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UNIVERSAL ALGEBRA (NMAG405)

Lecture: Mon 9:00 - 10:30 K4
Practicals run by Vojtech David: Mon 10:40 - 12:10 K4

Grading:

Practicals ("Z: Zapocet"): homeworks (60% from the sum of 12 best scores out of 15 homeworks). AI must not be used to generate mathematical proofs or text. It is appropriate to use AI for language editing of the text, consulting on mathematical concepts and results that are not the direct subject of the problem. AI may also be used, to a reasonable extent, to check the text from a mathematical point of view. See also guidelines.
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

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

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

19.10. Algebraic lattices and closure operators. Galois correspondences.
Pr.: Complete lattices, closure operators, Galois correspondences. Bergman 2.4, 2.5 lecture 3 Homework 1
due 2.11. 10:40

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

2.11. H,S,P operators, variety. Homomorphisms.
Pr.: HSP, homomorphisms. Bergman 1.1, 1.3, 3.1, 3.5 lecture 5 Homework 2
due 16.11. 10:40

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

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

23.11. Terms, identities, free algebras.
Pr.: Free algebras. Bergman 4.3, 4.4 lecture 8 Homework 3
due 7.12. 10:40

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

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

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

4.1. Mal'cev conditions: Mal'cev, majority.
Pr.: Mal'cev conditions Bergman 4.7 (part) lecture 12 Homework 5

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