Skip to main content
SpringerLink
Log in

Hilbert algebras with supremum

  • Published:
Algebra universalis Aims and scope Submit manuscript

Abstract

In this paper, we will study the class of Hilbert algebras with supremum, i.e., Hilbert algebras where the associated order is a join-semilattice. First, we will give a simplified topological duality for Hilbert algebras using sober topological spaces with a basis of open-compact sets satisfying an additional condition. Next, we will extend this duality to Hilbert algebras with supremum. We shall prove that the ordered set of all ideals of a Hilbert algebra with supremum has a lattice structure. We will also see that in this lattice, it is possible to define an implication, but the resulting structure is neither a Heyting algebra nor an implicative semilattice. Finally, we will give a dual description of the lattice of ideals of a Hilbert algebra with supremum.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Balbes, R., Dwinger, Ph.: Distributive Lattices. University of Missouri Press (1974)

  2. Celani S.A.: A note on homomorphisms of Hilbert algebras. Int. J. Math. Math. Sci 29, 55–61 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Celani S.A., Cabrer L.M.: Duality for finite Hilbert algebras. Discrete Math 305, 74–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Celani S.A., Cabrer L.M.: Topological duality for Tarski algebras. Algebra Universalis 58, 73–94 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Celani S.A., Cabrer L.M., Montangie D.: Representation and duality for Hilbert algebras. Cent. Eur. J. Math 7, 463–478 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chajda, I., Halas, R., Kühr, J.: Semilattice Structures. Research and Exposition in Mathematics, vol. 30. Heldermann (2007)

  7. Diego, A.: Sur les algèbres de Hilbert. Colléction de Logique Mathèmatique, ser. A, fasc. 21. Gouthier-Villars, Paris (1966)

  8. Figallo A.V., Ramón G., Saad S.: A note on Hilbert algebras with infimum. Mat. Contemp 24, 23–37 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Idziak P.M.: Lattice operations in BCK-algebras. Math. Japon 29, 839–846 (1984)

    MathSciNet  MATH  Google Scholar 

  10. Monteiro, A.: Sur les algèbres de Heyting symétriques. Portugal. Math. 39 (1980)

Download references

Author information

Authors and Affiliations

  1. CONICET and Departamento de Matemáticas, Facultad de Ciencias Exactas, Univ. Nac. del Centro, Pinto 399, 7000, Tandil, Argentina

    Sergio A. Celani

  2. Facultad de Economía y Administración, Departamento de Matemáticas, Universidad Nacional del Comahue, Buenos Aires, 1400 8300, Neuquén-Argentina

    Daniela Montangie

Authors
  1. Sergio A. Celani
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniela Montangie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sergio A. Celani.

Additional information

Presented by M. Ploscica.

The research of the first author was supported by the CONICET under grant no. 112-200801-02543.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Celani, S.A., Montangie, D. Hilbert algebras with supremum. Algebra Univers. 67, 237–255 (2012). https://doi.org/10.1007/s00012-012-0178-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00012-012-0178-z

2010 Mathematics Subject Classification

Keywords and phrases

Search

Navigation