# Semigroup Theory

Course category:
Masters
Module code:
0540503
Year:
2012/13
Term:
Spring
Credits:
10

## Aims

A semigroup is a set together with an associative binary operation. Fundamental examples abound: consider the set $\mathcal{T}_X$ of all functions from a set X to itself and $M_n(\mathbb{R})$ with operations composition of functions and matrix multiplication, respectively.

The aim of the course is to familiarise students with the elementary notions of semigroup theory. Abstract ideas will be illustrated by applying them to semigroups such as  $\mathcal{T}_X$, $M_n(\mathbb{R})$ and the bicyclic semigroup $B$. The course will move on to study Green's relations and how these may be used to develop structure theorems for semigroups

## Learning objectives

At the end of the module you should be familiar with:

• the basic ideas of the subject, including Green's relations, and to be able to handle the algebra of semigroups in a comfortable way;
• the role of structure theorems and to be able to use Rees' theorem for completely 0-simple semigroups;
• to have an appreciation of the place of semigroup theory in mathematics.

## Syllabus

•  Examples of semigroups and monoids.
• Semigroups, ideals, homomorphisms and congruences. The essential difference between semigroups and previously studied algebraic structures.
• Green's relations, regular D-classes, Green's theorem that any H-class containing an idempotent is a subgroup.
• Completely 0-simple semigroups; Rees's theorem.
• Regular and inverse semigroups.

## Recommended texts

J M Howie, Fundamentals of Semigroup Theory, Oxford: Clarendon Press (S 2.86 HOW).
Other texts will be discussed at the start of the course.

## Teaching

• Spring Term
• 2 Lectures and 1 Problems Class per week

## Assessment

Two hour closed examination in mid-Summer Term (90%)
Coursework (10%). Note that coursework submitted after the advertised deadlines will be given a mark of zero.

## Elective information

Available as an elective

Please check prerequisites carefully before asking to take this module as an elective.