Algebric properties of rough sets using topological characterisations and approximate equalities

The notion of rough sets [4] was used by Novotny and Pawlak [1, 2, 3] to introduce the concept of rough equality, in order to incorporate user knowledge in determining the equality of sets, which is a natural phenomenon in day to day life. Even this notion was shown to be deficient and was extended by Tripathy et al [6, 9] to define the notion of rough equivalence of sets, which is more suitable in real life situations. Tripathy [7] added two more approximate equalities to this list in order to complete the kinds of approximate equalities. Algebraic properties of sets with mathematical equality have very little meaning when sets are replaced with rough sets. As a consequence, replacing equality by rough equivalence the algebraic properties were studied by Tripathy et al [10]. In our present work, we have made an analysis of the validity of the algebraic properties with respect to the four kinds of approximate equalities and provide a comparative study to establish that rough equivalence is the best and rough equality is the worst among these approximate equalities with respect to the algebraic properties involving rough sets. The topological characterisations of rough sets leading to four types of rough sets and operations on them [8] are being used in the sequel. We support our descriptions with a running real life example of shares.