SWAC computes 126 distinct semigroups of order 4
George E. Forsythe · Proceedings of the American Mathematical Society · 1955
and history.A semigroup is a set of elements closed under an associative multiplication.Two semigroups are here considered distinct if they are neither isomorphic nor anti-isomorphic.Let/(w) be the number of distinct semigroups of order n.Prior work [1; 5] shows that/(1)=1,/(2)=4,/(3)=18.For » = 4, Poole [3] lists 55 distinct commutative semigroups.Carman, Harden, and Posey [l ] correct some errors in [3 ], add 66 distinct noncommutative semigroups for n-4, and list all 121 semigroups.The present work achieves three results believed to be new: A. The total number of semigroups of order 4, including isomorphs and anti-isomorphs, is found to be 3492.Of these, 126 are distinct, and 58 are distinct and commutative.