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Course, academic year 2023/2024
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Lattice Theory - NMAG435
Title: Teorie svazů
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2020
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Pavel Růžička, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG109
Interchangeability : NALG109
Is interchangeable with: NALG109
Annotation -
Last update: T_KA (09.05.2013)
Introduction to the lattice theory: structure and basic properties of distributive and modular lattices, structure of congruences of lattices, free lattices, lattice varieties.
Course completion requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (28.10.2019)

Students have to pass oral exam.

Literature -
Last update: doc. Mgr. Pavel Růžička, Ph.D. (10.10.2017)

1. Gratzer, G. General Lattice Theory (2nd ed.), Birkhauser Verlag, Basel, 1998.

2. Nation, J. B., Notes on Lattice Theory. Cambridge studies in advanced mathematics, 1998. Online:

Requirements to the exam -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (28.10.2019)

Students have to pass final oral exam. The requirements for the exam correspond to what has been done during lectures.

Syllabus -
Last update: T_KA (09.05.2013)

Basic properties of lattices:

lattices as ordered sets, algebraic concept, homomorphisms, congruences and ideals, join-irreducible elements

Distributive lattices:

characterization, free distributive lattices, congruences of distributive lattices, topological representation

Congruences and ideals:

weak projectivity and perspectivity, distributive, standard and neutral elements and ideals, congruences of a cartesian product, modular and weakly modular lattices, distributivity of the congruence lattice of a lattice

Modular and semimodular lattices:

characterization, Kurosh-Ore theorem, congruences in modular lattices, von Neumann theorem, Birghoff theorem, semimodular lattices, Jordan-Hölder theorem, geometric lattices, partition lattices, complemented modular lattices and projective geometries

Entry requirements -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (17.05.2019)

Basics of general algebra.

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