Rarity of the infinite chains in the tree of numerical semigroups

TL;DR

Problem: determine the frequency with which genus-$g$ numerical semigroups lie on infinite chains in the semigroup tree. Approach: leverage the gcd criterion that a semigroup lies in an infinite chain iff its nonzero left elements are not coprime, and combine this with KaplanYe asymptotics showing that, as $g\to\infty$, the multiplicity concentrates near $m\in ((\gamma-\varepsilon)g,(\gamma+\varepsilon)g)$ with $\gamma = (5+\sqrt{5})/10$, and the Frobenius number near $F\in ((2-\varepsilon)\gamma g,(2+\varepsilon)\gamma g)$. Key steps: define asymptotic subsets $A_g^m$, $A_g^F$, and $A_g^{m,F}$ and show they capture almost all semigroups; then use a bound ensuring two consecutive left elements, implying most $S_g$ lie in $S_g^{\bar{\infty}}$ and hence are not in infinite chains. Contributions: provides the first proof that the infinite-chain fraction tends to $0$ for fixed genus, solving an open problem since $2009$, and links left-element structure to genus-asymptotics to guide counting in the semigroup tree. Significance: confirms that typical numerical semigroups have finite descendants in the semigroup tree, aligning with computational observations and informing future counting strategies.

Abstract

We prove that, for each fixed genus, the portion of semigroups of that genus belonging to infinite chains in the semigroup tree approaches 0 as the genus grows to infinite. This means that most numerical semigroups have a finite number of descendants in the semigroup tree. This problem has been open since 2009.

Figures (1)

• Figure 1: Tree structure of the semigroup tree up to genus $11$.

Theorems & Definitions (13)

• Lemma 1
• Lemma 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6
• Lemma 7