Completely Compact Elements and Atoms of Rough Sets

Rough set theory established by Pawlak in 1982 plays an important role in dealing with uncertain information and to some extent overlaps fuzzy set theory. The key notions of rough set theory are approximation spaces of pairs (U,R) with R being an equivalence relation on U and approximation operators \(\underline{R}\) and \(\overline{R}\). Let \(\mathcal{R}\) be the family {( \(\underline{R}X,\overline{R}X)|X \subseteq U\)} of approximations endowed with the pointwise order of set-inclusion. It is known that \(\mathcal{R}\) is a complete Stone lattice with atoms and is isomorphic to the family of rough sets in the approximation space (U, R). This paper is devoted to investigate algebraicity and completely distributivity of \(\mathcal{R}\) from the view of domain theory. To this end, completely compact elements, compact elements and atoms of \(\mathcal{R}\) are represented. In terms of the representations established in this paper, it is proved that \(\mathcal{R}\) is isomorphic to a complete ring of sets, consequently \(\mathcal{R}\) is a completely distributive algebraic lattice. An example is given to show that \(\mathcal{R}\) is not atomic nor Boolean in general. Further, a sufficient and necessary condition for \(\mathcal{R}\) being atomic is thus given.