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Essential Completeness in Categories of Completely Regular Frames

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Abstract

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.

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References

  1. Banaschewski, B.: Compact regular frames and the Sikorski Theorem. Kyungpook Math. J. 28, 1–14 (1988)

    MathSciNet  MATH  Google Scholar 

  2. Banaschewski, B.: The real numbers in pointfree topology. Textos de Matemática, Série B, No. 12, Departamento de Matemática da Universidade de Coimbra (1997)

  3. Banaschewski, B.: The axiom of countable choice and pointfree topology. Appl. Categ. Struct. 9, 245–258 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  5. Banaschewski, B.: On the strong amalgamation of Boolean algebras. Alg. Univ. 63, 235–238 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banaschewski, B., Hager, A.W.: Injectivity of archimedean ℓ-groups with order unit. Alg. Univ. 62, 113–123 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Banaschewski, B., Pultr, A.: Paracompactness revisited. Appl. Categ. Struct. 1, 181–190 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gleason, A.: Projective topological spaces. Ill. J. Math. 2, 482–489 (1958)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Johnstone, P.T.: Stone spaces. Cambridge Stud. Adv. Math., vol. 3. Cambridge University Press (1982)

  12. LaGrange, R.: Amalgamation and epimorphisms in \({\mathfrak m}\)-complete Boolean algebras. Alg. Univ. 4, 277–279 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  13. Martinez, J., Zenk, E. R.: Epicompletion in frames with skeletal maps I: compact regular frames. Appl. Categ. Struct. 16, 521–533 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pultr, A., Hazewinkel, F.M. (eds.): Handbook of Algebra, vol. 3, 791–857. Elsevier Science B.V. (2003)

  15. Vickers, S.: Topology via Logic. Cambridge Tracts Theor. Comp. Sci., vol. 5. Cambridge University Press (1985)

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Authors and Affiliations

  1. Department of Mathematics & Statistics, McMaster University, Hamilton, ON L8S 4K1, Canada

    B. Banaschewski

  2. Department of Mathematics & Computer Science, Wesleyan University, Middletown, CT, 06459, USA

    A. W. Hager

Authors
  1. B. Banaschewski
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  2. A. W. Hager
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Correspondence to B. Banaschewski.

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Banaschewski, B., Hager, A.W. Essential Completeness in Categories of Completely Regular Frames. Appl Categor Struct 21, 167–180 (2013). https://doi.org/10.1007/s10485-011-9262-3

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  • DOI: https://doi.org/10.1007/s10485-011-9262-3

Keywords

Mathematics Subject Classifications (2010)