Course detail

Ordered Sets and Lattices

FSI-9UMSAcad. year: 2024/2025

Students will get acquainted with basic concepts and results of the theory of ordered sets and lattices used in many branches of mathematics and in other disciplines, e.g., in informatics.

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

The knowledge of the subjects General Algebra and Methods of Discrete Mathematics taught within the Bachelor's study programme is expected.

Rules for evaluation and completion of the course

The students will be assessed by means of a written and oral exam at the end of the semester.
The presence at lectures is not compulsory, it will therefore not be checked.

Aims

The goal of the subject is to get students acquainted with the theory of ordered sets with a stress to the lattice theory.
The students will learn basic concepts and results of the theory of orderd sets and lattices including their applications.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Jan Kopka, Svazy a Booleovy algebry, Univerzita J.E. Purkyně v Ústaí nad Labem, 1991 (CS)
Steve Roman, Lattices and ordered sets, Springer, New York 2008. (EN)
T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005 (EN)

Recommended reading

B.Davey, Introduction tolattices and order, Cambridge University Press 2012 (EN)
George Grätzer: Lattice Theory: Foundation, Birkhäuser, Basel, 2011 (EN)
L. Beran, Uspořádané množiny, Mladá fronta, Praha,1978 (CS)

Classification of course in study plans

  • Programme D-APM-P Doctoral 1 year of study, summer semester, recommended course
  • Programme D-APM-K Doctoral 1 year of study, summer semester, recommended course

Type of course unit

 

Lecture

20 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Basic concepts of the theory of ordered sets
2. Axiom of Choice and equivalent theorems
3. Duality and monotonne maps
4. Down-sets and up-sets, ascending and descending chain conditions
5. Well ordered sets and ordinal numbers
6. Cardinal numbers, cardinal and ordinal arithmetic
7. Closure operators on ordered sets
8. Ideals and filters
9. Modular and distributive lattices
10. Boolean lattices