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Dualizable Algebras with Parallelogram Terms gnes Szendrei CU Boulder/U Szeged Joint work with Keith Kearnes AAA88 Warsaw, Poland, June 1922, 2014 A. Szendrei (CU Boulder) Dualizable Algebras AAA88, June 2014 1 / 19 Motivating

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SLIDE 1

Dualizable Algebras with Parallelogram Terms

Ágnes Szendrei

CU Boulder/U Szeged

Joint work with Keith Kearnes AAA88 Warsaw, Poland, June 19–22, 2014

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 1 / 19

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SLIDE 2

Motivating Example: Stone Duality

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 3

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 4

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 5

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 6

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2) B − → B∂ := Hom(B, 2)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 7

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2) B − → B∂ := Hom(B, 2)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, 2)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 8

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2) B − → B∂ := Hom(B, 2)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, 2) T∂ := Hom(T, 2) ← − T

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 9

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2) B − → B∂ := Hom(B, 2)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, 2) T∂ := Hom(T, 2) ← − T ↑ β∗ ← − ↓ β U∂ := Hom(U, 2) ← − U

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 10

Motivating Example: Stone Duality

For 2 = ({0, 1}; {∧, ∨,′ , 0, 1}) 2 = ({0, 1}; {}; discrete top.) there exist functors

Boolean algebras Stone spaces

SP(2)

⇄

ScP+(2) B − → B∂ := Hom(B, 2)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, 2) T∂ := Hom(T, 2) ← − T ↑ β∗ ← − ↓ β U∂ := Hom(U, 2) ← − U For each B, the function eB : B − → B∂∂ = Hom(B∂, 2) b − → (χ → χ(b)), evaluation at b is an isomorphism; and dually.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 2 / 19

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SLIDE 11

Natural Duality, Dualizable Algebras

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 12

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 13

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 14

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 15

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 16

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, A)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 17

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, A) T∂ := Hom(T, A) ← − T

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 18

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, A) T∂ := Hom(T, A) ← − T ↑ β∗ ← − ↓ β U∂ := Hom(U, A) ← − U

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 19

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, A) T∂ := Hom(T, A) ← − T ↑ β∗ ← − ↓ β U∂ := Hom(U, A) ← − U For each B, the function eB : B − → B∂∂ = Hom(B∂, A) b − → (χ → χ(b)), evaluation at b is a 1–1 algebra homomorphism.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 20

Natural Duality, Dualizable Algebras

Given a finite algebra and a finite, discrete, relational structure A = (A; {f, g, . . .}) A = (A; { ρ, σ, . . .

compatible rels of A

}; discrete top.) there exist functors

algebras

  • top. rel. structures

SP(A)

⇄

ScP+(A) B − → B∂ := Hom(B, A)

α ↓

− →

α∗ ↑

C − → C∂ := Hom(C, A) T∂ := Hom(T, A) ← − T ↑ β∗ ← − ↓ β U∂ := Hom(U, A) ← − U For each B, the function eB : B − → B∂∂ = Hom(B∂, A) b − → (χ → χ(b)), evaluation at b is a 1–1 algebra homomorphism.

  • Definition. A is dualized by A if eB is onto for all B.

A is dualizable if it is dualized by some A.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 3 / 19

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SLIDE 21

The Dualizing Structure

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 22

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 23

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 24

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 25

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map ⇒ f preserves every compatible relation of A.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 26

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map ⇒ f preserves every compatible relation of A.

In particular, for B := F(k) = FV(A)(k) (k ≥ 1),

F(k)∂ = Hom(F(k), A) ∼ = Ak,

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 27

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map ⇒ f preserves every compatible relation of A.

In particular, for B := F(k) = FV(A)(k) (k ≥ 1),

F(k)∂ = Hom(F(k), A) ∼ = Ak, Hom(F(k)∂, A) ∼ = Hom(Ak, A) = {k-ary ops preserving each ρ ∈ R};

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 28

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map ⇒ f preserves every compatible relation of A.

