On minimal presentations of shifted affine semigroups with few generators

O’Neill, Christopher; White, Isabel

doi:10.2140/involve.2021.14.617

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On minimal presentations of shifted affine semigroups with few generators

Christopher O’Neill and Isabel White

Vol. 14 (2021), No. 4, 617–630

DOI: 10.2140/involve.2021.14.617

Abstract

An affine semigroup is a finitely generated subsemigroup of $(ℤ_{\geq 0}^{d}, +)$ , and a numerical semigroup is an affine semigroup with $d = 1$ . A growing body of recent work examines shifted families of numerical semigroups, that is, families of numerical semigroups of the form $M_{n} = ⟨ n + r_{1}, \dots, n + r_{k} ⟩$ for fixed $r_{1}, \dots, r_{k}$ , with one semigroup for each value of the shift parameter $n$ . It has been shown that within any shifted family of numerical semigroups, the size of any minimal presentation is bounded (in fact, this size is eventually periodic in $n$ ). We consider shifted families of affine semigroups, and demonstrate that some, but not all, shifted families of 4-generated affine semigroups have arbitrarily large minimal presentations.

Keywords

affine semigroup, factorization, shifted semigroup

Mathematical Subject Classification

Primary: 20M14

Milestones

Received: 1 June 2020

Revised: 5 April 2021

Accepted: 6 April 2021

Published: 23 October 2021

Communicated by Scott T. Chapman

Authors

Christopher O’Neill
Department of Mathematics and Statistics San Diego State University San Diego, CA United States

Isabel White
Department of Mathematics and Statistics San Diego State University San Diego, CA United States

Vol. 14, No. 4, 2021