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.