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

Title: $\widetilde{\mid}\hspace{1mm}$- 超滤子的可除性 II：全局图景

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. Finally, we show that every pattern with the closure property is realized by an ultrafilter. Show Chinese abstract

Abstract: 超滤子上的可除性关系定义如下：${\cal F}\hspace{1mm}\widetilde{\mid}\hspace{1mm}{\cal G}$当且仅当$\cal F$中的每个集合，对于可除性而言是向上封闭的，也属于$\cal G$。在描述了这个拟序的前$\omega$层级之后，本文我们推广了确定超滤子的基本除数的过程。首先我们描述这些基本除数，它们作为素超滤子的幂（等价类）得到。使用非标准分析的方法，我们确定了超滤子的模式：其基本除数的集合以及每个除数的重数。所有这样的模式在适当拓扑下具有某种闭包性质。我们分离出属于每个具有给定模式的超滤子的集合族。最后，我们证明每个具有闭包性质的模式都被某个超滤子实现。