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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
The Modular CommutatorApplication 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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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.
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Examples
Two Examples:
F2 = ({0, 1}; +, ·, 0, 1) G2 = ({0, 1}; +, 0)
F2 × F2: G2 ×G2:
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F2
F2
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uuu
G2
G2
Con(F2 × F2): Con(G2 ×G2):
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@@@
@@@
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uuu
u0
1
η1 η2�
��
@@@
@@@
���
uuu
u0
1
η1 η2u∆
No skew congruence ∆ is a skew congruence
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Diagonal Congruences
α, β ∈ Con(A); A(β) := β (≤ A × A)
A
A
βα
∆α,β := least ∆ ∈ Con(A(β)) s.t.(a, a) ∆ (b, b) for all a α b
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Diagonal Congruences
α, β ∈ Con(A); A(β) := β (≤ A × A)
A
A
βα
∆α,β := least ∆ ∈ Con(A(β)) s.t.(a, a) ∆ (b, b) for all a α b
M(α, β) := subalg. of A4 generated by{[a ab b
],
[c dc d
]: a α b, c β d
}
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Diagonal Congruences
α, β ∈ Con(A); A(β) := β (≤ A × A)
A
A
βα
∆α,β := least ∆ ∈ Con(A(β)) s.t.(a, a) ∆ (b, b) for all a α b
M(α, β) := subalg. of A4 generated by{[a ab b
],
[c dc d
]: a α b, c β d
}rows ∈ A(β)
M(α, β) is a reflexive, symmetriccompatible binary relation on A(β)
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Diagonal Congruences
α, β ∈ Con(A); A(β) := β (≤ A × A)
A
A
βα
∆α,β := least ∆ ∈ Con(A(β)) s.t.(a, a) ∆ (b, b) for all a α b
M(α, β) := subalg. of A4 generated by{[a ab b
],
[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)
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
u u u u
u u u u
u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
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u u u u
u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
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u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
I(∆, α̂) ↘
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
u u u u
u u u u
u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
x(∆∧η2)∨η1v
I(∆, α̂) ↘↗ I(•, α1∧β1)
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
⇐⇒ • < α1∧β1
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
u u u u
u u u u
u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
x(∆∧η2)∨η1v
I(∆, α̂) ↘↗ I(•, α1∧β1)
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
⇐⇒ • < α1∧β1��
��
@@
@@
u
u uu u
vu0
1
α β
α∧β
A A(β)� pr1
Con(A) I(η1, 1)�∼=
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
x(∆∧η2)∨η1v
I(∆, α̂) ↘↗ I(•, α1∧β1)
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
⇐⇒ • < α1∧β1��
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@@
@@
u
u uu u
vu0
1
α β
α∧βx[α, β]
A A(β)� pr1
Con(A) I(η1, 1)�∼=
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Definition of the Modular Commutator
η1, η2, ∆ := ∆α,β ∈ Con(A(β)); η1 ∧ η2 = 0Sublattice they generate is a homomorphic image of:
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u u u u
u u u u
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u u u uu u uu uu
u
0
1
η1 η2
∆
α̂α1∧β1
β̂α1 α2
x(∆∧η2)∨η1v
I(∆, α̂) ↘↗ I(•, α1∧β1)
∆ skew⇐⇒ ∆ < (∆∨η1)∧(∆∨η2)
= α̂
⇐⇒ • < α1∧β1
⇐⇒ [α, β] < α∧β�
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�
@@
@@
u
u uu u
vu0
1
α β
α∧βx[α, β]
A A(β)� pr1
Con(A) I(η1, 1)�∼=
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Properties of the Modular Commutator
HSP properties:
if Aφ� B is an onto homomorphism
with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ
���
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s
ss
s
ss sss
s ss0
1
θ 0
1
δ
γ
�
�
��
�
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Properties of the Modular Commutator
HSP properties:
if Aφ� B is an onto homomorphism
with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ
���
����
��CC
��
s
ss
s
ss sss
s ss0
1
θ 0
1
δ
γ
�
�
��
�
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Properties of the Modular Commutator
HSP properties:
if Aφ� B is an onto homomorphism
with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ
���
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��CC
��
s
ss
s
ss sss
s ss0
1
θ 0
1
δ
γ
�
�
��
�
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Properties of the Modular Commutator
HSP properties:
if Aφ� B is an onto homomorphism
with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ
���
����
��CC
��
s
ss
s
ss sss
s ss0
1
θ 0
1
δ
γ
�
�
��
�
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
Properties of the Modular Commutator
HSP properties:
if Aφ� B is an onto homomorphism
with kernel θ, thenφ−1([γ, δ]) = [φ−1(γ), φ−1(δ)] ∨ θ
���
����
��CC
��
s
ss
s
ss sss
s ss0
1
θ 0
1
δ
γ
�
�
��
�
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
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
sss
0
1
µ
Con(A)
Birkhoff’s SI Theorem.Every algebra is a subdirect product of SI algebras.Hence V = Psd(VSI) for every variety V.
