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

2020/09/13 by Gomes, Tara, O'Neill, Christopher, Davila, Eduardo Torres
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2009.05921

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_{≥ 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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