**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).



Structure theorem for a class of idempotent monoids (where $xy = x$ or $xy = y$ for all $x, y$)
The semigroups in the question seem to have first been introduced by Redei in his book Algebra. Vol 1. The book was originally published in Hungarian in 1954, and then German in 1959. The English edition is 1967. In the English version they are called breakable semigroups. The main result in the English edition about them is Theorem 50 of section 27 which is the result in the previous answers that you have a bunch of left/right zero semigroups indexed by a totally ordered set with the product of elements from different indices the one in the smaller indexed set. This is more or less immediate from the definition and the fact that a band is a semilattice of rectangular bands if you are familiar with semigroup structure theory. This is more or less what @YemonChoi explains in his answer in detail.


