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).