Vol. 14, No. 4, 2021

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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

An affine semigroup is a finitely generated subsemigroup of (0d,+), 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 Mn = n + r1,,n + rk for fixed r1,,rk, 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.

affine semigroup, factorization, shifted semigroup
Mathematical Subject Classification
Primary: 20M14
Received: 1 June 2020
Revised: 5 April 2021
Accepted: 6 April 2021
Published: 23 October 2021

Communicated by Scott T. Chapman
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