Commutator Theory Tutorial, Part 1

Á. Szendrei

Description

The Modular CommutatorApplication 1Commutator TheoryTutorial, Part 1Á. SzendreiDepartment of MathematicsUniversity of Colorado at BoulderConference on Order, Algebra, and LogicsNashville, June 12–16, 2007Á. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Algebras, VarietiesA, an algebraCongruences of A0= kernels of homomorphisms A→ A= equivalence relations on A that are subalgebras of A ACon(A) is a latticeVariety: equationally definable class of algebrasÁ. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Algebras, VarietiesA, an algebraCongruences of A0= kernels of homomorphisms A→ A= equivalence relations on A that are subalgebras of A ACon(A) is a latticeVariety: equationally definable class of algebrasÁ. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Congruence Modular VarietiesA varietyV iscongruence modular (CM) ifV |= → (∨)∧ = ∨(∧);Concongruence distributive (CD) ifV |= (∨)∧ = (∧)∨(∧);Concongruence permutable (CP) ifV |= = .ConCD =⇒ CM, CP =⇒ CMExamples of CM varieties: varieties oflattices, algebras with lattice reducts;implication algebras;groups, algebras with group reducts (rings, modules);quasigroups.Á. Szendrei Commutator Theory Tutorial, Part 1The Modular CommutatorApplication 1Congruence Modular VarietiesA varietyV iscongruence modular (CM) ifV |= → (∨)∧ = ∨(∧);Concongruence ...

Informations

• Publié par: Navug
• Nombre de lectures: 10
• Langue: English

Extrait

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Commutator Theory Tutorial, Part 1

Á. Szendrei

Department of Mathematics University of Colorado at Boulder

Conference on Order, Algebra, and Logics Nashville, June 12–16, 2007

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A, an algebra

Congruences ofA = kernels of homomorphismsA→A 0 = equivalence relations onAthat are subalgebras ofA×A

Con(A)is a lattice

Variety: equationally denable class of algebras

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Algebras, Varieties

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Algebras, Varieties

A, an algebra

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Variety: equationally denable class of algebras

Congruences ofA = kernels of homomorphismsA→A 0 = equivalence relations onAthat are subalgebras ofA×A

Con(A)is a lattice

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Examplesof CM varieties: varieties of lattices, algebras with lattice reducts; implication algebras; groups, algebras with group reducts (rings, modules); quasigroups.

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CD=⇒CM, CP=⇒CM

A varietyVis congruence modular(CM) if V |= Con α≤γ→(α∨β)∧γ=α∨(β∧γ); congruence distributive(CD) if V |= Con (α∨β)∧γ= (α∧γ)∨(β∧γ); congruence permutable(CP) if V |= Con α◦β=β◦α.

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CD=⇒CM, CP=⇒CM

Examplesof CM varieties: varieties of lattices, algebras with lattice reducts; implication algebras; groups, algebras with group reducts (rings, modules); quasigroups.

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A varietyVis congruence modular(CM) if V |= Con α≤γ→(α∨β)∧γ=α∨(β∧γ); congruence distributive(CD) if V |= Con (α∨β)∧γ= (α∧γ)∨(β∧γ); congruence permutable(CP) if V |= Con α◦β=β◦α.

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Examplesof CM varieties: varieties of lattices, algebras with lattice reducts; implication algebras; groups, algebras with group reducts (rings, modules); quasigroups.

A varietyVis congruence modular(CM) if V |= Con α≤γ→(α∨β)∧γ=α∨(β∧γ); congruence distributive(CD) if V |= Con (α∨β)∧γ= (α∧γ)∨(β∧γ); congruence permutable(CP) if V |= Con α◦β=β◦α.

CD=⇒CM, CP=⇒CM

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CD=⇒CM, CP=⇒CM

Examplesof CM varieties: varieties of lattices, algebras with lattice reducts; implication algebras; groups, algebras with group reducts (rings, modules); quasigroups.

A varietyVis congruence modular(CM) if V |= Con α≤γ→(α∨β)∧γ=α∨(β∧γ); congruence distributive(CD) if V |= Con (α∨β)∧γ= (α∧γ)∨(β∧γ); congruence permutable(CP) if V |= Con α◦β=β◦α.

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Examplesof CM varieties: varieties of lattices, algebras with lattice reducts; implication algebras; groups, algebras with group reducts (rings, modules); quasigroups.

A varietyVis congruence modular(CM) if V |= Con α≤γ→(α∨β)∧γ=α∨(β∧γ); congruence distributive(CD) if V |= Con (α∨β)∧γ= (α∧γ)∨(β∧γ); congruence permutable(CP) if V |= Con α◦β=β◦α.

CD=⇒CM, CP=⇒CM

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Commutator theory is a tool for understanding skew congruences in CM varieties.

Congruences ofA: product congruences:θ= Q i θ i ,θ i ∈Con(A i ) A/θ = ∼ Q A i /θ i skew congruences: all others

V(K)3A/θA≤ sd Q i A i ,A i =pr i (A)≤B i ∈ K

Corollary.The variety generated by a classKof algebras is V(K) =HSP(K).

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

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Congruences ofA: product congruences:θ= Q i θ i ,θ i ∈Con(A i ) A/θ = ∼ Q A i /θ i skew congruences: all others

V(K)3A/θA≤ sd Q i A i ,A i =pr i (A)≤B i ∈ K

Corollary.The variety generated by a classKof algebras is V(K) =HSP(K).

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

Commutator theory is a tool for understanding skew congruences in CM varieties.

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Congruences ofA: product congruences:θ= Q i θ i ,θ i ∈Con(A i ) A/θ = ∼ Q A i /θ i skew congruences: all others

V(K)3A/θA≤ sd Q i A i ,A i =pr i (A)≤B i ∈ K

Corollary.The variety generated by a classKof algebras is V(K) =HSP(K).

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

Commutator theory is a tool for understanding skew congruences in CM varieties.

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