Course details

Modern Mathematical Methods in Informatics

MID Acad. year 2021/2022 Winter semester

Current academic year

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). 

Guarantor

Šlapal Josef, prof. RNDr., CSc. (IM DAAG)

Course coordinator

Pavlík Jan, Mgr., Ph.D. (IM DAAG)

Language of instruction

Czech, English

Completion

Examination

Time span

  • 26 hrs lectures

Assessment points

  • 100 pts final exam

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

Lecturer

Šlapal Josef, prof. RNDr., CSc. (IM DAAG)

Instructor

Šlapal Josef, prof. RNDr., CSc. (IM DAAG)

Subject specific learning outcomes and competences

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.

Learning objectives

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.  

Recommended prerequisites

Prerequisite knowledge and skills

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

Study literature

  • G. Grätzer, Lattice Theory, Birkhäuser, 2003
  • K.Denecke and S.L.Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, 2002
  • G. Grätzer, Universal Algebra, Springer, 2008
  • P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982
  • N.M. Martin and S. Pollard, Closure Spaces and Logic, Kluwer, 1996
  • B.A. Davey, H.A. Pristley, Introduction to Lattices and Order, Cambridge University Press, 1990
  • S. Roman, Lattices and Ordered Sets, Springer, 2008.
  • V.K.Garg, Introduction to Lattice Theory with Computer Science Applications, Wiley, 2015
  • T. Y. Kong, Digital topology; in L. S. Davis (ed.), Foundations of Image Understanding, pp. 73-93. Kluwer, 2001

Syllabus of lectures

  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  

Progress assessment

Tests during the semester

Controlled instruction

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

Course inclusion in study plans

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