Generalized rough sets based on reflexive relations
Yu-Ru Syau, Lixing Jia · Communications in Information and Systems · 2012
We study the approximation theory of a special kind of neighborhood systems, called total pure reflexive neighborhood systems, which is a generalization of pretopological and topological neighborhood systems.In the framework of such neighborhood systems, the so-called lower and upper approximations are considered.For a pretopological neighborhood system on a fixed nonempty set U , we show that the family T of fixed points of its lower approximation is a topology for U , and establish a characterization (in terms of neighborhoods) of T -open sets.We then regard a reflexive relation R on U as the total pure reflexive neighborhood system x -→ {R(x) = { y ∈ U | (x, y) ∈ R}}, whose induced upper approximation is identical to the commonly used upper approximation R * : 2 U → 2 U based on R. We show that the family T R of all subsets X of U for which R * (U -X) = U -X is an Alexandroff topology for U , and that the pre-topologically maximal neighborhood system of the neighborhood system x -→ {R(x)} associated to R is exactly the pretopological neighborhood system whose induced upper approximation is identical to R * : 2 U → 2 U .Accordingly, we show that for each x ∈ U , its smallest T R -open neighborhood is the intersection of all T R -open sets containing R(x).In addition, we study the so-called R-definability.We establish a characterization of R-definable sets in terms of R * and its dual R * , and present a necessary condition for R-definability.