Cyclotomic exponent sequences of numerical semigroups

TL;DR

This paper investigates the cyclotomic exponent sequences of numerical semigroups, characterizing various families and establishing connections to cyclotomic numerical semigroups and complete intersection properties.

Contribution

It introduces methods to compute and analyze cyclotomic exponent sequences at specific elements, leading to characterizations of semigroup families and progress on a conjecture relating cyclotomic semigroups and complete intersections.

Findings

Characterization of Betti-sorted and Betti-divisible semigroups.

Identification of semigroups with a unique Betti element.

Progress towards the conjecture linking cyclotomic semigroups and complete intersections.

Abstract

We study the cyclotomic exponent sequence of a numerical semigroup $S,$ and we compute its values at the gaps of $S,$ the elements of $S$ with unique representations in terms of minimal generators, and the Betti elements $b\in S$ for which the set $\{a \in \operatorname{Betti}(S) : a \le_{S}b\}$ is totally ordered with respect to $\le_S$ (we write $a \le_S b$ whenever $a - b \in S,$ with $a,b\in S$). This allows us to characterize certain semigroup families, such as Betti-sorted or Betti-divisible numerical semigroups, as well as numerical semigroups with a unique Betti element, in terms of their cyclotomic exponent sequences. Our results also apply to cyclotomic numerical semigroups, which are numerical semigroups with a finitely supported cyclotomic exponent sequence. We show that cyclotomic numerical semigroups with certain cyclotomic exponent sequences are complete intersections, thereby making progress towards proving the conjecture of Ciolan, Garc\'ia-S\'anchez and Moree (2016) stating that $S$ is cyclotomic if and only if it is a complete intersection.