Locally finite varieties of Heyting algebras

Abstract

We show that for a variety $$ \mathcal{V} $$ of Heyting algebras the following conditions are equivalent: (1) $$ \mathcal{V} $$ is locally finite; (2) the $$ \mathcal{V} $$ -coproduct of any two finite $$ \mathcal{V} $$ -algebras is finite; (3) either $$ \mathcal{V} $$ coincides with the variety of Boolean algebras or finite $$ \mathcal{V} $$ -copowers of the three element chain $$ {\text{3}} \in \mathcal{V} $$ are finite. We also show that a variety $$ \mathcal{V} $$ of Heyting algebras is generated by its finite members if, and only if, $$ \mathcal{V} $$ is generated by a locally finite $$ \mathcal{V} $$ -algebra. Finally, to the two existing criteria for varieties of Heyting algebras to be finitely generated we add the following one: $$ \mathcal{V} $$ is finitely generated if, and only if, $$ \mathcal{V} $$ is residually finite.