We propose a trilevel framework for studying rough sets and granular rough sets by applying the principles of three-way decision as thinking in threes. The framework builds and interprets any model of rough sets at three levels: the binary relations level concerning the relationships between objects, the granular space level concerning granules of objects, namely, sets of objects called granular objects, and the approximation level concerning the approximations of sets of objects by granular objects. We identify and characterize eight classes of rough set models, including Pawlak, covering-based, and granular rough sets. By reviewing the existing studies within the framework, we find that there is a lack of investigations on three classes. To fill in these gaps, we investigate two types of granular spaces induced by any binary relations: neighborhood-induced granular spaces and maximal-clique-induced granular spaces. We examine the properties of the two types of granular space and the properties of rough set approximations in the corresponding two classes of models. We also consider a third class of models of granular rough sets based on granular spaces without referencing a binary relation.

Rough set approach for knowledge discovery in incomplete information systems has been extensively studied. This paper conduct a further study of valued tolerance relation based rough approximations. We make an analysis of the existing rough approximabilities and propose a new approach for lower (up per) approximability, which is a generalization of Pawlak approximation operators for complete information system. The approach has also been generalized to fuzzy cases. Some basic properties of the approximation operators are examined.