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Abstract
An affine semigroup is a finitely generated subsemigroup of
( ℤ ≥ 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 , … , n
+ r k ⟩ for
fixed
r 1 , … , 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