Semigroup Theory - MAT00110H
- Department: Mathematics
- Credit value: 20 credits
- Credit level: H
- Academic year of delivery: 2026-27
Module summary
An introduction to the algebraic theory of semigroups. We develop
algebraic techniques
that do not require the existence of
inverses, and apply these to natural examples.
Related modules
Pre-requisite modules
Prohibited combinations
Additional information
You cannot take both H and M-level versions of this module.
M-level to have four extra lectures and one extra seminar.
Recommended co-requisite: Groups, Actions & Galois Theory MAT00099H
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 2 2026-27 |
Module aims
An introduction to the algebraic theory of semigroups. Here we see techniques developed that do not require the existence of inverses, and apply these to natural examples.
Module learning outcomes
At the end of the module students should be familiar with and able to
handle the
following.
1. The basic ideas of the semigroup
theory, including Green’s relations.
2. The importance of natural
examples, such as full transformation semigroups
and free
semigroups.
3. The differences and similarities between the
theories of semigroups, inverse
semigroups, and groups.
4.
Students should also have an appreciation of the place of semigroup
theory in
mathematics.
Module content
Examples of semigroups and monoids.
Semigroups, ideals,
homomorphisms and congruences.
Green's relations, regular
D-classes, Green's theorem that any H-class
containing an
idempotent is a subgroup.
Free semigroups and free bands.
Finite semigroups, regular languages and automata.
Regular and inverse semigroups.
Representations of semigroups
by actions or linear actions.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100.0 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100.0 |
Module feedback
Current Department policy on feedback is available in the student
handbook. Coursework
and examinations will be marked and returned
in accordance with this policy.
The feedback will be focussed on the formative coursework handed in
during the module.
In line with current Departmental policy,
there will be 4 pieces. The coursework will be
different for the
H-level and M-level versions of the module.
Indicative reading
A.J. Cain, Nine Chapters on the Semigroup Art,
https://archive.org/details/cain_semigroups_ebook
B.J. Everitt,
The sympathetic sceptics guide to semigroup representations,
Expo.
Math. 39(2) (2021), 197-237.
J M Howie, Fundamentals
of Semigroup Theory, Oxford: Clarendon Press (S 2.86
HOW)
O.
Ganyushkin, V. Mazorchuk, Classical finite transformation semigroups,
Algebra and
its Applications, Springer.