Priestley duality for MV-algebras: A new perspective
Wesley Fussner
(joint work with Mai Gehrke, Sam van Gool, and Vincenzo Marra)
CNRS, France
Shanks Workshop on Ordered Algebras and Logic Nashville, TN, USA
7 March 2020
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DRAFT 2 1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
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DRAFT 2 2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
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DRAFT 2 3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
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DRAFT 2 4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
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DRAFT 2 5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
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DRAFT 2 6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
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DRAFT 2 7 Specializing this to MV-algebras.
Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
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DRAFT 2 Outline
1 Priestley duality as usually presented.
2 Canonical extensions of bounded distributive lattices.
3 Priestley duality via the canonical extension.
4 -algebras.
5 Priestley duality for -algebras by their canonical extensions.
6 Splitting of the operation on duals to obtain a more expressive environment.
7 Specializing this to MV-algebras.
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DRAFT 2 Part I: Priestley duality and canonical extensions
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DRAFT 2 Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
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DRAFT 2 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that:
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DRAFT 2 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and
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DRAFT 2 Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
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DRAFT 2 If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example:
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DRAFT 2 The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.
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DRAFT 2 , where
aˆ = {x ∈ X (L): a ∈ x}.
Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}
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DRAFT 2 Priestley spaces
Definition: A Priestley space is an ordered topological space (X , ≤, τ) such that: 1 (X , τ) is compact, and 2 for every x, y ∈ X with x 6≤ y, there exists a clopen down-set U ⊆ X such that x 6∈ U, y ∈ U.
Generic example: If L = (L, ∧, ∨, 0, 1) is a bounded distributive lattice, then the set of prime ideals X (L) of L is a Priestley space ordered by inclusion.The topology is generated by the clopen subbasis {aˆ, (ˆa)c : a ∈ L}, where
aˆ = {x ∈ X (L): a ∈ x}.
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DRAFT 2 The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products:
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
Priestley duality
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DRAFT 2 Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products:
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
Priestley duality
The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
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DRAFT 2 The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products:
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
Priestley duality
The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−].
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DRAFT 2 Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products:
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
Priestley duality
The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets.
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DRAFT 2 :
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
Priestley duality
The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products
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DRAFT 2 Priestley duality
The correspondence L 7→ X (L) may be lifted to (one functor of) an equivalence of categories between bounded distributive lattices and Priestley spaces (with continuous isotone maps).
Maps h: L1 → L2 are sent to continuous isotone maps −1 X (L2) → X (L1) given by h 7→ h [−]. The reverse functor takes a Priestley space to its bounded distributive lattice of clopen down-sets. Residuated operations · on BDLs can be captured by ternary relations that amount to the downward-closure (in X (L)) of their complex products:
R(x, y, z) ⇐⇒ x ⊆ ↓{a · b : a ∈ y, b ∈ z}.
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DRAFT 2 One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
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DRAFT 2 Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements.
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DRAFT 2 m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition:
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DRAFT 2 m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y.
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DRAFT 2 j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S.
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DRAFT 2 j is called completely join-irreducible if j = W S implies j ∈ S.
Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y.
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DRAFT 2 Meet- and join-irreducibles
One way of cashing out Priestley duality: It’s about meet/join irreducible elements. Recall: Definition: m is called meet-irreducible if m = x ∧ y implies m = x or m = y. m is called completely meet-irreducible if m = V S implies m ∈ S. j is called join-irreducible if j = x ∨ y implies j = x or j = y. j is called completely join-irreducible if j = W S implies j ∈ S.
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DRAFT 2 Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
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DRAFT 2 Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937):
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DRAFT 2 Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles).
7 / 36
DRAFT 2 Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting.
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DRAFT 2 But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all...
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DRAFT 2 , namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles
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DRAFT 2 Canonical extensions are a view of Priestley duality that exploits this.
Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones.
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DRAFT 2 Duality and irreducible elements
Key antecedent of Priestley duality (Birkhoff 1937): Each finite distributive lattice is determined by its poset of meet-irreducibles (likewise join-irreducibles). Not true in the infinite setting. Infinite distributive lattice need not have meet/join irreducibles at all... But some infinite distributive lattices are determined by their posets of meet-irreducibles, namely the doubly algebraic ones. Canonical extensions are a view of Priestley duality that exploits this.
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DRAFT 2 Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
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DRAFT 2 x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say:
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DRAFT 2 C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0.
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DRAFT 2 C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S.
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DRAFT 2 C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic.
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DRAFT 2 Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
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DRAFT 2 Doubly algebraic distributive lattices
Definition: Let C is a complete lattice. We say: x ∈ C is compact if whenever x ≤ W S there exists a finite S0 ⊆ S with x ≤ W S0. C is algebraic if for every x ∈ C there exists a set S ⊆ C of compact elements with x = W S. C is dually algebraic if its opposite lattice is algebraic. C is doubly algebraic if it is both algebraic and dually algebraic.
Every doubly algebraic distributive lattice C is determined by its poset M∞(C) of completely meet-irreducible elements.
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DRAFT 2 Let L be a bounded distributive lattice and C be a doubly algebraic distributive lattice with L be a sublattice of C. We say: L is compact in C if whenever S, T ⊆ L and V S ≤ W T in C, there exists finite subsets S0 ⊆ S and T 0 ⊆ T such that V S0 ≤ W T 0. L is separating in C if whenever x, y ∈ C with x 6≤ y, there exists a ∈ L with x 6≤ a and y ≤ a.
