Following the idea of L-fuzzy neighborhood system as introduced by F.-G. Shi, and its generalization to (L, M)-fuzzy neighborhood systems, we introduce the notion of (L, M)-fuzzy topological group, and obtain various properties, including some results on quotient groups of (L, M)-fuzzy topological groups. We show that the category of (L, M)-fuzzy topological groups is topological over the category of groups with respect to the forgetful functor.

In this paper, we point out that the proof of Theorem 2.4(5), Proposition 2.6(1) and Proposition 2.8(1) in the paper titled ?On (L,M)-fuzzy convex structures? (Filomat 33(13): 4151-4163, 2019) are not true in general. Then, we give three correct proofs of these results

As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved.

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.