SWAC computes 126 distinct semigroups of order 4

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.Thus/(4) turns out to be 126.B. The author has on punched cards multiplication tables for all 3492 semigroups of order 4.C. The table below includes multiplication tables for all 126 distinct semigroups of order 4.The table verifies that all 121 semigroups of [l] are correct, and adds five new ones.After the list was prepared, Professor Takayuki Tamura wrote Professor Hewitt that he also has prepared a list of distinct semigroups of order 4.2