Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the cardinality of the minimal generating set of $S$ (see the link for embedding dimension). My question is: for which integers $k>1$, can we say that all but finitely many positive integers occur as $g(S)$ for some $S$ with $e(S)=k$? Like what can we say for at least $k=2$ or $k=3$?
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The semigroup generated by $2$ and $2g+1$ has genus $g$, so every positive integer is the genus of a semigroup with two generators.
Assuming $b < 2a$ and $a < 2b+1$, the semigroup $\langle 3, 3a+1, 3b+2 \rangle$ has three generators and genus $a+b$.
answered Jun 19, 2018 at 15:15
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$\begingroup$ Ah right .. that's just Sylvester ... thanks. . How about with $e(S)=3$ ? $\endgroup$– user111492Jun 19, 2018 at 15:18
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$\begingroup$ Response regarding three generators added. $\endgroup$ Jun 20, 2018 at 0:15