Kleene and Stone algebras of rough sets induced by reflexive relations

TL;DR

This paper studies Kleene and Stone algebras on rough sets induced by reflexive relations, generalizing previous results from equivalence and tolerance relations.

Contribution

It characterizes conditions under which the completion of rough sets forms specific algebraic structures like regular pseudocomplemented Kleene and double Stone algebras.

Findings

DM(RS) forms a regular pseudocomplemented Kleene algebra under certain conditions.

DM(RS) becomes a completely distributive double Stone algebra for specific reflexive relations.

The results extend earlier work on rough sets from equivalence, quasiorder, and tolerance relations.

Abstract

We consider Kleene and Stone algebras defined on the completion DM(RS) of the ordered set of rough sets induced by a reflexive relation. We focus on cases where 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 the reflexive relations for which DM(RS) forms a regular double Stone algebra, which is the same structure as in the case of equivalences. Our results generalise earlier findings on algebras of rough sets induced by equivalences, quasiorders, and tolerance relations.