In particular, for B := F(k) = FV(A)(k) (k ≥ 1),

F(k)∂ = Hom(F(k), A) ∼ = Ak, Hom(F(k)∂, A) ∼ = Hom(Ak, A) = {k-ary ops preserving each ρ ∈ R}; for every f: Ak → A, f preserves each ρ ∈ R ⇒ f preserves every compatible relation of A.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 29

The Dualizing Structure

If A = (A; R) dualizes A = (A; F), then R has to contain ‘enough’ compatible relations of A:

For any B ∈ SP(A) and continuous map f: B∂ → A, f preserves each ρ ∈ R ⇔ f ∈ Hom(B∂, A)

A dualizes A

⇒ f is an evaluation map ⇒ f preserves every compatible relation of A.

In particular, for B := F(k) = FV(A)(k) (k ≥ 1),

F(k)∂ = Hom(F(k), A) ∼ = Ak, Hom(F(k)∂, A) ∼ = Hom(Ak, A) = {k-ary ops preserving each ρ ∈ R}; for every f: Ak → A, f preserves each ρ ∈ R ⇒ f preserves every compatible relation of A.

  • Corollary. A is dualizable ⇔ A is dualized by

A = (A; {all compatible relations of A}).

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 4 / 19

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SLIDE 30

Two Galois Connections

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 31

Two Galois Connections

Let A be a finite algebra, let R be the set of all (finitary) relations on A. R

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 32

Two Galois Connections

Let A be a finite algebra, let R be the set of all (finitary) relations on A. R

✲ ✛

F = {f: B∂ cont → A | B ∈ SP(A)}

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 33

Two Galois Connections

Let A be a finite algebra, let R be the set of all (finitary) relations on A. R

✲ ✛

F = {f: B∂ cont → A | B ∈ SP(A)} F0 = {f: F(k)∂

∼Ak

→ A | k = 0, 1, 2, . . .}

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 34

Two Galois Connections

Let A be a finite algebra, let R be the set of all (finitary) relations on A. R

✲ ✛

F = {f: B∂ cont → A | B ∈ SP(A)} F0 = {f: F(k)∂

∼Ak

→ A | k = 0, 1, 2, . . .} The compatibility of a function with a relation determines a Galois connection between R and F0, and between R and F.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 35

Two Galois Connections

Let A be a finite algebra, let R be the set of all (finitary) relations on A. R

✲ ✛

F = {f: B∂ cont → A | B ∈ SP(A)} F0 = {f: F(k)∂

∼Ak

→ A | k = 0, 1, 2, . . .} The compatibility of a function with a relation determines a Galois connection between R and F0, and between R and F.

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 5 / 19

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SLIDE 36

Clone Entailment (| =c) vs. Duality entailment (| =d)

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 6 / 19

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SLIDE 37

Clone Entailment (| =c) vs. Duality entailment (| =d)

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

R | =d γ ⇒ R | =c γ. (A; R) dualizes A ⇒ R | =d every compatible relation of A.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 6 / 19

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SLIDE 38

Clone Entailment (| =c) vs. Duality entailment (| =d)

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

R | =d γ ⇒ R | =c γ. (A; R) dualizes A ⇒ R | =d every compatible relation of A. The difference between | =c and | =d is identified by:

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 6 / 19

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SLIDE 39

Clone Entailment (| =c) vs. Duality entailment (| =d)

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

R | =d γ ⇒ R | =c γ. (A; R) dualizes A ⇒ R | =d every compatible relation of A. The difference between | =c and | =d is identified by:

  • Theorem. [BKKR]

R | =c γ ⇔ γ is constructible from R using =, permutation of coordinates, product, intersection and projection onto a subset of coordinates.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 6 / 19

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SLIDE 40

Clone Entailment (| =c) vs. Duality entailment (| =d)

  • Definition. For R ⊆ R and γ ∈ R,

R | =c γ if every f ∈ F0 preserving each ρ ∈ R also preserves γ, R | =d γ if every f ∈ F preserving each ρ ∈ R also preserves γ.

