Decidable varieties of p-algebras

TL;DR

This paper characterizes quasivarieties of p-algebras with decidable first-order theories, showing only trivial and Boolean algebras qualify, contrasting with Heyting algebras where decidability differs.

Contribution

It establishes a precise classification of decidable quasivarieties of p-algebras, highlighting the unique nature of these structures compared to Heyting algebras.

Findings

Only trivial and Boolean algebras have decidable theories among p-algebras.

Decidability of first-order theory coincides with that of finite members for these quasivarieties.

Contrasts with Heyting algebras where decidability properties differ.

Abstract

We show that for quasivarieties of p-algebras the properties of (i) having decidable first-order theory and (ii) having decidable first-order theory of the finite members, coincide. The only two quasivarieties with these properties are the trivial variety and the variety of Boolean algebras. This contrasts sharply, even for varieties, with the situation in Heyting algebras where decidable varieties do not coincide with finitely decidable ones.