Varieties of Regular Pseudocomplemented de Morgan Algebras

Abstract

In this paper, we investigate the varieties ${\mathbf{M}}_n$ and ${\mathbf{K}}_n$ of regular pseudocomplemented de Morgan and Kleene algebras of range $n$, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in ${\mathbf{M}}_n$ and explicitly describe the dual spaces of the simple algebras in ${\mathbf{M}}_1$ and ${\mathbf{K}}_1$. We show that the variety ${\mathbf{M}}_1$ is locally finite, but this property does not extend to ${\mathbf{M}}_n$ or even ${\mathbf{K}}_n$ for $n\ge 2$. We also show that the lattice of subvarieties of ${\mathbf{K}}_1$ is an $\omega +1$ chain and the cardinality of the lattice of subvarieties of either ${\mathbf{K}}_2$ or ${\mathbf{M}}_1$ is $2^{\omega }$. A description of the lattice of subvarieties of ${\mathbf{M}}_1$ is given.

29 pages; 2 figures