**Question.** Is there any structure theorem for the class of monoids $H$ with the property that $xy = x$ or $xy = y$ for all $x, y \in H$? Or does this look hopeless for some good reasons?

A monoid with the above property is idempotent and need not be commutative. Examples include (i) the unitization of a left (resp., right) singular semigroup, where one starts with a set $X$ and defines an associative binary operation on $X$ by taking $xy := x$ (resp., $xy := y$) for all $x, y \in X$; or a chain of subsets of a "universe" $V$ made into a monoid by the operation that maps two sets in the chain to their union.

I'm aware of the work of D. McLean in [*Idempotent semigroups*, Amer. Math. Monthly **61** (1954), 110-113] and N. Kimura (e.g., [*The structure of idempotent semigroups. I*, Pacific J. Math. **8** (1958), No. 2, 257-275]) on the structure of idempotent semigroups, but I'm wondering if something more precise/specific can be done/has been done for the monoids of this thread.

**Motivation.** The question arises from the study of the arithmetic of a certain monoid of sets, $P_{\text{fin},1}(H)$, naturally attached to an arbitrary monoid $H$ (hereby written multiplicatively). More precisely, $P_{\text{fin},1}(H)$ is obtained by endowing the family of all finite subsets of $H$ containing the identity $1_H$ with the operation of setwise multiplication induced by $H$ (in some circles, this thing is commonly referred to as the *reduced (finitary) power monoid* of $H$); and one can show that, if $H$ is commutative and satisfies the above property (that $xy = x$ or $xy = y$ for all $x, y \in H$), then $P_{\text{fin},1}(H)$ enjoys a form of unique factorization (into irreducibles).

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