Further study of multi-granulation fuzzy rough sets

The multi-granulation fuzzy rough sets (MGFRS), proposed by Xu [35], is a meaningful contribution in the view of the generalization of the classical rough set model. In this paper, the main objective is to make further studies based on reference [35]. It is shown that the MGFRS do not meet the operations of intersection and union. To solve these problems, we propose the concepts of the equivalence relations with minimum (maximum) element and find that the MGFRS based on the equivalence relations with minimum (maximum) element meet the operations of intersection and union. At the same time, some algebraic properties of the MGFRS are discussed.