Is there an equational theory which has non-trivial finite models but no infinite model?

Question

This is a dual question to my previous question, here: Is there an equational theory which has infinite models but no non-trivial finite models?. My current question is, does there exist an algebraic signature $\Omega$ and an equational theory $T$ of $\Omega$ such that $T$ has non-trivial finite models, but no infinite model? Non-trivial means cardinality greater than $1$.

Answer 1

If there is one result from universal algebra that you should know it is this:

A class of algebraic structures is an equational class if and only if it is closed under homomorphic images, subalgebras, and direct products (with arbitrary index).

Using this, let $\mathbf{A}$ be an algebraic structure with $\vert A\vert>1$. Choose your favorite infinite set $I$. Then $\displaystyle\Pi_{i\in I}\mathbf{A}$ satisfies the same equational identities as $\mathbf{A}$ and is infinite.

The original source for this result is

Birkhoff, Garrett. "On the structure of abstract algebras." Proceedings of the Cambridge Philosophical Society. 1935.