Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets

TL;DR

This paper introduces Kleene-Varlet spaces to represent regular pseudocomplemented Kleene algebras and demonstrates their isomorphism with rough set structures, providing a new algebraic and set-theoretic perspective.

Contribution

It establishes a novel representation of regular pseudocomplemented Kleene algebras using Kleene-Varlet spaces and rough set theory, extending the understanding of their structure.

Findings

Regular pseudocomplemented Kleene algebras are isomorphic to subalgebras of rough set algebras.

Kleene-Varlet spaces characterize these algebras satisfying the Stone identity.

The paper provides conditions for the correspondence between Kleene-Varlet spaces and algebraic structures.

Abstract

We introduce Kleene-Varlet spaces as partially ordered sets equipped with a polarity satisfying certain additional conditions. By applying Kleene-Varlet spaces, we prove that each regular pseudocomplemented Kleene algebra is isomorphic to a subalgebra of the rough set regular pseudocomplemented Kleene algebra defined by a tolerance induced by an irredundant covering. We also characterize the Kleene-Varlet spaces corresponding to the regular pseudocomplemented Kleene algebras satisfying the Stone identity.