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Exact Filters and Joins of Closed Sublocales

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Abstract

We prove, for a general frame, that the sublocales that can be represented as joins of closed ones are, somewhat surprisingly, in a natural one-to-one correspondence with the filters closed under exact meets, and explain some subfit facts from this perspective. Furthermore we discuss the filters associated in a similar vein with the fitted sublocales.

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Notes

  1. This may sound odd but it makes good sense; if L happens to have points P, they are sublocales of the form \(\{a,1\}\) with prime \(a\ne 1\). So \({\textsf {O}}\) is indeed smaller than any occupied sublocale.

References

  1. Balbes, R.: Projective and injective distributive lattices. Pac. J. Math. 21, 405–420 (1967)

    Article  MathSciNet  Google Scholar 

  2. Ball, R.N.: Distributive Cauchy lattices. Algebra Univers. 18, 134–174 (1984)

    Article  MathSciNet  Google Scholar 

  3. Ball, R.N., Picado, J., Pultr, A.: Notes on exact meets and joins. Appl. Categ. Struct. 22, 699–714 (2014)

    Article  MathSciNet  Google Scholar 

  4. Ball, R.N., Picado, J., Pultr, A.: On an aspect of scatteredness in the point-free setting. Port. Math. 73(2), 139–152 (2016)

    Article  MathSciNet  Google Scholar 

  5. Ball, R.N., Picado, J., Pultr, A.: Some aspects of (non)functoriality of natural discrete covers of locales. Quaest. Math. 42(6), 701–715 (2019)

    Article  MathSciNet  Google Scholar 

  6. Ball, R.N., Pultr, A.: Maximal essential extensions in the context of frames. Algebra Univers. 79(2), Article 32 (2018)

  7. Bruns, G., Lakser, H.: Injective hulls of semilattices. Can. Math. Bull. 13, 115–118 (1970)

    Article  MathSciNet  Google Scholar 

  8. Clementino, M.M., Picado, J., Pultr, A.: The other closure and complete sublocales. Appl. Categ. Struct. 26, 892–906 (2018). corr. 907–908

    MathSciNet  MATH  Google Scholar 

  9. Dube, T.: Submaximality in locales. Topol. Proc. 29, 431–444 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Ferreira, M.J., Picado, J.: On point-finiteness in point-free topology. Appl. Categ. Struct. 15, 185–198 (2007)

    Article  Google Scholar 

  11. Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)

    Article  MathSciNet  Google Scholar 

  12. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  13. MacNeille, H.M.: Extensions of partially ordered sets. Harvard dissertation (May 1935)

  14. MacNeille, H.M.: Extensions of partially ordered sets. Proc. Natl. Acad. Sci. USA 22, 45–50 (1936)

    Article  Google Scholar 

  15. Niefield, S., Rosenthal, K.: Spatial sublocales and essential primes. Topol. Appl. 26, 263–269 (1987)

    Article  MathSciNet  Google Scholar 

  16. Picado, J., Pultr, A.: Frames and Locales: Topology Without Points. Frontiers in Mathematics, vol. 28. Springer, Basel (2012)

    Book  Google Scholar 

  17. Picado, J., Pultr, A.: A Boolean extension of a frame and a representation of discontinuity. Quaest. Math. 40(8), 1111–1125 (2017)

    Article  MathSciNet  Google Scholar 

  18. Picado, J., Pultr, A.: New aspects of subfitness in frames and spaces. Appl. Categ. Struct. 24, 703–714 (2016)

    Article  MathSciNet  Google Scholar 

  19. Picado, J., Pultr, A., Tozzi, A.: Joins of closed sublocales. Houst. J. Math. 45(1), 21–38 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Plewe, T.: Sublocale lattices. J. Pure Appl. Algebra 168, 309–326 (2002)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Denver, Denver, CO, 80208, USA

    R. N. Ball

  2. Department of Mathematics, CECAT, Chapman University, Orange, CA, 92688, USA

    M. A. Moshier

  3. Department of Applied Mathematics and CE-ITI, MFF, Charles University, Malostranské nám. 25, 11800, Prague 1, Czech Republic

    A. Pultr

Authors
  1. R. N. Ball
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  2. M. A. Moshier
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  3. A. Pultr
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Correspondence to M. A. Moshier.

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Communicated by Jorge Picado.

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The third author gratefully acknowledges support from KAM at MFF, Charles University, Prague and from CECAT at Chapman University, Orange.

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Ball, R.N., Moshier, M.A. & Pultr, A. Exact Filters and Joins of Closed Sublocales. Appl Categor Struct 28, 655–667 (2020). https://doi.org/10.1007/s10485-020-09593-y

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  • DOI: https://doi.org/10.1007/s10485-020-09593-y

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