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.

1. $m^1 = m$
2. $(m^r)^s = m^{rs}$
3. $m^0 = 1$
4. $m^{r+s} = m^r m^s$

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

Examples.

1. An $\mathbb{N}$-monoid is just a monoid.
2. A $\mathbb{Z}$-monoid is just a group.
3. The monoid $(\mathbb{R}_{> 0},\times,1)$ can be regarded as an $\mathbb{R}$-monoid (or a $\mathbb{Q}$-monoid, if we wish.)

Questions.

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

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.

1. $m^1 = m$
2. $(m^r)^s = m^{rs}$
3. $m^0 = 1$
4. $m^{r+s} = m^r m^s$

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

Examples.

1. An $\mathbb{N}$-monoid is just a monoid.
2. A $\mathbb{Z}$-monoid is just a group.
3. 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.

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