Definition: The canonical extension of a bounded distributive lattice L is a doubly algebraic lattice Lδ that contains L as a compact, separating sublattice.
Canonical extensions
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DRAFT 2 L is compact in C if whenever S, T ⊆ L and V S ≤ W T in C, there exists finite subsets S0 ⊆ S and T 0 ⊆ T such that V S0 ≤ W T 0. L is separating in C if whenever x, y ∈ C with x 6≤ y, there exists a ∈ L with x 6≤ a and y ≤ a.
Definition: The canonical extension of a bounded distributive lattice L is a doubly algebraic lattice Lδ that contains L as a compact, separating sublattice.
Canonical extensions
Let L be a bounded distributive lattice and C be a doubly algebraic distributive lattice with L be a sublattice of C. We say:
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DRAFT 2 L is separating in C if whenever x, y ∈ C with x 6≤ y, there exists a ∈ L with x 6≤ a and y ≤ a.
Definition: The canonical extension of a bounded distributive lattice L is a doubly algebraic lattice Lδ that contains L as a compact, separating sublattice.
Canonical extensions
Let L be a bounded distributive lattice and C be a doubly algebraic distributive lattice with L be a sublattice of C. We say: L is compact in C if whenever S, T ⊆ L and V S ≤ W T in C, there exists finite subsets S0 ⊆ S and T 0 ⊆ T such that V S0 ≤ W T 0.
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DRAFT 2 Definition: The canonical extension of a bounded distributive lattice L is a doubly algebraic lattice Lδ that contains L as a compact, separating sublattice.
Canonical extensions
Let L be a bounded distributive lattice and C be a doubly algebraic distributive lattice with L be a sublattice of C. We say: L is compact in C if whenever S, T ⊆ L and V S ≤ W T in C, there exists finite subsets S0 ⊆ S and T 0 ⊆ T such that V S0 ≤ W T 0. L is separating in C if whenever x, y ∈ C with x 6≤ y, there exists a ∈ L with x 6≤ a and y ≤ a.
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DRAFT 2 Canonical extensions
Let L be a bounded distributive lattice and C be a doubly algebraic distributive lattice with L be a sublattice of C. We say: L is compact in C if whenever S, T ⊆ L and V S ≤ W T in C, there exists finite subsets S0 ⊆ S and T 0 ⊆ T such that V S0 ≤ W T 0. L is separating in C if whenever x, y ∈ C with x 6≤ y, there exists a ∈ L with x 6≤ a and y ≤ a.
Definition: The canonical extension of a bounded distributive lattice L is a doubly algebraic lattice Lδ that contains L as a compact, separating sublattice.
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DRAFT 2 1 • • 1 • 1’
4 4 5 • • 5 4 0 • ( 5 ) 4 00 • ( 5 )
1 • e 1 e ◦ 1 0 • ( e )
• 0 0 • • 0’
δ Q ∩ [0, 1] (Q ∩ [0, 1])
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An example of the canonical extension
DRAFT 2 • 1 • 1’
4 • 5 4 0 • ( 5 ) 4 00 • ( 5 )
1 • e 1 0 • ( e )
• 0 • 0’
δ (Q ∩ [0, 1])
An example of the canonical extension
1 •
4 5 •
1 e ◦
0 •
Q ∩ [0, 1]
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DRAFT 2 An example of the canonical extension
1 • • 1 • 1’
4 4 5 • • 5 4 0 • ( 5 ) 4 00 • ( 5 )
1 • e 1 e ◦ 1 0 • ( e )
• 0 0 • • 0’
δ Q ∩ [0, 1] (Q ∩ [0, 1])
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DRAFT 2 An example of the canonical extension
1 • • 1 • 1’
4 4 5 • • 5 4 0 • ( 5 ) 4 00 • ( 5 )
1 • e 1 e ◦ 1 0 • ( e )
• 0 0 • • 0’
δ Q ∩ [0, 1] (Q ∩ [0, 1])
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DRAFT 2 Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}. The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ). The structure (M∞(Lδ), ≤, τ) is a Priestley space And is isomorphic to the (usual) Priestley dual of L. And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
Priestley duality and canonical extensions
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DRAFT 2 The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ). The structure (M∞(Lδ), ≤, τ) is a Priestley space And is isomorphic to the (usual) Priestley dual of L. And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
Priestley duality and canonical extensions
Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}.
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DRAFT 2 The structure (M∞(Lδ), ≤, τ) is a Priestley space And is isomorphic to the (usual) Priestley dual of L. And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
Priestley duality and canonical extensions
Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}. The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ).
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DRAFT 2 And is isomorphic to the (usual) Priestley dual of L. And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
Priestley duality and canonical extensions
Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}. The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ). The structure (M∞(Lδ), ≤, τ) is a Priestley space
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DRAFT 2 And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
Priestley duality and canonical extensions
Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}. The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ). The structure (M∞(Lδ), ≤, τ) is a Priestley space And is isomorphic to the (usual) Priestley dual of L.
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DRAFT 2 Priestley duality and canonical extensions
Now if L is a bounded distributive lattice, then for any a ∈ L we define aˆ = {x ∈ M∞(Lδ): a 6≤ x}. The setsa ˆ, (ˆa)c form a subbase for a topology τ on M∞(Lδ). The structure (M∞(Lδ), ≤, τ) is a Priestley space And is isomorphic to the (usual) Priestley dual of L. And here the fact that the points in the dual space are idealized meet-irreducibles is much more explicit.
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DRAFT 2 Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension. This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by _ κ(a) = (A − ↑a).
In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
Working with completely join-irreducibles
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DRAFT 2 This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by _ κ(a) = (A − ↑a).
