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).

integral closureof a cancellative commutative monoid $H$ (written multiplicatively) is the set of all $x$ in the quotient group such that $x^n \in H$ for some non-negative integer $n$. (In particular, $(\mathbb N, +)$ is the integral closure of any numerical monoid.) As for your 2nd comment: Mea culpa, I've just edited and changed "homo" to "endo". $\endgroup$4more comments