Semigroup Theory - MAT00100M

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  • Department: Mathematics
  • Credit value: 20 credits
  • Credit level: M
  • Academic year of delivery: 2024-25

Module summary

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.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

Additional information

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.

Module will run

Occurrence Teaching period
A Semester 2 2024-25

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 subject, including Green’s relations.

  2. The importance of natural examples, such as full transformation semigroups.

  3. The role of structure theorems, and Rees' theorem for completely 0-simple semigroups.

  4. The differences and similarities between the theories of semigroups, inverse semigroups, and groups.

  5. 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.

  • Semigroup presentations.

  • Completely 0-simple semigroups; Rees' theorem.

  • Regular and inverse semigroups.

  • Idempotent semigroups (bands).

  • Monoid actions on sets or spaces.

Indicative assessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

Special assessment rules

None

Indicative reassessment

Task % of module mark
Closed/in-person Exam (Centrally scheduled) 100

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.

Indicative reading

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