On (O,G)-fuzzy rough sets based on overlap and grouping functions over complete lattices

Abstract

In rough set theory, upper and lower approximation operators are two crucial concepts. The study of these two approximation operators in the framework of lattice theory is an important generalization from the mathematical point of view. On the other hand, overlap and grouping functions, as two types of not necessarily associative binary aggregation functions different from the common binary aggregation functions triangular norms and triangular conorms, have not only rich theoretical results but also a wide range of practical applications. Therefore, based on overlap and grouping functions over complete lattices, this paper is devoted to proposing (O,G)-fuzzy rough sets as a further generalization of the notion of rough sets. Firstly, we define a pair of O-upper and G-lower L-fuzzy rough approximation operators and investigate basic properties of them. Then, the characterizations of (O,G)-fuzzy rough approximation operators are discussed by using different kinds of L-fuzzy relations. Meanwhile, we investigate the topological properties of (O,G)-fuzzy rough sets. Furthermore, we show a brief comparison of the (O,G)-fuzzy rough sets with other common rough set models. At the end of this paper, we further propose multigranulation (O,G)-fuzzy rough sets over complete lattices from the viewpoint of multigranulation structure.