Winter term 2021/22
Algebra 1 (NMAI062)

Please register to the !! Moodle course !!

Lecture notes

Lecture: Wednesday 10:40 - 12:10, lecture room S4
Exercise classes: Tuesdays 15:40 - 17:10, lecture room S10, given by Kevin Berg
(current covid regulations)

To get ''Zápočet'' (i.e. to pass the exercise classes) you need to score at least 60/100 points. These can be obtained from 3 homework assignments (3*30 points), or weekly quizzed (10 points), which will be posted on the Moodle course
The final grade will be determined by a written exam. Admission to the exam requires passing the exercise class.

Syllabus:This course aims to give an introduction to algebra for computer science students. It will cover the following topics:
  1. Number theory: prime factorization, congruences, Euler's theorem and RSA, the Chinese remainder theorem
  2. Polynomials: rings and integral domains, polynomial rings, irreducibility, GCD, the Chinese remainder theorem and interpolation, the construction of finite fields and applications (error correcting codes, secret sharing,...)
  3. Group theory: permutation groups, subgroups, Langrange's theorem, group actions and Burnsides's theorem, cyclic groups, discrete logarithm and applications in cryptography
Literature:The course has its own lecture notes, which are based on David Stanovsky's material from last year, and will be constantly updated during the semester. Complementary resources are for instance Consultation:
If you have open questions, do not hesitate to ask (either in person or via e-mail)! I have no official office hours, but if required, a personal meeting can be arranged. Please make also use of the exercise classes to discuss your questions.

Summer term 2020
Universal Algebra II (NMAG450)

Lecture notes

Exercises: homeworks (you need to score 60% on the 3 best out of 4 homeworks)
Lecture: oral examination (appointment by mail:
Not available from 17.07.-27.07, 17.08-28.08

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
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)

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

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
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.