Summer term 2020
Universal Algebra II (NMAG450)

Lecture notes

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).
Not available from 17.07.-27.07, 17.08-28.08

Additional literature:

• Clifford Bergman: Universal Algebra: Fundamentals and Selected Topics - our library has a few copies!
• Jaroslav Ježek: Universal Algebra
• (BKW) Barto, Krokhin, Willard: Polymorphisms, and how to use them.
• (BK) Barto, Kozik: Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem
• Burris, Sankappanavar: A Course in Universal Algebra (an alternative source of examples and exercises)

Date	Topics	Lecture notes	Exercises	Homework
24/02	Equational theories, fully invariant congruences; completeness theorem for equational logic.	Section 1.1	1.1-1.3
02/03	Convergent term rewriting systems	Section 1.2
09/03	Critical 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
16/03	Abelian algebras Herrmann's fundamental theorem	Section 2.1, 2.2	2.1-2.9; in particular 2.2, 2.6
23/03	Centralizer relation and commutator Example: groups	Section 2.3	2.11, 2.12	2.10, 2.13, 2.14 due on 07/04
30/03	Properties of the commutator Characterization of CD varieties	Section 2.3, 2.4	2.15-2.20; in particular 2.18,2.19
06/04	Nilpotent algebras and open questions	Section 2.5	Section 2.5
20/04	Birkhoff's theorem on Id_n(A) Example of a non-finitely based algebra	Chapter 3	3.1-3.4
27/04	McKenzies DPC theorem	Section 3.1	3.5-3.8
04/05	CSPs, 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
18/05	Taylor 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:

• Clifford Bergman: Universal Algebra: Fundamentals and Selected Topics - our library has a few copies!
• Jaroslav Jezek: Universal Algebra
• (BKW) Barto, Krokhin, Willard: Polymorphisms, and how to use them.
• (BK) Barto, Kozik: Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem
• Burris, Sankappanavar: A Course in Universal Algebra (an alternative source of examples and exercises)

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

Winter term 2018/19:

Exercises in Universal Algebra I (NMAG405)

see David Stanovsky's website.