In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
Working with completely join-irreducibles
Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension.
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DRAFT 2 _ κ(a) = (A − ↑a).
In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
Working with completely join-irreducibles
Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension. This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by
12 / 36
DRAFT 2 In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
Working with completely join-irreducibles
Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension. This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by _ κ(a) = (A − ↑a).
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DRAFT 2 I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
Working with completely join-irreducibles
Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension. This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by _ κ(a) = (A − ↑a).
In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
12 / 36
DRAFT 2 Working with completely join-irreducibles
Just like you can work with filters instead of ideals, you could work with set of completely join-irreducibles J∞(Lδ) of the canonical extension. This is because there is a poset isomorphism κ: J∞(C) → M∞(C) given by _ κ(a) = (A − ↑a).
In fact, you can work with a generic dual space X of L and present it in many ways using the obvious Priestley space isomorphisms...
I(−) : X → PrIdl(L),
F(−) : X → PrFil(L), µ: X → M∞(Lδ), ν : X → J∞(Lδ),
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DRAFT 2 These isomorphisms are connected via
Ix = L ∩ ↓µ(x), _ µ(x) = Ix ,
Fx = L ∩ ↑ν(x), ^ ν(x) = Fx , κ(ν(x)) = µ(x), c Fx = Ix .
Connecting the different presentations
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DRAFT 2 Connecting the different presentations
These isomorphisms are connected via
Ix = L ∩ ↓µ(x), _ µ(x) = Ix ,
Fx = L ∩ ↑ν(x), ^ ν(x) = Fx , κ(ν(x)) = µ(x), c Fx = Ix .
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DRAFT 2 A big advantage of the canonical extension: It is quite easy to see how to extend additional operations to duals because L is ‘dense’ in Lδ.
We call: the V-closure of L in Lδ the closed elements and denote them by K(L), and the W-closure of L in Lδ the open elements and denote them by O(L).
Open and closed elements
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DRAFT 2 We call: the V-closure of L in Lδ the closed elements and denote them by K(L), and the W-closure of L in Lδ the open elements and denote them by O(L).
Open and closed elements
A big advantage of the canonical extension: It is quite easy to see how to extend additional operations to duals because L is ‘dense’ in Lδ.
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DRAFT 2 the V-closure of L in Lδ the closed elements and denote them by K(L), and the W-closure of L in Lδ the open elements and denote them by O(L).
Open and closed elements
A big advantage of the canonical extension: It is quite easy to see how to extend additional operations to duals because L is ‘dense’ in Lδ.
We call:
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DRAFT 2 , and the W-closure of L in Lδ the open elements and denote them by O(L).
Open and closed elements
A big advantage of the canonical extension: It is quite easy to see how to extend additional operations to duals because L is ‘dense’ in Lδ.
We call: the V-closure of L in Lδ the closed elements and denote them by K(L)
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DRAFT 2 Open and closed elements
A big advantage of the canonical extension: It is quite easy to see how to extend additional operations to duals because L is ‘dense’ in Lδ.
We call: the V-closure of L in Lδ the closed elements and denote them by K(L), and the W-closure of L in Lδ the open elements and denote them by O(L).
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DRAFT 2 We extend operations to the canonical extension by ‘approximating’ with closed and open elements. First, we extend unary maps f : A → B to maps between the canonical extensions Aδ → Bδ This extends unary operations (maps f : A → A). Operations of higher arity can be extended by using the fact that (A × B)δ =∼ Aδ × Bδ.
Extending additional operations
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DRAFT 2 First, we extend unary maps f : A → B to maps between the canonical extensions Aδ → Bδ This extends unary operations (maps f : A → A). Operations of higher arity can be extended by using the fact that (A × B)δ =∼ Aδ × Bδ.
Extending additional operations
We extend operations to the canonical extension by ‘approximating’ with closed and open elements.
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DRAFT 2 This extends unary operations (maps f : A → A). Operations of higher arity can be extended by using the fact that (A × B)δ =∼ Aδ × Bδ.
Extending additional operations
We extend operations to the canonical extension by ‘approximating’ with closed and open elements. First, we extend unary maps f : A → B to maps between the canonical extensions Aδ → Bδ
15 / 36
DRAFT 2 Operations of higher arity can be extended by using the fact that (A × B)δ =∼ Aδ × Bδ.
Extending additional operations
We extend operations to the canonical extension by ‘approximating’ with closed and open elements. First, we extend unary maps f : A → B to maps between the canonical extensions Aδ → Bδ This extends unary operations (maps f : A → A).
15 / 36
DRAFT 2 Extending additional operations
We extend operations to the canonical extension by ‘approximating’ with closed and open elements. First, we extend unary maps f : A → B to maps between the canonical extensions Aδ → Bδ This extends unary operations (maps f : A → A). Operations of higher arity can be extended by using the fact that (A × B)δ =∼ Aδ × Bδ.
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DRAFT 2 There are two natural ways to extend a map f : A → B to the canonical extension. For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
Further: _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
The σ- and π-extensions
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DRAFT 2 For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
Further: _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
The σ- and π-extensions
There are two natural ways to extend a map f : A → B to the canonical extension.
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DRAFT 2 Further: _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
The σ- and π-extensions
There are two natural ways to extend a map f : A → B to the canonical extension. For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
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DRAFT 2 _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
The σ- and π-extensions
There are two natural ways to extend a map f : A → B to the canonical extension. For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
Further:
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DRAFT 2 and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
The σ- and π-extensions
There are two natural ways to extend a map f : A → B to the canonical extension. For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
Further: _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
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DRAFT 2 The σ- and π-extensions
There are two natural ways to extend a map f : A → B to the canonical extension. For p ∈ K(Aδ) and u ∈ O(Aδ), set:
[p, u] = {y ∈ Aδ : p ≤ y ≤ u}.
