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Satoh, S.; Yama, K.; Tokizawa, M.(J-CHUOE)

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.

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