R | =d γ ⇒ R | =c γ. (A; R) dualizes A ⇒ R | =d every compatible relation of A. The difference between | =c and | =d is identified by:

  • Theorem. [BKKR]

R | =c γ ⇔ γ is constructible from R using =, permutation of coordinates, product, intersection and projection onto a subset of coordinates.

  • Theorem. [Z, DHP]

R | =d γ ⇔ γ is constructible from R using =, permutation of coordinates, product, intersection and bijective projection onto a subset of coordinates.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 6 / 19

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SLIDE 41

Finitely Related Algebras

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 7 / 19

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SLIDE 42

Finitely Related Algebras

  • Theorem. [Willard, Zádori]

Assume that R is a finite set of compatible relations of A. If R | =d every compatible relation of A, then A is dualizable.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 7 / 19

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SLIDE 43

Finitely Related Algebras

  • Theorem. [Willard, Zádori]

Assume that R is a finite set of compatible relations of A. If R | =d every compatible relation of A, then A is dualizable.

  • Definition. Call A finitely related if there is a finite set R of compatible

relations of A such that R | =c every compatible relation of A.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 7 / 19

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SLIDE 44

Finitely Related Algebras

  • Theorem. [Willard, Zádori]

Assume that R is a finite set of compatible relations of A. If R | =d every compatible relation of A, then A is dualizable.

  • Definition. Call A finitely related if there is a finite set R of compatible

relations of A such that R | =c every compatible relation of A.

Theorem above concerns finitely related algebras only.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 7 / 19

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SLIDE 45

Finitely Related Algebras

  • Theorem. [Willard, Zádori]

Assume that R is a finite set of compatible relations of A. If R | =d every compatible relation of A, then A is dualizable.

  • Definition. Call A finitely related if there is a finite set R of compatible

relations of A such that R | =c every compatible relation of A.

Theorem above concerns finitely related algebras only. Most algebras that are known to be dualizable are finitely related.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 7 / 19

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SLIDE 46

Dualizable vs. Finitely Related Algebras

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 8 / 19

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SLIDE 47

Dualizable vs. Finitely Related Algebras

In general, ‘dualizable’ and ‘finitely related’ are independent properties.

A finitely related A dualizable

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 8 / 19

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SLIDE 48

Dualizable vs. Finitely Related Algebras

In general, ‘dualizable’ and ‘finitely related’ are independent properties.

A finitely related A dualizable ✛

There exist dualizable algebras that are not finitely related.

[Pitkethly]

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 8 / 19

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SLIDE 49

Dualizable vs. Finitely Related Algebras

In general, ‘dualizable’ and ‘finitely related’ are independent properties.

A finitely related A dualizable BA lattice implication alg ✛

There exist dualizable algebras that are not finitely related.

[Pitkethly]

A 2-element algebra is dualizable iff it is finitely related. [Clark–Davey]

✲ ✻ ❄

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 8 / 19

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SLIDE 50

Dualizable vs. Finitely Related Algebras

In general, ‘dualizable’ and ‘finitely related’ are independent properties.

A finitely related A dualizable BA lattice implication alg ✛

There exist dualizable algebras that are not finitely related.

[Pitkethly]

A 2-element algebra is dualizable iff it is finitely related. [Clark–Davey]

✛

Nonabelian nilpotent groups are finitely related, but not dualizable.

[Quackenbush–Szabó] ✲ ✻ ❄

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 8 / 19

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SLIDE 51

Parallelogram Terms

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 9 / 19

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SLIDE 52

Parallelogram Terms

✁ ✁ ✁ ✁-terms generalize Maltsev terms and near unanimity (NU) terms.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 9 / 19

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SLIDE 53

Parallelogram Terms

✁ ✁ ✁ ✁-terms generalize Maltsev terms and near unanimity (NU) terms.

