The semigroups of order 10

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. © 2012 Springer-Verlag.