Stone space partitions indexed by a poset

IF 0.6 4区 数学 Q3 MATHEMATICS
Andrew B. Apps
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引用次数: 2

Abstract

Stone space partitions \(\{X_{p}\mid p\in P\}\) satisfying conditions like \(\overline{X_{p}}=\bigcup _{q\leqslant p}X_{q}\) for all \(p\in P\), where P is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of \(\omega \)-categorical structures. A key concept for studying such partitions is that of a p-trim open set which meets precisely those \(X_{q}\) for which \(q\geqslant p\); for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition \(\{X_{r}\mid r\in [0,1]\}\) of the Cantor set such that \(\overline{X_{r}}=\bigcup _{s\leqslant r}X_{s}\text { for all }r\in [0,1]\).

Abstract Image

由偏序集索引的Stone空间分区
Stone空间分区\(\{X_{p}\ mid p\ in p\}\)满足所有\(p\ in p)的\(\ overline{X_{p}}=\ bigcup _{q\leqsplant p}X_{q}\的条件,其中p是偏序集或PO系统(具有可分辨子集的偏序集),在研究原始布尔代数和\(\ω\)-范畴结构时自然产生。研究这种划分的一个关键概念是p-边缘开集的概念,它恰好满足那些\(X_{q}\),其中\(q\geqslant p\);对于Stone空间,这是一个伪不可分解集的拓扑等价物。本文发展了由偏序集或PO系统索引的Stone空间的无限划分理论,其中修剪集形成拓扑的邻域基。我们研究了偏序集/PO系统的序性质和分区的拓扑性质之间的相互作用,检验了这些分区的扩张和完备,并导出了偏序集合/PO系统上存在所研究的各种类型分区的充要条件。我们还确定了具有由给定PO系统索引的修剪分区的第二可数Stone空间在同胚之前是唯一的情况,这取决于对孤立点结构和分区元素的有界性的选择。我们的结果的一个推论是,Cantor集存在一个分区\([0,1]\中的\{X_。
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来源期刊
Algebra Universalis
Algebra Universalis 数学-数学
CiteScore
1.00
自引率
16.70%
发文量
34
审稿时长
3 months
期刊介绍: Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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