Regular Double p-Algebras: A converse to a Katrinak Theorem, and Applications

Abstract

In 1973, Katri{n}´{a}k proved that regular double $p$-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heying implication and its dual in terms of pseudocomplement and its dual.We prove a converse to the Katri{n}´{a}k´s theorem, in the sense that in the variety $mathbb{RDPCH}$ of regular dually pseudocomplemented Heyting algebras, $o$ satisfies the Katrinak´s formula. As applications of this result together with the above-mentioned Katri{n}´{a}k´s theorem, we show that the varieties $mathcal{RDBLP}$, $mathcal{RDPCH}$ and $mathcal{RDBLH}$ of regular double p-algebras, regular dually pseudocomplemented Heyting algebras and regular double Heyting algebras, respectively, are term-equivalent to each other and that the varieties $mathcal{RDMP}$ and $mathcal{RDMH}$ of regular pseudocomlemented De Morgan algebras and regular Heyting algebras, respectively, are also term-equivalent to each other. From these results and recent results of cite{AdSaVc19} and cite{ AdSaVc20}, we deduce that the lattices of varieties $mathbb{RDPCH}$, $mathbb{RDBLH}$ and $mathbb{RDMH}$, respectively, of regular dually pseudocomplemented Heyting algebras, regular double Heyting algebras, regular De Morgan Heyting algebras all have cardinality $2^{aleph_0}$. These results, when combined with a result of cite{CoSa20}, in turn, lead us to define a new algebraizable logic, namely $mathcal{RDBLP}$, haviing $mathbb{RDBLP}$,as its equivalent algebraic semantics. It is also deduced that the lattices of extensions of logics $mathcal{RDBLP}$, $mathcal{RDPCH}$, $mathcal{RDBLH}$ and $mathcal{RDMH}$ have cardinality $2^{aleph_0}$.