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 other interesting examples 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).