V is
residually small if ∃ cardinality bound on the SIs in Vresidually large otherwise
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
sss
0
1
µ
Con(A)
Birkhoff’s SI Theorem.Every algebra is a subdirect product of SI algebras.Hence V = Psd(VSI) for every variety V.
V is
residually small if ∃ cardinality bound on the SIs in Vresidually large otherwise
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
sss
0
1
µ
Con(A)
Birkhoff’s SI Theorem.Every algebra is a subdirect product of SI algebras.Hence V = Psd(VSI) for every variety V.
V is
residually small if ∃ cardinality bound on the SIs in Vresidually large otherwise
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
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
sss
0
1
µ
Con(A)
Birkhoff’s SI Theorem.Every algebra is a subdirect product of SI algebras.Hence V = Psd(VSI) for every variety V.
V is
residually small if ∃ cardinality bound on the SIs in Vresidually large otherwise
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Jónsson-type Theorem
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Jónsson-type Theorem
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Jónsson-type Theorem
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Jónsson-type Theorem
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Jónsson-type Theorem
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
��ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
� η1 ≤ θ or η2 ≤ θ =⇒ (i)
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
���
CCC
����
DDD
ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
� η1 ≤ θ or η2 ≤ θ =⇒ (i)� η1 6≤ θ, η2 6≤ θ =⇒ ηi ∨ θ ≥ θ∗
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
���
CCC
����
DDD
ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
� η1 ≤ θ or η2 ≤ θ =⇒ (i)� η1 6≤ θ, η2 6≤ θ =⇒ ηi ∨ θ ≥ θ∗
[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]
≤ (η1 ∧ η2) ∨ θ = θ
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
���
CCC
����
DDD
ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
� η1 ≤ θ or η2 ≤ θ =⇒ (i)� η1 6≤ θ, η2 6≤ θ =⇒ ηi ∨ θ ≥ θ∗
[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]
≤ (η1 ∧ η2) ∨ θ = θ
=⇒ (η1 ∨ θ)/θ ≤ µc
Á. Szendrei Commutator Theory Tutorial, Part 1
The Modular CommutatorApplication 1
A Jónsson-type Theorem
A Special Case
Recall: V(K) 3 A/θ � A ≤sd∏
i A i , A i = pri(A) ≤ B i ∈ K
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).
--
-
--
���
CCC
����
DDD
ssss
s ss ssss
s sCon(A)
θ∗
θη1
η2
Con(A′)
µ
� η1 ≤ θ or η2 ≤ θ =⇒ (i)� η1 6≤ θ, η2 6≤ θ =⇒ ηi ∨ θ ≥ θ∗
[η1∨θ, θ∗] ≤ [η1∨θ, η2∨θ]
≤ (η1 ∧ η2) ∨ θ = θ
=⇒ (η1 ∨ θ)/θ ≤ µc
=⇒ A′/µc � A/(η1 ∨ θ) � B=⇒ (ii)
Á. Szendrei Commutator Theory Tutorial, Part 1
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