#### Vol. 14, No. 4, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
On minimal presentations of shifted affine semigroups with few generators

### Christopher O’Neill and Isabel White

Vol. 14 (2021), No. 4, 617–630
##### Abstract

An affine semigroup is a finitely generated subsemigroup of $\left({ℤ}_{\ge 0}^{d},+\right)$, 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
Primary: 20M14