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

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.