Abstract
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.
Similar content being viewed by others
References
Sloane, N.J.A.: The on-line encyclopedia of integer sequences. http://www.research.att.com/~njas/sequences/Seis.html (2008)
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)
The GAP Group: GAP—Groups, Algorithms, and Programming, version 4.4.10 (2007)
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)
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)
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)
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)
Grillet, P.: Semigroups: An Introduction to the Structure Theory. Marcel Dekker, New York (1995)
Plemmons, R.J.: There are 15973 semigroups of order 6. Math. Algorithms 2, 2–17 (1967)
Satoh, S., Yama, K., Tokizawa, M.: Semigroups of order 8. Semigroup Forum 49, 7–29 (1994)
Harary, F., Palmer, E.M.: Graphical Enumeration. Academic, New York (1973)
Distler, A., Mitchell, J.D.: smallsemi – a GAP package. http://turnbull.mcs.st-and.ac.uk/~jamesm/smallsemi/ (2008)
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)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Distler, A., Kelsey, T. The monoids of orders eight, nine & ten. Ann Math Artif Intell 56, 3–21 (2009). https://doi.org/10.1007/s10472-009-9140-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10472-009-9140-y