Permutation groups and transformation semigroups: results and problems

Summary

Abstract

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics.

This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation a. Our particular concern is the influence that properties of G (related to homogeneity, transitivity and primitivity) have on the structure of the semigroup. In the first part of the paper, we consider properties of S = 〈G, a〉 such as regularity and idempotent generation. The second is a brief report on the synchronization project, which aims to decide in what circumstances S contains an element of rank 1. The paper closes with a list of open problems on permutation groups and linear groups, and some comments about the impact on semigroups are provided.

These two research directions outlined above lead to very interesting and challenging problems on primitive permutation groups whose solutions require combining results from several different areas of mathematics, certainly fulfilling both of Howie's elegance and value tests in a new and fascinating way.

Regularity and generation

Introduction

How can group theory help the study of semigroups?

If a semigroup has a large group of units, we can apply group theory to it. But there may not be any units at all! According to a widespread belief, almost all finite semigroups have only one idempotent, which is a zero, not an identity (see [25] and [18]). This conjecture, however, should not deter us from the general goal of investigating how the group of units shapes the structure of the semigroup. Infinitely many families of finite semigroups, and the most interesting, are composed by semigroups with a group of units. Some of those families are interesting enough to keep many mathematicians busy their entire lives; in fact a unique family of finite semigroups, the endomorphism semigroups of vector spaces over finite fields, has been keeping experts in linear algebra busy for more than a century.