Preorders on Monads and Coalgebraic Simulations

  • Shin-ya Katsumata
  • Tetsuya Sato
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7794)


We study the construction of preorders on Set-monads by the semantic ⊤ ⊤-lifting. We show the universal property of this construction, and characterise the class of preorders on a monad as a limit of a Card op -chain. We apply these theoretical results to identifying preorders on some concrete monads, including the powerset monad, maybe monad, and their composite monad. We also relate the construction of preorders and coalgebraic formulation of simulations.


Equivalence Class Partial Order Modal Logic Binary Relation Monotone Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Aczel, P., Mendler, N.: A Final Coalgebra Theorem. In: Dybjer, P., Pitts, A.M., Pitt, D.H., Poigné, A., Rydeheard, D.E. (eds.) Category Theory and Computer Science. LNCS, vol. 389, pp. 357–365. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  2. 2.
    Balan, A., Kurz, A.: Finitary Functors: From Set to Preord and Poset. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 85–99. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Barr, M.: Relational Algebras. In: MacLane, S., Applegate, H., Barr, M., Day, B., Dubuc, E., Phreilambud, Pultr, A., Street, R., Tierney, M., Swierczkowski, S. (eds.) Reports of the Midwest Category Seminar IV. LNM, vol. 137, pp. 39–55. Springer, Heidelberg (1970)CrossRefGoogle Scholar
  4. 4.
    Benton, N., Hughes, J., Moggi, E.: Monads and Effects. In: Barthe, G., Dybjer, P., Pinto, L., Saraiva, J. (eds.) APPSEM 2000. LNCS, vol. 2395, pp. 42–122. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (February 2009)Google Scholar
  6. 6.
    Bílková, M., Kurz, A., Petrisan, D., Velebil, J.: Relation liftings on preorders and posets. CoRR, abs/1210.1433 (2012)Google Scholar
  7. 7.
    Cîrstea, C.: A modular approach to defining and characterising notions of simulation. Information and Computation 204(4), 469–502 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Goubault-Larrecq, J., Lasota, S., Nowak, D.: Logical Relations for Monadic Types. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 553–568. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Hasuo, I.: Generic Forward and Backward Simulations. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 406–420. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Hasuo, I., Jacobs, B., Sokolova, A.: Generic trace theory. Electr. Notes Theor. Comput. Sci. 164(1), 47–65 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hermida, C., Jacobs, B.: An Algebraic View of Structural Induction. In: Pacholski, L., Tiuryn, J. (eds.) CSL 1994. LNCS, vol. 933, pp. 412–426. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  12. 12.
    Hermida, C., Jacobs, B.: Structural induction and coinduction in a fibrational setting. Inf. Comput. 145(2), 107–152 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Hesselink, W.H., Thijs, A.: Fixpoint semantics and simulation. Theor. Comput. Sci. 238(1-2), 275–311 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Jacobs, B.: Categorical Logic and Type Theory. Elsevier (1999)Google Scholar
  15. 15.
    Jacobs, B., Hughes, J.: Simulations in coalgebra. Electr. Notes Theor. Comput. Sci. 82(1), 128–149 (2003)CrossRefGoogle Scholar
  16. 16.
    Katsumata, S.: A Semantic Formulation of ⊤ ⊤-Lifting and Logical Predicates for Computational Metalanguage. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 87–102. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Klin, B.: The Least Fibred Lifting and the Expressivity of Coalgebraic Modal Logic. In: Fiadeiro, J.L., Harman, N.A., Roggenbach, M., Rutten, J. (eds.) CALCO 2005. LNCS, vol. 3629, pp. 247–262. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  18. 18.
    Kurz, A.: Logics for Coalgebras and Applications to Computer Science. PhD thesis, Ludwig-Maximilians-Universität, Munchen (2000)Google Scholar
  19. 19.
    Levy, P.: Boolean precongruences. Manuscript (2009)Google Scholar
  20. 20.
    Levy, P.: Similarity Quotients as Final Coalgebras. In: Hofmann, M. (ed.) FOSSACS 2011. LNCS, vol. 6604, pp. 27–41. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Lindley, S.: Normalisation by Evaluation in the Compilation of Typed Functional Programming Languages. PhD thesis, University of Edinburgh (2004)Google Scholar
  22. 22.
    Lindley, S., Stark, I.: Reducibility and ⊤ ⊤-Lifting for Computation Types. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 262–277. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  23. 23.
    Mitchell, J.: Foundations for Programming Languages. MIT Press (1996)Google Scholar
  24. 24.
    Pitts, A.: Parametric polymorphism and operational equivalence. Mathematical Structures in Computer Science 10(3), 321–359 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Staton, S.: Relating coalgebraic notions of bisimulation. Logical Methods in Computer Science 7(1) (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shin-ya Katsumata
    • 1
  • Tetsuya Sato
    • 1
  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan

Personalised recommendations