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

TL;DR

This paper explores the relationship between the combinatorial structure of Kunz polyhedra and numerical semigroups, specifically how minimal presentations can be derived from associated posets and how face dimensions relate to these posets.

Contribution

It characterizes the extent to which minimal presentations of numerical semigroups are determined by Kunz posets and provides a method to compute face dimensions from these posets.

Findings

Numerical semigroups on the same face have identical minimal presentation cardinality.

A combinatorial method to determine face dimension from Kunz posets.

Faces of the Kunz polyhedron are indexed by finite posets derived from divisibility relations.

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.