Studia Logica

, Volume 100, Issue 1–2, pp 91–114 | Cite as

Frontal Operators in Weak Heyting Algebras



In this paper we shall introduce the variety FWHA of frontal weak Heyting algebras as a generalization of the frontal Heyting algebras introduced by Leo Esakia in [10]. A frontal operator in a weak Heyting algebra A is an expansive operator τ preserving finite meets which also satisfies the equation \({\tau(a) \leq b \vee (b \rightarrow a)}\), for all \({a, b \in A}\). These operators were studied from an algebraic, logical and topological point of view by Leo Esakia in [10]. We will study frontal operators in weak Heyting algebras and we will consider two examples of them. We will give a Priestley duality for the category of frontal weak Heyting algebras in terms of relational spaces \({\langle X, \leq, T, R \rangle}\) where \({\langle X, \leq, T \rangle}\) is a WH-space [6], and R is an additional binary relation used to interpret the modal operator. We will also study the WH-algebras with successor and the WH-algebras with gamma. For these varieties we will give two topological dualities. The first one is based on the representation given for the frontal weak Heyting algebras. The second one is based on certain particular classes of WH-spaces.


modal operators frontal operators weak Heyting algebras Priestley duality 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ardeshir M., Ruitenburg W.: ‘Basic Propositional Calculus I’. Mathematical Logic Quarterly 44, 317–343 (1998)CrossRefGoogle Scholar
  2. 2.
    Caicedo X., Cignoli R.: ‘An algebraic approach to intuitionistic connectives’. Journal of Symbolic Logic 66(4), 1620–1636 (2001)CrossRefGoogle Scholar
  3. 3.
    Castiglioni J.L., Sagastume M., San Martín H.J.: ‘On frontal Heyting algebras’. Reports on Mathematical Logic 45, 201–224 (2010)Google Scholar
  4. 4.
    Castiglioni J.L., San Martín H.J.: ‘On the variety of Heyting algebras with successor generated by all finite chains’. Reports on Mathematical Logic 45, 225–248 (2010)Google Scholar
  5. 5.
    Celani S. A., Jansana R.: ‘A closer look at some subintuitionistic logics’. Notre Dame Journal of Formal Logic 42, 225–255 (2003)Google Scholar
  6. 6.
    Celani S. A., Jansana R.: ‘Bounded distributive lattices with strict implication. Mathematical Logic Quarterly 51, 219–246 (2005)CrossRefGoogle Scholar
  7. 7.
    Celani, S. A., ‘Simple and subdirectly irreducibles bounded distributive lattices with unary operators’, International Journal of Mathematics and Mathematical Sciences, vol. 2006, Article ID 21835, 20 pages, doi: 10.1155/IJMMS/2006/21835, 2006.
  8. 8.
    Cignoli R., Lafalce S., Petrovich A.: ‘Remarks on Priestley duality for distributive lattices’. Order 8, 183–197 (1991)CrossRefGoogle Scholar
  9. 9.
    Epstein G., Horn A.: ‘Logics which are characterized by subresiduated lattices’. Zeitschrift fur Mathematiche Logik und Grundlagen der Mathematik 22, 199–210 (1976)CrossRefGoogle Scholar
  10. 10.
    Esakia L.: ‘The modalized Heyting calculus: a conservative modal extension of the Intuitionistic Logic’. Journal of Applied Non-Classical Logics 16(3-4), 349–366 (2006)CrossRefGoogle Scholar
  11. 11.
    Esakia L.: ‘Topological Kripke models’. Soviet Math. Dokl. 15, 147–151 (1974)Google Scholar
  12. 12.
    Esakia, L., Heyting Algebras I. Duality Theory (Russian), Metsniereba, Tbilisi, Georgia, 1985.Google Scholar
  13. 13.
    Goldblatt R.: ‘Varieties of complex algebras’. Annals Pure Appl. Logic 44, 173–242 (1989)CrossRefGoogle Scholar
  14. 14.
    Kuznetsov A.V.: ‘On the Propositional Calculus of Intuitionistic Provability’. Soviet Math. Dokl. 32, 18–21 (1985)Google Scholar
  15. 15.
    Kuznetsov A.V., Muravitsky A.Yu.: ‘On superintuitionistic logics as fragments of proof logic extensions’. Studia Logica 45(1), 77–99 (1986)CrossRefGoogle Scholar
  16. 16.
    Orlowska E., Rewitzky I.: ‘Discrete Dualities for Heyting algebras with Operators’. Fundamenta Informaticae 81, 275–295 (2007)Google Scholar
  17. 17.
    Priestley H.A.: ‘Representation of distributive lattices by means of ordered Stone spaces’. Bull. London Math Soc. 2, 186–190 (1970)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.CONICET and Departamento de Matemática, Facultad de Ciencias ExactasUniversidad Nacional del CentroTandilArgentina
  2. 2.CONICET and Departamento de Matemática, Facultad de Ciencias ExactasUniversidad Nacional de La PlataLa PlataArgentina

Personalised recommendations