Every commutative monoid $M$ is naturally equipped with its divisibility preorder, defined as follows.
$$x \mid y \leftrightarrow \exists a(ax=y)$$
Is there a name for those commutative monoids such that the above preorder is antisymmetric? In other words, I'm interested in those commutative monoids satisfying the following quasi-identity:
$$\frac{ax=y\quad by=x}{x=y}$$
Motivation. The category of all such structures is probably a reflective subcategory of the category of all commutative monoids, with the left-adjoint to the inclusion functor being the functor $F$ such that $F(M)$ is the commutative monoid obtained by identifying elements $x,y \in M$ satisfying $x \mid y$ and $y \mid x$. Now given a commutative monoid $M$, we are often interested in meets and joins with respect to the divisibility order, but uniqueness issues rear their annoying heads. They can be remedied by working not in $M$, but in $F(M).$
At least two different terms are used in the literature for a commutative monoid in which division is a partial order: holoid and naturally partially ordered. Another possibility would be $\mathcal{H}$-trivial since a commutative semigroup has the required property if and only if the Green's relation $\mathcal{H}$ is the equality in this monoid. See Grillet's book Commutative Semigroups (2001), pages 120 and 201. I was able to trace back the term "holoid" as early as 1942, but it might have been introduced long before.