The axiomatic characterization on fuzzy variable precision rough sets based on residuated lattice

Abstract

Axiomatization is a lively research direction in fuzzy rough set theory. Fuzzy variable precision rough set (FVPRS) incorporates fault-tolerant factors to fuzzy rough set, so its axiomatic description becomes more complicated and difficult to realize. In this paper, we present an axiomatic approach to FVPRSs based on residuated lattice (L-fuzzy variable precision rough set (LFVPRS)). First, a pair of mappings with three axioms is utilized to characterize the upper (resp., lower) approximation operator of LFVPRS. This is distinct from the characterization on upper (resp., lower) approximation operator of fuzzy rough set, which consists of one mapping with two axioms. Second, utilizing the notion of correlation degree (resp., subset degree) of fuzzy sets, three characteristic axioms are grouped into a single axiom. At last, various special LFVPRS generated by reflexive, symmetric and transitive L-fuzzy relation and their composition are also characterized by axiomatic set and single axiom, respectively.