Let $(H, \cdot)$ be a (multiplicative) monoid. Is there any consolidated name for the following Property $\text{(P)}$, or for the class of monoids for which it is satisfied?

$$\text{(P) If }\,xy = x\,\text{ or }\, yx=x\,\text{ for some }\,x,y \in H,\,\text{ then }\,y \in H^\times.$$

The property implies that $xy \in H^\times$, for some $x, y \in H$, iff $x, y \in H^\times$, and it is satisfied by cancellative monoids and a number of non-cancellative ones, including:

- the non-empty finite subsets of $H$ endowed with the operation of set multiplication $(X, Y) \mapsto \{xy: x \in X,\, y \in Y\}$, when $H$ is a linearly orderable monoid;
- the non-empty $r$-ideals of $H$ equipped with the operation of $r$-multiplication $(I, J) \mapsto (IJ)_r$, when $H$ is commutative and $r$ is a strictly $r$-noetherian ideal system on $H$;
- Möbius monoids (
*added on Mar 07, 2017*), a natural setting for generalizing Rota's Möbius inversion in posets.

The question has its origins in the factorization theory of atomic monoids, which is the reason for the tag ra.rings-and-algebras.

*Edit* (*Oct 29, 2016*). I've just noted Benjamin Steinberg's comment below, so let me ask the same questions for the following variant:

$$\text{(P') If }\,xay = a\,\text{ for some }\,a, x, y \in H,\,\text{ then }\,x,y \in H^\times.$$

The two properties coincide in the commutative case, and $\text{(P')}$ implies $\text{(P)}$ in general (I don't know about the converse, but believe that $\text{(P')}$ is strictly stronger).