Course details

Modern Mathematical Methods in Informatics

MID Acad. year 2012/2013 Summer 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)

Language of instruction

Czech, English

Completion

Examination

Time span

  • 26 hrs lectures

Department

Dept. of Algebra and Discrete Mathematics (ÚM OAAG)

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
  • S. Roman, Lattices and Ordered Sets, Springer, 2008 
  • J.L. Kelley, general Topology, Van Nostrand, 1955.

 

 

Fundamental literature

  • G. Grätzer, Universal Algebra, Springer, 2008
  • B.A. Davey, H.A. Pristley, Introduction to Lattices ad Order, Cambridge University Press, 1990
  • P.T. Johnstone, Stone Spaces, Cambridge University Press, 1982
  • S. Willard, General Topology, Dover Publications, Inc., 1970
  • N.M. Martin and S. Pollard, Closure Spaces and Logic, Kluwer, 1996
  • T. Y. Kong, Digital topology; in L. S. Davis (ed.), Foundations of Image Understanding, pp. 73-93. Kluwer, 2001
  • S. Roman, Lattices and Ordered Sets, Springer, 2008.     

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

Study evaluation is based on marks obtained for specified items. Minimimum number of marks to pass is 50.

Controlled instruction

Tests during the semester

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