Regular Double $p$-Algebras: A converse to a Katri\v{n}\'{a}k's Theorem, and Applications

TL;DR

This paper establishes a converse to Katriňák's theorem for regular double p-algebras, demonstrating their term-equivalence with related algebraic varieties, and introduces new algebraizable logics with large extension lattices.

Contribution

It proves a converse to Katriňák's theorem, shows term-equivalence among several algebraic varieties, and introduces new logics with extensive extension lattices.

Findings

Varieties RDBLP, RDPCH, RPCH^d, RDBLH are term-equivalent.

Varieties RDMP, RDMH, RDMDBLH, RDMDBLP are term-equivalent.

Lattices of subvarieties and extensions have cardinality 2^{}.

Abstract

In 1973, Katri\v{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. In this paper we prove a converse to the Katri\v{n}\'{a}k's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation $\to$ satisfies the Katri\v{n}\'{a}k's formula. As applications of this result together with the above-mentioned Katri\v{n}\'{a}k's theorem, we show that the varieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras, regular dually pseudocomplemented Heyting algebras, regular pseudocomplemented dual Heyting algebras, and regular double Heyting algebras, respectively, are term-equivalent to each other and also that the varieties RDMP, RDMH, RDMDBLH, RDMDBLP of regular De Morgan $p$-algebras, regular De Morgan Heyting algebras, regular De Morgan double Heyting algebras, and regular De Morgan double $p$-algebras, respectively, are also term equivalent to each other. From these results and recent results of Adams, Sankappanavar and vaz de Carvalho, we deduce that the lattices of subvarieties of all these varieties have cardinality $2^{\aleph_0}$. We then define new logics, RDPCH, RPCHd, and RDMH, and show that they are algebraizable with RDPCH, RPCH$^d$ and RDMH, respectively as their equivalent algebraic semantics. It is also deduced that the lattices of extensions of all of the above mentioned logics have cardinality $2^{\aleph_0}$.