Skip to main content
Springer Nature Link
Log in

Discrete Duality for Relation Algebras and Cylindric Algebras

  • Conference paper
Relations and Kleene Algebra in Computer Science (RelMiCS 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5827))

Included in the following conference series:

  • 464 Accesses

Abstract

Following the representation theorems for relation algebras and cylindric algebras presented in [5] and [7] we develop discrete duality for relation algebras and relation frames, and for cylindric algebras and cylindric frames.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Chin, L., Tarski, A.: Distributive and modular laws in the arithmetic of relation algebras. University of California Publications (1951)

    Google Scholar 

  2. De Morgan, A.: On the syllogism: IV, and on the logic of relations. Transactions of the Cambridge Philosophical Society 10, 331–358 (1864)

    Google Scholar 

  3. De Rijke, M., Venema, Y.: Salqvists theorem for Boolean algebras with operators with applications to cylindric algebras. Studia Logica 54, 61–78 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. Düntsch, I., Orłowska, E.: A discrete duality between the apartness algebras and apartness frames. Journal of Applied Non-classical Logics 18(2-3), 209–223 (2008)

    Article  Google Scholar 

  5. Düntsch, I., Orłowska, E., Radzikowska, A.: Lattice-based relation algebras II. In: de Swart, H., Orłowska, E., Schmidt, G., Roubens, M. (eds.) TARSKI 2006. LNCS (LNAI), vol. 4342, pp. 267–289. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  6. Dzik, W., Orłowska, E., van Alten, C.: Relational representation theorems for general lattices with negations. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 162–176. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  7. 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 

  8. Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras. Part I, Part II. North Holland, Amsterdam (1971/1985)

    Google Scholar 

  9. Järvinen, J., Orłowska, E.: Relational correspondences for lattices with operators. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 134–146. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  10. Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I: American Journal of Mathematics 73, 891–939 (1951), Part II: ibidem 74, 127–162 (1952)

    Article  MATH  Google Scholar 

  11. Maddux, R.: Some varieties containing relation algebras. Transactions of the American Mathematical Society 272, 501–526 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Maddux, R.: Finite Integral Relation Algebras. Lecture Notes in Mathematics, vol. 1149, pp. 175–197 (1985)

    Google Scholar 

  13. Maddux, R.: Introductory course on relation algebras, finite-dimensional cylindric algebras, and their interconnections. In: Andreka, H., Monk, J.D., Nemeti, I. (eds.) Algebraic Logic (Proc. Conf. Budapest 1988). Colloq. Math. Soc. J. Bolyai, vol. 54, pp. 361–392. North-Holland, Amsterdam (1991)

    Google Scholar 

  14. Maddux, R.: Relation algebras. In: Abramsky, S., Artemov, S., Gabbay, D.M., et al. (eds.). Studies in Logic and the Foundations of Mathematics, vol. 150. Elsevier, Amsterdam (1996)

    Google Scholar 

  15. Maddux, R.: Relation algebras. In: Brink, C., Kahland, W., Schmidt, G. (eds.) Relational Methods in Computer Sciences. Advances in Computer Science. Springer, New York (1997)

    Google Scholar 

  16. Maksimova, L.L.: Pretabular superintuitionistic logics. Algebra and Logic 11(5), 558–570 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Maksimova, L.L.: Pretabular extensions of the Lewis’ logic S4. Algebra and Logic 14(1), 28–55 (1975)

    Article  MathSciNet  Google Scholar 

  18. Orłowska, E., Golinska-Pilarek, J.: Dual Tableaux: Foundations, Methodolody, Case Studies, Draft of the book (2009)

    Google Scholar 

  19. Orłowska, E., Rewitzky, I., Düntsch, I.: Relational semantics through duality. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 17–32. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  20. Orłowska, E., Radzikowska, A.: Relational representability for algebras of substructural logics. In: MacCaull, W., Winter, M., Düntsch, I. (eds.) RelMiCS 2005. LNCS, vol. 3929, pp. 212–224. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  21. Orłowska, E., Radzikowska, A.: Representation theorems for some fuzzy logics based on residuated non-distributive lattices. Fuzzy Sets and Systems 159, 1247–1259 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  22. Orłowska, E., Rewitzky, I.: Duality via Truth: Semantic frameworks for lattice-based logics. Logic Journal of the IGPL 13(4), 467–490 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Orłowska, E., Rewitzky, I.: Context algebras, context frames and their discrete duality. In: Peters, J.F., Skowron, A., Rybiński, H. (eds.) Transactions on Rough Sets IX. LNCS, vol. 5390, pp. 212–229. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  24. Orłowska, E., Rewitzky, I.: Discrete duality and its applications to reasoning with incomplete information. In: Kryszkiewicz, M., Peters, J.F., Rybiński, H., Skowron, A. (eds.) RSEISP 2007. LNCS (LNAI), vol. 4585, pp. 51–56. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  25. Orłowska, E., Rewitzky, I.: Algebras for Galois-style connections and their discrete duality (submitted, 2008)

    Google Scholar 

  26. Peirce, C.S.: Note B: the logic of relatives. In: Peirce, C.S. (ed.) Studies in Logic by Members of the Johns Hopkins University, pp. 187–203. Little, Brown, and Co., Boston (1883)

    Google Scholar 

  27. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bulletin of the London Mathematical Society 2, 186–190 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  28. Sahlqvist, H.: Completeness and correspondence in the first and second order semantics for modal logics. In: Kanger, S. (ed.) 3rd Skandinavian Logic Symposium, Uppsala, Sweden, 1973, pp. 110–143. North-Holland, Amsterdam (1975)

    Chapter  Google Scholar 

  29. Stone, M.H.: The theory of representations for Boolean algebras. Transactions of the American Mathematical Society 40, 37–111 (1936)

    Article  MATH  MathSciNet  Google Scholar 

  30. Tarski, A.: On the calculus of relations. Journal of Symbolic Logic 6, 73–89 (1941)

    Google Scholar 

  31. Urquhart, A.: Duality for algebras of relevant logics. Studia Logica 56, 263–276 (1996)

    Google Scholar 

  32. Vakarelov, D., Orłowska, E.: Lattice-based modal algebras and modal logics. In: Hajek, P., Valds-Villanueva, L.M., Westerstahl, D. (eds.) Logic, Methodology and Philosophy of Science. Proceedings of the 12th International Congress, pp. 147–170. Kings College London Publications (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Institute of Telecommunications, Warsaw

    Ewa Orłowska

  2. Department of Mathematical Sciences, University of Stellenbosch, South Africa

    Ingrid Rewitzky

Authors
  1. Ewa Orłowska
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ingrid Rewitzky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Christian-Albrechts-Universität zu Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Rudolf Berghammer

  2. College of Engineering, Qatar University, P.O. Box 2713, CSE 209, Doha, Qatar,  

    Ali Mohamed Jaoua

  3. Institut für Informatik, Universität Augsburg, Universitätsstr. 6a, 86135, Augsburg, Germany

    Bernhard Möller

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Orłowska, E., Rewitzky, I. (2009). Discrete Duality for Relation Algebras and Cylindric Algebras. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2009. Lecture Notes in Computer Science, vol 5827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04639-1_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-04639-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-04638-4

  • Online ISBN: 978-3-642-04639-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics