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 $n \geq 2$. 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.