What does it mean to say that the quasi-equational theory of groups is not finitely based?

Question

In a MathOverflow question, I asked if the quasi-equational theory of groups in the signature $\{*\}$ is finitely based. The answer was negative. However, groups in the signature $\{*\}$ can definitely be finitely axiomatized, by the axioms for associativity and identity elements and inverse elements, and from those axioms, every quasi-identity that is valid for groups can be proven. So, was that answer incorrect? What does it really mean to say that the quasi-equational theory of groups in the signature $\{*\}$ is not finitely based?

Answer 1

Let $\mathcal G$ be the class of groups defined in the signature $\{\ast\}$. This question is about the relationship between the first-order theory $\Gamma$ of $\mathcal G$ and the quasi-equational theory $\Sigma$ of $\mathcal G$.

Some Facts.

$\Sigma\subseteq \Gamma$ (since quasi-identities are first-order). This inclusion is proper, since, for example, the sentence asserting the existence of inverses belongs to $\Gamma\setminus \Sigma$.

The class of models of $\Gamma$ is the class $\mathcal G$ of semigroup reducts of groups, while the class of models of $\Sigma$ is the class $\mathcal Q$ of semigroups that are embeddable in groups. (So ${\mathcal G}\subseteq {\mathcal Q}$, and the natural numbers under addition belongs to ${\mathcal Q}\setminus {\mathcal G}$.)

$\Gamma$ is finitely axiomatizable. (Just write down the familiar first-order axioms for groups in the signature $\{\ast\}$.)

$\Sigma$ is not finitely axiomatizable according to a famous result of Maltsev.

Now to turn back to the question asked. Since $\Gamma$ and $\Sigma$ are distinct theories, it is possible for one to be finitely axiomatizable while the other is not. (And, in fact, this is what happens.)