Say that a preorder (i.e., a reflexive and transitive binary relation) $\preceq$ on a set $X$ is

*artinian*if there is no sequence $(x_n)_{n \ge 1}$ of elements of $X$ with $x_{n+1} \prec x_n$ for each $n$, where $u \prec v$ means as usual that $u \preceq v$ and $v \not\preceq u$ (some authors prefer the term "well-founded", others the term "noetherian"; I'm going for the term "artinian" because it sounds natural in the light of certain applications);*noetherian*if its dual $\preceq^{\rm op}$ is artinian, where $x \preceq^{\rm op} y$ iff $y \preceq x$.

Next, let $H$ be a (commutative or non-commutative) monoid and denote

by $\mid_H$ the

*divisibility preorder*(on $H$), defined by $x \mid_H y$ iff $y = uxv$ for some $u, v \in H$;by $\dashv_H$ the "divides-from-the-right'' preorder, defined by $x \dashv_H y$ iff $y = ux$ for some $u \in H$;

by $\vdash_H$ the "divides-from-the-left" preorder, that is, the "divides-from-the-right" preorder in the opposite monoid $H^{\rm op}$ of $H$.

My question is **whether $\mid_H$ is artinian iff both $\dashv_H$ and $\vdash_H$ are artinian**. I'm sure this is well known, but I haven't been able so far to **find a reference**. (By the way, is there a more standard (relational) symbol for the preorders I'm denoting by $\dashv_H$ and $\vdash_H$?)

The duals of these preorders were thoroughly studied in

J.A. Green,

On the Structure of Semigroups, Annals of Math.54(1951) 163-172;

whence they are often referred to as the *Green preorders*. In particular, Theorem 4 in Green's paper implies that, if both $\dashv_H$ and $\vdash_H$ are ~~artinian~~ noetherian, then so also is $\mid_H$ (thanks to Benjamin Steinberg who made me notice in the comments below that I had misread Green's definitions and hence the conclusions of the theorem).

For the record, what I can prove is that the following are equivalent:

- $H$ is
*acyclic*(i.e., $uxv \ne x$ for all $u, v, x \in H$ with $u \notin H^\times$ or $v \notin H^\times$) and $\mid_H$ is artinian. - $H$ is
*unit-cancellative*(i.e., $xy \ne x$ and $x \ne yx$ for all $x, y \in H$ with $y \notin H^\times$) and both $\dashv_H$ and $\vdash_H$ are artinian.

Here, $H^\times$ is the group of units of the monoid $H$.

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