Further: _ n^ o f σ(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u
and ^ n_ o f π(x) = f ([p, u] ∩ A): p ∈ K(Aδ), u ∈ O(Aδ), and p ≤ x ≤ u .
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DRAFT 2 Note that we always have f σ ≤ f π.
But f σ 6= f π except in very special cases.
When f σ = f π we say that f is smooth (and write f δ).
The fact that the non-lattice operations of MV-algebras aren’t smooth is fundamental to the difficulty of MV.
Some properties of the σ- and π-extensions
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DRAFT 2 But f σ 6= f π except in very special cases.
When f σ = f π we say that f is smooth (and write f δ).
The fact that the non-lattice operations of MV-algebras aren’t smooth is fundamental to the difficulty of MV.
Some properties of the σ- and π-extensions
Note that we always have f σ ≤ f π.
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DRAFT 2 When f σ = f π we say that f is smooth (and write f δ).
The fact that the non-lattice operations of MV-algebras aren’t smooth is fundamental to the difficulty of MV.
Some properties of the σ- and π-extensions
Note that we always have f σ ≤ f π.
But f σ 6= f π except in very special cases.
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DRAFT 2 The fact that the non-lattice operations of MV-algebras aren’t smooth is fundamental to the difficulty of MV.
Some properties of the σ- and π-extensions
Note that we always have f σ ≤ f π.
But f σ 6= f π except in very special cases.
When f σ = f π we say that f is smooth (and write f δ).
17 / 36
DRAFT 2 Some properties of the σ- and π-extensions
Note that we always have f σ ≤ f π.
But f σ 6= f π except in very special cases.
When f σ = f π we say that f is smooth (and write f δ).
The fact that the non-lattice operations of MV-algebras aren’t smooth is fundamental to the difficulty of MV.
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DRAFT 2 Dual of a map f usually obtained by
I 7→ f −1[I ].
If f : A → B is a lattice homomorphism, then f is smooth.
And there exists a unique map f δ] : Bδ → Aδ such that
f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
We call the restriction of f δ] to M∞(Bδ) the dual of f .
The duals of homomorphisms
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DRAFT 2 If f : A → B is a lattice homomorphism, then f is smooth.
And there exists a unique map f δ] : Bδ → Aδ such that
f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
We call the restriction of f δ] to M∞(Bδ) the dual of f .
The duals of homomorphisms
Dual of a map f usually obtained by
I 7→ f −1[I ].
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DRAFT 2 And there exists a unique map f δ] : Bδ → Aδ such that
f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
We call the restriction of f δ] to M∞(Bδ) the dual of f .
The duals of homomorphisms
Dual of a map f usually obtained by
I 7→ f −1[I ].
If f : A → B is a lattice homomorphism, then f is smooth.
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DRAFT 2 f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
We call the restriction of f δ] to M∞(Bδ) the dual of f .
The duals of homomorphisms
Dual of a map f usually obtained by
I 7→ f −1[I ].
If f : A → B is a lattice homomorphism, then f is smooth.
And there exists a unique map f δ] : Bδ → Aδ such that
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DRAFT 2 We call the restriction of f δ] to M∞(Bδ) the dual of f .
The duals of homomorphisms
Dual of a map f usually obtained by
I 7→ f −1[I ].
If f : A → B is a lattice homomorphism, then f is smooth.
And there exists a unique map f δ] : Bδ → Aδ such that
f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
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DRAFT 2 The duals of homomorphisms
Dual of a map f usually obtained by
I 7→ f −1[I ].
If f : A → B is a lattice homomorphism, then f is smooth.
And there exists a unique map f δ] : Bδ → Aδ such that
f δ(x) ≤ y ⇐⇒ x ≤ f δ](y).
We call the restriction of f δ] to M∞(Bδ) the dual of f .
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DRAFT 2 Part II: MV-algebras, -algebras, and the duality
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DRAFT 2 Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
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DRAFT 2 In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction.
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DRAFT 2 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that:
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DRAFT 2 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid,
20 / 36
DRAFT 2 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A,
20 / 36
DRAFT 2 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and
20 / 36
DRAFT 2 The last condition is often called (MV6).
MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
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DRAFT 2 MV-algebras
Instead of the residuated lattice signature, it’s convenient for us to work with the MV-algebra addition/subtraction. In this language...
Definition: An MV-algebra is an algebraic structure (A, ⊕, 0, ¬) such that: 1 (A, ⊕, 0) is a commutative monoid, 2 ¬¬x = x for all x ∈ A, 3 x ⊕ 1 = 1 for all x ∈ A, where 1 := ¬0, and 4 ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x for all x, y ∈ A.
The last condition is often called (MV6).
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DRAFT 2 (MV6) gives MV-algebras most of their nice algebraic properties. In particular, the terms
x ∨ y := ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x
define the join operation ∨ of a lattice, so MV-algebras are lattice-ordered. If we set x y = ¬(¬x ⊕ y), we also obtain an operation that satisfies the (co-)residuation law
x ≤ y ⊕ z ⇐⇒ x z ≤ y.
