Infinite chains in the tree of numerical semigroups

TL;DR

This paper investigates the structure and rarity of infinite chains in the tree of numerical semigroups, providing new characterizations, fixing previous inaccuracies, and analyzing chains with fixed prime multiplicities.

Contribution

It offers a corrected characterization of infinite chains, explores their behavior in subtrees with fixed multiplicity, and derives formulas for chains with multiplicities 4 and 6.

Findings

Infinite chains are rare but essential for the tree’s infinitude.

More semigroups of a given genus do not belong to infinite chains for g ≥ 5.

Unique infinite chain exists for prime multiplicity in the semigroup tree.

Abstract

One major problem in the study of numerical semigroups is determining the growth of the semigroup tree. In the present work, infinite chains of numerical semigroups in the semigroup tree, firstly introduced by Bras-Amor\'os and Bulygin (Semigroup Forum, 79:561--574, 2009), are studied. Computational results show that these chains are rare, but without them the tree would not be infinite. It is proved that for each genus $g\geq 5$ there are more semigroups of that genus not belonging to infinite chains than semigroups belonging. Bras-Amor\'os and Bulygin (Semigroup Forum, 79:561--574, 2009) presented a characterization of the semigroups that belong to infinite chains in terms of the coprimality of the left elements of the semigroup as well as a result on the cardinality of the set of infinite chains to which a numerical semigroup belongs in terms of the primality of the greatest common divisor of these left elements. We revisit these results and fix an imprecision on the cardinality of the set of infinite chains to which a semigroup belongs in the case when the greatest common divisor of the left elements is a prime number. We then look at infinite chains in subtrees with fixed multiplicity. When the multiplicity is a prime number there is only one infinite chain in the tree of semigroups with such multiplicity. When the multiplicity is $4$ or $6$ we prove a self-replication behavior in the subtree and prove a formula for the number of semigroups in infinite chains of a given genus and multiplicity $4$ and $6$, respectively.