The title has it all. Is there any consolidated terminology for referring to a (multiplicative) monoid $H$ such that $xy \in H^\times$ (if and) only if $x, y \in H^\times$? Here is a short list of monoids with this property:


* Commutative monoids.
* [Unit-cancellative](https://mathoverflow.net/a/259790/16537) monoids (so, in particular, cancellative monoids).
* Monoids with at least one atom/irreducible element (as [proved by Benjamin Steinberg](https://mathoverflow.net/a/261870/16537)).


Incidentally, the three classes of monoids from this list have, of course, non-empty intersection, but none of them is contained in the union of the others.


*Edit.* In a previous version of this post, I was writing that the property in the title holds if $H^\times = \{1\_H\}$ (i.e., for reduced monoids). That's not true, as can be seen by looking at $\langle x, y \mid xy = 1 \rangle$.



Terminology for a monoid $H$ s.t. $xy \in H^\times$ only if $x, y \in H^\times$
The correct term is Dedekind finite. A monoid is Dedekind finite of $xy=1$ implies $yx=1$. This is clearly equivalent to your condition.


