SUBJECT
Universal algebra and lattice theory
lecture
master
3+3
Semesters 1-4
Autumn/Spring semester
Similarity type, algebra, clones, terms, polynomials. Subalgebra, direct product, homomorphism, identity, variety, free algebra, Birkhoff’s theorems. Subalgebra lattices, congruence lattices, Grätzer-Schmidt theorem. Mal’tsev-lemma. Subdirect decomposition, subdirectly irreducible algebras, Quackenbush-problem.
Mal’tsev-conditions, the characterization of congruence permutable, congruence distributive and congruence modular varieties. Jónsson’s lemma, Fleischer-theorem. Congruences of lattices, lattice varieties.
Partition lattices, every lattice is embeddable into a partition lattice. Free lattices, Whitman-condition, canonical form, atoms, the free lattice is semidistributive, the operations are continuous. There exists a fixed point free monotone map.
Closure systems. Complete, algebraic and geometric lattices. Modular lattices. The free modular lattice generated by three elements. Jordan-Dedekind chain condition. Semimodular lattices. Distributive lattices.
Lattices and geometry: subspace lattices of projective geometries. Desargues-identity, geomodular lattices. Coordinatization. Complemented lattices. The congruences of relatively complemented lattices.
The question of completeness, primal and functionally complete algebras, characterizations, discriminator varieties. Directly representable varieties.
The Freese-Lampe-Taylor theorem about the congruence lattice of algebras with a few operations. Abelian algebras, centrality, the properties of the commutator in modular varieties. Difference term, the fundamental theorem of Abelian algebras. Generalized Jónsson-theorem. The characterization of finitely generated, residually small varieties by Freese and McKenzie.
Congruence lattices of finite algebras: the results of McKenzie, Pálfy and Pudlak. Induced algebra, their geometry, relationship with the congruence lattice of the entire algebra. The structure of minimal algebras. Types, the labeling of the congruence lattice. Solvable algebras.
The behavior of free spectrum. Abelian varieties. The distribution of subdirectly irreducible algebras. Finite basis theorems. First order decidable varieties, undecidable problems.
- Burris-Sankappanavar: A course in universal algebra. Springer, 1981.
- Freese-McKenzie: Commutator theory for congruence modular varieties. Cambridge University Press, 1987.
- Hobby-McKenzie: The structure of finite algebras. AMS Contemporary Math. 76, 1996