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Universality Property and Dimension for Frames

  • D. GeorgiouEmail author
  • S. Iliadis
  • A. Megaritis
  • F. Sereti
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Abstract

The universality property plays an important role in the field of frames and the notion of saturated class of frames is combined with this property (see Dube et al. (Topology and its Applications 160, 2454–2464, 2013); Iliadis (Topology and its Applications 179, 99–110, 2015) and Iliadis (Topology and its Applications 201, 92–109, 18)). In this paper, we continue such a study, introducing and studying the notion of saturated class of bases for frames. Based on the notions of the small inductive dimension, frind, for frames, which is inserted in Georgiou et al. (2019), and the saturated class of bases, we define the base dimension like-function of the type frind for frames, and prove that in a class of bases which is characterized by this dimension there exist universal elements.

Keywords

Base dimension like-function Small inductive dimension Frame Saturated class of bases Universality property 

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Notes

Acknowledgements

The authors would like to thank the reviewer for the careful reading of the paper and the useful comments.

References

  1. 1.
    Banaschewski, B., Gilmour, G.: Stone-Čech compactification and dimension theory for regular σ-frames. J. London Math. Soc. 39(2), 1–8 (1989)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Banaschewski, B.: Universal zero-dimensional compactifications. Categorical topology and its relation to analysis, algebra and combinatorics (Prague, 1988), World Sci. Publ., Teaneck, NJ, 257–269 (1989)Google Scholar
  3. 3.
    Berghammer, R., Winter, M.: Order-and graph-theoretic investigation of dimensions of finite topological spaces and Alexandroff spaces. Monatshefte für Mathematik.  https://doi.org/10.1007/s00605-019-01261-1 (2019)
  4. 4.
    Brijlall, D., Baboolal, D.: Some aspects of dimension theory of frames. Indian J. Pure Appl. Math. 39(5), 375–402 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brijlall, D., Baboolal, D.: The katětov-morita theorem for the dimension of metric frames. Indian J. Pure Appl. Math. 41(3), 535–553 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Charalambous, M. G.: Dimension theory of σ-frames. J. London Math. Soc. 8(2), 149–160 (1974)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Dube, T., Iliadis, S., van Mill, J., Naidoo, I.: Universal frames. Topology and its Applications 160, 2454–2464 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Engelking, R.: Theory of dimensions, finite and infinite. Sigma series in pure mathematics, vol. 10. Heldermann Verlag, Lemgo (1995)Google Scholar
  9. 9.
    Español, L., Gutiérrez, G. J., Kubiak, T.: Separating families of locale maps and localic embeddings. Algebra Universalis 67, 105–112 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Georgiou, D., Iliadis, S., Megaritis, A., Sereti, F.: Small inductive dimension and universality on frames. Accepted for publication in Algebra Universalis (2019)Google Scholar
  11. 11.
    Georgiou, D., Kougias, I., Megaritis, A., Prinos, G., Sereti, F.: A study of a new dimension for frames. Accepted for publication in Topology and its Applications (2019)Google Scholar
  12. 12.
    Georgiou, D. N., Megaritis, A. C., Sereti, F.: A topological dimension greater than or equal to the classical covering dimension. Houst. J. Math. 43(1), 283–298 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gevorgyan, P. S., Iliadis, S. D., Sadovnichy, Y.V.: Universality on frames. Topology and its Applications 220, 173–188 (2017)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Iliadis, S.: A constuction of containing spaces. Topology and its Applications 107, 97–116 (2000)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Iliadis, S. D.: Universal spaces and mappings North-Holland mathematics studies, vol. 198. Elsevier Science B.V., Amsterdam (2005)Google Scholar
  16. 16.
    Iliadis, S. D.: Universal regular and completely regular frames. Topology and its Applications 179, 99–110 (2015)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Iliadis, S. D.: Dimension and universality on frames. Topology and its Applications 201, 92–109 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Isbell, J. R.: Graduation and Dimension in Locales. In: Aspects of Topology (in Memory of Hugh Dowker 1912–1982). London Math. Soc. Lecture Note Ser., Vol. 93, pp 195–210. Cambridge Univ. Press, Cambridge (1985)Google Scholar
  19. 19.
    Menger, K.: Über die Dimensionalität von Punktmengen, Erster. Teil Monatshefte für Mathematik und Physik 33, 148–160 (1923)CrossRefGoogle Scholar
  20. 20.
    Menger, K.: Über die Dimension von Punktmengen, II. Teil. Monatshefte für Mathematik und Physik 34, 137–161 (1926)CrossRefGoogle Scholar
  21. 21.
    Pears, A. R.: Dimension theory of general spaces. Cambridge University Press, Cambridge (1975)zbMATHGoogle Scholar
  22. 22.
    Picardo, J., Pultr, A.: Frames and Locales. Topology without points. Frontiers in mathematics. Birkhäuser/Springer, Basel (2012)Google Scholar
  23. 23.
    Sancho de Salas, J. B., Sancho de Salas, M. T.: Dimension of distributive lattices and universal spaces. Topology and its Applications 42, 25–36 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vinokurov, V.G.: A lattice method of defining dimension. Dokl. Akad. Nauk SSSR 168(3), 663–666 (1966). (Russian)MathSciNetzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  • D. Georgiou
    • 1
    Email author
  • S. Iliadis
    • 2
  • A. Megaritis
    • 3
  • F. Sereti
    • 1
  1. 1.Department of MathematicsUniversity of PatrasPatrasGreece
  2. 2.Department of General Topology and Geometry, (M. V. Lomonosov)Moscow State UniversityMoscowRussia
  3. 3.Department of Digital SystemsUniversity of PeloponneseSpartaGreece

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