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UNIVERSAL ALGEBRA (NMAG405)

Lecture: Thr 9:00 - 10:30 K433KNM
Practicals run by Max Hadek: Thr 10:40 - 12:10 K4

Grading:

Practicals ("Z: Zapocet"): homeworks (60% from the sum of 4 best scores out of 5 homeworks)
Lecture ("Zk: Zkouska"): written test + oral examination

Literature:

Clifford Bergman: Universal Algebra: Fundamentals and Selected Topics
Burris, Sankappanavar: A Course in Universal Algebra (an alternative source of examples and exercises)
Stanovsky: Exercises in algebra (in Czech) - chapter V

topics (future topics may change) recommended reading lecture notes homework

2.10. Motivation. Algebra (signature, type). Examples.
Pr.: Lattices vs. lattice ordered sets Bergman 1.1, 1.2 lecture 1 practical 1

9.10. Lattices, complete lattices, closure operators.
Pr.: Distributive and modular lattices. Bergman 2.1, 2.2, 2.3 lecture 2 practical 2

16.10. Algebraic lattices and closure operators. Galois correspondences.
Pr.: Complete lattices, closure operators, Galois correspondences. Bergman 2.4, 2.5 lecture 3 practical 3 Homework 1
due 30.10. 10:40

23.10. Subalgebras, products, quotients.
Pr.: Subalgebras, congruences. Bergman 1.3, 1.4, 1.5 lecture 4
corrected (lower quality)

30.10. H,S,P operators, variety. Homomorphisms.
Pr.: HSP, homomorphisms. Bergman 1.1, 1.3, 3.1, 3.5 lecture 5 Homework 2
due 20.11. 10:40

6.11. Direct and subdirect decomposition
Pr.: Direct and subdirect decomposition. Bergman 3.2, 3.3 lecture 6

13.11. Subdirect decomoposition, SIs in congruence distributive vaieties
Pr.: ? Bergman 3.4, 3.5, (5.2) lecture 7

20.11. Terms, identities, free algebras.
Pr.: Free algebras. Bergman 4.3, 4.4 lecture 8 Homework 3
due 4.12. 10:40

27.11. The syntax-semantics Galois correspondence, Birkhoff's theorem.
Pr.: Equational bases. Bergman 4.4, (4.6) lecture 9

4.12. Clones. Free algebras as clones of term operations.
Pr.: Clones. Bergman 4.1 lecture 10 Homework 4
due 18.12. 10:40

11.12. The operations-relations Galois correspondence.
Pr.: Algebraic and relational clones. Bergman 4.2 lecture 11

18.12. Mal'cev conditions: Mal'cev, majority.
Pr.: Mal'cev conditions Bergman 4.7 (part) lecture 12 Homework 5
due 8.1. 10:40

8.1. ? Tame Congruence Theory
Pr.: ?

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