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# Introduction to Semigroup Theory

When you'll study it
Semester 1
CATS points
15
ECTS points
7.5
Level
Level 7
Module lead
James Renshaw
Academic year
2021-22

## 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.

### Linked modules

Pre-requisites: MATH2003 AND MATH2049