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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).

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  • $\begingroup$ I know of no name $\endgroup$ – Benjamin Steinberg Oct 28 '16 at 20:44
  • $\begingroup$ The condition that xy is a unit of and only if x,y are units is true in all finite semigroups but not the stronger conditions which imply that only the identity is an idempotent $\endgroup$ – Benjamin Steinberg Oct 29 '16 at 14:38
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Sorry for answering my own question, but it's definitely clear that there is no consolidated terminology for the kind of properties mentioned in the OP. One reason could be that they have never been considered before, which is the impression I've drawn from talking to various semigroup theorists. So we resolved to use the term unit-cancellative (suggested by Pedro García-Sánchez) for a monoid with property (P), and strongly unit-cancellative for a monoid with the (stronger) property that if $xy = x$ or $yx = x$ for some $x, y \in H$, then $y = 1_H$.

Edit (March 03, 2017). It turns out that a similar notion has been considered by various authors in the context of commutative unital rings, where it is said that a commutative unital ring $R$ is présimplifiable if $ab = a$, for some $a, b \in R$, only if $a = 0_R$ or $b \in R^\times$.

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