# Minimal presentations of shifted numerical monoids

## TL;DR

This paper studies the structure of shifted numerical monoids, revealing periodic patterns in their minimal relations for large shifts, with applications across algebra, computation, and factorization.

## Contribution

It provides a detailed description of minimal relations in shifted monoids and demonstrates their periodic nature for large shifts, advancing understanding in algebraic and combinatorial contexts.

## Key findings

- Minimal relations become periodic in the shift parameter n.
- The structure of shifted monoids can be explicitly described for large n.
- Applications include improved computational methods and insights into factorization theory.

## Abstract

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.