discipline 
Pure
Mathematics

subject 
Universal Algebra II.

lecturers 
Emil Kiss, Csaba Szabó 
credits 
2 
period 

curriculum 
Applications of commutator theory: The GummMcKenzieWerner
Theorem: in a congruence permutable variety, every finite simple algebra is
either functionally complete, or polynomially
equivalent to a module. FreeseLampeTaylor Theorem
on the congruence lattice of algebras with a few basic operations. FreeseMcKenzie Theorem: the characterization of finitely
generated modular varieties. Simple algebras in a modular variety. Gumm's results on permutability.
The structure of nilpotent algebras, the finite basis theorem of Freese and VaughanLee. Tame congruence theory.
Induced algebra on a subset. The geometry of neighborhoods, the congruence
lattice of the induced algebra on a neighborhood. The structure of minimal
algebras: twin lemma, the five types. Relationship between the shape and the
labeling of the congruence lattice. Interpretation. Solvable algebras and
varieties. Higher dimensional minimal sets. Centrality and nilpotence in the general case (Kearnes). Applications: Results on
congruence lattices of finite algebras (McKenzie, Pálfy, Pudlak).
The structure of locally finite varieties whose first order theory is
decidable (Burris, McKenzie, Valeriote). The RS
conjecture in varieties that satisfy a nontrivial congruence identity, and in
general, undecidability of the type set (McKenzie,
Wood). Lattice of subvarieties. Free spectrum. Abelian varieties. 
literature 
FreeseMcKenzie: Commutator
theory for congruence modular varieties HobbyMcKenzie: The
structure of finite algebras (Tame congruence theory) 
form of tuition 
Lecture 
mode of assessment 
written/oral
exam 