On the system of length sets of power monoids

TL;DR

This paper investigates the structure of length sets in a monoid of finite subsets of natural numbers, demonstrating that for any rational number ≥ 1, there exists an element with a length ratio equal to that number, supporting a conjecture.

Contribution

It proves that for every rational number ≥ 1, there exists an element in the monoid with a length set ratio equal to that number, confirming a conjecture by Fan and Tringali.

Findings

Existence of elements with arbitrary rational length ratios

Supports the conjecture of Fan and Tringali

Advances understanding of factorization lengths in power monoids

Abstract

The set $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ of all finite subsets of $\mathbb{N}_0$ containing the zero element is a monoid with set addition as operation. If a set $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ can be written in the form $A=\sum_{i=1}^{\ell} A_i$ with $\ell\in\mathbb{N}_0$ and indecomposable elements $(A_i)_{i=1}^{\ell}$ of $\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$, then $\ell$ is a factorization length of $A$ and $\mathsf{L}(A)\subseteq\mathbb{N}_0$ denotes the set of all possible factorization lengths of $A$. We show that for each rational number $q\geq 1$, there is some $A\in\mathcal{P}_{{\rm fin},0}(\mathbb{N}_0)$ such that $q=\frac{\max(\mathsf{L}(A))}{\min(\mathsf{L}(A))}$. This supports a Conjecture of Fan and Tringali.