On the structure of length sets with maximal elasticity

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 $a=u_1\cdot \dots \cdot u_k$ where $k$ is the factorization length and $u_1,\dots ,u_k$ are atoms of $H$. The set $\mathsf{L}(a)$ of all possible factorizaton lengths is the length set of $a$, and $\rho (H)=\sup \{\max \mathsf{L}(a)/\min \mathsf{L}(a):a\in H\}$ 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.

arXiv comment15 pages