Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor and let $D\left(G\right)$ be the Davenport constant of $G$. Then a product of two atoms of $H$ can be written as a product of at most $D\left(G\right)$ atoms. We study this extremal case and consider the set ${\mathcal{U}}_{\{2,D\left(G\right)\}}\left(H\right)$ defined as the set of all $l\in \mathbb{N}$ with the following property: there are two atoms $u,v\in H$ such that $uv$ can be written as a product of $l$ atoms as well as a product of $D\left(G\right)$ atoms. If $G$ is cyclic, then ${\mathcal{U}}_{\{2,D\left(G\right)\}}\left(H\right)=\{2,D\left(G\right)\}$. If $G$ has rank two, then we show that (apart from some exceptional cases) ${\mathcal{U}}_{\{2,D\left(G\right)\}}\left(H\right)=[2,D\left(G\right)]\setminus \left\{3\right\}$. This result is based on the recent characterization of all minimal zero-sum sequences of maximal length over groups of rank two. As a consequence, we are able to show that the arithmetical factorization properties encoded in the sets of lengths of a rank $2$ prime power order group uniquely characterizes the group.

让 $H$ 成为具有有限类组的Krull monoid $G$ 这样每个类都包含一个主要除数， $d（G）$ 是Davenport常数 $G$。然后是两个原子的乘积$H$ 最多可以写为 $d（G）$原子。我们研究了这种极端情况并考虑了${\mathcal{ü}}_{\{\mathrm{2个}，d（G）\}}（H）$ 定义为全部 $升\in \mathbb{ñ}$ 具有以下性质：有两个原子 $ü，v\in H$ 这样 $üv$ 可以写成 $升$ 原子以及 $d（G）$原子。如果$G$ 是循环的，那么 ${\mathcal{ü}}_{\{\mathrm{2个}，d（G）\}}（H）=\{\mathrm{2个}，d（G）\}$。如果$G$ 排名第二，那么我们证明（除了一些例外情况） ${\mathcal{ü}}_{\{\mathrm{2个}，d（G）\}}（H）=[\mathrm{2个}，d（G）]\setminus \left\{3\right\}$。该结果基于最近对第二级组上最大长度的所有最小零和序列的表征。结果，我们能够证明算术分解性质编码在秩的长度集中$\mathrm{2个}$ 主力订单组是该组的唯一特征。