**Definition 0.** Let $R$ denote a commutative semiring with $0$ and $1$. By an $R$-monoid, I mean a monoid $M$ equipped with an action $R \times M \rightarrow M$ denoted $r,m \mapsto m^r$, satisfying the following axioms.

- $m^1 = m$
- $(m^r)^s = m^{rs}$
- $m^0 = 1$
- $m^{r+s} = m^r m^s$

*Remark.* If $R$ is a ring, then every $R$-monoid is a group, because $1 = m^{0} = m^{-1+1} = m^{-1} m.$

**Examples.**

- An $\mathbb{N}$-monoid is just a monoid.
- A $\mathbb{Z}$-monoid is just a group.
- The set $\mathbb{R}_{>0}$ can be regarded as an $\mathbb{R}$-monoid (or a $\mathbb{Q}$-monoid, if we wish), where the law of composition is $(p,q) \mapsto pq$ and the action is $x,p \mapsto p^x.$

Questions.

- What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids?
- What is the usual terminology for $R$-monoids?
- Is there any literature surrounding them? References appreciated.