David Stanovský    //   

UNIVERSAL ALGEBRA I 2018/19

This is an introductory course in universal algebra, essentially following Bergman's textbook. We will learn a general framework for elementary algebraic concepts such as factoring or direct decomposition. We will introduce free algebraic structures and learn the basics of equational logic. We will present the modern viewpoint via clones, that is, sets of operations closed with respect to composition.

Program (actual for past lectures, tentative for future lectures):

  1. Preliminaries - algebras, lattices, closure operators [1.1-1.2, 2.1, 2.3-2.5]
  2. Semantics - subalgebras, products, homomorphisms, quotients, isomorphism theorems, internal/external characterisation of products, subdirect irreducibility [1.3-1.5, 3.1-3.5]
  3. Syntax - terms, free algebras, equational classes, Birkhoff's theorem, fully invariant congruences [4.2-4.6]
  4. Clones - interpretations, algebraic vs. relational clones [4.1, 4.8 and other notes]
  5. Classification schemes - Maltsev conditions, abelianness [4.7 and other notes]

You can look at the 16/17 version or the 17/18 version of the lecture. This year will be similar.

covered topicsrecommended reading homework
4.10.Motivation. Examples of algebras and equational classes.
Ex.: Isomorphism. Lattices as ordered sets and as algebras.
Bergman 1.1, 1.2
11.10.Introduction to lattices, complete lattices, closure operators.
Ex.: The lattice of equivalence relations. Modularity and distributivity.
Bergman 2.1, 2.2, 2.3
18.10.Algebraic lattices, algebraic closure operators. Galois correspondences.
Ex.: Closure operators. Galois correspondences.
Bergman 2.4, 2.5 HOMEWORK
due on 1.11.
25.10.Basic constructions: subalgebras, products, homomorphisms. H,S,P operators. Subalgebra generation.
Ex.: Calculating Sg, Sub, properties of H,S,P.
Bergman 1.3, 1.4, 3.5
1.11.Finitely generated varieties are locally finite. Congruences and quotients. Congruence generation.
Ex.: Calculating congruences.
Bergman 1.5, 3.5 HOMEWORK
due on 15.11.
8.11.Isomorphism theorems. Direct decomposition.
Ex.: Direct decomposition.
Bergman 3.1, 3.2
15.11.Subdirect decomposition.
Ex.: -------
Bergman 3.3, 3.4
22.11.Examples of subdirectly irreducible algebras, SIs in congruence distributive varieties.
Ex.: Subdirect decomposition.
Bergman 3.4, 3.5 HOMEWORK
due on 6.12.
29.11.Terms, identities and free algebras.
Ex.: Free algebras.
Bergman 4.3
6.12.Free algebras in particular varieties. The syntax-semantics Galois connection and Birkhoff's theorem.
Ex.: Free algebras.
Bergman 4.4 HOMEWORK
due on 20.12.
13.12.Equational theories as fully invariant congruences. Functional clones. Free algebras as clones of term functions.
Ex.: Clones of term and polynomial operations.
Bergman 4.6, 4.1
look at Post's lattice
20.12.Galois connection between functions and relations.
Ex.: Pol, Inv, generating clones.
Bergman 4.2
(covers only part of it)
HOMEWORK
due on 10.1.
3.1.Relational clones. Maltsev conditions.
Ex.: Maltsev conditions.
Bergman 4.7
10.1.Maltsev conditions and affine representation.
Ex.: Maltsev conditions.

For exam, you shall submit HOMEWORKS. Homeworks will count for 15% of the grade. The exam test will count for the remaining 85% of the grade. There will be five or six series, I will count your four or five best scores.

Literature: