, Volume 28, Issue 2, pp 227–249 | Cite as

Distributive Lattices with a Generalized Implication: Topological Duality

  • Jorge E. Castro
  • Sergio Arturo Celani
  • Ramon Jansana


In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.


Generalized implication Quasi-modal operator Annihilator Weakly Heyting algebras Priestley’s duality 

Mathematics Subject Classifications (2010)

03G10 06B30 06B15 54H10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Castro, J., Celani, S.A.: Quasi-modal lattices. Order 21, 107–129 (2004)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Celani, S.A.: Quasi-modal algebras. Math. Bohem. 126, 721–736 (2001)MATHMathSciNetGoogle Scholar
  3. 3.
    Celani, S.A., Jansana R.: A closer look at some subintuitionistic logics. Notre Dame J. Form. Log. 42, 225–255 (2003)MathSciNetGoogle Scholar
  4. 4.
    Celani, S.A., Jansana R.: Bounded distributive lattices with strict implication. Math. Log. Q. 51, 219–246 (2005)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Mandelker, M.: Relative annihilators in lattices. Duke Math. J. 37, 377–386 (1970)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 3, 507–530 (1972)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Jorge E. Castro
    • 1
  • Sergio Arturo Celani
    • 2
  • Ramon Jansana
    • 3
  1. 1.Universidad Nacional de San JuanSan JuanArgentina
  2. 2.CONICET and Departamento de MatemáticasUniversidad Nacional del CentroTandilArgentina
  3. 3.Dept. Lògica, Història i Filsofia de la CiènciaUniversitat de BarcelonaBarcleonaSpain

Personalised recommendations