Skip to main content
Log in

Canonical extensions: an algebraic approach to Stone duality

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

This note aims to highlight some of the conceptual contributions to duality theory made by Bjarni Jónsson through the theory of canonical extensions.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Banaschewski, B.: Remarks on dual adjointness. In: Nordwestdeutsches Kategorienseminar Tagung, Bremen, 1976. Math.-Arbeitspapiere, vol. 7, Teil A: Math. Forschungspapiere, pp. 3–10. Univ. Bremen, Bremen (1976)

  2. Burris, S., Sankappanavar, H.P.: A Course in Universal Algebra, vol. 78. Graduate Texts in Mathematics, Springer, Berlin (1981) (Or, the Millenium Edition, see https://www.math.uwaterloo.ca/~snburris/htdocs/ualg.html)

    Book  Google Scholar 

  3. Conradie, W., Craig, A., Palmigiano, A., Zhao, Z.: Constructive canonicity for lattice-based fixed point logics. In: Kennedy, J., de Queiroz, R. (eds.) International Workshop on Logic, Language, Information, and Computation, WoLLIC 2017: Logic, Language, Information, and Computation. Theoretical Computer Science and General Issues, vol. 10388, pp. 92–109. Springer, Berlin (2017)

    Google Scholar 

  4. Conradie, W., Ghilardi, S., Palmigiano, A.: Unified correspondence. In: Baltag, A., Smets, S. (eds.) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol. 5, pp. 933–975. Springer, Basel (2014)

    Google Scholar 

  5. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Log. 70, 713–740 (2005)

    Article  MathSciNet  Google Scholar 

  6. Gehrke, M.: The order structure of Stone spaces and the T\(_D\) separation axiom. MLQ Math. Log. Q. 37, 5–15 (1991)

    Article  MathSciNet  Google Scholar 

  7. Gehrke, M.: Canonical extensions, Esakia spaces, and universal models. In: Bezhanishvili, G. (ed.) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol. 4, pp. 9–41. Springer, Dordrecht (2014)

    Google Scholar 

  8. Gehrke, M.: Stone duality, topological algebra, and recognition. J. Pure Appl. Algebra 220, 2711–2747 (2016)

    Article  MathSciNet  Google Scholar 

  9. Gehrke, M., van Gool, S.: Sheaves and duality. J. Pure Appl. Algebra 222, 2164–2180 (2018)

    Article  MathSciNet  Google Scholar 

  10. Gehrke, M., Grigorieff, S., Pin, J.-É.: Duality and equational theory of regular languages. In: Aceto, L., et al. (eds.) Automata, Languages and Programming: 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7–11, 2008, Proceedings, Part II. Theoretical Computer Science and General Issues, vol. 5126, pp. 246–257. Springer, Berlin (2008)

    Chapter  Google Scholar 

  11. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238, 345–371 (2001)

    Article  MathSciNet  Google Scholar 

  12. Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)

    Article  MathSciNet  Google Scholar 

  13. Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)

    Article  MathSciNet  Google Scholar 

  14. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japon. 40, 207–215 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 31, 65–102 (2005)

    Article  MathSciNet  Google Scholar 

  16. Gehrke, M., Vosmaer, J.: A view of canonical extension. In: Bezhanishvili, N., et al. (eds.) Logic, Language, and Computation. TbiLLC 2009. Lecture Notes in Computer Science, vol. 6618, pp. 77–100. Springer, Berlin (2009)

    Google Scholar 

  17. Goldblatt, R.: Varieties of complex algebras. Ann. Pure App. Logic 44, 173–242 (1989)

    Article  MathSciNet  Google Scholar 

  18. Goldblatt, R.: Mathematical modal logic: a view of its evolution. J. Appl. Log. 1, 309–392 (2003)

    Article  MathSciNet  Google Scholar 

  19. Goldblatt, R.: Canonical extensions and ultraproducts of polarities (preprint)

  20. Goldblatt, R., Hodkinson, I., Venema, Y.: Erdös graphs resolve Fine’s canonicity problem. Bull. Symb. Log. 10, 186–208 (2004)

    Article  Google Scholar 

  21. Hodkinson, I.: Hybrid formulas and elementarily generated modal logics. Notre Dame J. Form. Log. 47, 443–478 (2006)

    Article  MathSciNet  Google Scholar 

  22. Jónsson, B.: A survey of Boolean algebras with operators. In: Rosenberg, I., Sabiducci, G. (eds.) Algebras and Order. NATO ASI Series, pp. 239–286. Kluwer Academic Publishing, Norwell (1993)

    Chapter  Google Scholar 

  23. Jónsson, B.: On the canonicity of Sahlqvist identities. Stud. Log. 53, 473–491 (1994)

    Article  MathSciNet  Google Scholar 

  24. Jónsson, B., Tarski, A.: Boolean algebras with operators I. Am. J. Math. 73, 891–939 (1951)

    Article  MathSciNet  Google Scholar 

  25. Jónsson, B., Tarski, A.: Boolean algebras with operators II. Am. J. Math. 74, 127–162 (1952)

    Article  MathSciNet  Google Scholar 

  26. Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Časopis Mat. Fys. 67, 1–25 (1937)

    MATH  Google Scholar 

  27. Vosmaer, J.: Logic, algebra and topology. Investigations into canonical extensions, duality theory and point-free topology. PhD thesis, University of Amsterdam (2010)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mai Gehrke.

Additional information

In memoriam Bjarni Jónsson.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J. B. Nation.

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (Grant agreement no. 670624).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gehrke, M. Canonical extensions: an algebraic approach to Stone duality. Algebra Univers. 79, 63 (2018). https://doi.org/10.1007/s00012-018-0544-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00012-018-0544-6

Mathematics Subject Classification

Keywords

Navigation