David Stanovský
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UNIVERSAL ALGEBRA I 2016/17

Program (actual for past lectures, tentative for future lectures):
 Preliminaries  algebras, lattices, closure operators [1.11.2, 2.1, 2.32.5]
 Semantics  subalgebras, products, homomorphisms, quotients, isomorphism theorems, internal/external characterisation of products, subdirect irreducibility [1.31.5, 3.13.5]
 Syntax  terms, free algebras, equational classes, Birkhoff's theorem, fully invariant congruences [4.24.6]
 Clones  interpretations, algebraic vs. relational clones [4.1, 4.8, some other notes]
 Classification schemes (case study)  Maltsev conditions, abelianess [some other notes]
 covered topics  recommended reading  homework 
4.10.  Motivation. Examples of algebras and equational classes. Ex.: Term operations and term equivalence. Algebraic vs. ordertheoretic 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:
