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arXiv · 2207.00051

Reflective numerical semigroups

Caleb M. Shor
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Abstract

We define a reflective numerical semigroup of genus $g$ as a numerical semigroup that has a certain reflective symmetry when viewed within $\mathbb{Z}$ as an array with $g$ columns. Equivalently, a reflective numerical semigroup has one gap in each residue class modulo $g$. In this paper, we give an explicit description for all reflective numerical semigroups. With this, we can describe the reflective members of well-known families of numerical semigroups as well as obtain formulas for the number of reflective numerical semigroups of a given genus or given Frobenius number.

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Caleb M. Shor. 2022-06-30. Reflective numerical semigroups. https://arxiv.org/abs/2207.00051

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