# On the structure of length sets with maximal elasticity

## TL;DR

This paper investigates the structure of length sets in Krull monoids with finite class groups, showing that elements with maximal elasticity typically have length sets forming intervals, revealing insights into their factorization properties.

## Contribution

It characterizes the structure of length sets with maximal elasticity in Krull monoids, demonstrating they are generally intervals, which advances understanding of factorization theory.

## Key findings

- Length sets with maximal elasticity are intervals.
- The structure of these length sets is characterized.
- Insights into factorization properties of Krull monoids.

## Abstract

Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor. Then every non-unit $a \in H$ has a factorization into atoms, say $a=u_1 \cdot\ldots \cdot u_k$ where $k$ is the factorization length and $u_1, \ldots, u_k$ are atoms of $H$. The set $\mathsf L (a)$ of all possible factorizaton lengths is the length set of $a$, and $\rho (H) = \sup \{ \max \mathsf L (a)/\min \mathsf L (a) \colon a \in H \}$ is the elasticity of $H$. We study the structure of length sets of elements with maximal elasticity and show that, in general, these length sets are intervals.