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In 1973, Katri\v{n}\'{a}k proved that regular double $p$-algebras can beregarded as (regular) double Heyting algebras by ingeniously constructingbinary terms for the Heying implication and its dual in terms ofpseudocomplement and its dual. In this paper we prove a converse to theKatri\v{n}\'{a}k's theorem, in the sense that in the variety RDPCH of regulardually pseudocomplemented Heyting algebras, the implication operation $\to$satisfies the Katri\v{n}\'{a}k's formula. As applications of this resulttogether with the above-mentioned Katri\v{n}\'{a}k's theorem, we show that thevarieties RDBLP, RDPCH, RPCH$^d$ and RDBLH of regular double $p$-algebras,regular dually pseudocomplemented Heyting algebras, regular pseudocomplementeddual Heyting algebras, and regular double Heyting algebras, respectively, areterm-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 theseresults and recent results of Adams, Sankappanavar and vaz de Carvalho, wededuce that the lattices of subvarieties of all these varieties havecardinality $2^{\aleph_0}$. 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 thatthe lattices of extensions of all of the above mentioned logics havecardinality $2^{\aleph_0}$.

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