Title:

# Modern Mathematical Methods in Informatics

Code:MID
Ac.Year:2012/2013
Sem:Summer
Curriculums:
ProgrammeFieldYearDuty
CSE-PHD-4DVI4-Elective
Language of Instruction:Czech
Completion:examination
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:260000
ExamsTestsExercisesLaboratoriesOther
Points:1000000
Guarantor:Šlapal Josef, prof. RNDr., CSc. (DADM)
Lecturer:Šlapal Josef, prof. RNDr., CSc. (DADM)
Faculty:Faculty of Mechanical Engineering BUT
Department:Department of Algebra and Discrete Mathematics FME BUT
Prerequisites:
 Discrete Mathematics (IDA), DMAT Mathematical Structures in Computer Science (MAT), DADM

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.
Description:
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).
Knowledge and skills required for the course:
Basic knowledge of set theory, mathematical logic and general algebra.
Subject specific learning outcomes and competencies:
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 competencies:
The graduates will be able to use modrn and efficient mathematical methods in their scientific work.
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
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
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.
Progress assessment:
Tests during the semester