Semigroup Theory

Recommended Prerequisites

Knowledge of group theory obtained in a first undergraduate course of Algebra.   

Teaching Methods

The teaching/learning process will follow an interactive approach in which students will immediately apply the theory taught in solving exercises of various levels.The results will be discussed immediately.

Learning Outcomes

The aim of the unit is to equip students with the knowledge of semigroup theory that is necessary to conduct research in computational algebra. In particular the course will cover Green’s lemmas, Rees’ theorem, McAlister P-Theorem, Schein/Meakin construction of inverse semigroups and transformation semigroups. After completing this unit, the students should be

1.able to appreciate the importance of semigroup theory in abstract algebra;

2.familiar with the most important classes of semigroups and have an understanding of the structure of important examples, such as the most famous transformation semigroups: the full transformation monoid, the symmetric inverse semigroup, the endomorphism monoid of a vector space, and their ideals.   

Work Placement(s)

No

Syllabus

1- Green’s equivalences and regular semigroups

2- Completely 0-Simple semigroups.

3- Completely regular semigroups.

4- Inverse semigroups.

5- Bands

6- Free semigroups.

7- Transformation semigroups: their structure and properties.

Assessment Methods

Assessment
Exam: 30.0%
Research work: 35.0%
Resolution Problems: 35.0%

Bibliography

Higgins, Peter M.(4-ESSX)
Techniques of semigroup theory.
With a foreword by G. B. Preston. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992. x+258 pp.
Howie, J. M. An introduction to semigroup theory. L.M.S. Monographs, No. 7. Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1976. x+272 pp.
Howie, John M.
Fundamentals of semigroup theory.
London Mathematical Society Monographs. New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+351 pp.
Petrich, Mario Inverse semigroups. Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1984. x+674 pp.
Rhodes, John; Steinberg, Benjamin The q-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York, 2009. xxii+666 pp.