A semigroup $S$ is _duo_ if $aS = Sa$ for all $a \in S$, where $aS := \{ax: x \in S\}$ and similarly for $Sa$ (for instance, every commutative semigroup is duo, and so is every group). On the other hand, we say that $S$ is a _left/right cancellative_ semigroup if, for each $a \in S$, left/right multiplication by $a$ is an injective function on $S$. Then, $S$ is called _cancellative_ if it is both left and right cancellative.

I recently asked ([here][1]) if a cancellative semigroup embeds into a group. Pace Nielsen promptly answered the question in the affirmative ([here][2]). In a comment to Pace's answer ([here][3]), I then raised the following:

> **Q.** If a duo semigroup is left/right cancellative, is it cancellative? Any reference?

There is a certain abundance of non-commutative, cancellative duo semigroups in nature (see [here][4] for a short list). Yet, one-sided cancellative, duo semigroups appears to be 'rare' (of any exists). Among other things, note that every _finite_ left/right cancellative duo semigroup is a group. 


  [1]: https://mathoverflow.net/q/480390/16537
  [2]: https://mathoverflow.net/a/480395/16537
  [3]: https://mathoverflow.net/questions/480390/does-every-cancellative-duo-semigroup-embed-into-a-group#comment1250637_480395
  [4]: https://mathoverflow.net/a/468738/16537