Numerical semigroups, polyhedra, and posets II: locating certain families of semigroups

Abstract

Several recent papers have examined a rational polyhedron P m whose integer points are in bijection with the numerical semigroups (cofinite, additively closed substets of the non-negative integers) containing m. A combinatorial description of the faces of P m was recently introduced, one that can be obtained from the divisibility posets of the numerical semigroups a given face contains. In this paper, we study the faces of P m containing arithmetical numerical semigroups and those containing certain glued numerical semigroups, as an initial step towards better understanding the full face structure of P m . In most cases, such faces only contain semigroups from these families, yielding a tight connection to the geometry of P m .

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

“…Throughout the present project, we developed a Sage package KunzPoset that provides an abstract class for Kunz posets [14]. Much of the functionality discussed in this paper and its predecessors [2,10] has been implemented, including factorizations and minimal presentations, as well as more general Kunz polyhedron functionality, such as locating semigroups on a given face (equivalently, with a given Kunz poset).…”

“…For each m ≥ 2, the Kunz polyhedron P m is a pointed rational cone, translated from the origin, whose positive integer points are in bijection with the numerical semigroups of multiplicity m (we defer precise definitions to Section 2). This manuscript is the third in a series examining a combinatorial description of the faces of P m [2,10]. Given a numerical semigroup S with multiplicity m, the Kunz poset is the partially ordered set with ground set Z m obtained by replacing each element of the Apéry poset Ap(S; m) Date: September 15, 2020.…”