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Prompted by the comments to a recent answer by YCor to a related question (here), I'd like to ask the following:

Q. Does every cancellative duo semigroup embed into a group?

A (multiplicatively written) semigroup $S$ is cancellative if, for every $a \in S$, right and left multiplication by $a$ are both injective functions on $S$; and it is duo if $aS = Sa$ for all $a\in S$.

By a famous example of Mal'cev [Math. Ann. 113 (1937), No. 1, 686-691], not every cancellative semigroup is embeddable into a group; thus, one might preliminarily ask whether Mal'cev's example is duo. (EDIT: it follows from Pace Nielsen's answer below that Mal'cev's example is not duo.) On the other hand, it is folklore (and easy to prove) that a commutative semigroup embeds into a group if and only if it is cancellative. Duo semigroups lie halfway between the commutative and non-commutative worlds; it therefore seems natural to inquire whether they are closer to one or the other when it comes to embeddability.

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Let $S$ be a cancellative duo semigroup. Let $a,b\in S$ be arbitrary, and consider the element $x:=ba$. Clearly, $x\in Sa$ and by the duo property $x\in bS=Sb$. Thus, $S$ is right reversible, and so $S$ embeds in a group by Ore's theorem (see this wiki page).

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    $\begingroup$ Short n' sweet. $\endgroup$
    – Salvo Tringali
    Commented Oct 10 at 4:08
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    $\begingroup$ Is there a version of Ore's theorem that applies to duo semigroups that are cancellative only on one side (that is, left or right cancellative)? In other words, if a duo semigroup is left or right cancellative, then is it cancellative (on both sides)? I will eventually ask a separate question on this, unless the answer turns out to be 'trivial'. $\endgroup$
    – Salvo Tringali
    Commented Oct 10 at 13:19
  • $\begingroup$ Here is the link to the question alluded to in my previous comment: mathoverflow.net/q/480608/16537 $\endgroup$
    – Salvo Tringali
    Commented Nov 2 at 22:26

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