Is there a converse to Baker's theorem?

Question

Suppose that V is a variety of finite signature. Baker's Theorem says that every finite algebra is finitely based if V is congruence distributive. Conversely, if V is locally finite, then every finitely based algebra in V is finite. Are there other conditions on V which imply that finitely based algebras are finite?

Answer 1

Are there other conditions on V which imply that finitely based algebras are finite?

There are no nontrivial conditions like this.

Assume that $\mathcal V$ is a variety whose finitely based algebras are finite. Let $\bf A$ be any finitely based algebra in $\mathcal V$. The algebras $\bf A$ and ${\bf A}^{\omega}$ satisfy the same identities, so ${\bf A}^{\omega}$ is also a finitely based algebra in $\mathcal V$. The assumption on $\mathcal V$ forces ${\bf A}^{\omega}$ to be finite. This can only happen if $|{\bf A}|\leq 1$. Thus, if $\mathcal V$ is any variety whose finitely based algebras are finite, then the only finitely based algebras in $\mathcal V$ are the trivial algebras satisfying $|{\bf A}|\leq 1$. (Trivial algebras do have a finite basis, namely ${\mathcal B}=\{x\approx y\}$.)