The semigroups of order 9 and their automorphism groups

Andreas Distler, Tom Kelsey

Published 2013-01-25Version 1

We report the number of semigroups with 9 elements up to isomorphism or anti-isomorphism to be 52,989,400,714,478 and up to isomorphism to be 105,978,177,936,292. We obtained these results by combining computer search with recently published formulae for the number of nilpotent semigroups of degree 3. We further provide a complete account of the automorphism groups of the semigroups with at most 9 elements. We use this information to deduce that there are 148,195,347,518,186 distinct associative binary operations on an 8-element set and 38,447,365,355,811,944,462 on a 9-element set.