Title:

# Modern Mathematical Methods in Informatics

Code:MID
Ac.Year:2018/2019
Sem:Summer
Curriculums:
ProgrammeField/
Specialization
YearDuty
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, algebra and topology will be discussed.
Description:
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).
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. 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
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
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