Applied Categorical Structures

, Volume 26, Issue 5, pp 997–1013 | Cite as

Meet-Semilattice Congruences on a Frame

  • John FrithEmail author
  • Anneliese Schauerte


The congruence lattice of a frame has long been an object of considerable interest, not least because it turns out to be a frame itself. Perhaps more surprisingly congruence lattices of, for instance, \(\sigma \)-frames, \(\kappa \)-frames and some partial frames also turn out to be frames. The situation for congruences of a meet-semilattice is notably different. In this paper we analyze the meet-semilattice congruence lattices of arbitrary frames and compare them with the corresponding lattices of frame congruences. In the course of this, we provide a structure theorem as well as many examples and counter-examples.


Complete lattice Frame Partial frame \(\mathcal {S}\)-frame Frame congruence Meet-semilattice congruence Complements 

Mathematics Subject Classification

06B10 06D22 


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  1. 1.
    Adámek, J., Herrlich, H., Strecker, G.: Abstract and Concrete Categories. Wiley, New York (1990)zbMATHGoogle Scholar
  2. 2.
    Banaschewski, B.: Another look at the localic Tychonoff theorem. Comment. Math. Univ. Carolinae 29(4), 647–656 (1988)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Banaschewski, B.: Lectures on Frames. University of Cape Town, Cape Town (1988)zbMATHGoogle Scholar
  4. 4.
    Banaschewski, B.: \(\sigma \)-frames, unpublished manuscript. (1980). Accessed 25 Nov 2017
  5. 5.
    Banaschewski, B., Gilmour, C.R.A.: Realcompactness and the cozero part of a frame. Appl. Categ. Struct. 9, 395–417 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Erné, M., Zhao, D.: Z-join spectra of Z-supercompactly generated lattices. Appl. Categ. Struct. 9(1), 41–63 (2001)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Escardó, M.H.: Joins in the frame of nuclei. Appl. Categ. Struct. 11, 117–124 (2003)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Frith, J.L.: Structured frames. Ph.D. Thesis, University of Cape Town (1987)Google Scholar
  9. 9.
    Frith, J., Schauerte, A.: An asymmetric characterization of the congruence frame. Topol. Appl. 158(7), 939–944 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Frith, J., Schauerte, A.: Uniformities and covering properties for partial frames (I). Categ. Gen. Alg. Struct. Appl. 2(1), 1–21 (2014)zbMATHGoogle Scholar
  11. 11.
    Frith, J., Schauerte, A.: Uniformities and covering properties for partial frames (II). Categ. Gen. Alg. Struct. Appl. 2(1), 23–35 (2014)zbMATHGoogle Scholar
  12. 12.
    Frith, J., Schauerte, A.: The Stone–Čech compactification of a partial frame via ideals and cozero elements. Quaest. Math. 39(1), 115–134 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Frith, J., Schauerte, A.: Completions of uniform partial frames. Acta Math. Hung. 147(1), 116–134 (2015)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Frith, J., Schauerte, A.: Coverages give free constructions for partial frames. Appl. Categ. Struct 25(3), 303–321 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Frith, J., Schauerte, A.: Compactifications of partial frames via strongly regular ideals. Math. Slovaca (2016, accepted)Google Scholar
  16. 16.
    Frith, J., Schauerte, A.: One-point compactifications and continuity for partial frames. Categ. Gen. Algebr. Struct. Appl. 7, 57–88 (2017)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Frith, J., Schauerte, A.: The congruence frame and the Madden quotient for partial frames (submitted) Google Scholar
  18. 18.
    Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)zbMATHGoogle Scholar
  20. 20.
    Joyal, A., Tierney, M.: An Extension of the Galois Theory of Grothendieck, vol. 309. American Mathematical Society, Providence (1984)zbMATHGoogle Scholar
  21. 21.
    Klinke, O.: A presentation of the assembly of a frame by generators and relations exhibits its bitopological structure. Algebra Universalis 71, 55–64 (2014)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Mac Lane, S.: Categories for the Working Mathematician. Springer, Heidelberg (1971)CrossRefGoogle Scholar
  23. 23.
    Madden, J.J.: \(\kappa \)-frames. J. Pure Appl. Algebra 70, 107–127 (1991)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Papert, D.: Congruence relations in semi-lattices. J. Lond. Math. Soc. 39, 723–729 (1964)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Paseka, J.: Covers in generalized frames. In: Chajda, I. et al. (eds.) General Algebra and Ordered Sets (Horni Lipova 1994), pp. 84–99. Palacky University Olomouc, Olomouc.Google Scholar
  26. 26.
    Picado, J., Pultr, A.: Frames and Locales. Springer, Basel (2012)CrossRefGoogle Scholar
  27. 27.
    Plewe, T.: Higher order dissolutions and Boolean coreflections of locales. J. Pure Appl. Algebra 154, 273–293 (2000)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Plewe, T.: Sublocale lattices. J. Pure Appl. Algebra 168, 309–326 (2002)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Simmons, H.: A framework for topology. Stud. Logic Found. Math. 96, 239–251 (1978)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Simmons, H.: Spaces with Boolean assemblies. Colloq. Math. 43, 23–39 (1980)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Zenk, E.R.: Categories of partial frames. Algebra Univers. 54, 213–235 (2005)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Zhao, D.: Nuclei on \(Z\)-frames. Soochow J. Math. 22(1), 59–74 (1996)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Zhao, D.: On projective \(Z\)-frames. Can. Math. Bull. 40(1), 39–46 (1997)MathSciNetCrossRefGoogle Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Applied MathematicsUniversity of Cape TownRondeboschSouth Africa

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