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Course, academic year 2017/2018
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Lattice Theory 1 - NMAG435
Czech title: Teorie svazů 1
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2016
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Guarantor: Mgr. Pavel Růžička, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG109
Interchangeability : 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.
Literature -
Last update: T_KA (09.05.2013)

  • G. Grätzer, General Lattice Theory, Birkhäuser Verlag, Basel-Boston Berlin, 1998.
  • Garrett Birkhoff, Lattices theory, AMS, 1967.

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

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