When is a numerical semigroup a quotient?

TL;DR

This paper investigates when a numerical semigroup can be expressed as a quotient of another semigroup, providing necessary conditions, identifying families that cannot be quotients, and analyzing the likelihood of such quotients among random semigroups.

Contribution

It introduces necessary conditions for a numerical semigroup to be a quotient and identifies the first known families that are not quotients for each k ≥ 3.

Findings

Established a necessary condition for semigroup quotients.

Constructed families of semigroups that are not quotients for each k ≥ 3.

Analyzed the probability that a random semigroup with k generators is a quotient.

Abstract

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal S$ is a quotient of a numerical semigroup with $k$ generators, we call $\mathcal S$ a $k$-quotient. We give a necessary condition for a given numerical semigroup $\mathcal S$ to be a $k$-quotient, and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a $k$-quotient. We also examine the probability that a randomly selected numerical semigroup with $k$ generators is a $k$-quotient.