Generalized approximations defined by non-equivalence relations

The classical rough set theory, based on the conventional indiscernibility relation, is not much useful for analyzing incomplete information. Some successful generalized rough set models based on different non-equivalence relations have been proposed. Nowadays the generalized definitions of approximations have become one of the important research issues of the generalized rough set models. Although some different definitions of approximations have been presented, the relationships among them have not been analyzed adequately. In this paper, we investigate the relationships among 12 different basic definitions of approximations and suggest the suitable generalized definitions of approximations for each class of generalized indiscernibility relations. We also review eight classes of generalized indiscernibility relations and 12 different basic definitions of approximations as our foundations for discussion. Three examples are given to illustrate the relationships among the different generalized definitions of approximations.