The lattice and matroid representations of definable sets in generalized rough sets based on relations

Abstract

The definable set is a core concept in rough set theory. It plays an important role in the characterizations of rough sets. In this paper, we study the lattice and matroid representations of definable sets in generalized rough sets based on relations. First, we propose the lower definable lattice, consisting of all lower definable sets with set inclusion order. Then we give some conditions under which the lower definable lattice is distributive (or geometric, or Boolean). Furthermore, we discuss the relationship between distributive lattices and the lower definable lattices in generalized approximation spaces based on reflexive and transitive relations. On the one hand, we show that the lower definable lattice in a generalized approximation space based on reflexive and transitive relation is distributive. On the other hand, we obtain the result that a distributive lattice can induce a lower definable lattice in a generalized approximation space based on a reflexive and transitive relation. Finally, we investigate the combination of generalized rough sets and matroids in terms of the lower definable lattice. We show that if a lower definable lattice is a lattice of closed sets of a matroid, then it must be an open-closed set lattice of a matroid. In addition, we prove that some lower definable lattices are not the lattices of closed sets of matroids. These results of this paper will benefit to our understanding of the relationship between matroids and generalized rough sets based on relations.