The consequences of (MV6)
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DRAFT 2 In particular, the terms
x ∨ y := ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x
define the join operation ∨ of a lattice, so MV-algebras are lattice-ordered. If we set x y = ¬(¬x ⊕ y), we also obtain an operation that satisfies the (co-)residuation law
x ≤ y ⊕ z ⇐⇒ x z ≤ y.
The consequences of (MV6)
(MV6) gives MV-algebras most of their nice algebraic properties.
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DRAFT 2 If we set x y = ¬(¬x ⊕ y), we also obtain an operation that satisfies the (co-)residuation law
x ≤ y ⊕ z ⇐⇒ x z ≤ y.
The consequences of (MV6)
(MV6) gives MV-algebras most of their nice algebraic properties. In particular, the terms
x ∨ y := ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x
define the join operation ∨ of a lattice, so MV-algebras are lattice-ordered.
21 / 36
DRAFT 2 The consequences of (MV6)
(MV6) gives MV-algebras most of their nice algebraic properties. In particular, the terms
x ∨ y := ¬(¬x ⊕ y) ⊕ y = ¬(¬y ⊕ x) ⊕ x
define the join operation ∨ of a lattice, so MV-algebras are lattice-ordered. If we set x y = ¬(¬x ⊕ y), we also obtain an operation that satisfies the (co-)residuation law
x ≤ y ⊕ z ⇐⇒ x z ≤ y.
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DRAFT 2 MV-algebras have proven notoriously resistant to a useful duality-theoretic treatment. Gehrke and Priestley showed that the characteristic identity (MV6) is not canonical. This fact is behind a lot of the complexity. For example, Cabrer and Cignoli gave a duality for many algebras in the vicinity but could only find complicated second-order conditions that dualize (MV6).
(MV6) and duality theory
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DRAFT 2 Gehrke and Priestley showed that the characteristic identity (MV6) is not canonical. This fact is behind a lot of the complexity. For example, Cabrer and Cignoli gave a duality for many algebras in the vicinity but could only find complicated second-order conditions that dualize (MV6).
(MV6) and duality theory
MV-algebras have proven notoriously resistant to a useful duality-theoretic treatment.
22 / 36
DRAFT 2 This fact is behind a lot of the complexity. For example, Cabrer and Cignoli gave a duality for many algebras in the vicinity but could only find complicated second-order conditions that dualize (MV6).
(MV6) and duality theory
MV-algebras have proven notoriously resistant to a useful duality-theoretic treatment. Gehrke and Priestley showed that the characteristic identity (MV6) is not canonical.
22 / 36
DRAFT 2 (MV6) and duality theory
MV-algebras have proven notoriously resistant to a useful duality-theoretic treatment. Gehrke and Priestley showed that the characteristic identity (MV6) is not canonical. This fact is behind a lot of the complexity. For example, Cabrer and Cignoli gave a duality for many algebras in the vicinity but could only find complicated second-order conditions that dualize (MV6).
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DRAFT 2 But this generalizes to other order types. Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
(a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
2 For all a ∈ A we have 0 a = 0 and a 1 = 0. 3 For all a ∈ A, a 0 = a.
-algebras
We will actually work with algebras that model the MV without ⊕.
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DRAFT 2 Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
(a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
2 For all a ∈ A we have 0 a = 0 and a 1 = 0. 3 For all a ∈ A, a 0 = a.
-algebras
We will actually work with algebras that model the MV without ⊕. But this generalizes to other order types.
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DRAFT 2 (a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
2 For all a ∈ A we have 0 a = 0 and a 1 = 0. 3 For all a ∈ A, a 0 = a.
-algebras
We will actually work with algebras that model the MV without ⊕. But this generalizes to other order types. Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
23 / 36
DRAFT 2 2 For all a ∈ A we have 0 a = 0 and a 1 = 0. 3 For all a ∈ A, a 0 = a.
-algebras
We will actually work with algebras that model the MV without ⊕. But this generalizes to other order types. Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
(a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
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DRAFT 2 3 For all a ∈ A, a 0 = a.
-algebras
We will actually work with algebras that model the MV without ⊕. But this generalizes to other order types. Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
(a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
2 For all a ∈ A we have 0 a = 0 and a 1 = 0.
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DRAFT 2 -algebras
We will actually work with algebras that model the MV without ⊕. But this generalizes to other order types. Definition: A -algebra, (A, ), is a bounded distributive lattice A equipped with a binary operation satisfying: 1 For all a, b, c ∈ A,
(a ∧ b) c = (a c) ∧ (b c) (a ∨ b) c = (a c) ∨ (b c) a (b ∧ c) = (a b) ∨ (a c) a (b ∨ c) = (a b) ∧ (a c)
2 For all a ∈ A we have 0 a = 0 and a 1 = 0. 3 For all a ∈ A, a 0 = a.
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DRAFT 2 Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
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DRAFT 2 , then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras
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DRAFT 2 Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV.
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DRAFT 2 For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras.
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DRAFT 2 Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
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DRAFT 2 (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A:
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DRAFT 2 (ii) a ∧ b = a (a b).
MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c),
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DRAFT 2 MV-algebras as -algebras
Strategy: Develop a duality for -algebras, then specialize to MV. Success depends on being able to describe the -algebras coming from MV-algebras. For an -algebra (A, ), we set ¬a := 1 a and a ⊕ b := ¬(¬a b).
Proposition (F.): Let A = (A, ∨, ∧, , ¬, 0, 1) be a -algebra. Then with ⊕ and ¬ defined as above, (A, ⊕, 0, ¬) is an MV-algebra iff for all a, b, c ∈ A: (i)( a b) c = a ¬(¬b c), (ii) a ∧ b = a (a b).
