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Distributive Envelopes and Topological Duality for Lattices via Canonical Extensions

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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.

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References

  1. Birkhoff, G.: Lattice Theory, Colloquium Publications, vol. 25. American Mathematical Society (1967)

  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. Bruns, G., Lakser, H.: Injective hulls of semilattices. Canad. Math. Bull. 13(1), 115–118 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coumans, D.C.S.: Canonical Extensions in Logic: Some Applications and a Generalisation to Categories. Dissertation, Radboud University Nijmegen (2012)

  5. Császár, A.: D-completions of pervin-type quasi-uniformities. Acta. Sci. Math. (Szeged) 57, 329–335 (1993)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erné, M.: Ideal completions and compactifications. Appl. Cat. Struct. 9, 217–243 (2001)

    Article  MATH  Google Scholar 

  8. Erné, M.: Choiceless, pointless, but not useless: dualities for preframes. Appl. Cat. Struct. 15, 541–572 (2007)

    Article  MATH  Google Scholar 

  9. Erné, M., Palko, V.: Uniform ideal completions. Math. Slovaca. 48, 327–335 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Erné, M., Zhao, D.: Z-join spectra of Z-Supercompactly generated lattices. Appl. Cat. Struct. 9, 41–63 (2001)

    Article  MATH  Google Scholar 

  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. 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. 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. 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)

  15. Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 238(1), 345–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Japon. 40(2), 207–215 (1994)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  18. Goldblatt, R.: Maps and monads for modal frames. Stud. Logica. 83(1-3), 309–331 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harding, J.: Canonical completions of lattices and ortholattices. Tatra Mountains Math. Publ. 15, 85–96 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Hartung, G.: A topological representation of lattices. Algebra Univers. 29, 273–299 (1992). doi:10.1007/BF01.190610

    Article  MathSciNet  MATH  Google Scholar 

  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. Jónsson, B., Tarski, A.: Boolean algebras with operators, I. Amer. J. Math. 73(4), 891–939 (1951)

    Article  MATH  Google Scholar 

  23. Jónsson, B.: Boolean algebras with operators, II. Am. J. Math. 74(1), 127–162 (1952)

    Article  MATH  Google Scholar 

  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)

  25. Pervin, W.J.: Quasi-uniformization of topological spaces. Math. Ann. 147, 316–317 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  26. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  27. Stone, M.H.: The theory of representation for boolean algebras. Trans. Amer. Math. Soc. 74(1), 37–111 (1936)

    Google Scholar 

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

    Google Scholar 

  29. Urquhart, A.: A topological representation theory for lattices. Algebra Univers. 8(1), 45–58 (1978)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. LIAFA, CNRS and Université Paris Diderot, Case 7014, 75205, Paris Cedex 13, France

    Mai Gehrke

  2. IMAPP, Radboud Universiteit Nijmegen and LIAFA, Université Paris Diderot, P.O. Box 9010, 6500, GL, Nijmegen, The Netherlands

    Samuel J. van Gool

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  1. Mai Gehrke
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  2. Samuel J. van Gool
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Correspondence to Mai Gehrke.

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Gehrke, M., van Gool, S.J. Distributive Envelopes and Topological Duality for Lattices via Canonical Extensions. Order 31, 435–461 (2014). https://doi.org/10.1007/s11083-013-9311-7

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