Early on in "Nine Chapters on the Semigroup Art," by Alan J. Cain it is claimed that
there are $3{,}684{,}030{,}417$ different (non-isomorphic) semigroups with $8$ elements.
How was this number reached?
2026-03-28 07:37:14.1774683434
There are $3{,}684{,}030{,}417$ different semigroups of order $8$.
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You already found yourself an important reference. You may also read the following articles:
[1] Andreas Distler and Tom Kelsey. The monoids of orders eight, nine & ten. Ann. Math. Artif. Intell., 56(1):3–21, 2009.
[2] Andreas Distler and Tom Kelsey. The semigroups of order 9 and their automorphism groups, Semigroup Forum 88(1):93–112, 2014.
[3] S. Satoh, K. Yama, and M. Tokizawa. Semigroups of order 8. Semigroup Forum, 49(1):7–29, 1994.