利用谱空间实现正交补格的无选择对偶

IF 0.6 4区 数学 Q3 MATHEMATICS
Joseph McDonald, Kentarô Yamamoto
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引用次数: 5

摘要

通过Stone空间的clopen正交正则子集的正交补格的现有拓扑表示依赖于Alexander的子基定理,该定理断言拓扑空间X是紧致的,如果X的每个子基开覆盖都允许有限子覆盖。这是超滤定理的一个简单结果,其证明取决于Zorn引理,众所周知,它等价于选择公理。在这项工作中,我们通过谱空间的一个特殊子类给出了直补格的一个无选择拓扑表示;自由选择,因为我们的表示避免了使用亚历山大的子基定理及其相关的非结构化选择原则。然后,我们引入了一个新的谱空间子类,我们称之为上维托里斯正空间,以便刻画在我们的表示中使用的适当格滤波器的谱空间的同胚性(以及关于它们的正空间约简的同构性)。然后展示了我们的构造如何在直补格的范畴和上维托里正空间的对偶范畴之间产生范畴的无选择对偶等价。我们的对偶结合了Bezhanishvili和Holliday对布尔代数的Stone对偶的无选择谱空间方法,以及Goldblatt和Bimbó对直补格的Stone二重的选择相关正空间方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

Choice-free duality for orthocomplemented lattices by means of spectral spaces

Choice-free duality for orthocomplemented lattices by means of spectral spaces

The existing topological representation of an orthocomplemented lattice via the clopen orthoregular subsets of a Stone space depends upon Alexander’s Subbase Theorem, which asserts that a topological space X is compact if every subbasic open cover of X admits of a finite subcover. This is an easy consequence of the Ultrafilter Theorem—whose proof depends upon Zorn’s Lemma, which is well known to be equivalent to the Axiom of Choice. Within this work, we give a choice-free topological representation of orthocomplemented lattices by means of a special subclass of spectral spaces; choice-free in the sense that our representation avoids use of Alexander’s Subbase Theorem, along with its associated nonconstructive choice principles. We then introduce a new subclass of spectral spaces which we call upper Vietoris orthospaces in order to characterize up to homeomorphism (and isomorphism with respect to their orthospace reducts) the spectral spaces of proper lattice filters used in our representation. It is then shown how our constructions give rise to a choice-free dual equivalence of categories between the category of orthocomplemented lattices and the dual category of upper Vietoris orthospaces. Our duality combines Bezhanishvili and Holliday’s choice-free spectral space approach to Stone duality for Boolean algebras with Goldblatt and Bimbó’s choice-dependent orthospace approach to Stone duality for orthocomplemented lattices.

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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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