Lattice Structures of Probabilistic Rough Sets

This paper aims to examine the lattice structures of probabilistic rough sets. The concept of a rough membership function in a probabilistic approximation space is introduced. It is proved that the family of all rough membership functions in a probabilistic approximation space forms a stone lattice. In terms of rough membership function, the notions of   lower approximation operator and   upper approximation operator are obtained. Several properties of approximation operators and probabilistic approximation spaces are given. Then the notion of probabilistic rough sets in probabilistic approximation spaces is proposed. It is proved that the family of all probabilistic rough sets forms a complete Stone lattice, generalizing the corresponding result for traditional rough sets in Pawlak's sense.