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Semigroups of order 8
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Satoh, S.; Yama, K.; Tokizawa, M.(J-CHUOE)
Semigroups of order $8$.
(English. English summary)
Semigroup Forum 49 (1994), no. 1, 7--29.
Using the algorithm called the back-track method, the authors report
the construction of all non-equivalent (neither isomorphic nor
anti-isomorphic) semigroups of order 8. The back-track method has been
used by H. Jurgensen and P. Wick [Semigroup Forum 14
(1977), no. 1, 69--79; MR 56 #5764] and R. J. Plemmons [in
Computational problems in abstract algebra (Oxford, 1967),
223--228, Pergamon, Oxford, 1970; MR 41 #3639].
The results include showing that there are $1,843,120,128$
non-equivalent semigroups of order 8, about $99%$ of which are
nilpotent. A summary of semigroups of orders 2 through 8 is given,
specifying which are commutative, regular, inverse, and commutative
inverse. The semigroups of order 8 are classified by egg-box type,
that is, by their decompositions by Green's relations. Comparisons of
the numbers of semigroups of order 2 through 8 with theoretical
estimates are given.
Reviewed by Bernard L. Madison
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