Find all functions that compose with the successor function

Question

In Mac Lane/Birkhoff's Algebra, they spend some time discussing the natural numbers and give the Peano Axioms, roughly (from memory)

• $\sigma$ is injective
• 0 is not the successor of any element
• the principle of induction, which yields the natural numbers

My question is about an exercise at the end of the section which asks

For the usual successor function $\sigma$, find all functions $\phi : \mathbb{N} \to \mathbb{N}$ such that $\sigma \phi = \phi \sigma$.

My thought is that $\phi$ must commute with addition, which means that functions of the type $\phi(x) = x + a$ for some constant $a \in \mathbb{N}$. And then there are also inverses of $\sigma$ which would commute, and the identity function.

But are those the only functions? I'm not sure how to show that. I feel like the previous exercise is a hint:

For some $\tau : \mathbb{N} \to \mathbb{N}$ that satisfies the Peano Axioms, show $\tau \beta = \beta \sigma$ for some bijection $\beta : \mathbb{N} \to \mathbb{N}$.

I think (guessing here) this other exercise is showing that each model of $\mathbb{N}$ is unique (sorry, no model theory), for each function that satisfies the Peano Axioms, but I don't see how to apply this to the other exercise, especially since $\phi$ doesn't have to satisfy the Peano Axioms.

Answer 1

The condition $\sigma \phi = \phi \sigma$ induces a recurrence relation, namely : $$\forall n, \phi(n + 1) = \phi(n) + 1$$ Hence, $\phi$ is entirely determined by the value $\phi(0)$ since $\phi(n) = \phi(0) + n$.

The functions $\phi : \mathbb{N} \mapsto \mathbb{N}$ satisfying $\sigma \phi = \phi \sigma$ are hence precisely the functions of the form $x \mapsto x + a$, $a \in \mathbb{N}$.