Relational Correspondences for Lattices with Operators

  • Jouni Järvinen
  • Ewa Orłowska
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3929)


In this paper we present some examples of relational correspondences for not necessarily distributive lattices with modal-like operators of possibility (normal and additive operators) and sufficiency (co-normal and co-additive operators). Each of the algebras (P, ∨ , ∧ , 0, 1, f), where (shape P, ∨, ∧, 0,1) is a bounded lattice and shape f is a unary operator on shape P, determines a relational system (frame) \((X(P), \lesssim_1, \lesssim_2, R_f, S_f)\) with binary relations \(\lesssim_1\), \(\lesssim_2\), shape R f , shape S f , appropriately defined from shape P and shape f. Similarly, any frame of the form \((X, \lesssim_1, \lesssim_2, R, S)\) with two quasi-orders \(\lesssim_1\) and \(\lesssim_2\), and two binary relations shape R and shape S induces an algebra (shape L(shape X), ∨, ∧, 0,1, shape f R,S ), where the operations ∨, ∧, and shape f R,S and constants 0 and 1 are defined from the resources of the frame. We investigate, on the one hand, how properties of an operator shape f in an algebra shape P correspond to the properties of relations shape R f and shape S f in the induced frame and, on the other hand, how properties of relations in a frame relate to the properties of the operator shape f R,S of an induced algebra. The general observations and the examples of correspondences presented in this paper are a first step towards development of a correspondence theory for lattices with operators.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ackermann, W.: Untersuchungen uber des Eliminationsproblem der mathematischen Logik. Mathematische Annalen 110, 390–413 (1935)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Demri, S., Orłowska, E.: Incomplete Information: Structure, Inference, Complexity. EATCS Monographs in Theoretical Computer Science. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Düntsch, I., Orłowska, E.: Beyond modalities: sufficiency and mixed algebras. In: Orłowska, E., Szałas, A. (eds.) Relational Methods for Computer Science Applications. Studies in Fuzziness and Soft Computing, vol. 65, pp. 263–285. Physica-Verlag, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Düntsch, I., Orłowska, E., Radzikowska, A., Vakarelov, D.: Relational representation theorems for some lattice-based structures. Journal of Relational Methods in Computer Science 1, 132–160 (2005)Google Scholar
  5. 5.
    Nonnengart, A., Szałas, A.: A fixpoint approach to second-order quantifier elimination with applications to correspondence theory. In: Orłowska, E. (ed.) Logic at Work. Essays dedicated to the memory of Helena Rasiowa, pp. 89–108. Physica-Verlag, Heidelberg (1999)Google Scholar
  6. 6.
    Orłowska, E., Vakarelov, D.: Lattice-based modal algebras and modal logics. In: Hájek, P., Valdes-Villanueva, L., Westerstahl, D. (eds.) Proceedings of the 12th International Congress on Logic, Methodology and Philosophy of Science, pp. 147–170. KCL Publications, London (2005)Google Scholar
  7. 7.
    Orłowska, E., Rewitzky, I.: Duality via truth: semantic frameworks for lattice-based logics. Logic Journal of the IGPL 13, 467–490 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Orłowska, E.: Relational semantics through duality. In: Düntsch, I., Winter, M. (eds.) Proceedings of the Eighth International Conference on Relational Methods in Computer Science (RelMiCS), vol. 8, p. 187. Department of Computer Science, Brock University (2005)Google Scholar
  9. 9.
    Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logics. In: Kanger, S. (ed.) Third Scandinavian Logic Symposium, Uppsala, Sweden, 1973, pp. 110–143. North Holland, Amsterdam (1975)Google Scholar
  10. 10.
    Szałas, A.: On the correspondence between modal and classical logic: an automated approach. Journal of Logic and Computation 3, 605–620 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Urquhart, A.: A topological representation theory for lattices. Algebra Universalis 8, 45–58 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    van Benthem, J.: Correspondence theory. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. II, pp. 167–247. Reidel, Dordrecht (1984)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Jouni Järvinen
    • 1
  • Ewa Orłowska
    • 2
  1. 1.Turku Centre for Computer ScienceTurkuFinland
  2. 2.National Institute of TelecommunicationsWarszawaPoland

Personalised recommendations