Is there a HSP-like theorem for algebras that can be axiomatized by a finite number of equations?

Question

It is a famous classic theorem that classes of algebras which can be axiomatized by equations are precisely those which are closed under homomorphisms, subalgebras, and products. Is there a corresponding theorem for those classes of algebras which can be axiomatized by finitely many equations?

Answer 1

Here's a weak negative observation:

Since any infinite equational theory is the union of its finite subtheories, we dually have that the variety corresponding to an infinite equational theory is the intersection of the family of varieties corresponding to its finite subtheories. Basically, this implies that any 'simple closure property' holding of each of the varieties in the latter family will also hold a fortiori of the variety of the whole infinite equational theory. In particular, suppose $\mathbb{F}$ is any monotone map between classes of isomorphism types of algebras (such as the $\mathbb{H}$, $\mathbb{S}$, and $\mathbb{P}$ of the original HSP theorem). Then if every finitely axiomatizable variety is closed with respect to $\mathbb{F}$, every variety whatsoever is closed with respect to $\mathbb{F}$.

Of course this doesn't rule out more intricate conditions of higher quantifier complexity than simply "is closed with respect to [thing]," but at that point we are straying a bit from the "flavor" of the original HSP theorem.