On the system of length sets of power monoids

Abstract

The set ${\mathcal{P}}_{\mathrm{fin},0}({\mathbb{N}}_0)$ of all finite subsets of ${\mathbb{N}}_0$ containing the zero element is a monoid with set addition as operation. If a set $A\in {\mathcal{P}}_{\mathrm{fin},0}({\mathbb{N}}_0)$ 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 ${\mathcal{P}}_{\mathrm{fin},0}({\mathbb{N}}_0)$, 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 $q\ge 1$, there is some $A\in {\mathcal{P}}_{\mathrm{fin},0}({\mathbb{N}}_0)$ such that $q=\frac{\max (\mathsf{L}(A))}{\min (\mathsf{L}(A))}$. This supports a conjecture of Fan and Tringali.