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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?

user107952
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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.

Alex Kruckman
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Noah Schweber
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    [Here](http://www.math.unl.edu/~jmeakin2/groups%20and%20semigroups.pdf) is a nice article. Apparently, in 1951, Lambek gave a different but equivalent list of axioms, described as "more geometric". There seems to be quite a lot of reading to do on this subject. – Andrew Dudzik Jun 18 '15 at 01:23