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Lattice-equivalence of convex spaces

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Abstract

The links between order and topology have been extensively studied. The objective of this paper is to study the similar links between convex spaces and order structures via the lattices of convex sets. In particular, we prove a characterization of convex spaces uniquely determined by means of their lattices of convex sets, they are precisely sober and \(S_D\). One adjunction between the category of convex spaces and that of continuous lattices is constructed, revealing the connection between convex structures and domain theory.

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Acknowledgements

We are every grateful to the referee for carefully checking the original draft and give us a lot of helpful suggestions for improvements.

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Authors and Affiliations

  1. Beijing Key Laboratory on MCAACI, School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Chong Shen & Fu-Gui Shi

  2. The Fujian Provincial Key Laboratory of Network Security and Cryptology, College of Mathematics and Informatics, Fujian Normal University, Fuzhou, 350117, China

    Shao-Jun Yang

  3. National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore, 637616, Singapore

    Dongsheng Zhao

Authors
  1. Chong Shen
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  2. Shao-Jun Yang
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  3. Dongsheng Zhao
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  4. Fu-Gui Shi
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Correspondence to Shao-Jun Yang.

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The project is supported by the National Natural Science Foundation of China (11871097, 11701089, 61822202, 61872089, 61872087), Joint Ph.D. Program of Beijing Institute of Technology, Science and Technology Program of Fujian Province, China (2019J01428), and Nanyang Technological University Research Scholarship, and NIE ACRF project (RI 3/16 ZDS).

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Shen, C., Yang, SJ., Zhao, D. et al. Lattice-equivalence of convex spaces. Algebra Univers. 80, 26 (2019). https://doi.org/10.1007/s00012-019-0600-x

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  • DOI: https://doi.org/10.1007/s00012-019-0600-x

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