A Survey on Varieties Generated by Small Semigroups and a Companion Website

TL;DR

This survey compiles known identity bases for varieties generated by small semigroups and groups, providing a comprehensive online resource with automated reasoning tools, and addresses gaps in existing classifications for semigroups of order up to 4 and 5.

Contribution

It offers the first complete list of identity bases for semigroups of order up to 4 and an online companion website with extensive classification and computational tools.

Findings

Complete identity bases for semigroups of order up to 4.

Online database includes varieties generated by semigroups of order up to 5.

Tools for deciding properties of finite semigroups and generating multiplication tables.

Abstract

The aim of this paper is to provide an atlas of identity bases for varieties generated by small semigroups and groups. To help the working mathematician easily find information, we provide a companion website that runs in the background automated reasoning tools, finite model builders, and GAP, so that the user has an automatic \textit{intelligent} guide on the literature. This paper is mainly a survey of what is known about identity bases for semigroups or groups of small orders, and we also mend some gaps left unresolved by previous authors. For instance, we provide the first complete and justified list of identity bases for the varieties generated by a semigroup of order up to~$4$, and the website contains the list of varieties generated by a semigroup of order up to~$5$. The website also provides identity bases for several types of semigroups or groups, such as bands, commutative groups, and metabelian groups. On the inherently non-finitely based finite semigroups side, the website can decide if a given finite semigroup possesses this property or not. We provide some other functionalities such as a tool that outputs the multiplication table of a semigroup given by a $C$-presentation, where~$C$ is any class of algebras defined by a set of first order formulas. The companion website can be found here \url{http://sgv.pythonanywhere.com} Please send any comments/suggestions to \url{jj.araujo@fct.unl.pt}