Skip to main content
SpringerLink
Log in

Choice-Free Dualities for Domains

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

A basic tool in domain theory and point-free topology are (Scott) open filters in a partially ordered set. A systematic investigation of that concept shows that central notions and facts like Lawson’s famous self-duality of the category of continuous domains may be established without invoking any choice principles, if only continuous domains are replaced by so-called δ-domains, which coincide with the former in the presence of dependent choices. Many of the conclusions remain valid for the more flexible notion of ζ-domains, comprising important variants such as algebraic or hypercontinuous domains.

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. Alexandroff, P.: Diskrete Raumë. Mat. Sb. (N.S.) 2, 501–518 (1937)

    MATH  Google Scholar 

  2. Bandelt, H.-J.: Regularity and complete distributivity. Semigroup Forum 19, 123–126 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bandelt, H.-J., Erné, M.: The category of Z-continuous posets. J. Pure Appl. Algebra 30, 219–226 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bandelt, H.-J., Erné, M.: Representations and embeddings of M-distributive lattices. Houston J. Math. 10, 315–324 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Erné, M.: Scott convergence and Scott topologies on partially ordered sets II. In: Hoffmann, R.-E., Hofmann, K. H. (eds.) Continuous Lattices. Lecture notes in math, vol. 871, pp 61–96. Springer, Berlin (1981)

  6. Erné, M.: The ABC of order and topology. In: Herrlich, H., Porst, H. (eds.) Category theory at work, pp 57–83. Heldermann Verlag, Berlin (1991)

  7. Erné, M.: The Dedekind-MacNeille completion as a reflector. Order 8, 159–173 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Erné, M.: Algebraic ordered sets and their generalizations. In: Rosenberg, I., Sabidussi, G. (eds.) Algebras and orders, Proc. Montreal 1992, pp 113–192. Kluwer, Amsterdam (1994)

  9. Erné, M.: General stone duality. In: Clementino, M.M. et al. (eds.) Proceedings of IV Iberoamerican Conf. On Topology, Coimbra 2001. Top. Appl., vol. 137, pp 125–158 (2004)

  10. Erné, M.: Minimal bases, ideal extensions, and basic dualities. Topology Proceedings 29, 445–489 (2005)

    MathSciNet  MATH  Google Scholar 

  11. Erné, M.: Choiceless, pointless, but not useless: dualities for categories of preframes. Appl. Categor. Struct. 15, 541–572 (2007)

    Article  MATH  Google Scholar 

  12. Erné, M.: Infinite distributive laws versus local connectedness and compactness properties. Top. Appl. 156, 2054–2069 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Erné, M.: Finiteness conditions and distributive laws for Boolean algebras. Math. Logic Quarterly 55, 572–586 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Erné, M.: The strength of prime separation, sobriety and compactness principles. Preprint, Leibniz University Hannover (2013)

  15. Erné, M.: Patch, web, core, sector and fan spaces. Preprint, Leibniz University Hannover (2016)

  16. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: A compendium of continuous lattices. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  17. Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., Scott, D.S.: Continuous lattices and domains. Oxford University Press (2003)

  18. Gierz, G., Lawson, J.D.: Generalized continuous lattices and hypercontinuous lattices. Rocky Mountain J. Math. 11, 271–296 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  19. Herrlich, H.: Axiom of Choice. Lecture Notes in Math. Springer, Berlin (2006)

    Google Scholar 

  20. Hoffmann, R.-E.: Topological spaces admitting a ‘Dual’. In: Categorical topology, lecture notes in math, vol. 719, pp 157–166. Springer, Berlin (1979)

  21. Hoffmann, R.-E.: Continuous posets, prime spectra of completely distributive complete lattices, and Hausdorff compactifications. In: Hoffmann, R.-E., Hofmann, K.H. (eds.) Continuous lattices. Lecture Notes in Math. 871, pp 159–208. Springer, Berlin (1981)

  22. Howard, P.: The finiteness of compact Boolean algebras. Math. Logic Quarterly 57, 14–18 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Howard, P., Rubin, J.E.: Consequences of the axiom of choice. Mathematical Surveys and Monographs, vol. 59. American Mathematical Society, Providence (1998)

  24. Isbell, J.R.: Completion of a construction of Johnstone. Proc. Amer. Math. Soc. 85, 333–334 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  25. Johnstone, P.T.: Continuous Lattices, Lecture notes in math. In: Hoffmann, R.-E., Hofmann, K.H. (eds.) , vol. 871, pp 282–283. Springer, Berlin (1981)

  26. Johnstone, P.T.: Stone spaces. Cambridge University Press, Cambridge (1982)

    MATH  Google Scholar 

  27. Kou, H., Liu, M.-Y., Luo, M.-K.: On Meet-Continuous dcpos. In: Lawson, J., Zhang, G.-Q., Liu, Y.-M., Luo, M.-K. (eds.) Domains and Processes II, semantic structures in computation, p 2001. Kluwer

  28. Lawson, J.D.: The duality of continuous posets. Houston J. Math. 5, 357–386 (1979)

    MathSciNet  MATH  Google Scholar 

  29. Lawson, J.D.: Order and strongly sober compactifications. In: Reed, G.M., Roscoe, A.W., Wachter, R.F. (eds.) Topology and category theory in computer science, pp 179–205. Clarendon Press, Oxford (1991)

  30. Levine, N.: Strongly connected sets in topology. Amer. Math. Monthly 72, 1098–1101 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mislove, M.: Algebraic posets, algebraic cpo’s and models of concurrency. In: Reed, G.M., Roscoe, A.W., Wachter, R.F. (eds.) Topology and category theory in computer science, pp 75–111. Clarendon Press, Oxford (1991)

  32. Moore, G.H.: Zermelo’s axiom of choice. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  33. Picado, J., Pultr, A.: Frames and locales. Birkhäuser, Basel (2012)

    Book  MATH  Google Scholar 

  34. Pincus, D.: Adding dependent choices to the prime ideal theorem, vol. 76. Logic Colloquium, North-Holland (1977)

  35. Raney, G.N.: A subdirect-union representation for completely distributive complete lattices. Proc. Amer. Math. Soc. 4, 518–522 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rudin, M., Rao, M.M.: Directed sets which converge. In: McAuley, L.F. (ed.) General topology and modern analysis, University of California, Riverside, 1980, pp 305–307. Academic (1981)

  37. Scott, D.S.: Continuous Lattices. In: Toposes, algebraic geometry and logic. Lecture notes in math, p 274. Springer, Berlin (1972)

  38. Wyler, O.: Dedekind complete posets and Scott topologies. In: Hoffmann, R.-E., Hofmann, K.H. (eds.) Continuous Lattices. Lecture Notes in Math, vol. 871, pp 384–389. Springer, Berlin (1981)

Download references

Author information

Authors and Affiliations

  1. Faculty for Mathematics and Physics, Leibniz University, Welfengarten 1, 30167, Hannover, Germany

    Marcel Erné

Authors
  1. Marcel Erné
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcel Erné.

Additional information

To the memory of Horst Herrlich, who opened our eyes for miracles caused by the axiom of choice and for disasters caused by its absence

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Erné, M. Choice-Free Dualities for Domains. Appl Categor Struct 24, 471–496 (2016). https://doi.org/10.1007/s10485-016-9444-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-016-9444-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation