Introduction to Semigroup Theory

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

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

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.

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.

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.