Subvarieties of Pseudocomplemented Kleene Algebras

Abstract

In this paper we study the subdirectly irreducible algebras in the variety $\mathcal{P}\mathcal{C}\mathcal{D}\mathcal{M}$ of pseudocomplemented De Morgan algebras by means of their De Morgan $p$-spaces. We introduce the notion of $body$ of an algebra $\mathbf{L}\in \mathcal{P}\mathcal{C}\mathcal{D}\mathcal{M}$ and determine $Body(\mathbf{L})$ when $\mathbf{L}$ is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, three special subvarieties arise naturally, for which we give explicit identities that characterize them. We also introduce a subvariety $\mathcal{B}\mathcal{P}\mathcal{K}$ of $\mathcal{P}\mathcal{C}\mathcal{D}\mathcal{M}$, namely the variety of $bundle$ $pseudocomplemented$ $Kleene$ $algebras$, determine the whole subvariety lattice and find explicit equational bases for each of the subvarieties. In addition, we study the subvariety ${\mathcal{B}\mathcal{P}\mathcal{K}}_0$ of $\mathcal{B}\mathcal{P}\mathcal{K}$ generated by the simple members of $\mathcal{B}\mathcal{P}\mathcal{K}$, determine the structure of the free algebra over a finite set and their finite weakly projective algebras.