Commutator Theory Tutorial, Part 2

Á. Szendrei

Description

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 ...

Informations

• Publié par: Pheyf
• Nombre de lectures: 14
• Langue: English

Extrait

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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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A(β) :=β(≤A×A) Δ α,β :=leastΔ∈Con(A(β))s.t. (a,a) Δ (b,b)for allaαb

α, β∈Con(A);

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Application 2 Abelian Algebras and Congruences Application 3 Reminder: Denition of the Modular Commutator η 1 ,η 2 ,Δ := Δ α,β ∈Con(A(β));η 1 ∧η 2 =0 Sublattice they generate is a homomorphic image of: r 1 ✉ 1 A ✛ p ∼ A(β) ✉ 1 Con(A) ✛ = I(η 1 ,1) Δskew ✉ ① ✉ ① ①① ✉     ✉ ✉ ❅ ˆ ❅  ✉ α ⇐⇒Δ<(Δ∨η 1 )∧ [ α α α ✉ ,  ❅ ∧ ① β β ❅  ] ✉ ✈ ✉ ❅ 0  ❅ β ✉ (Δ∧ ① η η 2 ✉ 1 ) α ∨ ❅ 1 η ∧ 1 ✉ ✉ ✈ ❅   ❅ α β 1 1   ❅ ❅ Δ ✉ ✉ ✉ ✉   ❅ ❅  ❅  ❅ ❅  ❅ ❅ ❅ ❅ ❅ ✉ ✉ ✉ ❅  ❅  ❅ β αˆ ❅ ❅  ❅  ✉ ✉ ✉  ❅ ❅ 2  ❅ ❅ ✉ ✉ ❅ ❅  η ✉ 2 I ⇐ ⇐ (Δ ⇒ ⇒ , [ • ˆα α ) < , = β & ] α α ˆ 1 < ∧ % α β I 1 ∧ ( β •, α ( 1 Δ ∧ ∨ β η 2 1 ) )  ❅       ✉ Á. Szendrei Commutator Theory Tutorial, Part 2

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Order theoretical properties: monotonicity:α 0 ≤α,β 0 ≤β=⇒[α 0 , β 0 ]≤[α, β] [α, β]≤α∧β commutativity:[α, β] = [β, α] additivity:[ W α i , β] = W i [α i , β]

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HSP properties: ifA φ Bis an onto homomorphism with kernelθ, then φ −1 ([γ, δ]) = [φ −1 (γ), φ −1 (δ)]∨θ

1 s s δ s γ ✂ ✂s C on 0 s (B)

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Jónsson’s Theorem.For a CD varietyV=V(K),

A∈ V SI =⇒A∈HSP u (K). Note:HSP u (K) =HS(K)ifKis a nite set of nite algebras.

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Theorem.For a CM varietyV=V(K), A∈ V SI with monolithµ=⇒A/µ c ∈HSP u (K).

Thecentralizer,α c , ofαis the largestγsuch that[γ, α] =0.

Ifµis the monolith of an SI, µ c 6=0⇐⇒µ c ≥µ⇐⇒[µ, µ] =0⇐⇒µabelian

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Assume:

Ais an SI with abelian monolithµin a CM variety µ≤µ c , since[µ, µ] =0 µ≤µ cc , since[µ, µ c ] = [µ c , µ] =0 µ cc ≤µ c , since[µ cc , µ]≤[µ cc , µ c ] =0 µ cc < µ c (that is,µ c is nonabelian)

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Ais an SI with abelian monolithµin a CM variety µ≤µ c , since[µ, µ] =0 µ≤µ cc , since[µ, µ c ] = [µ c , µ] =0 µ cc ≤µ c , since[µ cc , µ]≤[µ cc , µ c ] =0 µ cc < µ c (that is,µ c is nonabelian)

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Assume:

Ais an SI with abelian monolithµin a CM variety µ≤µ c since[µ, µ] =0 , cc µ≤µ, since[µ, µ c ] = [µ c , µ] =0 µ cc ≤µ c , since[µ cc , µ]≤[µ cc , µ c ] =0 µ cc < µ c (that is,µ c is nonabelian)

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Assume:

Ais an SI with abelian monolithµin a CM variety µ≤µ c , since[µ, µ] =0 µ≤µ cc , since[µ, µ c ] = [µ c µ] =0 , µ cc ≤µ c , since[µ cc , µ]≤[µ cc , µ c ] =0 µ cc < µ c (that is,µ c is nonabelian)

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Assume:

Ais an SI with abelian monolithµin a CM variety µ≤µ c , since[µ, µ] =0 µ≤µ cc , since[µ, µ c ] = [µ c , µ] =0 µ cc ≤µ c , since[µ cc , µ]≤[µ cc , µ c ] =0 µ cc < µ c (that is,µ c is nonabelian)

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