Locally finite varieties of Heyting algebras (Q818713)

For a variety \(\mathcal V\) of Heyting algebras it is shown that the following are equivalent: (1) \(\mathcal V\) is locally finite; (2) the \(\mathcal V\)-coproduct of two finite \(\mathcal V\)-algebras is finite; (3) either \(\mathcal V\) is the variety of Boolean algebras or finite \(\mathcal V\)-copowers of the 3-element chain are finite. It is also shown that \(\mathcal V\) is generated by its finite members if and only if it is generated by a locally finite \(\mathcal V\)-algebra, and that \(\mathcal V\) is finitely generated if and only if it is residually finite.