Title:  Modern Mathematical Methods in Informatics 

Code:  MID 

Ac.Year:  2015/2016 

Sem:  Summer 

Curriculums:  

Language of Instruction:  Czech 

Completion:  examination 

Type of instruction:  Hour/sem  Lectures  Seminar Exercises  Laboratory Exercises  Computer Exercises  Other 

Hours:  26  0  0  0  0 

 Exams  Tests  Exercises  Laboratories  Other 

Points:  100  0  0  0  0 



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:  

 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 (ZermeloFraenkel) set theories, finite and countable sets, cardinal arithmetic, continuum hypothesis and axiom of choice. Partially and wellordered sets and ordinals. Varieties of universal algebras, Birkhoff theorem. Lattices and lattice homomorphisms. Adjunctions, fixedpoint 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: 


 Naive and axiomatic (ZermeloFraenkel) set theories, finite and countable sets.
 Cardinal arithmetic, continuum hypothesis and axiom of choice.
 Partially and wellordered sets, isotone maps, ordinals.
 Varieties of universal algebras, Birkhoff theorem.
 Lattices and lattice homomorphisms
 Adjunctions of ordered sets, fixpoint theorems and their applications
 Partially ordered sets with suprema of directed sets (DCPO) and their applications in informatics
 Scott information systems and domains, category of domains
 Closure operators, their basic properties and applications (in logic)
 Basics og topology: topological spaces and continuous maps, separation axioms
 Connectedness and compactness in topological spaces
 Special topologies in informatics: Scott and Lawson topologies
 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. 7393. 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  
