Regular Double p-Algebras: A Converse to a Katriňák Theorem and Applications

Abstract

ABSTRACT In 1973, Katriňák proved that regular double p-algebras can be regarded as (regular) double Heyting algebras by ingeniously constructing binary terms for the Heyting implication and its dual in terms of pseudocomplement and its dual. In this paper, we prove a converse to Katriňák’s theorem, in the sense that in the variety ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] of regular dually pseudocomplemented Heyting algebras, the implication operation → satisfies Katriňák’s formula. As applications of this result together with the above-mentioned Katriňák’s theorem, we show that the varieties ℝ D B L ℙ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] , ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] 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 ℝ D M ℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{P}\] , ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , ℝ D M D B L ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{H}\] , ℝ D M D B L ℙ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{D}\mathbb{B}\mathbb{L}\mathbb{P}\] 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 on varieties of regular double p-algebras and regular pseudocomplemented De Morgan algebras, we deduce that the lattices of subvarieties of all these varieties have cardinality 2 ℵ 0 \[{{2}^{{{\aleph }_{0}}}}\] . We then define new logics, ℛ D P C ℋ \[\mathcal{R}\mathcal{D}\mathcal{P}\mathcal{C}\mathcal{H}\] , ℛ P C ℋ d \[\mathcal{R}\mathcal{P}\mathcal{C}{{\mathcal{H}}^{d}}\] and ℛ D ℳ ℋ \[\mathcal{R}\mathcal{D}\mathcal{M}\mathcal{H}\] , and show that they are algebraizable with ℝ D ℙ ℂ ℍ \[\mathbb{R}\mathbb{D}\mathbb{P}\mathbb{C}\mathbb{H}\] , ℝ ℙ ℂ ℍ d \[\mathbb{R}\mathbb{P}\mathbb{C}{{\mathbb{H}}^{d}}\] and ℝ D M ℍ \[\mathbb{R}\mathbb{D}\mathbb{M}\mathbb{H}\] , 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 ℵ 0 \[{{2}^{{{\aleph }_{0}}}}\] .