Skip to main content
SpringerLink
Log in

Notions of Local Compactness and Smallest Compactifications of Biframes

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

The category of coherent biframes is shown to be equivalent to that of the coupled lattices, and dually equivalent to the spectral bispaces. Stably continuous biframes arise as the retracts of the coherent biframes. The coherent and the stably continuous biframes are coreflective in all biframes. Weak local compactness is introduced, and in conjunction with regularity, is shown to be sufficient for the construction of smallest compactifications.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Banaschewski, B.: Coherent Frames, Continuous Lattices, Proceedings Bremen 1979. Springer Lect. Notes Math. 871, 1–11 (1981)

  2. Banaschewski, B.:Universal Zero-dimensional Compactifications, Categorical Topology and its Relations to Modern Analysis, Algebra and Combinatorics (Prague, 1988), pp. 257–269. World Sci., Teaneck, New Jersey (1989)

    Google Scholar 

  3. Banaschewski, B.: Compactification of frames. Math. Nachr. 149, 105–116 (1990)

    MathSciNet  MATH  Google Scholar 

  4. Banaschewski, B., Brümmer, G.C.L.: Stably continuous frames. Math. Proc. Camb. Philos. Soc. 104(1), 7–19 (1988)

    Article  MATH  Google Scholar 

  5. Banaschewski, B., Brümmer, G.C.L., Hardie, K.A.: Biframes and bispaces. Quaestiones Math. 6, 13–25 (1983)

    MathSciNet  MATH  Google Scholar 

  6. Banaschewski, B., Hoffmann, R.-E. (eds.): Continuous Lattices, Proceedings of the Conference on Topological and Categorical Aspects of Continuous Lattices, Bremen, Germany, 1979. Vol. 871 Lecture Notes in Mathematics. Springer, Berlin Heidelberg New York (1981)

  7. Bourbaki, N.: General Topology. Addison-Wesley, Reading, MA (1966)

    Google Scholar 

  8. Clementino, M.M., Tholen, W.: A note on local compactness. Rend. Istit. Mat. Univ. Trieste XXXII(Suppl. 2), 17–31 (2001)

    MathSciNet  Google Scholar 

  9. Escardó, M.H.: The regular-locally compact coreflection of a stably locally compact locale. J. Pure Appl. Algebra 157, 41–55 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: A Compendium of Continuous Lattices. Springer, Berlin Heidelberg New York (1980)

  11. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains, Encyclopedia of Mathematics and its Applications 93. Cambridge University Press (2003)

  12. Ginsberg, M.L.: Multivalued logics: A uniform approach to reasoning in artificial intelligence. Comput. Intelligence 4, 265–316 (1988)

    Article  Google Scholar 

  13. Hofmann, K.H.: Stably continuous frames and their topological manifestations. Categorical Topology, Sigma Series in Pure Math, pp. 282–307. Heldermann Verlag Berlin (1984)

  14. Johnstone, P.T.: Stone Spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  15. Kelly, J.C.: Bitopological spaces. Proc. London Math. Soc. 13(3), 71–89 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mac Lane, S.: Categories for the Working Mathematician, Graduate Texts in Math. no.5. Springer, Berlin Heidelberg New York (1971)

  17. Mobasher, B., Pigozzi, D., Slutzki, G., Voutsadakis, G.: A duality theory for bilattices. Algebra Universalis 43, 109–125 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Paseka, J., Šmarda, B.: On some notions related to compactness for locales. Acta Univ. Carolin., Math. Phys. 29(2), 51–65 (1988)

    MathSciNet  MATH  Google Scholar 

  19. Schauerte, A.: Biframe compactifications. Comment. Math. Univ. Carolin. 34(3), 567–574 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Schauerte, A.: Normality for biframes. Appl. Categ. Struct. 3, 1–9 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Stone, M.H.: Boolean algebras and their applications to topology. Proc. Nat. Acad. Sci. U.S.A. 20, 197–202 (1934)

    Article  Google Scholar 

  22. Stone, M.H.: Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc. 41, 375–481 (1937)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701, South Africa

    Anneliese Schauerte

Authors
  1. Anneliese Schauerte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anneliese Schauerte.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schauerte, A. Notions of Local Compactness and Smallest Compactifications of Biframes. Appl Categor Struct 14, 259–272 (2006). https://doi.org/10.1007/s10485-006-9023-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-006-9023-x

Key words

Mathematics Subject Classifications (2000)

Search

Navigation