Rough sets by reflexive relations and their algebras
This is an incremental theoretical contribution for researchers in rough set theory and algebraic logic, extending known results from equivalence relations to reflexive relations.
The paper studies the algebraic properties of rough sets induced by reflexive relations, characterizing when their completion forms specific algebraic structures like regular pseudocomplemented Kleene algebras and completely distributive double Stone algebras. It identifies conditions under which these algebras match the properties of those from equivalence relations.
We consider various types of algebras defined on the completion DM(RS) of the partially ordered set of rough sets induced by a reflexive relation. We restrict ourselves to the cases in which the completion forms a spatial and completely distributive lattice. We derive the conditions under which DM(RS) is a regular pseudocomplemented Kleene algebra and a completely distributive double Stone algebra. Finally, we describe reflexive relations for which DM(RS) has the same properties as in the case of an equivalence relation: it forms a completely distributive and spatial regular double Stone algebra.