The semigroups of order 10

Authors:
Andreas Distler;Chris Jefferson;Tom Kelsey;Lars Kotthoff
Affiliations:
Centro de Álgebra da Universidade de Lisboa, Lisboa, Portugal;School of Computer Science, University of St. Andrews, UK;School of Computer Science, University of St. Andrews, UK;School of Computer Science, University of St. Andrews, UK
Venue:
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
Year:
2012

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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.