Minimal presentations of shifted numerical monoids

A numerical monoid is an additive submonoid of the non-negative integers. Given a numerical monoid $S$, consider the family of “shifted” monoids $M_n$ obtained by adding $n$ to each generator of $S$. In this paper, we examine minimal relations among the generators of $M_n$ when $n$ is sufficiently large, culminating in a description that is periodic in the shift parameter $n$. We explore several applications to computation, combinatorial commutative algebra, and factorization theory.