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.