  • Definition. A k-✁

✁ ✁ ✁-term for an algebra A is a (k + 3)-ary term t (k ≥ 2) s.t.

A | = t           x x y . . . x x y y x x . . . y x x z y · · · y y · · · y y . . . ... . . . y y z y y y y y y z y y . . . ... . . . y y · · · y y · · · y z           =           y . . . y y . . . y           .

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 9 / 19

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SLIDE 54

Parallelogram Terms

✁ ✁ ✁ ✁-terms generalize Maltsev terms and near unanimity (NU) terms.

  • Definition. A k-✁

✁ ✁ ✁-term for an algebra A is a (k + 3)-ary term t (k ≥ 2) s.t.

A | = t           x x y . . . x x y y x x . . . y x x z y · · · y y · · · y y . . . ... . . . y y z y y y y y y z y y . . . ... . . . y y · · · y y · · · y z           =           y . . . y y . . . y           . A Maltsev term for A is a k-✁

✁ ✁ ✁-term independent of its last k variables. Example: xy−1z for any group.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 9 / 19

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SLIDE 55

Parallelogram Terms

✁ ✁ ✁ ✁-terms generalize Maltsev terms and near unanimity (NU) terms.

  • Definition. A k-✁

✁ ✁ ✁-term for an algebra A is a (k + 3)-ary term t (k ≥ 2) s.t.

A | = t           x x y . . . x x y y x x . . . y x x z y · · · y y · · · y y . . . ... . . . y y z y y y y y y z y y . . . ... . . . y y · · · y y · · · y z           =           y . . . y y . . . y           . A Maltsev term for A is a k-✁

✁ ✁ ✁-term independent of its last k variables. Example: xy−1z for any group.

A k-NU term for A is a k-✁

✁ ✁ ✁-term independent of its first 3 variables. Example: (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) for any lattice.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 9 / 19

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SLIDE 56

Dualizable vs. Finitely Related Algebras in CD Varieties

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 10 / 19

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SLIDE 57

Dualizable vs. Finitely Related Algebras in CD Varieties

A finitely related V(A) CD

  • A. Szendrei (CU Boulder)

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SLIDE 58

Dualizable vs. Finitely Related Algebras in CD Varieties

A finitely related V(A) CD A has NU term

  • Theorem. (1) ⇔ (2)

for any finite algebra A. (1) (a) A is finitely related & (b) V(A) is CD. (2) A has an NU term.

[(2)⇒(b): Mitschke; (2)⇒(1)(a): Baker–Pixley; (1)⇒(2): Barto;

  • A. Szendrei (CU Boulder)

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SLIDE 59

Dualizable vs. Finitely Related Algebras in CD Varieties

A finitely related A dualizable V(A) CD A has NU term

  • Theorem. (1) ⇔ (2) ⇔ (3) for any

finite algebra A. (1) (a) A is finitely related & (b) V(A) is CD. (2) A has an NU term. (3) (a) A is dualizable & (b) V(A) is CD.

[(2)⇒(b): Mitschke; (2)⇒(1)(a): Baker–Pixley; (1)⇒(2): Barto; (2)⇒(3)(a): Davey–Werner; (3)⇒(2): Davey–Heindorf–McKenzie.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 10 / 19

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SLIDE 60

Dualizable vs. Finitely Related Algebras in CM Varieties

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 11 / 19

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SLIDE 61

Dualizable vs. Finitely Related Algebras in CM Varieties

A finitely related V(A) CD V(A) CM

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 11 / 19

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SLIDE 62

Dualizable vs. Finitely Related Algebras in CM Varieties

A finitely related V(A) CD A has NU term V(A) CM A has ✁ ✁ ✁ ✁-term

  • Theorem. [1] ⇔ [2]

for any finite algebra A. [1] (a) A is finitely related & (b) V(A) is CM. [2] A has a ✁

