Algebra universalis

, Volume 78, Issue 2, pp 193–213 | Cite as

Spectral-like duality for distributive Hilbert algebras with infimum

  • Sergio A. CelaniEmail author
  • María Esteban


Distributive Hilbert algebras with infimum, or DH^-algebras for short, are algebras with implication and conjunction, in which the implication and the conjunction do not necessarily satisfy the residuation law. These algebras do not fall under the scope of the usual duality theory for lattice expansions, precisely because they lack residuation. We propose a new approach, that consists of regarding the conjunction as the additional operation on the underlying implicative structure. In this paper, we introduce a class of spaces, based on compactly-based sober topological spaces. We prove that the category of these spaces and certain relations is dually equivalent to the category of DH^-algebras and \({\wedge}\)-semi-homomorphisms. We show that the restriction of this duality to a wide subcategory of spaces gives us a duality for the category of DH^-algebras and algebraic homomorphisms. This last duality generalizes the one given by the author in 2003 for implicative semilattices. Moreover, we use the duality to give a dual characterization of the main classes of filters for DH^-algebras, namely, (irreducible) meet filters, (irreducible) implicative filters and absorbent filters.

Key words and phrases

Hilbert algebras topological representation spectral spaces distributive meet-semilattices 

2000 Mathematics Subject Classification

Primary: 03G25 Secondary: 06F35 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Banaschewski B., Erné M.: On Krull’s separation lemma. Order 10, 253–260 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bezhanishvili G., Jansana R.: Priestley style duality for distributive meet-semilattices. Studia Logica 98, 83–123 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Busneag D., Ghita M.: Some latticial properties of Hilbert algebras. Bull. Math. Soc. Sci. Math. Roumanie 53, 87–107 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Celani S.A.: A note on homomorphisms of Hilbert algebras. International Journal of Mathematics and Mathematical Sciences 29, 55–61 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Celani S.A.: Representation of Hilbert algebras and implicative semilattices. Cent. Eur. J. Math. 4, 561–572 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Celani S.A.: Topological representation of distributive semilattices. Scientiae Mathematicae Japonicae 8, 41–51 (2003)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Celani S.A., Cabrer L.M., Montangie D.: Representation and duality for Hilbert algebras. Cent. Eur. J. Math. 7, 463–478 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Celani S.A., Calomino I.: Some remarks on distributive semilattices. Commentationes Mathematicae Universitatis Carolinae 54, 407–428 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Celani S.A., Montangie D.: Hilbert algebras with supremum. Algebra Universalis 67, 237–255 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chajda, I., Halaš, R., Kühr, J.: Semilattice Structures, volume 30 of Research and Exposition in Mathematics. Heldermann Verlag (2007)Google Scholar
  11. 11.
    David E., Erné M.: Ideal completion and Stone representation of ideal-distributive ordered sets. Topology and its Applications 44, 95–113 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Diego, A.: Sur les algèbres de Hilbert, volume 21 of A. Gouthier-Villars, Paris, (1966)Google Scholar
  13. 13.
    Erné, M.: Algebraic ordered sets and their generalizations, in: Rosenberg, I. and Sabidussi, G. (eds.), Algebras and Orders, pp. 113–192, Kluwer, Amsterdam, (1994)Google Scholar
  14. 14.
    Erné M.: Minimal bases, ideal extensions, and basic dualities. Topology Proc. 29, 445–489 (2005)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Erné M.: Algebraic models for T 1-spaces. Topology and its Applications 158, 945–962 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Figallo, A. Jr., Ramón, G., Saad, S.: iH-propositional calculus. Bulletin of the Section of Logic 35, 157–162, (2006)Google Scholar
  17. 17.
    Figallo, A.V., Ramón, G.Z., Saad, S.: A note on the Hilbert algebras with infimum. 8th Workshop on Logic, Language, Informations and Computation, WoLLIC’2001 (Brasília), Mat. Contemp. 24, 23–37 (2003)Google Scholar
  18. 18.
    Gehrke M., Priestley H.A.: Duality for double Quasioperator Algebra via their Canonical Extensions. Studia Logica 86, 31–68 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Goldblatt G.: Varieties of complex algebras. Ann. Pure App. Logic 44, 173–242 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Grätzer, G.: General lattice theory. Academic Press, Birkhäuser, (1978)Google Scholar
  21. 21.
    Hofmann, K.H., Watkins, F.: The spectrum as a functor. In: R.E. Hoffmann and K.H. Hofmann (Eds.), Continuous Lattices. Lecture Notes in Math. 871, pp. 249–263. Springer, Berlin (1981)Google Scholar
  22. 22.
    Idziak P.M.: Lattice operations in BCK-algebras. Mathematica Japonica 29, 839–846 (1984)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Jónsson B., Tarski A.: Boolean algebras with operators. part I. Amer. J. of Math. 73, 891–939 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Monteiro, A.: Sur les algèbres de Heyting symétriques. Portugaliae Mathematica 39, (1980)Google Scholar
  25. 25.
    Priestley H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. London Math. Soc. 2, 186–190 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Stone M.H.: Topological representation of distributive lattices and Brouwerian logics. Časopis pešt. mat. fys. 67, 1–25 (1937)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CONICET and Departamento de Matemáticas, Facultad de Ciencias Exactas. Univ. Nac. del Centro.TandilArgentina
  2. 2.Departament de Lógica, História i Filosofia de la Ciéncia, Facultat de FilosofíaUniversitat de Barcelona (UB).BarcelonaSpain

Personalised recommendations