Course: Universal Algebra
Basic knowledge in algebra, e.g. acquired in at least one
of the following lectures: Linear algebra, Algebra
and Discrete Mathematics, Algebra for Informatics.
The language of instruction will be English.
Universal algebra is a discipline that studies all
algebraic structures (rings, groups, semigroups) in one
common framework and focusses on questions that
are best treated in this generality.
Among the classic results are
the completness of the equational calculus (Birkhoff 1935)
or Birkhoff's subdirect representation theorem.
In this year, a selection of the following topics is planned:
- Lattices and ordered sets
- The language of universal algebra:
congruence relations,
homomorphisms, subalgebras, direct and subdirect products
- Birkhoff's HSP-Theorem on equationally defined classes and the completeness of the equational calculus
- Clones: polymorphisms and invariant relations
- Polynomial functions and polynomial completeness
- Classification of varieties: Mal'cev conditions
There will be exercises during the course. The final exam
is an oral exam. Solved exercises will be taken into account.
Exercise Problems
Particularly recommended exercises from Burris-Sakkappanavar:
for this course:
I § 1 (4),(5); § 2 (2); § 3 (7).
II § 1 (1); § 4 (1), (3); § 5 (1) or (2).
Course material
-
S. Burris and H. P. Sankappanavar. A course in universal algebra.
Springer New York Heidelberg Berlin, 1981.
[book].
- EA et al. Basics of Clone Theory.
[pdf].
- Some topics in equational logic. [pdf].
- Lecture on Tarski's irredundant basis theorem.
[pdf].
Web-page maintained by Erhard Aichinger