Ordered Sets and Lattices (FSI-9UMS)

Academic year 2022/2023
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech or English
Aims of the course unit:
The goal of the subject is to get students acquainted with the theory of ordered sets with a stress to the lattice theory.
Learning outcomes and competences:
The students will learn basic concepts and results of the theory of orderd sets and lattices including their applications.
Prerequisites:
The knowledge of the subjects General Algebra and Methods of Discrete Mathematics taught within the Bachelor's study programme is expected.
Course contents:
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.
Teaching methods and criteria:
Regular lectures focused on basic principles and methods of the theory of ordered sets and lattices including examples..
Assesment methods and criteria linked to learning outcomes:
The students will be assessed by means of a written and oral exam at the end of the semester.
Controlled participation in lessons:
The presence at lectures is not compulsory, it will therefore not be checked.
Type of course unit:
    Lecture  10 × 2 hrs. optionally                  
Course curriculum:
    Lecture 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
Literature - fundamental:
1. Steve Roman, Lattices and ordered sets, Springer, New York 2008.
2. Jan Kopka, Svazy a Booleovy algebry, Univerzita J.E. Purkyně v Ústaí nad Labem, 1991
3. T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005
Literature - recommended:
2. B.Davey, Introduction tolattices and order, Cambridge University Press 2012
3. L. Beran, Uspořádané množiny, Mladá fronta, Praha,1978
4. George Grätzer: Lattice Theory: Foundation, Birkhäuser, Basel, 2011
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
D-APM-K combined study --- -- DrEx 0 Recommended course 3 1 S
D-APM-P full-time study --- -- DrEx 0 Recommended course 3 1 S