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DRAFT 2 We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a.
Duality for -algebras by canonical extensions
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DRAFT 2 We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras?
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DRAFT 2 Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces.
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DRAFT 2 , so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A
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DRAFT 2 We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i.
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DRAFT 2 σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
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DRAFT 2 σ σ] +
π π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
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DRAFT 2 σ] +
π[ ?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ
π
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DRAFT 2 +
?
Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ]
π π[
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DRAFT 2 Duality for -algebras by canonical extensions
How do we get a duality for -algebras? We ‘split’ the operation into three pieces. Convenient to consider the operation ¬ defined by ¬a = 1 a. We can view this as a lattice homomorphism Aop → A, so it has a dual map that we denote by i. We also look at the σ- and π-extensions of and extract two partial operations on the Priestley dual:
σ σ] +
π π[ ?
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DRAFT 2 To follow the process explicitly: For each completely join-irreducible j ∈ Aδ, the map
u 7→ u π j
has range [0, ¬δj]. And we may show that this map has an adjoint π[ determined by the property that for all j ∈ J∞(Aδ), v ∈ Aδ, u ∈ [0, ¬δj],
u ≤ v π j ⇐⇒ u π[ j ≤ v.
The π-extension of
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DRAFT 2 For each completely join-irreducible j ∈ Aδ, the map
u 7→ u π j
has range [0, ¬δj]. And we may show that this map has an adjoint π[ determined by the property that for all j ∈ J∞(Aδ), v ∈ Aδ, u ∈ [0, ¬δj],
u ≤ v π j ⇐⇒ u π[ j ≤ v.
The π-extension of
To follow the process explicitly:
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DRAFT 2 And we may show that this map has an adjoint π[ determined by the property that for all j ∈ J∞(Aδ), v ∈ Aδ, u ∈ [0, ¬δj],
u ≤ v π j ⇐⇒ u π[ j ≤ v.
The π-extension of
To follow the process explicitly: For each completely join-irreducible j ∈ Aδ, the map
u 7→ u π j
has range [0, ¬δj].
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DRAFT 2 The π-extension of
To follow the process explicitly: For each completely join-irreducible j ∈ Aδ, the map
u 7→ u π j
has range [0, ¬δj]. And we may show that this map has an adjoint π[ determined by the property that for all j ∈ J∞(Aδ), v ∈ Aδ, u ∈ [0, ¬δj],
u ≤ v π j ⇐⇒ u π[ j ≤ v.
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DRAFT 2 Similar remarks show that σ has an adjoint determined for all u, v ∈ Aδ and m ∈ M∞(Aδ) by
u σ m ≤ v ⇐⇒ u ≤ v σ] m.
These maps can be restricted to partial binary operations on the dual of A, i.e., on M∞(Aδ) (Fussner–Palmigiano, Gehrke–Priestley). σ] manifests as a partial operation that we call +, and π[ as an operation that we call ?.
The σ-extension of
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DRAFT 2 These maps can be restricted to partial binary operations on the dual of A, i.e., on M∞(Aδ) (Fussner–Palmigiano, Gehrke–Priestley). σ] manifests as a partial operation that we call +, and π[ as an operation that we call ?.
The σ-extension of
Similar remarks show that σ has an adjoint determined for all u, v ∈ Aδ and m ∈ M∞(Aδ) by
u σ m ≤ v ⇐⇒ u ≤ v σ] m.
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DRAFT 2 σ] manifests as a partial operation that we call +, and π[ as an operation that we call ?.
The σ-extension of
Similar remarks show that σ has an adjoint determined for all u, v ∈ Aδ and m ∈ M∞(Aδ) by
u σ m ≤ v ⇐⇒ u ≤ v σ] m.
These maps can be restricted to partial binary operations on the dual of A, i.e., on M∞(Aδ) (Fussner–Palmigiano, Gehrke–Priestley).
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DRAFT 2 The σ-extension of
Similar remarks show that σ has an adjoint determined for all u, v ∈ Aδ and m ∈ M∞(Aδ) by
u σ m ≤ v ⇐⇒ u ≤ v σ] m.
These maps can be restricted to partial binary operations on the dual of A, i.e., on M∞(Aδ) (Fussner–Palmigiano, Gehrke–Priestley). σ] manifests as a partial operation that we call +, and π[ as an operation that we call ?.
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DRAFT 2 Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
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DRAFT 2 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where:
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DRAFT 2 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function,
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DRAFT 2 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)},
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DRAFT 2 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y},
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DRAFT 2 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates,
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DRAFT 2 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y}
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DRAFT 2 Abstract -spaces
Definition: A -space is a Priestley space X expanded by operations i, +, ?, where: 1 i : X → X is a continuous order-reversing function, 2 + is an upper continuous partial function with dom(+) = {(x, y) ∈ X 2 | y ≤ i(x)}, 3 ? is a lower continuous partial function with 2 dom(?) = {(x, y) ∈ X | i(x) y}, 4 + and ? are order preserving in both coordinates, 5 for any (x, y) ∈ dom(?),
x ? y = inf{x + w | (x, w) ∈ dom(+) and w y} 6 for any x ∈ X , the image of the left translation y 7→ x + y is a totally-ordered subset of ↑x, and moreover this function has an upper adjoint.
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DRAFT 2 As always, the core of a duality is in its morphisms. Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
Morphisms
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DRAFT 2 Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
Morphisms
As always, the core of a duality is in its morphisms.
