距离集的一个实现定理
A realization theorem for sets of distances
摘要：
设$H$是一个原子幺半群。$H$的距离集$Δ(H)$是所有具有以下性质的$d \in \mathbb{N}$的集合：存在不可约元$u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$，使得对于任何具有$k\lt \ell \lt k+d$的$\ell \in \mathbb{N}$，$u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$但$u\_1 \cdot \ldots \cdot u\_k$不能写成$\ell$不可约元的乘积。众所周知（并且很容易证明），如果$Δ(H)$非空，则$\min Δ(H) = \gcd Δ(H)$。本文反过来证明了：对每个有限非空集$Δ\subset \mathbb{N}$，若有$\min Δ= \gcd Δ$，则存在一个Krull生成幺半群$H$，使得$Δ(H)=Δ$ .
英文摘要：
Let $H$ be an atomic monoid. The set of distances $Δ(H)$ of $H$ is the set of all $d \in \mathbb{N}$ with the following property: there are irreducible elements $u\_1, \ldots, u\_k, v\_1 \ldots, v\_{k+d}$ such that $u\_1 \cdot \ldots \cdot u\_k=v\_1 \cdot \ldots \cdot v\_{k+d}$ but $u\_1 \cdot \ldots \cdot u\_k$ cannot be written as a product of $\ell$ irreducible elements for any $\ell \in \mathbb{N}$ with $k\lt \ell \lt k+d$. It is well-known (and easy to show) that, if $Δ(H)$ is nonempty, then $\min Δ(H) = \gcd Δ(H)$. In this paper we show conversely that for every finite nonempty set $Δ\subset \mathbb{N}$ with $\min Δ= \gcd Δ$ there is a finitely generated Krull monoid $H$ such that $Δ(H)=Δ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
