Applied Categorical Structures

, Volume 21, Issue 2, pp 167–180 | Cite as

Essential Completeness in Categories of Completely Regular Frames

  • B. Banaschewski
  • A. W. Hager


The basic result here is that certain easily described coreflective subcategories S of the category CRFrm of completely regular frames have unique essential completions determined for each L ∈ S by the S-coreflection of the Booleanization of L. Next, this is shown to apply to several familiar subcategories of CRFrm, and concrete descriptions of the essential completions as well as internal characterizations of essential completeness are then given for these cases. Finally, back to the subcategories S in general, the essential completions in any of these are proved to become the epicomplete reflections in the category derived from S by considering only skeletal maps.


Completely regular frame Essential monomorphism Essential completion Strongly monocoreflective subcategory Booleanization of a frame 

Mathematics Subject Classifications (2010)

06F25 54C30 54H10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Banaschewski, B.: Compact regular frames and the Sikorski Theorem. Kyungpook Math. J. 28, 1–14 (1988)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática, Série B, No. 12, Departamento de Matemática da Universidade de Coimbra (1997)Google Scholar
  3. 3.
    Banaschewski, B.: The axiom of countable choice and pointfree topology. Appl. Categ. Struct. 9, 245–258 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Banaschewski, B.: Uniform completion in pointfree topology. In: Rodabaugh, S.E., Klement, E.P. (eds.), Topological and Algebraic Structures in Fuzzy Sets, pp. 19–56. Kluwer Academic Publishers (2003)Google Scholar
  5. 5.
    Banaschewski, B.: On the strong amalgamation of Boolean algebras. Alg. Univ. 63, 235–238 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Banaschewski, B., Hager, A.W.: Injectivity of archimedean ℓ-groups with order unit. Alg. Univ. 62, 113–123 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Banaschewski, B., Pultr, A.: Paracompactness revisited. Appl. Categ. Struct. 1, 181–190 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gleason, A.: Projective topological spaces. Ill. J. Math. 2, 482–489 (1958)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Halmos, P.R.: Injective and projective Boolean algebras. In: Proc. Symp. Pure Math. vol. 11, pp. 114–122. Amer. Math. Soc., Providence, RI (1961)Google Scholar
  10. 10.
    Isbell, J.R.: Atomless parts of spaces. Math. Scand. 31, 5–32 (1972)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Johnstone, P.T.: Stone spaces. Cambridge Stud. Adv. Math., vol. 3. Cambridge University Press (1982)Google Scholar
  12. 12.
    LaGrange, R.: Amalgamation and epimorphisms in \({\mathfrak m}\)-complete Boolean algebras. Alg. Univ. 4, 277–279 (1974)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Martinez, J., Zenk, E. R.: Epicompletion in frames with skeletal maps I: compact regular frames. Appl. Categ. Struct. 16, 521–533 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Pultr, A., Hazewinkel, F.M. (eds.): Handbook of Algebra, vol. 3, 791–857. Elsevier Science B.V. (2003)Google Scholar
  15. 15.
    Vickers, S.: Topology via Logic. Cambridge Tracts Theor. Comp. Sci., vol. 5. Cambridge University Press (1985)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Mathematics & StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of Mathematics & Computer ScienceWesleyan UniversityMiddletownUSA

Personalised recommendations