The monoids of orders eight, nine & ten

Authors:
Andreas Distler;Tom Kelsey
Affiliations:
School of Mathematics and Statistics, Mathematical Institute, University of St Andrews, St Andrews, UK KY16 9SS;School of Computer Science, Jack Cole Building, University of St Andrews, St Andrews, UK KY16 9SX
Venue:
Annals of Mathematics and Artificial Intelligence
Year:
2009

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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.