# Two numerical monoids are isomorphic iff they are equal

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).

• 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 $S\subset\mathbf{N}$.)
– YCor
Jan 2 at 10:53
• @YCor, I think it is usually called the normalization. Jan 2 at 12:58
• Salvo, I think what you call "homomorphism" is usually called "endomorphism" (= homomorphism from something to itself).
– YCor
Jan 2 at 13:37
• @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". Jan 2 at 15:11
• 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. Jan 2 at 20:07

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 $$K'$$.