A *numerical monoid* (or *numerical semigroup*) is a submonoid $S$ of the additive monoid $(\mathbb N, +)$ of non-negative integers with the property that the set $\mathbb N \setminus S$ is finite.


It is folklore that two numerical monoids are (monoid-)isomorphic [if and] only if they are equal. I know at least a couple of proofs of this result, but **what about a reference?** For instance, the conclusion follows from the following facts:


* In the category of cancellative commutative monoids, every homomorphism extends to a (group) homomorphism of the corresponding quotient groups (see, e.g., Lemma 11.20 in J.C. Rosales and P.A. García-Sánchez, *Numerical Semigroups*, Dev. Math. **20**, Springer, 2009).
* The quotient group of a numerical monoid is the additive group $(\mathbb Z, +)$ of the integers.
* The group endomorphisms of $(\mathbb Z, +)$ are the dilation maps.


A more direct proof is using that any two coprime elements in a numerical monoid $S$ generate every sufficiently large element of $S$ (by a corollary of [Bézout's identity](https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity)).



Two numerical monoids are isomorphic iff they are equal
The earliest reference for this seems to be Theorem 3 of Higgins, John C. Representing N-semigroups. Bull. Austral. Math. Soc. 1 (1969), 115–125. In this theorem, he proves an essentially equivalent result. He proves if $K$ and $L$ are submonoids of N and there is as surjective homomorphism from $K$ to $L$, then $K$ and $L$ are both integral multiples of a numerical semigroup $K'$. Hence if $K$ and $L$ are numerical, they are both equal to $K'$.


