Geodesic
Numerical semigroups, polyhedra, and posets. III: Minimal presentations and face dimension
Tara Gomes1 ; Christopher O'Neill2 ; Eduardo Torres Davila1
1University of Minnesota
2San Diego State University
The electronic journal of combinatorics, Tome 30 (2023) no. 2
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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.
DOI : 10.37236/10380
Classification : 20M14

Tara Gomes  1   ; Christopher O'Neill  2   ; Eduardo Torres Davila  1

1 University of Minnesota
2 San Diego State University 
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Tara Gomes; Christopher O'Neill; Eduardo Torres Davila. Numerical semigroups, polyhedra, and posets. III: Minimal presentations and face dimension. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/10380

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