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Topological representations of distributive hypercontinuous lattices

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Abstract

The concept of locally strong compactness on domains is generalized to general topological spaces. It is proved that for each distributive hypercontinuous lattice L, the space SpecL of nonunit prime elements endowed with the hull-kernel topology is locally strongly compact, and for each locally strongly compact space X, the complete lattice of all open sets \( \mathcal{O} \)(X) is distributive hypercontinuous. For the case of distributive hyperalgebraic lattices, the similar result is given. For a sober space X, it is shown that there is an order reversing isomorphism between the set of upper-open filters of the lattice \( \mathcal{O} \)(X) of open subsets of X and the set of strongly compact saturated subsets of X, which is analogous to the well-known Hofmann-Mislove Theorem.

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Authors and Affiliations

  1. Department of Mathematics, Jiangxi Normal University, Nanchang, 330022, China

    Xiaoquan Xu & Jinbo Yang

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  1. Xiaoquan Xu
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  2. Jinbo Yang
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Correspondence to Xiaoquan Xu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10331010, 10861007), the Foundation for the Author of National Excellent Doctoral Dissertation of China (No. 2007B14), the Jiangxi Provincial Natural Science Foundation of China (Nos. 0411025, 2007GZS0179), the Foundation of the Education Department of Jiangxi Province (No. GJJ08162) and the Doctoral Fund of Jiangxi Normal University.

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Xu, X., Yang, J. Topological representations of distributive hypercontinuous lattices. Chin. Ann. Math. Ser. B 30, 199–206 (2009). https://doi.org/10.1007/s11401-007-0316-7

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  • DOI: https://doi.org/10.1007/s11401-007-0316-7

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