超滤子可除性的综述

AI总结：

本文综述了自然数可除性关系向超滤子集合扩展的研究，总结了分区、碎片化及同余扩展等核心成果。

摘要：

在过去十年中，若干论文定义并研究了 $\mathbb{N}$ 上的可除性关系向 $\mathbb{N}$ 上的超滤子集合 $β\mathbb{N}$ 的扩展。本文对迄今取得的成果进行了综述，并补充了若干将不同研究阶段的主题联系起来的结果。亮点包括：将 $β\mathbb{N}$ 划分为下部 $L$（及其按层级的划分）和上部；识别基本要素（素数的幂）以及每个超滤子向这些要素的碎片化；寻找具有给定基本要素的超滤子所对应的向上封闭集；给定可除性类的后继与前驱的存在性及数量；扩展同余关系（以两种方式）并检验所得关系的性质。

英文摘要：

An extension of the divisibility relation on $\mathbb{N}$ to the set $β\mathbb{N}$ of ultrafilters on $\mathbb{N}$ was defined and investigated in several papers during the last ten years. Here we make a survey of results obtained so far, adding several results connecting the themes of different stages of the research. The highlights include: separation of $β\mathbb{N}$ into the lower part $L$ (with its division into levels) and the upper part; identifying basic ingredients (powers of primes) and fragmentation of each ultrafilter into them; finding the corresponding upward closed sets belonging to an ultrafilter with given basic ingredients; existence and number of successors and predecessors of a given divisibility class; extending the congruence relation (in two ways) and checking properties of the obtained relations.

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