✁ ✁ ✁-term. [[2]⇒(b): Kearnes–Sz & BIMMVW; [1]⇒[2]: Barto; [2]⇒[1](a): Aichinger–Mayr–McK;

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 11 / 19

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SLIDE 63

Dualizable vs. Finitely Related Algebras in CM Varieties

A finitely related A dualizable V(A) CD A has NU term V(A) CM A has ✁ ✁ ✁ ✁-term

  • Theorem. [1] ⇔ [2] ⇐ [3] for any

finite algebra A. [1] (a) A is finitely related & (b) V(A) is CM. [2] A has a ✁

✁ ✁ ✁-term.

[3] (a) A is dualizable & (b) V(A) is CM.

[[2]⇒(b): Kearnes–Sz & BIMMVW; [1]⇒[2]: Barto; [2]⇒[1](a): Aichinger–Mayr–McK; [3]⇒[2]: Moore.]

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 11 / 19

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SLIDE 64

Dualizable vs. Finitely Related Algebras in CM Varieties

A finitely related A dualizable V(A) CD A has NU term V(A) CM A has ✁ ✁ ✁ ✁-term

  • Theorem. [1] ⇔ [2] ⇐ [3] for any

finite algebra A. [1] (a) A is finitely related & (b) V(A) is CM. [2] A has a ✁

✁ ✁ ✁-term.

[3] (a) A is dualizable & (b) V(A) is CM.

[[2]⇒(b): Kearnes–Sz & BIMMVW; [1]⇒[2]: Barto; [2]⇒[1](a): Aichinger–Mayr–McK; [3]⇒[2]: Moore.]

[2]⇒[3](a) (e.g., nonabelian nilpotent groups)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 11 / 19

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SLIDE 65

Dualizability vs. Residual Smallness

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 66

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 67

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ). A dualizable ??? ⇒ V(A) RS

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 68

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

V(A) RS A dualizable

A dualizable ??? ⇒ V(A) RS These properties are independent,

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 69

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

V(A) RS A dualizable A has ✁ ✁ ✁ ✁-term

A dualizable ??? ⇒ V(A) RS These properties are independent, even for expanded groups.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 70

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

V(A) RS A dualizable A has ✁ ✁ ✁ ✁-term ✛

  • Z4; +, (xy)2, 0, 1
  • [Davey–Pitkethly–Willard]

A dualizable ??? ⇒ V(A) RS These properties are independent, even for expanded groups.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 71

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

V(A) RS A dualizable A has ✁ ✁ ✁ ✁-term ✛

  • Z4; +, (xy)2, 0, 1
  • [Davey–Pitkethly–Willard]

✛

For a group A, A dualizable ⇔ V(A) RS

(discussed later)

A dualizable ??? ⇒ V(A) RS These properties are independent, even for expanded groups.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 72

Dualizability vs. Residual Smallness

  • Definition. A variety V is residually small (RS) if there is a cardinal κ such

that every B ∈ V embeds in a product of algebras of size < κ. (I.e., every SI in V has size < κ).

V(A) RS A dualizable A has ✁ ✁ ✁ ✁-term ✛

  • Z4; +, (xy)2, 0, 1
  • [Davey–Pitkethly–Willard]

✛

For a group A, A dualizable ⇔ V(A) RS

(discussed later) ✛

S3 expanded by all constants

[Idziak]

A dualizable ??? ⇒ V(A) RS These properties are independent, even for expanded groups.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 12 / 19

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SLIDE 73

Main Theorem

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 74

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .
  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 75

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .

A is dualizable if the following split centralizer condition holds in every subalgebra S of A:

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 76

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .

A is dualizable if the following split centralizer condition holds in every subalgebra S of A:

s s

1 Con(S)

s s δ θ ∀ δ ≺ θ s.t. δ is ∧-irred. and [θ, θ] ≤ δ

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 77

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .

