A result on single valued neutrosophic refined rough approximation operators 1

Abstract

Smarandache (1998) initiated neutrosophic sets as a new mathematical tool for dealing with problems involving incomplete, indeterminant and inconsistent knowledge. By simplifying neutrosophic sets, Smarandache (1998) and Wang et al. (2010) proposed the concept of single valued neutrosophic sets and studied some properties of single valued neutrosophic sets. Recently, Bao and Yang (2017) introduced n-dimension single valued neutrosophic refined rough sets by combining single valued neutrosophic refined sets with rough sets and further studied the hybrid model from two perspectives–constructive viewpoint and axiomatic viewpoint. A natural problem is: Can the supremum and infimum of n-dimension single valued neutrosophic refined rough approximation operators be given? Following the idea of Bao and Yang, in this paper, let X be a set, H _n (X) and L _n (X) denote the family of all n-dimension single valued neutrosophic refined upper and lower approximation operators in X, respectively. We can define appropriate order relation ≦ on H _n (X) (resp., L _n (X)) such that both (H _n (X), ≦) and (L _n (X), ≦) are complete lattices. In particular, both (H, ≦) and (L, ≦) are complete lattices, where H and L denote the family of single valued neutrosophic upper and lower approximation operators in X, respectively.