Subvarieties of Pseudocomplemented Kleene Algebras

TL;DR

This paper characterizes subvarieties of pseudocomplemented Kleene algebras using De Morgan spaces, introduces the notion of 'body' of an algebra, and explores the structure and identities of these subvarieties.

Contribution

It introduces the concept of 'body' for algebras in ${ m PCDM}$, characterizes subvarieties of pseudocomplemented Kleene algebras, and determines their subvariety lattice and free algebra structures.

Findings

Identified three natural subvarieties with explicit identities.

Determined the subvariety lattice of bundle pseudocomplemented Kleene algebras.

Analyzed the structure of free algebras over finite sets.

Abstract

In this paper we study the subdirectly irreducible algebras in the variety ${\cal PCDM}$ of pseudocomplemented De Morgan algebras by means of their De Morgan $p$-spaces. We introduce the notion of $body$ of an algebra ${\bf L} \in {\cal PCDM}$ and determine $Body({\bf L})$ when ${\bf L}$ 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 ${\cal BPK}$ of ${\cal PCDM}$, 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 ${\cal BPK}_0$ of ${\cal BPK}$ generated by the simple members of ${\cal BPK}$, determine the structure of the free algebra over a finite set and their finite weakly projective algebras.