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DRAFT 2 1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
Morphisms
As always, the core of a duality is in its morphisms. Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
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DRAFT 2 2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
Morphisms
As always, the core of a duality is in its morphisms. Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
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DRAFT 2 3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
Morphisms
As always, the core of a duality is in its morphisms. Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
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DRAFT 2 Morphisms
As always, the core of a duality is in its morphisms. Definition:
A morphism from a -space (X1, i1, +1,?1) to a -space (X2, i2, +2,?2) is a continuous isotone function f : X1 → X2 such that
1 for all x ∈ X1, f (i1(x)) = i2(f (x)),
2 for all x, y ∈ X1, if (x, y) ∈ dom(+1), then f (x) +2 f (y) ≤ f (x +1 y),
3 for all x ∈ X1 and z ∈ X2, if (f (x), z) ∈ dom(+2), then there 0 0 0 exists w ∈ X1 such that (x, w ) ∈ dom(+1), z ≤ f (w ), and 0 f (x +1 w ) = f (x) +2 z.
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DRAFT 2 Considering the category of -spaces with morphisms as defined above, we obtain...
Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
This forms the basis of our duality for MV-algebras. In particular, we specialize to those -algebras that give MV-algebras. And this class can be transparently described on the duals.
Priestley duality for -algebras
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DRAFT 2 Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
This forms the basis of our duality for MV-algebras. In particular, we specialize to those -algebras that give MV-algebras. And this class can be transparently described on the duals.
Priestley duality for -algebras
Considering the category of -spaces with morphisms as defined above, we obtain...
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DRAFT 2 This forms the basis of our duality for MV-algebras. In particular, we specialize to those -algebras that give MV-algebras. And this class can be transparently described on the duals.
Priestley duality for -algebras
Considering the category of -spaces with morphisms as defined above, we obtain...
Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
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DRAFT 2 In particular, we specialize to those -algebras that give MV-algebras. And this class can be transparently described on the duals.
Priestley duality for -algebras
Considering the category of -spaces with morphisms as defined above, we obtain...
Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
This forms the basis of our duality for MV-algebras.
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DRAFT 2 And this class can be transparently described on the duals.
Priestley duality for -algebras
Considering the category of -spaces with morphisms as defined above, we obtain...
Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
This forms the basis of our duality for MV-algebras. In particular, we specialize to those -algebras that give MV-algebras.
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DRAFT 2 Priestley duality for -algebras
Considering the category of -spaces with morphisms as defined above, we obtain...
Theorem (F., Gehrke, van Gool, Marra): The category of -algebras is dually equivalent to the category of -spaces.
This forms the basis of our duality for MV-algebras. In particular, we specialize to those -algebras that give MV-algebras. And this class can be transparently described on the duals.
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DRAFT 2 (i) i(i(x)) = x. (ii) x + y = y + x. (iii) x + (y + z) = (x + y) + z. (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x. (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X ,
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DRAFT 2 (ii) x + y = y + x. (iii) x + (y + z) = (x + y) + z. (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x. (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X , (i) i(i(x)) = x.
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DRAFT 2 (iii) x + (y + z) = (x + y) + z. (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x. (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X , (i) i(i(x)) = x. (ii) x + y = y + x.
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DRAFT 2 (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x. (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X , (i) i(i(x)) = x. (ii) x + y = y + x. (iii) x + (y + z) = (x + y) + z.
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DRAFT 2 (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X , (i) i(i(x)) = x. (ii) x + y = y + x. (iii) x + (y + z) = (x + y) + z. (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x.
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DRAFT 2 Specializing to MV-algebras
Theorem (F., Gehrke, van Gool, Marra): The category of MV-algebras with MV-algebra homomorphisms is dually equivalent to the category of -spaces (X , i, +,?) such that for all x, y, z ∈ X , (i) i(i(x)) = x. (ii) x + y = y + x. (iii) x + (y + z) = (x + y) + z. (iv) i(x) + y ≤ i(z) if and only if z + y ≤ x. (v) If there exists w y such that z + w ≤ x ? y, then z ≤ x.
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DRAFT 2 Note that all of the conditions we obtain when specializing to MV-algebras are simple first-order properties. In particular, the troublesome axiom (MV6) is dualized by the condition that for all (x, y) ∈ dom(?),
w y and z + w ≤ x ? y =⇒ z ≤ x.
This axiom did not even have a first-order equivalent in previously-known dualities, but in the language of these two partial operations it is transparent.
Remarks on the duality
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DRAFT 2 In particular, the troublesome axiom (MV6) is dualized by the condition that for all (x, y) ∈ dom(?),
w y and z + w ≤ x ? y =⇒ z ≤ x.
This axiom did not even have a first-order equivalent in previously-known dualities, but in the language of these two partial operations it is transparent.
Remarks on the duality
Note that all of the conditions we obtain when specializing to MV-algebras are simple first-order properties.
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DRAFT 2 w y and z + w ≤ x ? y =⇒ z ≤ x.
This axiom did not even have a first-order equivalent in previously-known dualities, but in the language of these two partial operations it is transparent.
Remarks on the duality
Note that all of the conditions we obtain when specializing to MV-algebras are simple first-order properties. In particular, the troublesome axiom (MV6) is dualized by the condition that for all (x, y) ∈ dom(?),
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DRAFT 2 This axiom did not even have a first-order equivalent in previously-known dualities, but in the language of these two partial operations it is transparent.
Remarks on the duality
Note that all of the conditions we obtain when specializing to MV-algebras are simple first-order properties. In particular, the troublesome axiom (MV6) is dualized by the condition that for all (x, y) ∈ dom(?),
w y and z + w ≤ x ? y =⇒ z ≤ x.