A is dualizable if the following split centralizer condition holds in every subalgebra S of A:

s s

1 Con(S)

s s δ θ ∀ δ ≺ θ s.t. δ is ∧-irred. and [θ, θ] ≤ δ s ν for ν = (δ : θ)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 78

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .

A is dualizable if the following split centralizer condition holds in every subalgebra S of A:

s s

1 Con(S)

s s δ θ ∀ δ ≺ θ s.t. δ is ∧-irred. and [θ, θ] ≤ δ s ν for ν = (δ : θ) RS,CM ⇒ [ν, ν] ≤ δ]

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 79

Main Theorem

  • Theorem. Let A be a finite algebra such that

(i) A has a ✁

✁ ✁ ✁-term

  • ⇔ A fin rel & V(A) CM
  • , and

(ii) V(A) is RS

  • ⇔ Con(S(A)) |

= x ∧ [y, y] ≈ [x ∧ y, y]

  • .

A is dualizable if the following split centralizer condition holds in every subalgebra S of A:

s s

1 Con(S)

s s δ θ ∀ δ ≺ θ s.t. δ is ∧-irred. and [θ, θ] ≤ δ s ν for ν = (δ : θ) RS,CM ⇒ [ν, ν] ≤ δ]

  • κ

❝ β s s ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅

  • α

∃κ ∈ Q-Con(S), Q = SP(A) ∃β ≤ δ ∃α with [α, α] ≤ κ s.t. α ∨ β = ν α ∧ β = κ

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 13 / 19

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SLIDE 80

Idziak’s Example

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 81

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term,

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 82

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 83

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 84

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A:

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 85

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 86

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

r r r Con(A)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 87

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

r r r Con(A) δ θ = ν

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

slide-88
SLIDE 88

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

r r r Con(A) δ θ = ν κ = 0;

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

slide-89
SLIDE 89

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

r r r Con(A) δ θ = ν κ = 0; α, β ≤ δ,

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

slide-90
SLIDE 90

Idziak’s Example

Example (Idziak): For A = (S3; all constants), A has a Maltsev term, V(A) is RS, but A is not dualizable. The split centralizer condition fails for A: S = A

r r r Con(A) δ θ = ν κ = 0; α, β ≤ δ, α ∨ β = ν

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 14 / 19

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SLIDE 91

Applications: 1. Algebras with NU Terms

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 15 / 19

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SLIDE 92

Applications: 1. Algebras with NU Terms

  • 1. [Davey–Werner]

A has an NU term ⇒ A is dualizable. Proof:

s s δ θ 1

Con(S)

r s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 15 / 19

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SLIDE 93

Applications: 1. Algebras with NU Terms

  • 1. [Davey–Werner]

A has an NU term ⇒ A is dualizable. Proof:

s s δ θ 1

Con(S)

r s

No δ ≺ θ to check.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 15 / 19

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SLIDE 94

Applications: 2. Modules

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 16 / 19

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SLIDE 95

Applications: 2. Modules

  • 2. [NEW]

A is a module ⇒ A is dualizable. Proof:

s s δ θ 1 = ν = α

Con(S) 0 = κ = β

r ❞ s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 16 / 19

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SLIDE 96

Applications: 3. Groups

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Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 97

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 98

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó])

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 99

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó]) Proof (⇒):

s s s δ θ ν 1

Con(S)

s s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 100

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó]) Proof (⇒):

s s s δ θ ν 1

Con(S)

s s [ν, ν] s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 101

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó]) Proof (⇒):

s s s δ θ ν 1

Con(S)

s s [ν, ν] s

  • Lemma. ∃ endomorphism ǫ: S → S

s.t. [ν, ν] ≤ ker ǫ ≤ δ.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 102

