Semigroup Theory - MAT00050M

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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.

Module will run

Occurrence	Teaching period
A	Autumn Term 2022-23

Module aims

• To familiarise students with the elementary notions of semigroup theory.

• To illustrate abstract ideas by applying them to a range of concrete examples of semigroups.

• To study Green's relations and how these may be used to develop structure theorems for semigroups.

Module learning outcomes

At the end of the module you should be familiar with:

• The basic ideas of the subject, including Green’s relations, and be able to handle the algebra of semigroups in a comfortable way.

• The role of structure theorems, and be able to use Rees' theorem for completely 0-simple semigroups.

• Have an appreciation of the place of semigroup theory in mathematics.

Module content

Syllabus

• Examples of semigroups and monoids.

• Semigroups, ideals, homomorphisms and congruences.

• The essential difference between semigroups and previously studied algebraic structures.

• Green's relations, regular D-classes, Green's theorem that any H-class containing an idempotent is a subgroup.

• Completely 0-simple semigroups; Rees' theorem.

• Regular and inverse semigroups.

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)