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DRAFT 2 Remarks on the duality
Note that all of the conditions we obtain when specializing to MV-algebras are simple first-order properties. In particular, the troublesome axiom (MV6) is dualized by the condition that for all (x, y) ∈ dom(?),
w y and z + w ≤ x ? y =⇒ z ≤ x.
This axiom did not even have a first-order equivalent in previously-known dualities, but in the language of these two partial operations it is transparent.
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DRAFT 2 Consider the Chang MV-algebra as the rotation of the cancellative hoop {0, −1, −2,... }. The following describes + and ? when defined:
↓(0, a) + ↓(0, b) = ↓(0, b) + ↓(0, a) = ↓(0, a + b)
C + ↓(0, a) = ↓(0, a) + C = C, and ↓(0, a) + ↓(1, b) = ↓(1, b) + ↓(0, a) = ↓(1, a − b) for a < b, ↓(0, a) ? ↓(0, b) = ↓(0, b) ? ↓(0, a) = ↓(0, a + b − 1), ↓(0, a) ? C = C ? ↓(0, a) = C, ↓(0, a) ? ↓(1, b) = ↓(1, a − b − 1) if b < a, and ↓(1, a) ? ↓(0, b) = ↓(1, a − b + 1) if b < a,
An example: The Chang algebra
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DRAFT 2 An example: The Chang algebra
Consider the Chang MV-algebra as the rotation of the cancellative hoop {0, −1, −2,... }. The following describes + and ? when defined:
↓(0, a) + ↓(0, b) = ↓(0, b) + ↓(0, a) = ↓(0, a + b)
C + ↓(0, a) = ↓(0, a) + C = C, and ↓(0, a) + ↓(1, b) = ↓(1, b) + ↓(0, a) = ↓(1, a − b) for a < b, ↓(0, a) ? ↓(0, b) = ↓(0, b) ? ↓(0, a) = ↓(0, a + b − 1), ↓(0, a) ? C = C ? ↓(0, a) = C, ↓(0, a) ? ↓(1, b) = ↓(1, a − b − 1) if b < a, and ↓(1, a) ? ↓(0, b) = ↓(1, a − b + 1) if b < a,
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DRAFT 2 Let FMV (1) be the free MV-algebra on one generator, i.e., the subalgebra of [0, 1][0,1] whose members are piecewise linear with integer coefficients.
The Chang algebra is the quotient of FMV (1) by the prime MV-ideal consisting of all f ∈ FMV (1) such that f [0,)= 0 for some > 0
One could similarly compute the duals of quotients at other prime MV-ideals.
The duals of free MV-algebras
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DRAFT 2 , i.e., the subalgebra of [0, 1][0,1] whose members are piecewise linear with integer coefficients.
The Chang algebra is the quotient of FMV (1) by the prime MV-ideal consisting of all f ∈ FMV (1) such that f [0,)= 0 for some > 0
One could similarly compute the duals of quotients at other prime MV-ideals.
The duals of free MV-algebras
Let FMV (1) be the free MV-algebra on one generator
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DRAFT 2 The Chang algebra is the quotient of FMV (1) by the prime MV-ideal consisting of all f ∈ FMV (1) such that f [0,)= 0 for some > 0
One could similarly compute the duals of quotients at other prime MV-ideals.
The duals of free MV-algebras
Let FMV (1) be the free MV-algebra on one generator, i.e., the subalgebra of [0, 1][0,1] whose members are piecewise linear with integer coefficients.
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DRAFT 2 One could similarly compute the duals of quotients at other prime MV-ideals.
The duals of free MV-algebras
Let FMV (1) be the free MV-algebra on one generator, i.e., the subalgebra of [0, 1][0,1] whose members are piecewise linear with integer coefficients.
The Chang algebra is the quotient of FMV (1) by the prime MV-ideal consisting of all f ∈ FMV (1) such that f [0,)= 0 for some > 0
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DRAFT 2 The duals of free MV-algebras
Let FMV (1) be the free MV-algebra on one generator, i.e., the subalgebra of [0, 1][0,1] whose members are piecewise linear with integer coefficients.
The Chang algebra is the quotient of FMV (1) by the prime MV-ideal consisting of all f ∈ FMV (1) such that f [0,)= 0 for some > 0
One could similarly compute the duals of quotients at other prime MV-ideals.
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DRAFT 2 These quotients are the stalks in the sheaf representation of FMV (1) (Gehrke–van Gool–Marra 2014) over its space of prime MV-ideals.
So examples like the above give you the dual of FMV (1) (and even FMV (n)).
The duals of free MV-algebras
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DRAFT 2 So examples like the above give you the dual of FMV (1) (and even FMV (n)).
The duals of free MV-algebras
These quotients are the stalks in the sheaf representation of FMV (1) (Gehrke–van Gool–Marra 2014) over its space of prime MV-ideals.
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DRAFT 2 (and even FMV (n)).
The duals of free MV-algebras
These quotients are the stalks in the sheaf representation of FMV (1) (Gehrke–van Gool–Marra 2014) over its space of prime MV-ideals.
So examples like the above give you the dual of FMV (1)
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DRAFT 2 The duals of free MV-algebras
These quotients are the stalks in the sheaf representation of FMV (1) (Gehrke–van Gool–Marra 2014) over its space of prime MV-ideals.
So examples like the above give you the dual of FMV (1) (and even FMV (n)).
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DRAFT 2 Thank you!
Thank you!
Preprint at: arXiv 2002.12715
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DRAFT 2