Let $S$ be a [numerical semigroup](https://en.m.wikipedia.org/wiki/Numerical_semigroup). Let $\mathbb N$ denote the monoid of non-negative integers under addition. Let $F(S)=\max (\mathbb N \setminus S)$ be the *Frobenius number* of $S$; let $g(S)=|\mathbb N \setminus S|$ be the *genus* of $S$; and let $m(S)=\min (S \setminus \{0\})$ be the *multiplicity* of $S$. If $m(S)=3$, then is it true that $S$ is the submonoid of $\mathbb N$ generated by $3, 3g(S)- F(S)$ and $3+F(S)$?



On a generating set of numerical semigroups of multiplicity three
Yes, this is **true** and it follows immediately from



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> > **[1, Corollary 4]** Two numerical semigroups with multiplicity three are equal if and only if they have the same Froebenius number and the same gender.
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together with



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> > **[1, Lemma 6]** Let $S$ be a numerical semigroup with multiplicity three, Frobenius
> >  number $F(S)$ and gender $G(S)$. Then $\frac{F(S) + 1}{2} \le G(S) < \frac{2F(S) + 3}{3}$.
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and



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> > **[1, Theorem 7]** Let $F$ be a positive integer greater than or equal to four that is
> >  not a multiple of three. Let $G$ be a positive integer such that $\frac{F + 1}{2} \le G < \frac{2F + 3}{3}$.
> >  Then $S = \langle 3, 3G − F, F + 3 \rangle$ is a numerical semigroup with multiplicity three, Frobenius number $F$ and gender $G$.
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[1] J. C. Rosales, "Numerical semigroups with multiplicity three and four", 2005. 


