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The Semigroups of Order 10

  • Andreas Distler
  • Chris Jefferson
  • Tom Kelsey
  • Lars Kotthoff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7514)

Abstract

The number of finite semigroups increases rapidly with the number of elements. Since existing counting formulae do not give the complete number of semigroups of given order up to equivalence, the remainder can only be found by careful search. We describe the use of mathematical results combined with distributed Constraint Satisfaction to show that the number of non-equivalent semigroups of order 10 is 12,418,001,077,381,302,684. This solves a previously open problem in Mathematics, and has directly led to improvements in Constraint Satisfaction technology.

Keywords

Constraint Satisfaction Mathematics semigroup Minion symmetry breaking distributed search 

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References

  1. 1.
    Amazon Elastic Compute Cloud, Amazon EC2 (2008), http://aws.amazon.com/ec2/
  2. 2.
    Bilous, R.T., Van Rees, G.H.J.: An enumeration of binary self-dual codes of length 32. Des. Codes Cryptography 26(1-3), 61–86 (2002), http://dx.doi.org/10.1023/A:1016544907275CrossRefzbMATHGoogle Scholar
  3. 3.
    Cohen, D., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M.: Symmetry Definitions for Constraint Satisfaction Problems. In: van Beek, P. (ed.) CP 2005. LNCS, vol. 3709, pp. 17–31. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A.: Symmetry-breaking predicates for search problems. In: Aiello, L.C., Doyle, J., Shapiro, S. (eds.) KR 1996: Principles of Knowledge Representation and Reasoning, pp. 148–159. Morgan Kaufmann, San Francisco (1996)Google Scholar
  5. 5.
    Distler, A.: Classification and Enumeration of Finite Semigroups. Shaker Verlag, Aachen (2010), also PhD thesis, University of St Andrews (2010), http://hdl.handle.net/10023/945
  6. 6.
    Distler, A., Kelsey, T.: The Monoids of Order Eight and Nine. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC/Calculemus/MKM 2008. LNCS (LNAI), vol. 5144, pp. 61–76. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Distler, A., Kelsey, T.: The monoids of orders eight, nine & ten. Ann. Math. Artif. Intell. 56(1), 3–21 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Distler, A., Mitchell, J.D.: The number of nilpotent semigroups of degree 3. Electron. J. Combin. 19(2), Research Paper 51 (2012) Google Scholar
  9. 9.
    Forsythe, G.E.: SWAC computes 126 distinct semigroups of order 4. Proc. Amer. Math. Soc. 6, 443–447 (1955)MathSciNetGoogle Scholar
  10. 10.
    Frisch, A.M., Hnich, B., Kiziltan, Z., Miguel, I., Walsh, T.: Propagation algorithms for lexicographic ordering constraints. Artificial Intelligence 170, 834 (2006)CrossRefMathSciNetGoogle Scholar
  11. 11.
    The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.4.12 (2008), http://www.gap-system.org
  12. 12.
    Gecode: Generic constraint development environment, http://www.gecode.org/
  13. 13.
    Gent, I.P., Jefferson, C., Miguel, I.: Minion: A fast scalable constraint solver. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) The European Conference on Artificial Intelligence 2006 (ECAI 2006), pp. 98–102. IOS Press (2006)Google Scholar
  14. 14.
    Gent, I.P., Jefferson, C., Miguel, I.: Watched Literals for Constraint Propagation in Minion. In: Benhamou, F. (ed.) CP 2006. LNCS, vol. 4204, pp. 182–197. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  15. 15.
    Howie, J.M.: Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series, vol. 12. The Clarendon Press, Oxford University Press, New York (1995), Oxford Science PublicationsGoogle Scholar
