commutator theory tutorial, part 1

The Modular CommutatorApplication 1

Commutator TheoryTutorial, Part 1

Á. Szendrei

Department of MathematicsUniversity of Colorado at Boulder

Conference on Order, Algebra, and LogicsNashville, June 12–16, 2007

Á. Szendrei Commutator Theory Tutorial, Part 1

Algebras, Varieties

A, an algebra

Congruences of A= kernels of homomorphisms A → A′

= equivalence relations on A that are subalgebras of A × A

Con(A) is a lattice

Variety: equationally definable class of algebras

Algebras, Varieties

A, an algebra

Congruences of A= kernels of homomorphisms A → A′

= equivalence relations on A that are subalgebras of A × A

Con(A) is a lattice

Variety: equationally definable class of algebras

Congruence Modular Varieties

A variety V is

congruence modular (CM) ifV |=Con α ≤ γ → (α ∨ β) ∧ γ = α ∨ (β ∧ γ);

congruence distributive (CD) ifV |=Con (α ∨ β) ∧ γ = (α ∧ γ) ∨ (β ∧ γ);

congruence permutable (CP) ifV |=Con α ◦ β = β ◦ α.

CD =⇒ CM, CP =⇒ CM

Examples of CM varieties: varieties of

lattices, algebras with lattice reducts;

implication algebras;

groups, algebras with group reducts (rings, modules);

quasigroups.

A variety V is

quasigroups.

A variety V is

quasigroups.

A variety V is

quasigroups.

A variety V is

quasigroups.

The HSP Theorem and Skew Congruences

Birkhoff’s HSP Theorem. V is a variety ⇐⇒ HSP(V) = V

Corollary. The variety generated by a class K of algebras isV(K) = HSP(K).

V(K) 3 A/θ � A ≤sd∏

i A i , A i = pri(A) ≤ B i ∈ K

Congruences of A:

product congruences: θ =∏

i θi , θi ∈ Con(A i)A/θ ∼=

∏A i/θi

skew congruences: all others

Commutator theory is a tool for understanding skewcongruences in CM varieties.

V(K) 3 A/θ � A ≤sd∏

Congruences of A:

∏A i/θi

V(K) 3 A/θ � A ≤sd∏

Congruences of A:

∏A i/θi

V(K) 3 A/θ � A ≤sd∏

Congruences of A:

∏A i/θi

Examples

Two Examples:

F2 = ({0, 1}; +, ·, 0, 1) G2 = ({0, 1}; +, 0)

F2 × F2: G2 ×G2:

��u

Con(F2 × F2): Con(G2 ×G2):

��

η1 η2�

��

η1 η2u∆

No skew congruence ∆ is a skew congruence

Diagonal Congruences

α, β ∈ Con(A); A(β) := β (≤ A × A)

∆α,β := least ∆ ∈ Con(A(β)) s.t.(a, a) ∆ (b, b) for all a α b

α, β ∈ Con(A); A(β) := β (≤ A × A)

M(α, β) := subalg. of A4 generated by{[a ab b

[c dc d

]: a α b, c β d

α, β ∈ Con(A); A(β) := β (≤ A × A)

[c dc d

]: a α b, c β d

}rows ∈ A(β)

M(α, β) is a reflexive, symmetriccompatible binary relation on A(β)

α, β ∈ Con(A); A(β) := β (≤ A × A)

[c dc d

]: a α b, c β d

}rows ∈ A(β)

M(α, β) is a reflexive, symmetriccompatible binary relation on A(β)

Hence: ∆α,β = transitive cl. of M(α, β)

Definition of the Modular Commutator

η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:

��

��@

��

��@

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

��

��@

��

��@

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

��

��@

��

��@

��

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

I(∆, α̂) ↘

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

��

��@

��

��@

��

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

x(∆∧η2)∨η1v

I(∆, α̂) ↘↗ I(•, α1∧β1)

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

⇐⇒ • < α1∧β1

��

��@

��

��@

��

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

x(∆∧η2)∨η1v

I(∆, α̂) ↘↗ I(•, α1∧β1)

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

⇐⇒ • < α1∧β1��

��

u uu u

α∧β

A A(β)� pr1

Con(A) I(η1, 1)�∼=

��

��@

��

��@

��

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

x(∆∧η2)∨η1v

I(∆, α̂) ↘↗ I(•, α1∧β1)

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

⇐⇒ • < α1∧β1��

��

u uu u

α∧βx[α, β]

A A(β)� pr1

Con(A) I(η1, 1)�∼=

��

��@

��

��@

��

u u u u

u u u uu u uu uu

η1 η2

α̂α1∧β1

β̂α1 α2

x(∆∧η2)∨η1v

I(∆, α̂) ↘↗ I(•, α1∧β1)

∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)

= α̂

⇐⇒ • < α1∧β1

⇐⇒ [α, β] < α∧β�

��

u uu u

α∧βx[α, β]

A A(β)� pr1

Con(A) I(η1, 1)�∼=

Alternative Definition of the Modular Commutator

α, β, δ, . . . ∈ Con(A)

Definition. α centralizes β modulo δ, C(α, β; δ), if

for all[

t uv w

]∈ M(α, β), t δ u ⇐⇒ v δ w

It follows:

C(α, β; δi) (i ∈ I) =⇒ C(α, β;∧

i∈I δ)

there is a least δ such that C(α, β; δ)

Theorem. [α, β] is the least δ such that C(α, β; δ).

