- Department: Mathematics
- Credit value: 20 credits
- Credit level: M
- Academic year of delivery: 2024-25
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.
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
Pre-requisite knowledge for MSc students: familiarity with and maturity in handling sets, functions, algebraic structures such as groups, rings and fields; knowledge of ideals and notions of divisibility in rings; knowledge of group actions.
Occurrence | Teaching period |
---|---|
A | Semester 2 2024-25 |
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.
At the end of the module students should be familiar with and able to handle the following.
The basic ideas of the subject, including Green’s relations.
The importance of natural examples, such as full transformation semigroups.
The role of structure theorems, and Rees' theorem for completely 0-simple semigroups.
The differences and similarities between the theories of semigroups, inverse semigroups, and groups.
Students should also have an appreciation of the place of semigroup theory in mathematics.
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.
Semigroup presentations.
Completely 0-simple semigroups; Rees' theorem.
Regular and inverse semigroups.
Idempotent semigroups (bands).
Monoid actions on sets or spaces.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
J M Howie, Fundamentals of Semigroup Theory, Oxford: Clarendon Press (S 2.86 HOW)
Olexandr Ganyushkin, Volodymyr Mazorchuk, Classical finite transformation semigroups, Algebra and its Applications, Springer