I’m recently delving into abstract algebra, and I’ve attempted to devise a monoid action on the natural numbers. I think I must be missing something here—is there a better way to represent these same operations as an algebra?
I’ve defined a transformation monoid of $\mathbb{N}$ such that the elements are the nth-prime function $p$, the “curried” multiply-by-$k$ function for all $k \in \mathbb{N}$ (which includes the identity as $k=1$), and all compositions of these. So the basis is $(n \mapsto p_n, n \mapsto n, n \mapsto 2n, n \mapsto 3n, \ldots)$. The monoid operation is function composition.
Including each separate multiply-by-$k$ function seems like a mistake, or at least inelegant—especially if I let the natural numbers also be a multiplicative monoid $(\mathbb{N}, \times)$. It seems like a redundancy.
Is there a better way to represent this? What am I missing?
If I understand correctly, you are defining a submonoid $T$ of the monoid $T_{\Bbb N}$ of all transformations on $\Bbb N$. Your monoid $T$ is generated by the transformation $\pi: n \to p_n$ and by the transformations $\rho_k: n \to kn$, for all $k > 0$. It turns out that the set $M = \{\rho_k \mid k > 0\}$ is closed under composition and hence forms a submonoid of $T_{\Bbb N}$, namely the monoid of all non-zero translations of $(\Bbb N, \times)$. Note that $M$ is generated by the set $P = \{\rho_p \mid \text{$p$ is prime}\}$ and is actually the free commutative monoid of base $P$. I guess that is what you mean when you talk about the redundancy of $M$.
Describing $T$ is more complicated, since it involves the powers of $\pi$. But as the submonoid $N$ of $T$ generated by $\pi$ is free and isomorphic to $({\Bbb N}, 0, +)$, I would guess that $T$ is isomorphic to the free product of $N$ and $M$.