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó]) Proof (⇒):

s s s δ θ ν 1

Con(S)

s s [ν, ν] s

  • Lemma. ∃ endomorphism ǫ: S → S

s.t. [ν, ν] ≤ ker ǫ ≤ δ. κ := ker ǫ ❞

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 103

Applications: 3. Groups

  • 3. [Nickodemus]

A is a group whose Sylow subgroups of A are abelian (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ by [Quackenbush–Szabó]) Proof (⇒):

s s s δ θ ν 1

Con(S)

s s [ν, ν] s

  • Lemma. ∃ endomorphism ǫ: S → S

s.t. [ν, ν] ≤ ker ǫ ≤ δ. κ := ker ǫ ❞ = β = α

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 17 / 19

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SLIDE 104

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

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SLIDE 105

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • 4. [NEW]

A is a K-algebra (comm or not, has 1 or not) s.t. J(B)2 = 0 for all B ∈ HS(A) (⇔ V(A) is RS) ⇒ A is dualizable.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

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SLIDE 106

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • 4. [NEW]

A is a K-algebra (comm or not, has 1 or not) s.t. J(B)2 = 0 for all B ∈ HS(A) (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ for rings by [Szabó], ⇔ for comm rings with 1 by [CISSzW])

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

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SLIDE 107

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • 4. [NEW]

A is a K-algebra (comm or not, has 1 or not) s.t. J(B)2 = 0 for all B ∈ HS(A) (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ for rings by [Szabó], ⇔ for comm rings with 1 by [CISSzW]) Proof (⇒):

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

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SLIDE 108

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • 4. [NEW]

A is a K-algebra (comm or not, has 1 or not) s.t. J(B)2 = 0 for all B ∈ HS(A) (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ for rings by [Szabó], ⇔ for comm rings with 1 by [CISSzW]) Proof (⇒): ⊲ [McKenzie] R ∈ V(A) SI ⇒ (1) J(R) = R (i.e., R is a zero ring), or (2) J(R) = 0, R is simple, or (3) J(R) is the monolith of R, and R/J(R) ∼ = F or F × F′ (F, F′ fields)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

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SLIDE 109

Applications: 4. Rings and K-Algebras (K comm with 1)

ring = Z-algebra

  • 4. [NEW]

A is a K-algebra (comm or not, has 1 or not) s.t. J(B)2 = 0 for all B ∈ HS(A) (⇔ V(A) is RS) ⇒ A is dualizable. (⇐ for rings by [Szabó], ⇔ for comm rings with 1 by [CISSzW]) Proof (⇒): ⊲ [McKenzie] R ∈ V(A) SI ⇒ (1) J(R) = R (i.e., R is a zero ring), or (2) J(R) = 0, R is simple, or (3) J(R) is the monolith of R, and R/J(R) ∼ = F or F × F′ (F, F′ fields) ⊲ B ∈ V(A) finite ⇒ B = B/B2

zero ring

× B/(0 : B)

  • subdir. pr. of (2)–(3) SIs

, and every congruence of B is a product congruence.

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 18 / 19

slide-110
SLIDE 110

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ

s s r s δ θ 1

Con(S)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-111
SLIDE 111

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-112
SLIDE 112

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B) =

ν

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-113
SLIDE 113

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B) =

ν sθJ(S)

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-114
SLIDE 114

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B) =

ν sθJ(S) s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-115
SLIDE 115

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B) =

ν sθJ(S) s s

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19

slide-116
SLIDE 116

Applications: 4. Rings and K-Algebras, cont’d

⊲ To check the split centralizer condition for S ∈ S(A) there are two possibilities: (i) θS2 ≤ δ, (ii) θ(0:S) ≤ δ A special case of (ii): θ(0:S) = 0

s s r s δ θ 1

Con(S) 0 = θ(0:B) =

ν sθJ(S) s s

=

β α ❞

0 = θ(0:B) = κ

  • A. Szendrei (CU Boulder)

Dualizable Algebras AAA88, June 2014 19 / 19