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 monoids (so, in particular, cancellative monoids).
- Monoids with at least one atom/irreducible element (as proved by Benjamin Steinberg).

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 $\langle x, y \mid xy = 1 \rangle$.

Edit.OK, I guess you mean the bicyclic monoid from p. 32 in the 2003 reprint of Howie'sFundamentals of Semigroup Theory. $\endgroup$