Summer term 2020
Universal Algebra II (NMAG450)

Lecture notes

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Lecture: Mondays 17:20- 18:50 KA9
Practicals: Tuesdays 10:40 - 12:10 KA12 (only odd semester weeks = even calendar weeks)

Due to the Corona lockdown: starting from 11.03 there will be no lectures and no practicals until further notice. I will however continue to regularly update the lecture notes, so that you can follow the material in self-study. Further I'm happy to answer your questions via e-mail. There will still be homework assignments; please send me your homework via e-mail.

Grading:
Exercises: homeworks (you need to score 60% on the 3 best out of 4 homeworks)
Lecture: oral examination (appointment by mail: michael@logic.at).

Additional literature:
Date Topics Lecture notes Exercises Homework
24/02Equational theories, fully invariant congruences;
completeness theorem for equational logic.		 Section 1.1 1.1-1.3
02/03Convergent term rewriting systems Section 1.2
09/03Critical pairs, Knuth-Bendix algorithm;
Affine algebras 		Section 1.2, 1.3
Section 2.1		 1.4, 1.8,1.9 1.5,1.6,1.7
due on 24/03
16/03Abelian algebras
Herrmann's fundamental theorem 		Section 2.1, 2.2 2.1-2.9;
in particular 2.2, 2.6
23/03Centralizer relation and commutator
Example: groups		 Section 2.3 2.11, 2.12 2.10, 2.13, 2.14
due on 07/04
30/03Properties of the commutator
Characterization of CD varieties		 Section 2.3, 2.4 2.15-2.20;
in particular 2.18,2.19
06/04Nilpotent algebras
and open questions		 Section 2.5 Section 2.5
20/04Birkhoff's theorem on Id_n(A)
Example of a non-finitely based algebra 		Chapter 3 3.1-3.4
27/04McKenzies DPC theorem Section 3.1 3.5-3.8
04/05CSPs, pp-definable relations
Pol-Inv 		Section 4.1 4.1-4.5 3.5,3.6,4.3 due on 19/05
11/05 Clone and minion homomorphisms
 Section 4.2 4.6-4.8
18/05Taylor operations
the CSP dichotomy conjecture/theorem		 Section 4.3 4.9-4.11 4.6,4.7,4.8
until your exam





Winter term 2019/20:

Exercises in Universal Algebra I (NMAG405)

See Libor's website.

Summer term 2019:

Universal Algebra II (NMAG450)

The course will roughly follow Libor Barto's lecture from 17/18.

Lecture: Thurday 9:00 - 10:30 Seminar room of KA
Exercises: Thurday 10:40 - 12:10 Seminar room of KA (only odd semester weeks = even calendar weeks)

Grading:
Exercises: homeworks (60% from 3 best scores out of 4 homeworks)
Lecture: oral examination (appointment by mail: michael@logic.at).
next available dates 27/05-07/06; 17/06-20/06; 04/07-

Literature:
Date Topics Recommended reading Exercises Homework
28/02Abelian and affine algebras, Fundamental theorem. Bergman 7.3 Ex. 1
07/03Checking identities, Relational description of Abelianness;
Centralizer relation (in general and in groups) 		Bergman 7.4
14/03Properties of the commutator
Characterization of CD varieties		 Bergman 7.4 Ex. 2 HW 1
due 28/03
21/03Equational theories, fully invariant congruences;
completeness theorem for equational logic.		Bergman 4.6
Jezek 13
28/03Reduction order, critical pairs
Knuth-Bendix algorithm		Jezek 13Ex. 3
04/04Examples of finitely based and non finitely based algebrasBergman 5.4HW 2
due 25/04
11/04 McKenzie's result on definable principal congruencesBergman 5.5Ex. 4
18/04Constraint satisfaction problems over finite templates
Pol-Inv revisited 		BKW
25/04(h1-)clone homomorphisms
Taylor terms		BKWEx. 5HW 3
due 09/05
02/05Taylor's theoremBergman 8.4.
09/05Smooth digraphs, algebraic length 1,absorptionBK
16/05Absorption, transitive termsBKEx. 6HW 4
23/05Absorption theorem, LLL (loop lemma 'light', for linked digraphs)BK


Winter term 2018/19:

Exercises in Universal Algebra I (NMAG405)

see David Stanovsky's website.