论文标题
通过光谱空间的矫形晶格的无选择二元性
Choice-free duality for orthocomplemented lattices by means of spectral spaces
论文作者
论文摘要
通过石材空间的clropen矫置子集对正面晶格的现有拓扑表示取决于亚历山大的子基本定理，如果$ x $的每个子基本封面都允许使用有限的子套件。这是超级定理的简单结果 - 其证明取决于Zorn的引理，众所周知，这相当于选择的公理。在这项工作中，我们通过特殊的光谱空间子类给出了无调拓扑的拓扑表示。从我们的代表避免使用亚历山大的子基础定理以及其相关的非构造选择原理的意义上，无选择。然后，我们引入了一个新的光谱空间子类，我们称之为\ emph {上越野孔孔}，以表征（直至（直至同构和同构））在我们表示中使用的适当晶格过滤器的光谱空间。然后，展示了我们的构造如何产生无调的双重等效性，类别之间的晶格类别与上部越野距离旁观空间的双重类别之间。我们的二元性结合了Bezhanishvili和Holliday的无选择的光谱空间方法，用于布尔代数的石材双重性与Goldblatt和Bimbó和Bimbó的选择依赖性的矫形器方法，用于用于矫形器的石材二重性。
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 \emph{upper Vietoris orthospaces} in order to characterize (up to homeomorphism and isomorphism) the spectral space 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.
