Pure Mathematics


Universal Algebra II.


Emil Kiss, Csaba Szabó






Applications of commutator theory: The Gumm-McKenzie-Werner Theorem: in a congruence permutable variety, every finite simple algebra is either functionally complete, or polynomially equivalent to a module. Freese-Lampe-Taylor Theorem on the congruence lattice of algebras with a few basic operations. Freese-McKenzie 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 Vaughan-Lee.

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.


Freese-McKenzie: Commutator theory for congruence modular varieties

Hobby-McKenzie: The structure of finite algebras (Tame congruence theory)

form of tuition


mode of assessment

written/oral exam