> **Q1.** Is there any standard name for a 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 are 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.