REGULAR DOUBLE p-ALGEBRAS: A CONVERSE TO A KATRINAK THEOREM AND APPLICATIONS

1973, Katrinak 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 Katrinak's theorem, in the sense that in the variety RDPCH of regular dually pseudocomplemented Heyting algebras, the implication operation -> satisfies Katrinak's formula. As applications of this result together with the above-mentioned Katrinak's theorem, we show that the varieties RDBLP, RDPCH, RPCHd 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 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(N0). We then define new logics, RDPCH, RPCHd and RDMH, and show that they are algebraizable with RDPCH, RPCHd 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(N0). (C) 2023 Mathematical Institute Slovak Academy of Sciences