Novel results on G -lower L -fuzzy rough approximation operator in ( O,G ) -fuzzy rough set model 1

Abstract

Lately, Jiang and Hu (H.B. Jiang, B.Q. Hu, On $(\mathfrak{O},\mathfrak{G})$ -fuzzy rough sets based on overlap and grouping functions over complete lattices, Int. J. Approx. Reason. 144 (2022) 18-50.) put forward $(\mathfrak{O},\mathfrak{G})$ -fuzzy rough sets via overlap and grouping functions over complete lattices. Meanwhile, they showed the characterizations of $\mathfrak{O}$ -upper and $\mathfrak{G}$ -lower $\mathcal{L}$ -fuzzy rough approximation operators in $(\mathfrak{O},\mathfrak{G})$ -fuzzy rough set model based on some of specific $\mathcal{L}$ -fuzzy relations and studied the topological properties of the proposed model. Nevertheless, we discover that the partial results given by Jiang and Hu could be further optimized. So, as a replenish of the above article, in this paper, based on $\mathfrak{G}$ -lower $\mathcal{L}$ -fuzzy rough approximation operator in $(\mathfrak{O},\mathfrak{G})$ -fuzzy rough set model, we further explore several new conclusions on the relationship between $\mathfrak{G}$ -lower $\mathcal{L}$ -fuzzy rough approximation operator and different $\mathcal{L}$ -fuzzy relations. In particular, the equivalent descriptions of relationship between $\mathfrak{G}$ -lower $\mathcal{L}$ -fuzzy rough approximation operator and $\mathfrak{O}$ -transitive ( $\mathfrak{O}$ -Euclidean) $\mathcal{L}$ -fuzzy relations are investigated, which are not involved in above literature and can make the theoretical results of this newly fuzzy rough set model more perfect.