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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).

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  • $\begingroup$ I would have seen more directly as seeing $\mathbf{N}$ as a completion (e.g., it consists of $\{0\}$ those $n$ in the enveloping group such that $mn\in S$ for some $m\ge 1$). And $\mathbf{N}$ has a trivial automorphism group. (I don't know the name of the analogue of the integral closure of a commutative ring, which here gives the inclusion $S\subset\mathbf{N}$.) $\endgroup$
    – YCor
    Jan 2 at 10:53
  • $\begingroup$ @YCor, I think it is usually called the normalization. $\endgroup$
    – Benjamin Steinberg
    Jan 2 at 12:58
  • $\begingroup$ Salvo, I think what you call "homomorphism" is usually called "endomorphism" (= homomorphism from something to itself). $\endgroup$
    – YCor
    Jan 2 at 13:37
  • $\begingroup$ @YCor As for your comment about the name of the analogue of the integral closure of a commutative ring: In some circles, the integral closure of 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". $\endgroup$
    – Salvo Tringali
    Jan 2 at 15:11
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    $\begingroup$ This is essentially in dml.cz/bitstream/handle/10338.dmlcz/101056/… but not explicit. It follows from Corollary 4 and see the discussion after Prop 6. $\endgroup$
    – Benjamin Steinberg
    Jan 2 at 20:07

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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 $K'$.

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  • $\begingroup$ Fantastic! Many thanks. $\endgroup$
    – Salvo Tringali
    Jan 2 at 21:58

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