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Which Distributive Lattices are Lattices of Open Sets of P-Spaces?

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Abstract

Representing lattices by topologies has been studied to a great extent. In this paper, we prove that a complete lattice L is isomorphic to the lattice of open subsets of a P-space iff L is order generated by its countably prime elements. We establish the dual equivalence between the category of complete lattices order generated by their countably prime elements with morphisms preserving arbitrary sups and countable infs, and the category of countably sober P-spaces with morphisms of continuous maps. Finally, we show that the category of countably sober P-spaces is reflective in the category of P-spaces.

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References

  1. Banaschewski, B., Gilmour, C.R.A.: Stone-Cěch compactification and dimension theory for regular σ-frames. J. Lond. Math. Soc. 39, 1–8 (1989)

    Article  Google Scholar 

  2. Banaschewski, B., Matutu, P.: Remarks on the frame envelope of a σ-frame. J. Pure Appl. Algebra 177, 231–236 (2003)

    Article  MathSciNet  Google Scholar 

  3. Büchi, J.R.: Representation of complete lattices by sets. Portugaliae Math. 11, 151–167 (1952)

    MathSciNet  MATH  Google Scholar 

  4. Drake, D., Thron, W.J.: On the representation of an abstract lattice as the family of closed subsets of a topological space. Trans. Am. Math. Soc. 120, 57–71 (1965)

    Article  Google Scholar 

  5. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous Lattices and Domains. Cambridge University Press, Cambridge (2003)

    Book  Google Scholar 

  6. Gillman, L., Jerrison, M.: Rings of Continuous Functions. Graduate Texts in Mathmatics, pp. 62–65. D. Van Nostrand Publ. Co., New York (1976)

    Google Scholar 

  7. Han, Y.H., Hong, S.S., Lee, C.K., Park, P.U.: A generalization of posets. Commun. Korean Math. Soc. 4(1), 129–138 (1989)

    MATH  Google Scholar 

  8. Hofmann, K.H., Lawson, J.D.: The spetral theory of distributive continuous lattices. Trans. Am. Math. Soc. 246, 285–310 (1978)

    Article  Google Scholar 

  9. Lee, S.O.: Countably approximating frames. Commun. Korean Math. Soc. 17, 295–308 (2002)

    Article  MathSciNet  Google Scholar 

  10. McGovern, W.W.: Free topologcial groups of weak P-spaces. Topol. Appl. 112, 175–180 (2001)

    Article  Google Scholar 

  11. Papert, S.: Which distributive lattices are lattice of closed sets? Proc. Camb. Philos. Soc. 55, 172–176 (1959)

    Article  MathSciNet  Google Scholar 

  12. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Am. Math. Soc. 40, 37–111 (1936)

    MathSciNet  MATH  Google Scholar 

  13. Stone, M.H.: Topological representation of distributive lattices and Brouwerian logics. Časopis pro Pěstovánĭ Matematiky a Fysiky 67, 1–25 (1937)

    MATH  Google Scholar 

  14. Yang, J., Shi, J.: Countably sober spaces. Electron. Notes Theor. Comput. Sci. 333, 143–151 (2017)

    Article  Google Scholar 

Download references

Acknowledgements

We thank the referees for carefully checking the original manuscript and providing us with valuable suggestions for improvement.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Jiangxi Normal University, Nanchang, 330022, China

    Jinbo Yang

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, China

    Xiaoyong Xi

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  1. Jinbo Yang
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  2. Xiaoyong Xi
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Correspondence to Jinbo Yang.

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Supported by the NSF of China (12071188, 11361028, 11671008, 11761034) and Science and Technology Project from Jiangxi Education Department (GJJ150344)

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Yang, J., Xi, X. Which Distributive Lattices are Lattices of Open Sets of P-Spaces?. Order 38, 391–399 (2021). https://doi.org/10.1007/s11083-020-09547-y

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