David Stanovský    //   

UNIVERSAL ALGEBRA I 2016/17

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, some other notes]
  5. Classification schemes (case study) - Maltsev conditions, abelianess [some other notes]

covered topicsrecommended reading homework
4.10.Motivation. Examples of algebras and equational classes.
Ex.: Term operations and term equivalence. Algebraic vs. order-theoretic lattices. 		Bergman 1.1, 1.2
11.10.Introduction to lattices, complete lattices.
Ex.: Lattices. The lattice of equivalence relations. 		Bergman 2.1, 2.3
18.10.Algebraic lattices, closure operators.
Ex.: Galois correspondences. 		Bergman 2.4, 2.5 HOMEWORK
due on 1.11.
25.10.Basic constructions: subalgebras, products, homomorphisms. HSP operators.
Ex.: properties of HSP.		 Bergman 1.3, 3.5
1.11.Subalgebra generation. Congruences and quotients. Congruence generation.
Ex.: calculating subalgebras.		 Bergman 1.4, 1.5 HOMEWORK
due on 22.11.
8.11.No lecture.
Ex.: Calculating congruences.
15.11.Isomorphism theorems. Direct decomposition.
No Exercises. You can read more on infinite decompositions in the book (3.2).		 Bergman 3.1, 3.2
22.11.Subdirect decomposition.
Ex.: Direct and subdirect decomposition.		 Bergman 3.3, 3.4
29.11.Applications of subdirect decomposition. Finitely generated varieties (local finiteness, SIs under congruence distributivity).
Ex.: Subdirect decomposition.		 Bergman 3.5 HOMEWORK
due on 15.12.
6.12.Terms, identities and free algebras.
Ex.: Free algebras.		 Bergman 4.3
13.12.Free algerbas and Birkhoff's theorem.
Ex.: Free algebras.		 Bergman 4.4 HOMEWORK
due on 5.1.
20.12.Functional clones. Free algebras as clones of term functions.
Ex.: Clones of term and polynomial operations.		 Bergman 4.1
look at Post's lattice
3.1.Galois connection between functional and relational clones.
Ex.: Pol, Inv, generating clones.		 Bergman 4.2
(covers only part of it)		 HOMEWORK
due on 27.1.
10.1.Maltsev conditions.
Ex.: 		Bergman 4.7

For exam, you shall submit HOMEWORKS. Homeworks will count for 20% of the grade. The exam test will count for the remaining 80% of the grade. There will be five series, I will count your four best scores.

Homework results

Literature: