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arXiv · 2212.08285

When is a numerical semigroup a quotient?

Tristram Bogart·Christopher O'Neill·Kevin Woods
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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.

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Tristram Bogart, Christopher O'Neill, Kevin Woods. 2022-12-16. When is a numerical semigroup a quotient?. https://arxiv.org/abs/2212.08285

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