Skip to main content
SpringerLink
Log in

Rigid Binary Relations on a 4-Element Domain

  • Published:
Order Aims and scope Submit manuscript

Abstract

An n-ary relation ρ on a set U is strongly rigid if it is preserved only by trivial operations. It is projective if the only idempotent operations in P o l ρ are projections. Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) characterized all strongly rigid relations on a set with two elements and found a strongly rigid binary relation on every domain U of at least 3 elements. Larose and Tardif (Mult.-Valued Log. 7(5-6), 339–362, 2001) studied the projective and strongly rigid graphs and constructed large families of strongly rigid graphs. Łuczak and Nešetřil (J. Graph Theory. 47, 81–86, 2004) settled in the affirmative a conjecture of Larose and Tardif that most graphs on a large set are projective, and characterized all homogenous graphs that are projective. Łuczak and Nešetřil (SIAM J. Comput. 36(3), 835–843, 2006) confirmed a conjecture of Rosenberg that most relations on a big set are strongly rigid. In this paper, we characterize all strongly rigid relations on a set with at least three elements to answer an open question by Rosenberg, (Rocky Mt. J. Math. 3, 631–639, 1973) and we classify the binary relations on the 4-element domain by rigidity and demonstrate that there are merely 40 pairwise nonisomorphic rigid binary relations on the same domain (among them 25 are pairwise nonisomorphic strongly rigid).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Post, E.L.: The two-valued iterative system of mathematical logic, Annals of Mathematical Studies 5, pp 1–122. Princeton University Press (1941)

  2. Swierczkowski, S.: Algebras which are independently generated by every n elements. Fund. Math. 49, 93–104 (1960)

    MathSciNet  MATH  Google Scholar 

  3. Geiger, D.: Closed systems of functions and predicates. Pacific J. Math. 27, 95–100

  4. Bodnarčuk, V.G., Kalužhnin, L.A., Kotov, V.N., Romov, B.A.: Galois theory for Post algebras I-II, Kibernetika, 3 (1969), pp. 1–10 and 5 (1969), pp. 1–9 (in Russian); Cybernetics, (1969), pp. 243–252, 531–539 (English version), 1969

  5. Rosenberg, I.G.: Strongly rigid relations. Rocky Mt. J. Math. 3, 631–639 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Schaefer, T.J.: The complexity of the satisfiability problems. Proc. 10th ACM Symp. Theory Comput. (STOC) 15, 216–226 (1978)

    MATH  Google Scholar 

  7. Csákány, B.: All minimal clones on the three element set. Acta Cybernet. 6, 227–238 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Rosenberg, I.G.: Minimal clones I: the five types, Lectures in universal algebra (Szeged, 1983). Colloquia Mathematical Society Janos Bolyai, Janos Bolyai, 43, North-Holland, Amsterdam, 1986, pp. 405–427

  9. Szendrei, Á.: Clones in Universal Algebra. Presses de l’Université de Montréal, Montreal (1986)

  10. Bang-Jensen, J., Hell, P.: The effect of two cycles on the complexity of colourings by directed graphs. Discret. Appl. Math. 26(1), 1–23 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Szczepara, B.: Minimal Clones Generated by Groupoids. Ph.D. Thesis, Université de Montréal, Montréal (1995)

  12. Fearnley, A.: A strongly rigid binary relation. Acta Sci. Math. (Szeged) 61, 35–41 (1995)

    MathSciNet  MATH  Google Scholar 

  13. Berman, J., Burris, S.: A computer study of 3-element groupoids. In: Logic and Algebra (Pontignano, 1994), Lecture Notes in Pure and Applied Mathematics, 180, pp 379–429, Dekker (1996)

  14. Feder, T., Vardi, M.Y.: The computational structure of monotone monadic SNP and constraint satisfaction: A study through Datalog and group theory. SIAM J. Comput. 28(1), 57–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jeavons, P.G.: On the algebraic structure of combinatorial problems. Theor. Comput. Sci. 200, 185–204 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bulatov, A., Jeavons, P., Krokhin, A.: Constraint satisfaction problems and finite algebras. In: Proceedings of 27th International Colloquim on Automata, Languages and Programming (ICALP00), vol. 1853, pp 272–282. Lecture Notes in Computer Science, Geneva, Switzerland (2000)

  17. Larose, B., Tardif, C.: Strongly rigid graphs and projectivity. Mult.-Valued Log. 7(5-6), 339–362 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Bulatov, A., Jeavons, P., Krokhin, A.: The complexity of maximal constraint languages. In: Proceedings of the 33rd Annual ACM Simposium on Theory of Computing, pp 667–674. ACM Press, Crete, Greece (2001)

  19. Łuczak, T., Nešetřil, J.: A note on projective graphs. J. Graph Theory 47, 81–86 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Csákány, B.: Minimal clones a minicourse. Algebra Univ. 54, 73–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Larose, B.: Taylor operations on finite reflexive structures. Int. J. Math. Comput. Sci. 1(1), 1–26 (2006)

    MathSciNet  MATH  Google Scholar 

  22. Łuczak, T., Nešetřil, J.: A probabilistic approach to the dichotomy problem. SIAM J. Comput. 36(3), 835–843 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Larose, B., Tesson, P.: Universal algebra and hardness results for constraint satisfaction problems. Theor. Comput. Sci. 410(18), 1629–1647 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Barto, L., Kozik, M., Niven, T.: The CSP dichotomy holds for digraphs with no sources and no sinks (a positive answer to a conjecture of Bang-Jensen and Hell). SIAM J. Comput. 38(5), 1782–1802 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Barto, L., Kozik, M.: Constraint satisfaction problems of bounded width. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pp 595–603 (2009)

  26. Barto, L., Stanovský, D.: Polymorphisms of small digraphs. Novi. Sad J. Math. 2(40), 95–109 (2010)

    MathSciNet  MATH  Google Scholar 

  27. Machida, H., Rosenberg, I.G. Centralizing Monoids on a Three-Element Set: 2012 IEEE 42nd International Symposium on Multiple-Valued Logic, pp 274–280

  28. Kazda, A.: Complexity of the homomorphism extension problem in the random case. Chic. J. Theor. Comput. Sci. 2013(9) (2013)

  29. Jovanović, J.: On optimal strong Mal’cev conditions for congruence meet-semidistributivity in a locally finite variety. Novi Sad J. Math. 44(2), 207–224 (2014)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques et de Statistique, Université de Montréal, Montréal, Québec, Canada

    Qinghe Sun

Authors
  1. Qinghe Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qinghe Sun.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sun, Q. Rigid Binary Relations on a 4-Element Domain. Order 34, 165–183 (2017). https://doi.org/10.1007/s11083-016-9394-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11083-016-9394-z

Keywords

Search

Navigation