Commutator Theory Tutorial, Part 1

The Modular CommutatorApplication 1Commutator TheoryTutorial, Part 1Á. SzendreiDepartment of MathematicsUniversity of Colorado at BoulderConference on Order, Algebra, and LogicsNashville, June 12–16, 2007Á. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Algebras, VarietiesA, an algebraCongruences of A0= kernels of homomorphisms A→ A= equivalence relations on A that are subalgebras of A ACon(A) is a latticeVariety: equationally definable class of algebrasÁ. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Algebras, VarietiesA, an algebraCongruences of A0= kernels of homomorphisms A→ A= equivalence relations on A that are subalgebras of A ACon(A) is a latticeVariety: equationally definable class of algebrasÁ. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Congruence Modular VarietiesA varietyV iscongruence modular (CM) ifV |= → (∨)∧ = ∨(∧);Concongruence distributive (CD) ifV |= (∨)∧ = (∧)∨(∧);Concongruence permutable (CP) ifV |= = .ConCD =⇒ CM, CP =⇒ CMExamples of CM varieties: varieties oflattices, algebras with lattice reducts;implication algebras;groups, algebras with group reducts (rings, modules);quasigroups.Á. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Congruence Modular VarietiesA varietyV iscongruence modular (CM) ifV |= → (∨)∧ = ∨(∧);Concongruence ...