On minimal presentations of shifted affine semigroups with few generators

TL;DR

This paper investigates the minimal presentations of shifted affine semigroups, revealing that unlike numerical semigroups, some 4-generated affine semigroups can have arbitrarily large minimal presentations, indicating more complex behavior.

Contribution

It extends the study of shifted families from numerical to affine semigroups, showing that the boundedness property does not always hold for 4-generated affine semigroups.

Findings

Some shifted families of 4-generated affine semigroups have unbounded minimal presentations.

Boundedness of minimal presentations holds for numerical semigroups but not universally for affine semigroups.

The behavior of minimal presentations varies depending on the structure of the affine semigroup.

Abstract

An affine semigroup is a finitely generated subsemigroup of $(\mathbb Z_{\ge 0}^d, +)$, 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 $M_n = \langle n + r_1, \ldots, n + r_k \rangle$ for fixed $r_1, \ldots, r_k$, 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 $n$). In this paper, 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.