On the system of length sets of 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.