Course detail

Mathematical Structures

FSI-SSR-AAcad. year: 2021/2022

The course will familiarise students with basic concepts and results of the theory of mathematical structures. A number of examples of concrete structures which students know from previously passed mathematical subjects will be used to demonstrate the exposition.

Language of instruction

English

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students will acquire the ability of viewing different mathematical structures from a unique, categorical point of view. This will help them to realize new relationships and links between different branches of mathematics. The students will also be able to apply their knowledge of the theory of mathematical structures, e.g. in computer science.

Prerequisites

Students are expected to know the following subjects taught within the bachelor's study programme: Mathermatical Analysis I-III, Functional Analysis, both Linear and General Algebra, and Methods of Discrete Mathematics. Concerning the the master's study programme, knowledge of Graph Theory is required.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

The graded-course unit credit is awarded on condition of having passed a written test assessing the knowledge of the theory presented..

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to show the students possibility of a unified perspective on seemingly different mathematical subjects.

Specification of controlled education, way of implementation and compensation for absences

Since the attendance at lectures is not compulsory, it will not be checked, and compensation of possible absence will not be required.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A.Adámek, H.Herrlich. G.E.Strecker: Abstract and Concrete Categories, John Willey & Sons, New York, 1990 (EN)
Jiří Adámek, Theory of Mathematical Structures, D. Reidel Publ. Company, Dordrecht, 1983. (EN)
Steve Awodey: Category Theory, Oxford University Press Inc. 2006. (EN)

Recommended reading

H.Herrlich. G.E.Strecker: Category Theory, Allyn and Bacon Inc., Boston 1973 (EN)
Jiří Adámek, Matematické struktury a kategorie, SNTL Praha, 1982 (CS)

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-MAI , 2 year of study, summer semester, compulsory

  • Programme M2A-A Master's

    branch M-MAI , 2 year of study, summer semester, compulsory

  • Programme N-MAI-A Master's 2 year of study, summer semester, compulsory
  • Programme N-AIM-A Master's 2 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Sets and classes
2. Mathematical structures
3. Isomorphisms
4. Fibres
5. Subobjects
6. Quotient objects
7. Free objects
8. Initial structures
9. Final structures
10.Cartesian product
11.Cartesian completeness
12.Functors
13.Reflection and coreflection