A realization theorem for sets of lengths in numerical monoids

TL;DR

This paper proves that for any finite set of integers greater than or equal to 2, there exists a numerical monoid and a squarefree element whose set of lengths matches that set, demonstrating a realization theorem.

Contribution

It establishes a realization theorem linking finite sets of integers to sets of lengths in numerical monoids, specifically for squarefree elements.

Findings

For any finite set L of integers ≥ 2, a corresponding numerical monoid H exists.

A squarefree element a in H can be constructed with set of lengths exactly L.

The result bridges finite sets of integers with algebraic properties of numerical monoids.

Abstract

We show that for every finite nonempty set L of integers greater than or equal to 2 there are a numerical monoid H and a squarefree element a $\in$ H whose set of lengths L(a) is equal to L.