The semigroups of order 9 and their automorphism groups

TL;DR

This paper provides exact counts of semigroups of order 9 up to isomorphism and anti-isomorphism, details their automorphism groups, and computes the number of distinct associative binary operations on sets of sizes 8 and 9.

Contribution

It offers the first comprehensive enumeration of semigroups of order 9 and analyzes their automorphism groups, using computational methods and recent formulae.

Findings

Number of semigroups of order 9 up to isomorphism: 105,978,177,936,292

Number of semigroups of order 9 up to anti-isomorphism: 52,989,400,714,478

Count of automorphism groups for semigroups of size up to 9

Abstract

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.