Faculty of Information Technology, BUT

Course details

Modern Mathematical Methods in Informatics

MID Acad. year 2012/2013 Summer semester

Partially ordered sets, Axiom of choice and its equivalents, well-ordered sets, ordinal and cardinal rithmetic. Semilattices, lattices, complete lattices and lattice homomorphisms. Meet structures and closure operators, ieals and filters, Galois correspondence and Dedekind-MacNeille completion. 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

Language of instruction

Czech

Completion

Examination

Time span

26 hrs lectures

Assessment points

100 exam

Department

DADM FME BUT

Lecturer

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.

Generic learning outcomes and competences

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   and on topology.  

Prerequisites

Prerequisite kwnowledge 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     

Syllabus of lectures

  1. Partially ordered sets, Axiom of choice and its equivalents.
  2. Well ordered sets, ordinal and cardinal numbetrs.
  3. Semilattices, lattices and complete lattices.
  4. Meet structures and closure operators.
  5. Lattice homomorphisms.
  6. Ideals and filters.
  7. Galois correspondence and Dedekind-McNeille completion.
  8. Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
  9. Scott information systems and domains, category of domains
  10. Closure operators, their basic properties and applications (in logic)
  11. Basics og topology: topological spaces and continuous maps, separation axioms
  12. Connectedness and compactness in topological spaces
  13. Special topologies in informatics: Scott and Lawson topologies
  14. Basics of digital topology, Khalimsky topology  

Progress assessment

Tests during the semester

Course inclusion in study plans

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