Question
What are all the functions $\alpha , \beta : \mathbb{N} \to \mathbb{N}$ satisfying the following functional equations?
$\bullet ~~~~ \alpha(0)=0, \quad \beta(0)=0\\ \bullet ~~~ \alpha^n(m+m')=\alpha^n(m) + \alpha^{\beta^m(n)}(m')\\ \bullet ~~~ \beta^m(n+n') = \beta^m(n) + \beta^{\alpha^n(m)}(n')$
Here, $\alpha^n$ denotes the $n$-fold composition of $\alpha$. So far I have only found the following solutions:
1) $\alpha(n)=n$, $\beta(n)=pn$, where $p \in \mathbb{N}$
2) $\alpha(n)=pn$, $\beta(n)=n$, where $p \in \mathbb{N}$
3) $\alpha(n)=\beta(n)=\max(n-1,0)$
I am not even sure if there are more solutions. Of course, 1) and 2) agree for $p=1$. Notice that $p=0$ is allowed in both 1) and 2).
Background
This is not necessary to understand the question, but I hope that it shows why the question is interesting.
The Zappa-Szép product $A \bowtie_{\sigma} B$ of two monoids $A,B$, written additively for our purposes, with respect to a map $\sigma=(\sigma_1,\sigma_2) : U(B) \times U(A) \to U(A) \times U(B)$ (here $U(A)$ denotes the underlying set of $A$) is a monoid with underlying set $U(A) \times U(B)$, unit $(0,0)$, and multiplication $(a,b) +_{\sigma} (a',b') = (a + \sigma_1(b,a'),\sigma_2(b,a') + b')$. For the monoid axioms to be satisfied, the maps $\sigma_1,\sigma_2$ have to satisfy the following requirements:
- $\sigma_1$ is a left action of $B$ on the set $U(A)$
- $\sigma_2$ is a right action of $A$ on the set $U(B)$
- $\sigma_1(b,0)=0, \quad \sigma_2(0,a)=0$
- $\sigma_1(b,a + a')=\sigma_1(b,a) + \sigma_1(\sigma_2(b,a),a')$
- $\sigma_2(b + b' , a) = \sigma_2(b,\sigma_1(b',a)) + \sigma_2(b',a)$
My goal is to classify all Zappa-Szép products of $(\mathbb{N},+,0)$ with itself (has this already been done in the literature?). Since here an action is simply determined by a map, this comes down to the system of functional equations as described above: We have $\alpha^n(m)=\sigma_1(n,m)$ and $\beta^m(n)=\sigma_2(n,m)$. The monoids corresponding to the three solutions so far have the following presentations as monoids:
1) $\langle X,Y : YX = X Y^p \rangle$
2) $\langle X,Y : YX = X^p Y \rangle$
3) $\langle X,Y : YX=1 \rangle$
