Commutator Theory  Tutorial, Part 2
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Commutator Theory Tutorial, Part 2

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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 ...

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Á. Szendrei
Commutator Theory Tutorial, Part 2
Conference on Order, Algebra, and Logics Nashville, June 12–16, 2007
Department of Mathematics University of Colorado at Boulder
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Application 2 Abelian Algebras and Congruences Application 3 Reminder: Denition of the Modular Commutator η1,η2,Δ := Δα,βCon(A(β));η1η2=0 Sublattice they generate is a homomorphic image of: r1 1ApA(β)1 Con(A)=I(η1,1)Δskew ˆα⇐⇒Δ<η1)[ααα,ββ]0βηη21)α1η1αβ11Δβαˆ2η2I,[ˆαα)<,=β&]ααˆ1<%αβI1(β, α(1Δβη21)) Á. Szendrei Commutator Theory Tutorial, Part 2
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Order theoretical properties: monotonicity:α0α,β0β=[α0, β0][α, β] [α, β]αβ commutativity:[α, β] = [β, α] additivity:[Wαi, β] =Wi[αi, β]
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HSP properties: ifAφBis an onto homomorphism with kernelθ, then φ1([γ, δ]) = [φ1(γ), φ1(δ)]θ
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Jónsson’s Theorem.For a CD varietyV=V(K),
A∈ VSI=AHSPu(K). Note:HSPu(K) =HS(K)ifKis a nite set of nite algebras.
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Theorem.For a CM varietyV=V(K), A∈ VSIwith monolithµ=AcHSPu(K).
Thecentralizer,αc, ofαis the largestγsuch that[γ, α] =0.
Ifµis the monolith of an SI, µc6=0⇐⇒µcµ⇐⇒[µ, µ] =0⇐⇒µabelian
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Assume:
Ais an SI with abelian monolithµin a CM variety µµc, since[µ, µ] =0 µµcc, since[µ, µc] = [µc, µ] =0 µccµc, since[µcc, µ][µcc, µc] =0 µcc< µc(that is,µcis nonabelian)
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Ais an SI with abelian monolithµin a CM variety µµc, since[µ, µ] =0 µµcc, since[µ, µc] = [µc, µ] =0 µccµc, since[µcc, µ][µcc, µc] =0 µcc< µc(that is,µcis nonabelian)
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Assume:
Ais an SI with abelian monolithµin a CM variety µµcsince[µ, µ] =0 , cc µµ, since[µ, µc] = [µc, µ] =0 µccµc, since[µcc, µ][µcc, µc] =0 µcc< µc(that is,µcis nonabelian)
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Assume:
Ais an SI with abelian monolithµin a CM variety µµc, since[µ, µ] =0 µµcc, since[µ, µc] = [µcµ] =0 , µccµc, since[µcc, µ][µcc, µc] =0 µcc< µc(that is,µcis nonabelian)
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Assume:
Ais an SI with abelian monolithµin a CM variety µµc, since[µ, µ] =0 µµcc, since[µ, µc] = [µc, µ] =0 µccµc, since[µcc, µ][µcc, µc] =0 µcc< µc(that is,µcis nonabelian)
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