Coclass theory for nilpotent semigroups via their associated algebras

TL;DR

This paper extends coclass theory from finite nilpotent groups to semigroups by incorporating associated algebras, proposing conjectures that could simplify classification to finite calculations, supported by initial classifications and computational evidence.

Contribution

It introduces a novel coclass framework for nilpotent semigroups using associated algebras, with conjectures that aim to streamline their classification process.

Findings

Classification of coclass 0 and 1 semigroups supports conjectures.

Computational evidence suggests conjectures hold for coclass 2 and 3.

Proposes a reduction of classification to finite calculations if conjectures are proven.

Abstract

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in that we additionally use certain algebras associated to the considered semigroups. We propose a series of conjectures on our suggested approach. If these become theorems, then this would reduce the classification of nilpotent semigroups of a fixed coclass to a finite calculation. Our conjectures are supported by the classification of nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.