不是每个可数完全分配格都是清醒的

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2023-07-28 DOI:10.1017/s0960129523000269
Hualin Miao, Xiaoyong Xi, Qingguo Li, Dongsheng Zhao
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引用次数: 0

摘要

在领域理论中,对斯科特空间的清醒性的研究已经有了较长的历史。Lawson和Hoffmann分别证明了每个连续有向完全偏序集(通常称为定域)的Scott空间是清醒的。Johnstone构造了第一个斯科特空间是非清醒的有向完全偏序集。不久之后,Isbell给出了一个具有非清醒斯科特空间的完备格。荣格接着问是否每一个可数完备格都有一个清醒的斯科特空间。本文的主要目的是通过构造一个斯科特空间是非清醒的可数完备格来回答荣格的问题。然后对该格进行修正,得到一个具有非清醒Scott空间的可数分布完备格。此外,我们证明了积空间$\Sigma P\乘以∑Q$的拓扑与积偏序集$P\乘以Q$的斯科特拓扑重合,如果所有偏序集P和Q的增量理想集合Id(P)和Id(Q)都是可数的。在此基础上,推导出如果Id(P)是可数的,且$\Sigma P$是相干且滤除良好的,则有向完备偏序集P具有清醒的Scott空间。特别地,每一个Id(L)可数的完备格L都有一个清醒的Scott空间。
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Not every countable complete distributive lattice is sober
The study of the sobriety of Scott spaces has got a relatively long history in domain theory. Lawson and Hoffmann independently proved that the Scott space of every continuous directed complete poset (usually called domain) is sober. Johnstone constructed the first directed complete poset whose Scott space is non-sober. Soon after, Isbell gave a complete lattice with a non-sober Scott space. Based on Isbell’s example, Xu, Xi, and Zhao showed that there is even a complete Heyting algebra whose Scott space is non-sober. Achim Jung then asked whether every countable complete lattice has a sober Scott space. The main aim of this paper is to answer Jung’s problem by constructing a countable complete lattice whose Scott space is non-sober. This lattice is then modified to obtain a countable distributive complete lattice with a non-sober Scott space. In addition, we prove that the topology of the product space $\Sigma P\times \Sigma Q$ coincides with the Scott topology of the product poset $P\times Q$ if the set Id(P) and Id(Q) of all incremental ideals of posets P and Q are both countable. Based on this, it is deduced that a directed complete poset P has a sober Scott space, if Id(P) is countable and $\Sigma P$ is coherent and well filtered. In particular, every complete lattice L with Id(L) countable has a sober Scott space.
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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