Advertisement

Order

, Volume 31, Issue 3, pp 435–461 | Cite as

Distributive Envelopes and Topological Duality for Lattices via Canonical Extensions

  • Mai Gehrke
  • Samuel J. van Gool
Article
  • 119 Downloads

Abstract

We establish a topological duality for bounded lattices. The two main features of our duality are that it generalizes Stone duality for bounded distributive lattices, and that the morphisms on either side are not the standard ones. A positive consequence of the choice of morphisms is that those on the topological side are functional. Towards obtaining the topological duality, we develop a universal construction which associates to an arbitrary lattice two distributive lattice envelopes with a Galois connection between them. This is a modification of a construction of the injective hull of a semilattice by Bruns and Lakser, adjusting their concept of ‘admissibility’ to the finitary case. Finally, we show that the dual spaces of the distributive envelopes of a lattice coincide with completions of quasi-uniform spaces naturally associated with the lattice, thus giving a precise spatial meaning to the distributive envelopes.

Keywords

Lattice Non-distributivity Distributive envelope Canonical extension Priestley duality Pervin spaces Bicompletions 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. 25. American Mathematical Society (1967)Google Scholar
  2. 2.
    Bourbaki, N.: General topology. In: Chapters 1–4, Elements of Mathematics (Berlin), 1st edn. 1974, reprinted 2nd edn. Springer-Verlag, Berlin (1998)Google Scholar
  3. 3.
    Bruns, G., Lakser, H.: Injective hulls of semilattices. Canad. Math. Bull. 13(1), 115–118 (1970)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Coumans, D.C.S.: Canonical Extensions in Logic: Some Applications and a Generalisation to Categories. Dissertation, Radboud University Nijmegen (2012)Google Scholar
  5. 5.
    Császár, A.: D-completions of pervin-type quasi-uniformities. Acta. Sci. Math. (Szeged) 57, 329–335 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions of ordered algebraic structures and relational completeness of some substructural logics. J. Symb. Log. 70(3), 713–740 (2005)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Erné, M.: Ideal completions and compactifications. Appl. Cat. Struct. 9, 217–243 (2001)CrossRefMATHGoogle Scholar
  8. 8.
    Erné, M.: Choiceless, pointless, but not useless: dualities for preframes. Appl. Cat. Struct. 15, 541–572 (2007)CrossRefMATHGoogle Scholar
  9. 9.
    Erné, M., Palko, V.: Uniform ideal completions. Math. Slovaca. 48, 327–335 (1998)MathSciNetMATHGoogle Scholar
  10. 10.
    Erné, M., Zhao, D.: Z-join spectra of Z-Supercompactly generated lattices. Appl. Cat. Struct. 9, 41–63 (2001)CrossRefMATHGoogle Scholar
  11. 11.
    Fletcher, P., Lindgren, W.F.: Quasi-uniform spaces. In: Lectures Notes in Pure and Applied Mathematics, vol. 77. Marcel Dekker Inc., New York (1982)Google Scholar
  12. 12.
    Ganter, B., Wille, R.: Formal concept analysis. Mathematical Foundations. Translated from the 1996 German original by Cornelia Franzke. Springer-Verlag, Berlin (1999)Google Scholar
  13. 13.
    Gehrke, M.: Canonical extensions, Esakia spaces, and universal models. In: Leo Esakia on Duality in Modal and Intuitionistic Logics, Trends in Logic: Outstanding Contributions. Springer. preprint available at http://www.liafa.univ-paris-diderot.fr/mgehrke/Ge12.pdf (2012)
  14. 14.
    Gehrke, M., Grigorieff, S., Pin, J.-É.: A topological approach to recognition, automata, languages and programming. In: Abramsky, S., et al. (eds.) 37th International Colloquium (ICALP 2010), vol. 6199, no. 2, pp. 151–162. LNCS, Springer (2010)Google Scholar
  15. 15.
    Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japon. 40(2), 207–215 (1994)MathSciNetMATHGoogle Scholar
  17. 17.
    Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Math. Scand. 94, 13–45 (2004)MathSciNetMATHGoogle Scholar
  18. 18.
    Goldblatt, R.: Maps and monads for modal frames. Stud. Logica. 83(1-3), 309–331 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Harding, J.: Canonical completions of lattices and ortholattices. Tatra Mountains Math. Publ. 15, 85–96 (1998)MathSciNetMATHGoogle Scholar
  20. 20.
    Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992). doi: 10.1007/BF01.190610 MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Hartung, G.: An extended duality for lattices. In: Denecke, K., Vogel, H.-J. (eds.) General Algebra and Applications, pp 126–142. Heldermann-Verlag, Berlin (1993)Google Scholar
  22. 22.
    Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Amer. J. Math. 73(4), 891–939 (1951)CrossRefMATHGoogle Scholar
  23. 23.
    Jónsson, B.: Boolean algebras with operators, II. Am. J. Math. 74(1), 127–162 (1952)CrossRefMATHGoogle Scholar
  24. 24.
    Jung, A., Moshier, M.A., Vickers, S.: Presenting dcpos and dcpo algebras. In: Proceedings of the 24th Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIV), Electronic Notes in Theoretical Computer Science, vol. 218, pp. 209–229 (2008)Google Scholar
  25. 25.
    Pervin, W.J.: Quasi-uniformization of topological spaces. Math. Ann. 147, 316–317 (1962)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Stone, M.H.: The theory of representation for boolean algebras. Trans. Amer. Math. Soc. 74(1), 37–111 (1936)Google Scholar
  28. 28.
    Stone, M.H.: Topological representations of distributive lattices and Brouwerian logics. Čas. Mat. Fys. 67, 1–25 (1937)Google Scholar
  29. 29.
    Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8(1), 45–58 (1978)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.LIAFACNRS and Université Paris DiderotParis Cedex 13France
  2. 2.IMAPPRadboud Universiteit Nijmegen and LIAFA, Université Paris DiderotNijmegenThe Netherlands

Personalised recommendations