The monoids of orders eight, nine & ten

  • Andreas DistlerEmail author
  • Tom Kelsey


We describe the use of symbolic algebraic computation allied with AI search techniques, applied to the problem of the identification, enumeration and storage of all monoids of order ten or less. Our approach is novel, using computer algebra to break symmetry and constraint satisfaction search to find candidate solutions. We present new results in algebraic combinatorics: up to isomorphism and anti-isomorphism, there are 858,977 monoids of order eight; 1,844,075,697 monoids of order nine and 52,991,253,973,742 monoids of order ten.


Monoids Enumeration combinatorics AI search 

Mathematics Subject Classifications (2000)

Primary 05E15 Secondary 05-04 


  1. 1.
    Sloane, N.J.A.: The on-line encyclopedia of integer sequences. (2008)
  2. 2.
    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.) Intelligent Computer Mathematics, 9th International Conference, AISC 2008, Proceedings. Lecture Notes in Artificial Intelligence, vol. 5144, pp. 61–76. Springer, New York (2008)Google Scholar
  3. 3.
    The GAP Group: GAP—Groups, Algorithms, and Programming, version 4.4.10 (2007)Google Scholar
  4. 4.
    Linton, S.: Finding the smallest image of a set. In: ISSAC ’04: Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, pp. 229–234. ACM, New York (2004)CrossRefGoogle Scholar
  5. 5.
    Gent, I.P., Jefferson, C., Miguel, I.: Minion: a fast scalable constraint solver. In: Brewka, G., Coradeschi, S., Perini, A., Traverso, P. (eds.) ECAI, pp. 98–102. IOS, Amsterdam (2006)Google Scholar
  6. 6.
    Cohen, D.A., Jeavons, P., Jefferson, C., Petrie, K.E., Smith, B.M.: Symmetry definitions for constraint satisfaction problems. Constraints 11(2–3), 115–137 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Crawford, J.M., Ginsberg, M.L., Luks, E.M., Roy, A.: Symmetry-breaking predicates for search problems. In: Proceedings of the International Conference on Principles of Knowledge Representation and Reasoning, pp. 148–159 (1996)Google Scholar
  8. 8.
    Grillet, P.: Semigroups: An Introduction to the Structure Theory. Marcel Dekker, New York (1995)Google Scholar
  9. 9.
    Plemmons, R.J.: There are 15973 semigroups of order 6. Math. Algorithms 2, 2–17 (1967)MathSciNetGoogle Scholar
  10. 10.
    Satoh, S., Yama, K., Tokizawa, M.: Semigroups of order 8. Semigroup Forum 49, 7–29 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Harary, F., Palmer, E.M.: Graphical Enumeration. Academic, New York (1973)zbMATHGoogle Scholar
  12. 12.
    Distler, A., Mitchell, J.D.: smallsemi – a GAP package. (2008)
  13. 13.
    Jefferson, C., Kelsey, T., Linton, S., Petrie, K.: Gaplex: generalised static symmetry breaking. In: Benhamou, F., Jussien, N., O’Sullivan, B. (eds.) Trends in Constraint Programming. ISTE, pp. 191–205 (2007)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Mathematical InstituteUniversity of St AndrewsSt AndrewsUK
  2. 2.School of Computer Science, Jack Cole BuildingUniversity of St AndrewsSt AndrewsUK

Personalised recommendations