456-0038/01 – Relational and Algebraic Structures (RAS)

Gurantor departmentDepartment of Computer ScienceCredits8
Subject guarantorprof. RNDr. Václav Snášel, CSc.Subject version guarantorprof. RNDr. Václav Snášel, CSc.
Study levelundergraduate or graduateRequirementCompulsory
Year3Semestersummer
Study languageCzech
Year of introduction1992/1993Year of cancellation2008/2009
Intended for the facultiesFEIIntended for study typesMaster
Instruction secured by
LoginNameTuitorTeacher giving lectures
MAR23 Ing. Jan Martinovič, Ph.D.
SNA57 prof. RNDr. Václav Snášel, CSc.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+2

Subject aims expressed by acquired skills and competences

Teaching methods

Summary

The course deals with fundamentals of system theory and modern algebra. It covers the following main topics: general relational systems, mapping, equivalence relation, ordering relation, general algebraic systems, morphisms, congruence relation and quotient algebras, grupoids, semigroups, monoids, groups, lattices, Boolean lattices, rings, sigma algebras and abstract data types.

Compulsory literature:

Recommended literature:

Way of continuous check of knowledge in the course of semester

E-learning

Další požadavky na studenta

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Přednášky: Rekapitulace potřebných pojmů z matematické logiky a teorie množin. Obecný pojem relace. Homogenní a heterogenní (vícedruhové) relace. Binární relace. Operace s relacemi. Relace typu zobrazení. Zobrazení úplné, parciální, surjektivní, injektivní, bijektivní. Operace se zobrazeními. Homogenní binární relace. Reflexivita, antireflexivita, symetrie, antisymetrie, asymetrie, tranzitivita, souvislost. Typy binárních relací. n-ární relace. Uzávěrové systémy. Relace typu ekvivalence. Rozklad množiny indukovaný ekvivalencí na množině, faktorová množina. Vytvoření ekvivalence pomocí zobrazení. Ekvivalenční uzávěr relace. Direktní součin ekvivalencí. Relace typu uspořádání. Uspořádání ostré, neostré, parciální, úplné, kvaziuspořádání. Relace pokrytí (bezprostředního předcházení). Direktní součin uspořádání. Lexikografické uspořádání. Svazy jako uspořádané množiny. Obecný pojem operace. Operace homogenní a heterogenní (vícedruhové). Signatura operace. Binární homogenní operace a jejich časté vlastnosti. Obecný pojem algebry. Morfismy (mono-, epi-,iso-) mezi algebrami. Kongruence na algebře a faktorová algebra. Direktní součin algeber. Algebry s jednou binární operací: grupoidy, pologrupy, monoidy, grupy, Abelovy grupy. Struktura grup. Grupa (podgrupa) generovaná množinou generátorů, cyklická (pod)grupa, symetrická grupa. Některé věty teorie grup. Algebry se dvěma binárními duálními operacemi: svazy, úplné svazy, distributivní a komplementární svazy, Booleovy svazy (algebry). Algebraická a relační definice svazu a jejich rovnocennost. Některé věty teorie Booleových algeber. Reprezentace libovolné konečné Booleovy algebry množinovou Booleovou algebrou a nebo vhodnou direktní mocninou logické Booleovy algebry. Koncepty, konceptuální svazy. Algebry se dvěma základními neduálními operacemi: okruhy, obory integrity, tělesa, pole. Algebry s číselnými nosiči a jejich význam v kvantitativní matematice. Vektorové prostory. Vícedruhové (heterogenní) algebry. Specifikace abstraktního datového typu pomocí signatury a systému axiómů.

Conditions for subject completion

Full-time form (validity from: 1960/1961 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (145) 51
        Examination Examination 100  0
        Exercises evaluation Credit 45  0
Mandatory attendence parzicipation:

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Occurrence in study plans

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2003/2004 (M2612) Electrical Engineering and Computer Science (3902T023) Computer Science P Czech Ostrava 3 Compulsory study plan
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2003/2004 (M2612) Electrical Engineering and Computer Science (2642T004) Electrical Machines, Apparatus and Drives (10) Elektrické stroje a přístroje P Czech Ostrava Choice-compulsory study plan
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2001/2002 (M2612) Electrical Engineering and Computer Science (2612T018) Electronics and Communication Technology P Czech Ostrava Choice-compulsory study plan
2001/2002 (M2612) Electrical Engineering and Computer Science (2642T004) Electrical Machines, Apparatus and Drives (10) Elektrické stroje a přístroje P Czech Ostrava Choice-compulsory study plan
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2001/2002 (M2612) Electrical Engineering and Computer Science (3902T023) Computer Science P Czech Ostrava Choice-compulsory study plan
2001/2002 (M2612) Electrical Engineering and Computer Science (3907T001) Electrical Power Engineering P Czech Ostrava Choice-compulsory study plan
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2000/2001 (M2612) Electrical Engineering and Computer Science (3902T023) Computer Science P Czech Ostrava 3 Compulsory study plan

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