Course detail

Modern Mathematical Methods in Informatics

FIT-MIDAcad. year: 2025/2026

Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets, cardinal arithmetic, continuum hypothesis and axiom of choice. Partially and well-ordered sets and ordinals. Varieties of universal algebras, Birkhoff theorem. Lattices and lattice homomorphisms. Adjunctions, fixed-point theorems and their applications. Partially ordered sets with suprema of directed sets,  (DCPO), Scott domains. Closure spaces and topological spaces, applications in informatics (Scott, Lawson and Khalimsky topologies). 

Language of instruction

Czech

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

Entry knowledge

Basic knowledge of set theory, mathematical logic and general algebra.

Rules for evaluation and completion of the course

Tests during the semester
The subject is evaluated according to the result of the final exam, the minimum for passing the exam is 50/100 points.

Aims

The aim of the subject is to acquaint students with modern mathematical methods used in informatics. In particular, methods based on the theory of ordered sets and lattices, algebra and topology will be discussed.  
Students will learn about modern mathematical methods used in informatics and will be able to use the methods in their scientific specializations.
The graduates will be able to use modrn and efficient mathematical methods in their scientific work.

Study aids

Not applicable.

Prerequisites and corequisites

Basic literature

Not applicable.

Recommended reading

G. Grätzer, Lattice Theory, Birkhäuser, 2003
N.M. Martin and S. Pollard, Closure Spaces and Logic, Kluwer, 1996
P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982
S. Roman, Lattices and Ordered Sets, Springer, 2008.
T. Y. Kong, Digital topology; in L. S. Davis (ed.), Foundations of Image Understanding, pp. 73-93. Kluwer, 2001

V.K.Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, 2015

Classification of course in study plans

  • Programme DIT Doctoral 0 year of study, winter semester, compulsory-optional
  • Programme DIT Doctoral 0 year of study, winter semester, compulsory-optional
  • Programme DIT-EN Doctoral 0 year of study, winter semester, compulsory-optional
  • Programme DIT-EN Doctoral 0 year of study, winter semester, compulsory-optional

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

  1. Naive and axiomatic (Zermelo-Fraenkel) set theories, finite and countable sets.
  2. Cardinal arithmetic, continuum hypothesis and axiom of choice.
  3. Partially and well-ordered sets, isotone maps, ordinals.
  4. Varieties of universal algebras, Birkhoff theorem.
  5. Lattices and lattice homomorphisms
  6. Adjunctions of ordered sets, fix-point theorems and their applications
  7. Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
  8. Scott information systems and domains, category of domains
  9. Closure operators, their basic properties and applications (in logic)
  10. Basics og topology: topological spaces and continuous maps, separation axioms
  11. Connectedness and compactness in topological spaces
  12. Special topologies in informatics: Scott and Lawson topologies
  13. Basics of digital topology, Khalimsky topology  

Guided consultation in combined form of studies

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.