Introduction to the lattice theory: structure and basic properties of distributive and modular lattices, structure of congruences
of lattices, free lattices, lattice varieties.
Last update: T_KA (09.05.2013)
Úvod do teorie svazu: struktura a základní vlastnosti distributivních, modulárních a semimodulárních svazu, struktura
kongruencí svazu.
Literature -
Last update: doc. Mgr. Pavel Růžička, Ph.D. (10.10.2017)
G. Grätzer, General Lattice Theory, Birkhäuser Verlag, Basel-Boston Berlin, 1998.
Garrett Birkhoff, Lattices theory, AMS, 1967.
Last update: doc. Mgr. Pavel Růžička, Ph.D. (10.10.2017)
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
Last update: T_KA (09.05.2013)
Základní vlasnosti svazu:
svazy jako usporádané množiny, algebraická definice svazu, homomorfismy, kongruence a ideály, spojove nerozložitelné prvky
