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Not every countable complete distributive lattice is sober

Published online by Cambridge University Press:  28 July 2023

Hualin Miao ,
Xiaoyong Xi Open the ORCID record for Xiaoyong Xi [Opens in a new window] ,
Qingguo Li Open the ORCID record for Qingguo Li [Opens in a new window]  and
Dongsheng Zhao
Hualin Miao
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, China
Xiaoyong Xi*
Affiliation:
School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, Jiangsu, China
Qingguo Li
Affiliation:
School of Mathematics, Hunan University, Changsha, Hunan, China
Dongsheng Zhao
Affiliation:
Mathematics and Mathematics Education, National Institute of Education, Nanyang Technological University, Singapore
*
Corresponding author: Xiaoyong Xi; Email: littlebrook@jsnu.edu.cn
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Abstract

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.

Keywords

CountabilitySobrietyScott topologyComplete lattices
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 33 , Issue 9 , October 2023 , pp. 809 - 831
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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References

Goubault-Larrecq, J. (2013). Non-Hausdorff Topology and Domain Theory , New Mathematical Monographs, vol. 22, Cambridge, Cambridge University Press.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003). Continuous Lattices and Domains , Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge, Cambridge University Press.Google Scholar
Hochster, M. (1969). Prime ideal structure in commutative rings. Transactions of the American Mathematical Society 142 43–60.CrossRefGoogle Scholar
Hoffmann, R.-E. (1981). Continuous posets, prime spectra of a completely distributive complete lattice, and Hausdorff compactifications. In: Continuous Lattices, Springer LNNM, vol. 871, 159–208.CrossRefGoogle Scholar
Isbell, J. (1982). Completion of a construction of Johnstone. Proceedings of the American Mathematical Society 85 333–334.CrossRefGoogle Scholar
Jia, X., Jung, A. and Li, Q. (2016). A note on coherence of dcpos. Topology and Its Applications 209 235–238.CrossRefGoogle Scholar
Jia, X. (2018). Meet-Continuity and Locally Compact Sober Dcpos. Phd thesis, University of Birmingham.Google Scholar
Johnstone, P. (1981). Scott is not always sober. In: Continuous Lattices, Lecture Notes in Mathematics, vol. 871, Springer-Verlag, 282–283.CrossRefGoogle Scholar
Jung, A. (2018). Four dcpos, a theorem, and an open problem, an invited talk in Hunan University, Hunan Province, China, 1 December 2018.Google Scholar
Lawson, J. (1979). The duality of continuous posets, Houston Journal of Mathematics 5 357–386.Google Scholar
Xi, X. and Lawson, J. D. (2017) On well-filtered spaces and ordered sets. Topology and Its Applications 228 139–144.CrossRefGoogle Scholar
Xu, X., Xi, X. and Zhao, D. (2021). A complete Heyting algebra whose Scott space is non-sober. Fundamenta Mathematicae 252 315–323.CrossRefGoogle Scholar
Xu, X. and Zhao, D. (2020). Some open problems on well-filtered spaces and sober spaces. Topology and Its Applications 301 107540.CrossRefGoogle Scholar