Algebra universalis

, Volume 77, Issue 4, pp 375–393 | Cite as

Kleene algebras with implication

  • José Luis CastiglioniEmail author
  • Sergio Arturo Celani
  • Hernán Javier San Martín


Inspired by an old construction due to J. Kalman that relates distributive lattices and centered Kleene algebras, in this paper we study an equivalence for certain categories whose objects are algebras with implication \({(H, \bigwedge, \bigvee, \rightarrow, 0,1)}\) which satisfy the following property for every \({a,b,c\, \in\, H}\): if \({a \leq b \rightarrow c}\), then \({a \bigwedge b \leq c}\).

Key words and phrases

involutive distributive lattices centered Kleene algebras lattices with implication 

2010 Mathematics Subject Classification

Primary: 06D99 Secondary: 06D05 08A30 08A62 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balbes, R., Dwinger P.: Distributive Lattices. University of Missouri Press (1974)Google Scholar
  2. 2.
    Bezhanishvili N., Gehrke M.: Finitely generated free Heyting algebras via Birkhoff duality and coalgebra. Logical Methods in Computer Science 7, 1–24 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Busaniche M., Cignoli R.: Constructive Logic with Strong Negation as a Substructural Logic. Journal of Logic and Computation 20, 761–793 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Castiglioni J.L., Menni M., Sagastume M.: On some categories of involutive centered residuated lattices. Studia Logica 90, 93–124 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Castiglioni J.L., Lewin R., Sagastume M.: On a definition of a variety of monadic l-groups. Studia Logica 102, 67–92 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Celani S.: Bounded distributive lattices with fusion and implication. Southeast Asian Bull. Math. 27, 1–10 (2003)MathSciNetGoogle Scholar
  7. 7.
    Celani S., Jansana R.: Bounded distributive lattices with strict implication. Mathematical Logic Quarterly 51, 219–246 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cignoli R.: The class of Kleene algebras satisfying an interpolation property and Nelson algebras. Algebra Universalis 23, 262–292 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Epstein G., Horn A.: Logics which are characterized by subresiduated lattices. Z. Math. Logik Grundlagen Math. 22, 199–210 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fidel M. M.: An algebraic study of a propositional system of Nelson. In: Arruda, A.I., Da Costa, N.C.A., Chuaqui, R. (eds.) Mathematical Logic. Proceedings of the First Brazilian Conference. Lectures in Pure and Applied Mathematics, vol. 39, pp. 99–117. Marcel Dekker, New York (1978)Google Scholar
  11. 11.
    Kalman J.A.: Lattices with involution. Trans. Amer. Math. Soc. 87, 485–491 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Sagastume M.: Kalman’s construction (2007, preprint).Google Scholar
  13. 13.
    Spinks M., Veroff R.: Constructive logic with strong negation is a substructural logic. I. Studia Logica 88, 325–348 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Vakarelov D.: Notes on \({\mathcal{N}}\)-lattices and constructive logic with strong negation. Studia Logica 36, 109–125 (1977)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  • José Luis Castiglioni
    • 1
    Email author
  • Sergio Arturo Celani
    • 2
  • Hernán Javier San Martín
    • 1
  1. 1.Departamento de Matemática, Facultad de Ciencias ExactasUNLP and CONICETLa PlataArgentina
  2. 2.Departamento de Matemática, Facultad de Ciencias ExactasUNCPBA and CONICETTandilArgentina

Personalised recommendations