发表机构
University of New Hampshire; Susquehanna University（新罕布什尔大学; 萨斯奎哈纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在无限集合划分格中研究极大滤子，将其分为I型和II型，给出II型滤子由心超滤子诱导的条件，证明纤维拓扑的三分法，并指出纤维不编码滤子，但超幂子结构表示成功。
AI 中文摘要
我们研究无限集合~$X$ 的划分的完全格中的极大（真）滤子。在一致空间的术语中，这些正是由 Pelant 和 Reiterman 引入、并由 Pelant、Reiterman、Rödl 和 Simon 进一步研究的零维一致性格的原子。本文前半部分以纯划分论术语恢复、精化并推广了他们的分类。{若一个划分的极大滤子不包含所有有限划分，则称其为 I 型；若包含，则称其为 II 型。} I 型滤子本质上由两两不相交的双元素集族上的超滤子唯一诱导。{一个 II 型滤子}确定 $X$ 上的一个非主超滤子，即\textit{心}；当{前者}在 Rudin--Keisler 序中极小之时，心精确确定该滤子。{一个 II 型滤子}的每个成员 $F$ 在 $X^{*} = \beta X \backslash X$ 中产生一个闭的\textit{纤维}，{由}与心在 $F$ 上一致的超滤子集合{组成}。我们证明了一个三分法来描述任意纤维的拓扑。然后我们证明纤维并不编码滤子：纤维在交集下不构成半格，一个单一 Rudin--Keisler 类型的无限离散超滤子集合（按我们的术语，即\textit{稀疏}集）的闭包不必是纤维，且与滤子某成员不相容的划分可能具有{严格大于该成员的纤维}。然而，在另一个范畴中仍存在一个成功的表示：心为 $\mathfrak{u}$ 的 {II 型滤子}对应于 $X$ 上全结构的超幂 $X^X\\!/\mathfrak{u}$ 的极大真子结构。拓扑表示问题仍然开放。
英文摘要
We study maximal (proper) filters in the complete lattice of partitions of an infinite set~$X$. In the language of uniform spaces, these are precisely the atoms of the lattice of zero-dimensional uniformities on $X$, introduced by Pelant and Reiterman and studied further by Pelant, Reiterman, Rödl and Simon. The first half of this paper recovers, sharpens, and extends their classification in purely partition-theoretic terms. {Call a maximal filter of partitions {\em type I} if it does not contain all finite partitions, and {\em type II} if it does.} Type I filters are induced, in an essentially unique way, by ultrafilters on families of pairwise disjoint doubletons. {A type II filter} determines a non-principal ultrafilter on $X$, the \emph{heart}; the heart determines the filter precisely when {the former} is minimal in the Rudin--Keisler order. Each member $F$ of {a type II filter} gives rise to a closed \emph{fiber} in $X^{*} = βX \setminus X$, {consisting of} the set of ultrafilters agreeing with the heart on $F$. We prove a trichotomy describing the topology of arbitrary fibers. We then show that the fibers do not encode the filter: fibers do not form a semilattice under intersection, the closure of an infinite discrete set of ultrafilters of a single Rudin--Keisler type (a \emph{sparse} set, in our terminology) need not be a fiber, and a partition incompatible with a member of the filter may have a {fiber strictly larger than that member.} A representation that does succeed is nevertheless available in another category: the {type-II filters} with heart $\mathfrak{u}$ correspond to the maximal proper substructures of the ultrapower $X^X\!/\mathfrak{u}$ of the full structure on $X$. The topological representation problem remains open.
Comments21 pages, LEAN 4 verification of all the new results
