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:
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 $F(M).$