Title: $\widetilde{\mid}\hspace{1mm}$-divisibility of ultrafilters II: The big picture Show Chinese title

Title: $\widetilde{\mid}\hspace{1mm}$-超滤器的可分性 II：整体图景

Abstract: A divisibility relation on ultrafilters is defined as follows: ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$ if and only if every set in $\cal F$ upward closed for divisibility also belongs to $\cal G$. After describing the first $\omega$ levels of this quasiorder, in this paper we generalize the process of determining the basic divisors of an ultrafilter. First we describe these basic divisors, obtained as (equivalence classes of) powers of prime ultrafilters. Using methods of nonstandard analysis we determine the pattern of an ultrafilter: the collection of its basic divisors as well as the multiplicity of each of them. All such patterns have a certain closure property in an appropriate topology. We isolate the family of sets belonging to every ultrafilter with a given pattern. We show that every pattern with the closure property is realized by an ultrafilter. Finally, we apply patterns to obtain an equivalent condition for an ultrafilter to be self-divisible. Show Chinese abstract

Abstract: 关于超滤器的整除关系定义如下：当且仅当属于 $\cal F$ 的每个向上封闭于整除性的集合也属于 $\cal G$ 时，有 ${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$。 在描述了这个拟序关系的前 $\omega$ 个层次之后，本文推广了确定超滤器基本整除元的过程。 首先我们描述这些基本整除元，它们被表示为（等价类中的）素超滤器的幂。 利用非标准分析的方法，我们确定了超滤器的模式：即其基本整除元的集合以及每个元的多重性。 所有这些模式在适当的拓扑中有某种闭合性质。 我们分离出属于具有给定模式的所有超滤器的集合族。 我们证明具有闭合性质的每个模式都被某个超滤器实现。 最后，我们将模式应用于得到一个超滤器为自整除的等价条件。