrosophic relation mappings and inverse single valued neutrosophic relation mappings are introduced, and some interesting properties are also obtained.

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