The number of nilpotent semigroups of degree 3

TL;DR

This paper derives formulas to count nilpotent semigroups of degree 3, including commutative cases, providing a comprehensive enumeration up to various equivalences.

Contribution

It introduces explicit formulas for counting nilpotent semigroups of degree 3 and their commutative variants, advancing enumeration in semigroup theory.

Findings

Formulas for counting nilpotent semigroups of degree 3 up to isomorphism and anti-isomorphism.

Explicit counts for nilpotent commutative semigroups up to isomorphism.

Results confirm the folklore that most finite semigroups are nilpotent of degree 3.

Abstract

A semigroup is \emph{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 with $n\in\N$ elements up to equality, isomorphism, and isomorphism or anti-isomorphism. Likewise, we give formulae for the number of nilpotent commutative semigroups with $n$ elements up to equality and up to isomorphism.