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LINEARNI ALGEBRA A GEOMETRIE 1 (NMAG101)
Stranka kurzu
UNIVERSAL ALGEBRA 1 (NMAG405)
Tutorial: Monday 10:40  12:10 K7 (instructor: Pavel Ruzicka)
Lecture: Thursday 14:00  15:30 K7
LECTURES
 3.10. Introduction. Algebra (formalisms using types and signatures). Homomorfism, isomorphism.
 10.10. Subuniverse, subalgebra. Product, power, subpower. Congruence, quotient.
 17.10. Variety, V(K)=HSP(K). Homomorphisms and constructions. 3 Isomorphism Theorems, Correspondence theorem.
 24.10. Lattice <> lattice ordered set. Modular lattice. Thm: modular <=> no N5 <=> isomorphisms of intervals.
Thm: modular => max. chains have the same length. Thms: congruences in groups permute => Con(group) is modular.
 31.10. Distributive lattice. Thm: distributive <=> two majorities coincide <=> no N5, M3 <=> at most one complement.
Thm: Con(lattice) is distributive. Thm: Distributive lattices = sublattice of P(X).
 7.11. Complete lattice, complete sublattice, compact elements, algebraic lattice. Closure operator, algebraic closure operator.
Correspondence: (algebraic) closure operators ... (algebraic) complete lattices.
 14.11. Galois connection, induced closure operators, induced dual lattice isomorphism.
 21.11. Direct decomposition, internal characterization. Subdirect decomposition, SI algebra, internal characterization,
completely meet irreducible element, monolith. Thm: every algebra is (isomorphic to) a subdirect product of SI algebras.
Crlr: Each variety is determined by SIs in it.
 28.11. Term, evaluation, term operation, identity, satisfaction. Absolutely free algebra, free algebra for a class.
Thm: Free algebra for K is in SP(K) and is equal to terms modulo identities true in K.
 5.12. Galois connection between classes of algebras and sets of identities (over a countable infinite set X of variables).
Thm (Birkhoff): Closed classes of algebras = varieties. Thm: Closed classes of identities = fully invariant congruences of the absolutely free algebra over X.
 12.12. Clone, clone of an algebra, free algebras via clones.
Galois connection between sets of operations and sets of relations  invariant relations, polymorphisms.
 19.12. Thm: Closed sets of operations = clones. PPdefinability, relational clone. Thm: Closed classes of relations = relational clones.
 2.1. canceled
 9.1. (not required for the exam) CSP
LITERATURE
 C. Bergmann: Universal Algebra, CRC Press, 2012. (Available in the library)
 S. Burris, H.P. Sankappanavar: A course in universal algebra, 1981. Online version (update 2012) here
 D. Stanovsky, Priklady z algebry, in Czech (exercises on general algebra are on pages 4149)
SEMINAR K PROBLEMU CSP (NALG118)
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