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.