Ordered Sets and Lattices
FSI-9UMSAcad. year: 2020/2021
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.
Learning outcomes of the course unit
The students will learn basic concepts and results of the theory of orderd sets and lattices including their applications.
The knowledge of the subjects General Algebra and Methods of Discrete Mathematics taught within the Bachelor's study programme is expected.
Recommended optional programme components
Recommended or required reading
Steve Roman, Lattices and ordered sets, Springer, New York 2008. (EN)
Jan Kopka, Svazy a Booleovy algebry, Univerzita J.E. Purkyně v Ústaí nad Labem, 1991 (CS)
B.Davey, Introduction tolattices and order, Cambridge University Press 2012 (EN)
T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005 (EN)
L. Beran, Uspořádané množiny, Mladá fronta, Praha,1978 (CS)
George Grätzer: Lattice Theory: Foundation, Birkhäuser, Basel, 2011 (EN)
Planned learning activities and teaching methods
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.
Language of instruction
The goal of the subject is to get students acquainted with the theory of ordered sets with a stress to the lattice theory.
Specification of controlled education, way of implementation and compensation for absences
The presence at lectures is not compulsory, it will therefore not be checked.
Type of course unit
20 hours, optionally
Teacher / Lecturer
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