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arXiv · 2009.05921

Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension

Tara Gomes·Christopher O'Neill·Eduardo Torres Davila
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Abstract

This paper is the third in a series of manuscripts that examine the combinatorics of the Kunz polyhedron $P_m$, whose positive integer points are in bijection with numerical semigroups (cofinite subsemigroups of $\mathbb Z_{\ge 0}$) whose smallest positive element is $m$. The faces of $P_m$ are indexed by a family of finite posets (called Kunz posets) obtained from the divisibility posets of the numerical semigroups lying on a given face. In this paper, we characterize to what extent the minimal presentation of a numerical semigroup can be recovered from its Kunz poset. In doing so, we prove that all numerical semigroups lying on the interior of a given face of $P_m$ have identical minimal presentation cardinality, and we provide a combinatorial method of obtaining the dimension of a face from its corresponding Kunz poset.

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Tara Gomes, Christopher O'Neill, Eduardo Torres Davila. 2020-09-13. Numerical semigroups, polyhedra, and posets III: minimal presentations and face dimension. https://arxiv.org/abs/2009.05921

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