Finite nilpotent semigroups of small coclass

TL;DR

This paper introduces a new concept of coclass for nilpotent semigroups, classifies those with small coclass values, and provides presentations and enumeration formulas, enriching the understanding of their algebraic structure.

Contribution

It defines coclass for nilpotent semigroups and classifies all such semigroups with coclass 0, 1, and 2, including presentations and enumeration formulas.

Findings

Classified nilpotent semigroups of coclass 0, 1, and 2.

Provided presentations and enumeration formulas for these semigroups.

Identified commutative and self-dual semigroups within the classification.

Abstract

The parameter coclass has been used successfully in the study of nilpotent algebraic objects of different kinds. In this paper a definition of coclass for nilpotent semigroups is introduced and semigroups of coclass 0, 1, and 2 are classified. Presentations for all such semigroups and formulae for their numbers are obtained. The classification is provided up to isomorphism as well as up to isomorphism or anti-isomorphism. Commutative and self-dual semigroups are identified within the classification.