Under which conditions are finitely presented algebras finite?

Question

Let V be a nontrivial variety with a finite signature. Under which conditions (if any) is every algebra finite which is finitely presentable in V?

Answer 1

Let $\mathcal V$ be a variety. The following are equivalent:

1. All algebras that are finitely presentable in $\mathcal V$ are finite.
2. All finitely generated $\mathcal V$-free algebras are finite.
3. All finitely generated algebras in $\mathcal V$ are finite. ($\mathcal V$ is locally finite.)