When is a numerical semigroup 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.