discipline

Pure Mathematics

subject

Universal Algebra I.

lecturers

Emil Kiss, Csaba Szabó

credits

2

period

 

curriculum

Introduction. Similarity type, algebra, clone, term, polynomial. Subalgebra, homomorphism, direct product, identity, variety, Birkhoff's theorems. Congruence lattice, subalgebra lattice, Mal'tsev's Lemma. Subdirect decomposition, subdirectly irreducible algebras. Quackenbush's Conjecture. Graph algebras.

Mal'tsev conditions, the characterization of congruence permutable, congruence distributive and congruence modular varieties. Jónsson's Lemma, Fleischer's Theorem. Baker's finite basis theorem.

Completeness questions, primal and functionally complete algebras and their characterizations (Foster-Pixley, Baker-Pixley, Rosenberg, Murskii). Discriminator, Werner's Theorem. Boolean product and its application to describe algebras in discriminator varieties. Quackenbush's characterization. Directly representable varieties.

Commutator Theory. Abelian algebras, centrality, the properties of the commutator in modular varieties. Gumm's shifting lemma, the semantic definition of the commutator, the proof of the commutator properties. Difference term, the Fundamental Theorem of Abelian Algebras (Taylor-Herrmann). The structure of Abelian congruences, the module associated with them. Gumm's characterization ofcentrality. Generalized Jónsson Theorem.

literature

Burris-Sankappanavar: A course in Universal Algebra

Freese-McKenzie: Commutator theory for congruence modular varieties

form of tuition

Lecture

mode of assessment

written/oral exam