The Number of Nilpotent Semigroups of Degree 3

Abstract

A semigroup is nilpotent of degree $3$ if it has a zero, every product of $3$ elements equals the zero, and some product of $2$ elements is non-zero. It is part of the folklore of semigroup theory that almost all finite semigroups are nilpotent of degree $3$.

We give formulae for the number of nilpotent semigroups of degree $3$ on a set with $n\in\mathbb{N}$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups on a set with $n$ elements up to equality and up to isomorphism.