  16. 16.
    Jefferson, C.: Quicklex - a case study in implementing constraints with dynamic triggers. In: Proceedings of the ERCIM Workshop on Constraint Solving and Constraint Logic Programming, CSCLP 2011 (2011)Google Scholar
  17. 17.
    Jürgensen, H., Wick, P.: Die Halbgruppen der Ordnungen ≤ 7. Semigroup Forum 14(1), 69–79 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Katritzky, A., Hall, C., El-Gendy, B., Draghici, B.: Tautomerism in drug discovery. Journal of Computer-Aided Molecular Design 24, 475–484 (2010), http://dx.doi.org/10.1007/s10822-010-9359-z, doi:10.1007/s10822-010-9359-zCrossRefGoogle Scholar
  19. 19.
    Klee Jr., V.L.: The November meeting in Los Angeles. Bull. Amer. Math. Soc. 62(1), 13–23 (1956), http://dx.doi.org/10.1090/S0002-9904-1956-09973-2CrossRefMathSciNetGoogle Scholar
  20. 20.
    Kotthoff, L., Moore, N.C.: Distributed solving through model splitting. In: 3rd Workshop on Techniques for Implementing Constraint Programming Systems (TRICS), pp. 26–34 (2010)Google Scholar
  21. 21.
    Kreher, D., Stinson, D.: Combinatorial Algorithms: Generation, Enumeration, and Search. CRC Press (1998)Google Scholar
  22. 22.
    Lecoutre, C., Sais, L., Tabary, S., Vidal, V.: Nogood recording from restarts. In: Proceedings of the 20th International Joint Conference on Artifical Intelligence, pp. 131–136 (2007)Google Scholar
  23. 23.
    McKay, B.D.: Transitive graphs with fewer than twenty vertices. Math. Comp. 33(147), 1101–1121 (1979), contains microfiche supplementCrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    McKay, B.D., Royle, G.F.: The transitive graphs with at most 26 vertices. Ars Combin. 30, 161–176 (1990)zbMATHMathSciNetGoogle Scholar
  25. 25.
    Milletti, F., Storchi, L., Sforna, G., Cross, S., Cruciani, G.: Tautomer enumeration and stability prediction for virtual screening on large chemical databases. Journal of Chemical Information and Modeling 49(1), 68–75 (2009), http://pubs.acs.org/doi/abs/10.1021/ci800340jCrossRefGoogle Scholar
  26. 26.
    Motzkin, T.S., Selfridge, J.L.: Semigroups of order five. Presented in [19] (1955)Google Scholar
  27. 27.
    Petrie, K.E., Smith, B.M.: Symmetry Breaking in Graceful Graphs. In: Rossi, F. (ed.) CP 2003. LNCS, vol. 2833, pp. 930–934. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  28. 28.
    Plemmons, R.J.: There are 15973 semigroups of order 6. Math. Algorithms 2, 2–17 (1967)MathSciNetGoogle Scholar
  29. 29.
    Rossi, F., van Beek, P., Walsh, T.: Handbook of Constraint Programming (Foundations of Artificial Intelligence). Elsevier Science Inc., New York (2006)Google Scholar
  30. 30.
    Satoh, S., Yama, K., Tokizawa, M.: Semigroups of order 8. Semigroup Forum 49(1), 7–29 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Thain, D., Tannenbaum, T., Livny, M.: Distributed computing in practice: The Condor experience. Concurrency – Practice and Experience 17(2-4), 323–356 (2005)CrossRefGoogle Scholar
  32. 32.
    Yamanaka, K., Otachi, Y., Nakano, S.-I.: Efficient Enumeration of Ordered Trees with k Leaves (Extended Abstract). In: Das, S., Uehara, R. (eds.) WALCOM 2009. LNCS, vol. 5431, pp. 141–150. Springer, Heidelberg (2009), http://dx.doi.org/10.1007/978-3-642-00202-1_13CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Andreas Distler
    • 1
  • Chris Jefferson
    • 2
  • Tom Kelsey
    • 2
  • Lars Kotthoff
    • 2
  1. 1.Centro de Álgebra da Universidade de LisboaLisboaPortugal
  2. 2.School of Computer ScienceUniversity of St. AndrewsUK

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