What are some examples of non-commutative $\mathbb{Q}$-monoids and/or $\mathbb{R}$-monoids? 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.
  

 A: A nice example.  Goes back to Cayley, 1860 [1].
Formal power series of the form
$$
f(x) = x + a_1x^2+a_2x^3+a_3x^4+\cdots
$$
with real coefficients.  Under composition.  Cayley showed how to do "fractional" composites, of real (or even complex) order.  The series need not converge, even if $f$ does.  That's why I said "formal" power series.  
plug: I did a generalization for transseries [2].
[1] A. Cayey, On some numerical expansions.  Quarterly Journal of Pure and Applied Mathematics 3 (1860) 366--369
[2] G. Edgar, Fractional iteration of series and transseries.  Transactions of the American Mathematical Society 365 (2013) 5805--5832
A: I would guess that an $\mathbb R$-monoid is the same thing (once you put extra smoothness axioms on the $\mathbb R$-action) as a Lie group whose exponential map $\mathfrak g\to G$ is a diffeomorphism. The Heisenberg group
$$
\left(\begin{matrix}
1 & * & * \\
0 & 1 & * \\
0 & 0 & 1 \\
\end{matrix}\right)
$$
is the simplest non-commutative example of such a group. Take the same example but restrict $*\in \mathbb Q$ to get an example of a $\mathbb Q$-monoid which is not an $\mathbb R$-monoid.
