Every infinite cyclic semigroup is isomorphic to the positive natural numbers under addition

Prove that every infinite cyclic semigroup is isomorphic to the semigroup (\mathbb{N}^+, +) of positive natural numbers under addition.


First, let’s clarify some terminology. If S is a semigroup and s \in S, then there is a semigroup homomorphism \varphi : \mathbb{N}^+ \rightarrow S given by k \mapsto s^k. The image of \varphi is called the cyclic subsemigroup of S generated by s, and if \varphi is surjective, then we say S is cyclic with s as a generator. If S has an (unique) identity element e, then the homomorphism \varphi extends to a mapping \psi : \mathbb{N} \rightarrow S by 0 \mapsto e and this is also a semigroup homomorphism. The image of \psi is called the cyclic submonoid of S generated by s.

First, it is clear that (\mathbb{N}^+, +) is an infinite cyclic semigroup.

Suppose now that S is an infinite cyclic semigroup with s a generator. Now \varphi : \mathbb{N}^+ \rightarrow S given by k \mapsto s^k is a surjective semigroup homomorphism. We only need to show that \varphi is injective. To that end, suppose we have \varphi(a) = \varphi(b) for some a,b \in \mathbb{N}^+, so that s^a = s^b. Suppose, without loss of generality, that a < b; say b = a+k, so that s^{a+k} = s^{a}. But now for all t \geq 0, by the division algorithm we have t = qk + r for some q and 0 \leq r < k. Then s^{a+t} = s^{a+qk+r} = s^{a+r}. Indeed, for all positive natural numbers \ell, we have \varphi(\ell) \in \{s^1, s^2, \ldots, s^a, s^{a+1}, \ldots, s^{a+k-1}\}. In particular, \mathsf{im}\ \varphi, hence S, is finite- a contradiction. Thus a = b, and so \varphi is injective. So S \cong \mathbb{N}^+.

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