The number of commutative semigroups of order \(n\) (Q1891651)

An explicit upper bound is given for the number of commutative semigroups of \(n\) elements. The basic principle of the computation is based on Ponizovskij's theorem: A finite commutative semigroup \(S\) is a subdirect product of a group, a nilsemigroup and \(q - 1\) elementary semigroups in which \(q (\geq 1)\) is the number of idempotents of \(S\). By an elementary semigroup we mean a commutative semigroup \(S\) with identity element and zero such that \(S\) is the disjoint union \(G \cup N\) of the group of units \(G\) and a nilsemigroup \(N\) which is an ideal of \(S\). The result is still large and complicated.