, Volume 25, Issue 4, pp 299–320 | Cite as

Comparison of MacNeille, Canonical, and Profinite Completions

  • Guram Bezhanishvili
  • Jacob Vosmaer
Open Access


Using duality theory, we give necessary and sufficient conditions for the MacNeille, canonical, and profinite completions of distributive lattices, Heyting algebras, and Boolean algebras to be isomorphic.


MacNeille completion Canonical completion Profinite completion Duality theory 

Mathematics Subject Classifications (2000)

06B23 06D20 06D50 


  1. 1.
    Banaschewski, B., Bruns, G.: Categorical characterization of the MacNeille completion. Arch. Math. 18, 369–377 (1967)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bezhanishvili, G., Bezhanishvili, N.: Profinite Heyting algebras. Order 25, 211–227 (2008). Google Scholar
  3. 3.
    Bezhanishvili, G., Gehrke, M., Mines, R., Morandi, P.J.: Profinite completions and canonical extensions of Heyting algebras. Order 23, 143–161 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bezhanishvili, N.: Lattices of intermediate and cylindric modal logics. PhD thesis, University of Amsterdam (2006)Google Scholar
  5. 5.
    Erné, M.: Bigeneration in complete lattices and principal separation in ordered sets. Order 8(2), 197–221 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Erné, M.: The Dedekind-MacNeille completion as a reflector. Order 8(2), 159–173 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Esakia, L.L.: Topological Kripke models. Sov. Math., Dokl. 15, 147–151 (1974)zbMATHGoogle Scholar
  8. 8.
    Gehrke, M., Harding, J.: Bounded lattice expansions. J. Algebra 239, 345–371 (2001)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Gehrke, M., Jónsson, B.: Bounded distributive lattices with operators. Math. Jpn. 2, 207–215 (1994)Google Scholar
  11. 11.
    Goldblatt, R.: Varieties of complex algebras. Ann. Pure Appl. Logic 44, 173–242 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Harding, J.: On profinite completions and canonical extensions. Algebra Univers. 55, 293–296 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Harding, J., Bezhanishvili, G.: MacNeille completions of Heyting algebras. Houst. J. Math. 30, 937–952 (2004)zbMATHMathSciNetGoogle Scholar
  14. 14.
    Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73, 891–939 (1951)zbMATHCrossRefGoogle Scholar
  15. 15.
    McKenzie, R.: Equational bases and nonmodular lattice varieties. Trans. Am. Math. Soc. 174, 1–43 (1972)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Ono, H.: Algebraic semantics for predicate logics and their completeness. In: Logic at work. Stud. Fuzziness Soft Comput., vol. 24, pp. 637–650. Physica, Heidelberg (1999)Google Scholar
  17. 17.
    Ono, H.: Completions of algebras and completeness of modal and substructural logics. In: Advances in modal logic, vol. 4, pp. 335–353. King’s Coll. Publ., London (2003)Google Scholar
  18. 18.
    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rasiowa, H., Sikorski, R.: The Mathematics of Metamathematics. Monografie Matematyczne, Tom 41. Państwowe Wydawnictwo Naukowe, Warsaw (1963)Google Scholar
  20. 20.
    Vosmaer, J.: MacNeille completion and profinite completion can coincide on finitely generated modal algebras. Algebra Univers. (2008, in press)Google Scholar
  21. 21.
    Whitman, P.M.: Splittings of a lattice. Am. J. Math. 65, 179–196 (1943)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© The Author(s) 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA
  2. 2.Institute for Logic, Language and ComputationUniversity of AmsterdamAmsterdamThe Netherlands

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