α, β, δ, . . . ∈ Con(A)

for all[

t uv w

]∈ M(α, β), t δ u ⇐⇒ v δ w

It follows:

C(α, β; δi) (i ∈ I) =⇒ C(α, β;∧

i∈I δ)

α, β, δ, . . . ∈ Con(A)

for all[

t uv w

]∈ M(α, β), t δ u ⇐⇒ v δ w

It follows:

C(α, β; δi) (i ∈ I) =⇒ C(α, β;∧

i∈I δ)

Interpretation of the Modular Commutator

Groups: [M, N] = [M, N] (actually: [θM , θN ] = θ[M,N])

Rings: [I, J] = I · J + J · ILie algebras: [I, J] = [I, J]

Modules: [U, V ] = 0

Lattices: [α, β] = α ∧ β

Properties of the Modular Commutator

Order theoretical properties:

monotonicity: α′ ≤ α, β′ ≤ β =⇒ [α′, β′] ≤ [α, β]

[α, β] ≤ α ∧ β

commutativity: [α, β] = [β, α]

additivity: [∨

αi , β] =∨

i [αi , β]

Consequence: TFAE for a CM variety V:

(1) V is CD

(2) S ≤sd A × B (A, B ∈ V) =⇒ S has no skew congruences

(3) V |=Con [α, β] = α ∧ β

[α, β] ≤ α ∧ β

additivity: [∨

αi , β] =∨

i [αi , β]

(1) V is CD

(3) V |=Con [α, β] = α ∧ β

[α, β] ≤ α ∧ β

additivity: [∨

αi , β] =∨

i [αi , β]

(1) V is CD

(3) V |=Con [α, β] = α ∧ β

[α, β] ≤ α ∧ β

additivity: [∨

αi , β] =∨

i [αi , β]

(1) V is CD

(3) V |=Con [α, β] = α ∧ β

[α, β] ≤ α ∧ β

additivity: [∨

αi , β] =∨

i [αi , β]

(1) V is CD

(3) V |=Con [α, β] = α ∧ β

HSP properties:

if Aφ� B is an onto homomorphism

with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ

��

��CC

��

ss sss

��

φ−1

Con(A)

Con(B)

[α|B , β|B] ≤ [α, β]|B if B ≤ A, α, β ∈ Con(A)

[α× α′, β × β′] = [α, β]× [α′, β′]for product congruences of S ≤sd A × A′

Abelian algebras and congruences:α is abelian if [α, α] = 0; A is abelian if [1, 1] = 0

Abelian algebras (blocks of abelian congruences)are essentially modules

HSP properties:

��

��CC

��

ss sss

��

φ−1

Con(A)

Con(B)

HSP properties:

��

��CC

��

ss sss

��

φ−1

Con(A)

Con(B)

HSP properties:

��

��CC

��

ss sss

��

φ−1

Con(A)

Con(B)

HSP properties:

��

��CC

��

ss sss

��

φ−1

Con(A)

Con(B)

A Jónsson-type Theorem

Subdirectly Irreducible Algebras

A is subdirectly irreducible (SI) ifA

ι↪→

∏i B i =⇒ some pri ◦ ι is 1–1

A SI ⇐⇒ Con(A) is monolithic, i.e., of the form

Con(A)

Birkhoff’s SI Theorem.Every algebra is a subdirect product of SI algebras.Hence V = Psd(VSI) for every variety V.

residually small if ∃ cardinality bound on the SIs in Vresidually large otherwise

ι↪→

Con(A)

ι↪→

Con(A)

ι↪→

Con(A)

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

A ∈ VSI =⇒ A ∈ HSPu(K).

Note: HSPu(K) = HS(K) if K is a finite set of finite algebras.

The centralizer, αc, of α is the largest γ such that [γ, α] = 0.

If µ is the monolith of an SI,µc 6= 0 ⇐⇒ µc ≥ µ ⇐⇒ [µ, µ] = 0 ⇐⇒ µ abelian

Theorem. For a CM variety V = V(K),

A ∈ VSI with monolith µ =⇒ A/µc ∈ HSPu(K).

A Special Case

Recall: V(K) 3 A/θ � A ≤sd∏

Special Case of Theorem. For a CM variety V = V(K),

A′ = A/θ ∈ VSI with monolith µ, A ≤sd B × C (B, C ∈ K)=⇒ (i) A′ ∈ H(B, C) or

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

s ss ssss

s sCon(A)

Con(A′)

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

��ssss

s ss ssss

s sCon(A)

Con(A′)

� η1 ≤ θ or η2 ≤ θ =⇒ (i)

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

��

s ss ssss

s sCon(A)

Con(A′)

� η1 ≤ θ or η2 ≤ θ =⇒ (i)� η1 6≤ θ, η2 6≤ θ =⇒ ηi ∨ θ ≥ θ∗

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

��

s ss ssss

s sCon(A)

Con(A′)

[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]

≤ (η1 ∧ η2) ∨ θ = θ

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

��

s ss ssss

s sCon(A)

Con(A′)

[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]

≤ (η1 ∧ η2) ∨ θ = θ

=⇒ (η1 ∨ θ)/θ ≤ µc

A Special Case

(ii) [µ, µ] = 0, A′/µc ∈ H(B, C).

��

s ss ssss

s sCon(A)

Con(A′)

[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]

≤ (η1 ∧ η2) ∨ θ = θ

=⇒ (η1 ∨ θ)/θ ≤ µc

=⇒ A′/µc � A/(η1 ∨ θ) � B=⇒ (ii)

commutator theory tutorial, part 1

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