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 topics recommended 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 HOMEWORKdue 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 HOMEWORKdue 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 HOMEWORKdue 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 HOMEWORKdue 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) HOMEWORKdue 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.

Literature: