Q1.Is there any standard name for a (multiplicatively written) monoid $H$ with the property that, for all $x, y \in H \setminus H^\times$, there exist $m, n \in \mathbf N^+$ and $u, v \in H^\times$ such that $x^m = uy^n v$? Here, $H^\times$ is, as usual, the set of units (or invertible elements) of $H$.

A few examples of monoids with the above property: (a) groups; (b) *numerical monoids*, that is, submonoids of $(\mathbf N, +)$; (c) *Puiseux monoids*, i.e., submonoids of $(\mathbf Q_{\ge 0}, +)$.

Q2.What about otherinterestingexamples from the literature?

Of course, Puiseux monoids are more general than numerical monoids, but there are some good reasons for distinguishing them.

*Edit.* Benjamin Steinberg points out that, if we require that, for all $x, y \in H$, there exist $m,n \in \mathbf N^+$ such that $x^m \mid_H y^n$ (i.e., if we drop the restriction about non-units), then what we get are the *archimedean monoids*. So, I should probably add that the reason for requiring $x, y \in H \setminus H^\times$ and $u,v \in H^\times$ in the above definitions, comes from the fact that I'm interested in the "structure" of the *system of sets of lengths* of $H$, that is, the family of sets
$$
\mathscr{L}(H) := \{\mathsf L_H(x): x \in H\} \setminus \{\emptyset\} \subseteq \mathcal P(\mathbf N),
$$
where for a fixed $x\in H\setminus\{1_H\}$ we denote by $\mathsf L_H(x)$ the set of all $k \in \mathbf N^+$ for which there exist atoms $a_1,\ldots,a_k\in H$ such that $x=a_1\cdots a_k$, while $\mathsf L_H(1_H):=\{0\}$. So, in this context, $H$ is *interesting* only if $\mathscr L(H)$ is "rich" (in particular, $H$ should have at least one atom).