Is the class of group-embeddable monoids an axiomatizable class?

Question

Is the class of monoids which can be embedded in a group a first-order axiomatizable class? And if it is, is it finitely axiomatizable?

Answer 1

See https://en.wikipedia.org/wiki/Cancellative_semigroup#Embeddability_in_groups and When a semigroup can be embedded into a group. A commutative monoid (or more generally a semigroup) can be embedded in a group if and only if it satisfies a cancellation property, which is expressible by a single first-order sentence.

On the other hand, for non-commutative semigroups, the situation is vastly more complicated. It appears (via Wiki) that in 1939 Mal'cev (http://www.ams.org/mathscinet-getitem?mr=0002152) found an infinite family of first-order sentences characterizing the semigroups which are so embeddable, and then in 1940 (http://www.ams.org/mathscinet-getitem?mr=0002895) showed that no finite set would suffice. However, I can't get access to these papers, so I'm not sure that's accurate.