Commutator Theory Tutorial, Part 2

Application 2Abelian Algebras and CongruencesApplication 3Commutator TheoryTutorial, Part 2Á. SzendreiDepartment of MathematicsUniversity of Colorado at BoulderConference on Order, Algebra, and LogicsNashville, June 12–16, 2007Á. Szendrei Commutator Theory Tutorial, Part 2Application 2Abelian Algebras and CongruencesApplication 3Reminder: Diagonal Congruences,∈ Con(A); A() := ( A A) := least ∈ Con(A()) s.t. ,A (a, a) (b, b) for all a bAÁ. Szendrei Commutator Theory Tutorial, Part 2Application 2Abelian Algebras and CongruencesApplication 3Reminder: Definition of the Modular Commutator , , := ∈ Con(A()); ∧ = 01 2 , 1 2Sublattice they generate is a homomorphic image of:pr1A A()u u1 1=Con(A) I( , 1) skew1u u⇐⇒ < ( ∨ )∧( ∨ )1 2@ @ ˆx u @u 1u u@u 2 = ˆ@ @ @ ⇐⇒ < ∧1 1x∧@u 1 1u@u@u∧ ˆ ⇐⇒ [,] < ∧@ @x x( ∧ )∨2 1x[,]v uv u u u@ @@ @ @@u u @u@u@u @u0@ @ @ 1 2@u @u @u@ @ I( ,ˆ)&% I(, ∧ )1 1@u @u@@u0Á. Szendrei Commutator Theory Tutorial, Part 2Application 2Abelian Algebras and CongruencesApplication 3Reminder: Properties of the Modular CommutatorOrder theoretical properties:0 0 0 0monotonicity: , =⇒ [, ] [,][,] ∧commutativity: [,] = [,]W Wadditivity: [ ,] = [ ,]i iiÁ. Szendrei Commutator Theory Tutorial, Part 2Application 2Abelian Algebras and CongruencesApplication 3Reminder: Properties of the Modular Commutator1 ...