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Acta Mathematica Hungarica

, Volume 147, Issue 1, pp 116–134 | Cite as

Completions of uniform partial frames

  • J. Frith
  • A. Schauerte
Article

Abstract

We address classical questions concerning the existence and properties of completions in a new context, namely, that of uniform partial frames.

A partial frame is a meet-semilattice in which certain joins exist and finite meets distribute over these joins. We specify these joins by means of a so-called selection function, which must satisfy certain axioms to produce a useful theory. The axioms we use here were introduced in [9] and are sufficiently general to encompass in the resulting theory frames, \({\kappa}\)-frames and \({\sigma}\)-frames.

Using covers to describe uniform structures on partial frames, we develop the notion of completeness for a uniform partial frame, using the frame-theoretic version of the well-known fact that a complete uniform space is isomorphic to any uniform space in which it is densely embedded.

In constructing a completion, we make substantial use of the functor which takes \({\mathcal{S}}\)-ideals and the functor which takes \({\mathcal{S}}\)-cozero elements, as well as the category equivalence that these functors induce. Our strategy involves the transfer of important properties concerning the completion from the category of uniform frames to that of uniform partial frames.

As a final application, we provide two constructions of the Samuel compactification. One involves the completion of the totally bounded coreflection; the other uses the functors mentioned above to transfer the corresponding compactification from uniform frames.

Key words and phrases

frame uniform \({\mathcal{S}}\)-frame uniform partial frame \({\sigma}\)-frame \({\kappa}\)-frame meet-semilattice complete completion compact Samuel compactification cozero \({\mathcal{S}}\)-cozero ideal \({\mathcal{S}}\)-ideal \({\mathcal{S}}\)-Lindelöf \({\mathcal{S}}\)-separable selection function 

Mathematics Subject Classification

06A12 06B10 06D22 18A40 54B30 54D20 54D35 54E15 

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References

  1. 1.
    J. Adámek, H. Herrlich and G. Strecker, Abstract and Concrete Categories John Wiley & Sons Inc. (New York, 1990).Google Scholar
  2. 2.
    B. Banaschewski, Uniform completion in pointfree topology, in: Topological and Algebraic Structures in Fuzzy Sets S. E. Rodabaugh and E. P. Klement (Ed.s), Kluwer Academic Publishers (2003), pp. 19–56.Google Scholar
  3. 3.
    Banaschewski B., Gilmour C. R. A.: Realcompactness and the cozero part of a frame. Appl. Categ. Struct. 9, 395–417 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banaschewski B., Pultr A.: Paracompactness revisited. Appl. Categ. Struct. 1, 181–190 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    B. Banaschewski and A. Pultr, Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. 108 (1990), 63–78.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    B. Banaschewski and J. L. Walters-Wayland, On Lindelöf uniform frames and the axiom of countable choice, Quaestiones Math., 30 (2007), 115–121.Google Scholar
  7. 7.
    J. Frith and A. Schauerte, The Samuel compactification for quasi-uniform biframes, Topology and its Applications 156 (2009), 2116–2122.Google Scholar
  8. 8.
    J. Frith and A. Schauerte, Uniformities and covering properties for partial frames, accepted by Categ. Gen. Alg. Struct. Appl. (May 2014).Google Scholar
  9. 9.
    J. Frith and A. Schauerte, The Stone–Čech compactification of a partial frame via ideals and cozero elements, accepted by Quaestiones Math. (Oct. 2014).Google Scholar
  10. 10.
    S. Ginsburg and J. R. Isbell, Some operators on uniform spaces, Trans. Amer. Math. Soc., 93 (1959), 145–168.Google Scholar
  11. 11.
    J. R. Isbell, Atomless parts of spaces, Math. Scand., 31 (1972), 5–32.Google Scholar
  12. 12.
    P. T. Johnstone, Stone Spaces Cambridge University Press (Cambridge, 1982).Google Scholar
  13. 13.
    I. Kříž, A direct description of uniform completion and a characterization of LT groups, Cah. Top. Géom. Diff. Cat., 27 (1986), 19–34.Google Scholar
  14. 14.
    S. Mac Lane, Categories for the Working Mathematician Springer-Verlag (Heidelberg, 1971).Google Scholar
  15. 15.
    J. J. Madden, \({\kappa}\)-frames, J. Pure Appl Algebra 70 (1991), 107–127.Google Scholar
  16. 16.
    I. Naidoo, Aspects of nearness in \({\sigma}\)-frames, Quaestiones Math., 30 (2007), 133–145.Google Scholar
  17. 17.
    I. Naidoo, On completion in the category SSN\({\sigma}\)FRM, Math. Slovaka 63 (2013), 201–214.Google Scholar
  18. 18.
    J. Paseka, Covers in generalized frames, in: General Algebra and Ordered Sets (Horni Lipova, 1994), Palacky Univ. Olomouc, Olomouc, pp. 84–99.Google Scholar
  19. 19.
    J. Picado and A. Pultr, Frames and Locales Springer (Basel, 2012).Google Scholar
  20. 20.
    G. Reynolds, Alexandroff algebras and complete regularity, Proc. Amer. Math. Soc., 76 (1979), pp. 322–326.Google Scholar
  21. 21.
    P. Samuel, Ultrafilters and compactification of uniform spaces, Transactions Amer. Math. Soc., 64 (1948), 100–132.Google Scholar
  22. 22.
    J. L. Walters, Compactifications and uniformities on \({\sigma}\)-frames, Comment. Math. Univ. Carolinae 32 (1991), 189–198.Google Scholar
  23. 23.
    J. L. Walters, Uniform Sigma Frames and the Cozero Part of Uniform Frames Masters Dissertation, University of Cape Town (1990).Google Scholar
  24. 24.
    E. R. Zenk, Categories of partial frames, Algebra Univers., 54 (2005), 213–235.Google Scholar
  25. 25.
    D. Zhao, Nuclei on Z-frames, Soochow J. Math., 22 (1996), 59–74.Google Scholar
  26. 26.
    D. Zhao, On projective Z-frames, Canad. Math. Bull., 40 (1997), 39–46.Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2015

Authors and Affiliations

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

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