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The University of Southampton

MATH6095 Introduction to Semigroup Theory

Module Overview

This is a structured module, partly delivered by self-study and partly by lectures, designed for MMath students in their fourth year. A semigroup is a non-empty set on which is defined an associative binary operation. Unlike a group, a semigroup needn't contain an identity element nor inverses for each element. For example, the natural numbers N with the operation of addition + is a semigroup as is the set T(X) of all maps from a set X to itself with operation of composition of maps. As another example consider the set of all nxn matrices with real coefficients with the binary operation of matrix multiplication. We already know that matrix multiplication is associative and this set forms a semigroup. In some respects we can think of a semigroup as an abstraction of a group but on the other hand it is sometimes useful to compare the theory of semigroups with that of rings (the 'multiplicative part' of a ring is just a semigroup) and many of the historical developments in the theory of semigroups owe much to these two theories. However recent work has highlighted strong connections with, for example, many aspects of theoretical computer science (automata theory, theory of codes and formal language theory) as well as with other areas of mathematics such as the theory of ordered structures and (partial) symmetries.

Aims and Objectives

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

  • the basic definitions in semigroup theory
  • the basic properties of Green's relations and use these in an appropriate way
  • the definitions of (completely) (0)-simple semigroups and the proofs of some of the main theorems in this section
  • Construct new semigroups using congruences


Introduction and Basic Concepts • basic definitions, subsemigroups, left (right) zeros, idempotents, ideals • semigroup morphisms, isomorphisms, direct products. • full transformation semigroups, representations, monogenic semigroups, generators. • ordered sets and semilattices. • binary relations, equivalences, congruences. Green's Equivalences • Green's relations R • the structure of D-classes • regular elements and regular semigroups 0-simple semigroups • simple and 0-simple semigroups, principal factors • Ree's theorem • completely simple semigroups In addition, time permitting, a selection will be made from the following topics • completely regular semigroups • inverse semigroups • free semigroups and codes • applications in automata theory • combinatorial problems in semigroup theory.

Learning and Teaching

Teaching and learning methods

The lecturer will provide a structured week-by-week programme of self-study and lectures, based mainly on Howie’s book, but with references to other material as appropriate. There will be three timetabled hours per week. A high standard of explanation will be expected in the solutions to the coursework assessments, and this will form part of the assessment criteria in addition to their mathematical correctness and the scope of the problems submitted.

Independent Study114
Total study time150

Resources & Reading list

HIGGINS, Peter M. Techniques of semigroup theory. 

HOWIE, JM. Fundamentals of semigroup theory. 

LAWSON, MV,. Inverse Semigroups, The theory of partial symmetries. 



MethodPercentage contribution
Assignment 30%
Assignment 30%
Exam  (120 minutes) 40%


MethodPercentage contribution
Coursework assignment(s) 100%

Repeat Information

Repeat type: Internal & External

Linked modules

Pre-requisites: MATH2003 AND MATH2049


Costs associated with this module

Students are responsible for meeting the cost of essential textbooks, and of producing such essays, assignments, laboratory reports and dissertations as are required to fulfil the academic requirements for each programme of study.

In addition to this, students registered for this module typically also have to pay for:

Books and Stationery equipment

Course texts are provided by the library and there are no additional compulsory costs associated with the module.

Please also ensure you read the section on additional costs in the University’s Fees, Charges and Expenses Regulations in the University Calendar available at

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