An Elementary Proof of a Theorem of Graham on Finite Semigroups

Adolfo Ballester-Bolinches,Vicent Pérez-Calabuig

doi:10.3390/math8010105

Abstract

The purpose of this note is to give a very elementary proof of a theorem of Graham that provides a structural description of finite 0-simple semigroups and its idempotent-generated subsemigroups.

Highlights

The aim of this short note is to present a very elementary proof of a classical theorem of Graham that gives a complete and useful description of a finite 0-simple semigroup and its idempotent-generated subsemigroup.all semigroups considered in the sequel will be finite.0-simple semigroups were first introduced and studied by Rees in his seminar paper [1]
They play a significant role in semigroup theory: every regular J -class can be seen as a 0-simple semigroup, and every semigroup may be obtained from 0-simple semigroups by a sequence of ideal extensions
In [1], Rees proved a theorem that gives to the class of 0-simple semigroups a transparent characterization

Summary

Introduction

The aim of this short note is to present a very elementary proof of a classical theorem of Graham that gives a complete and useful description of a finite 0-simple semigroup and its idempotent-generated subsemigroup. A detailed study of the idempotent-generated subsemigroup of a given semigroup turns out to be crucial for understanding complexity, which is one of the most famous problems in semigroup theory In this context, Graham’s theorem has become one of the most important basic results. Up to isomorphism, 0-simple semigroups are precisely the Rees matrix semigroups, so called because they were introduced by him in the same paper The proof of this theorem relies on a bunch of important structural results concerning 0-simple semigroups. In 1968, Graham published an influential contribution to the structural study of a 0-simple semigroup He showed how to apply graph theory to obtain a description of the idempotent-generated subsemigroup of a 0-simple semigroup [2]. Our concern here is in applying the basic results on regularity to the method used by Rees to prove his isomorphism theorem

Preliminaries

Two Key Lemmas and Their Corollaries

Proof of Graham’s Theorem

Conclusions