Covering based rough set approximations

We propose a framework for the study of covering based rough set approximations. Three equivalent formulations of the classical rough sets are examined by using equivalence relations, partitions, and σ-algebras, respectively. They suggest the element based, the granule based and the subsystem based definitions of approximation operators. Covering based rough sets are systematically investigated by generalizing these formulations and definitions. A covering of universe of objects is used to generate different neighborhood operators, neighborhood systems, coverings, and subsystems of the power set of the universe. They are in turn used to define different types of generalized approximation operators. Within the proposed framework, we review and discuss covering based approximation operators according to the element, granule, and subsystem based definitions.