duality for the logic of quantum actions · duality for the logic of quantum actions jort martinus...

Duality for the logic of quantum actions

Jort Martinus Bergfeld

Joint work with: A. Baltag, K. Kishida, J. Sack, S. Smets, S. ZhongInstitute for logic, language and computation

Universiteit van Amsterdam

Saturday 30 November 2013Whither Quantum Structures?

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 1 / 19

Two approaches

Piron lattices

I Algebraic approachI Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames

s

t u

P? Q?

I Spatial approachI Based on Propositional Dynamic Logic

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

Two approaches

Piron latticesI Algebraic approachI Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames

s

t u

P? Q?

I Spatial approachI Based on Propositional Dynamic Logic

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

Two approaches

Piron latticesI Algebraic approachI Every Piron lattice (with rank ≥ 4) is realizable by a generalized

Hilbert space.

Dynamic quantum frames

s

t u

P? Q?

I Spatial approachI Based on Propositional Dynamic Logic

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 2 / 19

DualityA category consists of objects and morphisms

A functor acts on objects and morphismsA duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2

Dynamic Quantum Frame

f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

DualityA category consists of objects and morphismsA functor acts on objects and morphisms

A duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2

Dynamic Quantum Frame

f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

DualityA category consists of objects and morphismsA functor acts on objects and morphismsA duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2

Dynamic Quantum Frame

f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

DualityA category consists of objects and morphismsA functor acts on objects and morphismsA duality means:

G ◦ F ' IdDQF

F ◦G ' IdPL

Σ1

Σ2

Dynamic Quantum Frame

f

PΣ1

PΣ2

f−1

L1

L2

Piron Lattice

h

F

G

⇐⇒ ⇐⇒

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 3 / 19

References

A. Baltag and S. Smets (2005), “Complete Axiomatization for Quantum Actions”,Int. J. Theor. Phys. 44, 2267–2282.B. Coecke and D. Moore (2000), “Operational Galois Adjunction”, in B. Coecke,et al., eds., Current Research in Operational Quantum Logic, pp. 195–218,Kluwer.B. Coecke and I. Stubbe (2000), “State Transitions as Morphisms for CompleteLattices”, Int. J. Theor. Phys. 39, 601–610.C.-A. Faure and A. Frölicher (1995), “Dualities for Infinite- Dimensional ProjectiveGeometries”, Geom. Ded. 56, 225–236.D. Moore (1995), “Categories of Representations of Physical Systems”, HelvetiaPhysica Acta 68, 658–678.C. Piron (1976), Foundations of Quantum Physics, W. A. Benjamin.I. Stubbe and B. Van Steirteghem (2007), “Propositional Systems, HilbertLattices and Generalized Hilbert Spaces”, in K. Engesser, et al., eds., Handbookof Quantum Logic and Quantum Structures: Quantum Structures, Elsevier, pp.477–524.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 4 / 19

Outline

1 Introduction

2 Piron latticesPL-morphisms

3 Dynamic Quantum FramesDQF-morphisms

4 Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 5 / 19

Outline

1 Introduction

2 Piron latticesPL-morphisms

3 Dynamic Quantum FramesDQF-morphisms

4 Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 6 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ L

atomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ p

orthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atom

irreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron latticesA Piron lattice L = (L,≤, (·)′,0,1) is a complete, orthomodular, atomicand irreducible lattice satisfying the covering law.

complete: X ⊆ L =⇒∧

X ∈ Latomic: for all p > 0 there exists an atom a such that a ≤ porthomodular: there is an orthocomplementation (·)′ such that

1 (p′)′ = p2 p ∧ p′ = 0 and p ∨ p′ = 13 p ≤ q =⇒ q′ ≤ p′

4 weakly modular: p ≤ q =⇒ q ∧ (q′ ∨ p) = p

covering law: if a is an atom and p a proposition such thata ∧ p = 0, then (a ∨ p) ∧ p′ is an atomirreducible if a1 6= a2, then there exists an a3 such thata1 6= a3 6= a2 and a1 ∨ a2 = a1 ∨ a3 = a2 ∨ a3

Definitiona 6= 0 is an atom if for all p ≤ a we have p = 0 or p = a.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 7 / 19

Piron Lattice morphism

Given two Piron lattices L1 = (L1,≤1, (·)′) and L2 = (L2,≤2, (·)∗), aPL-morphism is a function h : L1 → L2 such that

h(∧

S) =∧

p∈S h(s) for all S ⊆ L1.h(p′) = h(p)∗

for all atoms b ∈ L2 there exists an atom a ∈ L1 such that

b ≤ h(a).

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 8 / 19

Outline

1 Introduction

2 Piron latticesPL-morphisms

3 Dynamic Quantum FramesDQF-morphisms

4 Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 9 / 19

Dynamic quantum frame

A dynamic quantum frame is a tuple F = (Σ, { P?−→}P∈L) with:Σ a set of statesL ⊆ P(Σ) a set of testable propertiesP?−→ represent projections

→:=⋃L

P?−→ is non-orthogonality∼P := {s ∈ Σ : s 9 t for all t ∈ P}

that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frame

A dynamic quantum frame is a tuple F = (Σ, { P?−→}P∈L) with:Σ a set of statesL ⊆ P(Σ) a set of testable propertiesP?−→ represent projections

→:=⋃L

P?−→ is non-orthogonality

∼P := {s ∈ Σ : s 9 t for all t ∈ P}

that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frame

A dynamic quantum frame is a tuple F = (Σ, { P?−→}P∈L) with:Σ a set of statesL ⊆ P(Σ) a set of testable propertiesP?−→ represent projections

→:=⋃L

P?−→ is non-orthogonality∼P := {s ∈ Σ : s 9 t for all t ∈ P}

that satisfies the following properties:

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 10 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ L

atomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → s

covering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 s

proper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

Dynamic quantum frames

intersection: if X ⊆ L, then⋂

X ∈ Lorthocomplement: if P ∈ L, then ∼P ∈ Latomicity: {s} ∈ L for all s ∈ Σ

adequacy: if s ∈ P, then s P?−→ s

repeatability: if s P?−→ t , then t ∈ P

self-adjointness: if s P?−→ v → t , then there is a w such thatt P?−→ w → scovering law: if s P?−→ t 6= v ∈ P, then there is a w ∈ P such thatv → w 9 sproper superposition: for all s, t there is w such that s → w → t

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 11 / 19

DQF-morphisms

Given two DQFs F1 = (Σ1, {P?−→}P∈L1) and F2 = (Σ2, {

P?−→}P∈L2), aDQF-morphism is a function f : Σ1 → Σ2 such that

f is a bounded morpism:I if s → t , then f (s)→ f (t)I if f (s)→ w , then there exists a t ∈ Σ1 such that s → t and f (t) = w .

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 12 / 19

Outline

1 Introduction

2 Piron latticesPL-morphisms

3 Dynamic Quantum FramesDQF-morphisms

4 Duality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 13 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

DQF→ PLTheoremThe set of testable properties L is a Piron lattice with ⊆ as partial orderand ∼ as the orthocomplement.

intersection

orthocomplement

atomicity

adequacy

repeatability

self-adjointness

covering law

proper superposition

complete

atomic

orthomodular

covering law

irreducible

symmetry of→

partial functionality

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 14 / 19

PL→ DQF

Let L = (L,≤, (·)′) be a Piron lattice.Σ = atoms of LL = {atoms[p] | p ∈ L}

ap?−→ b iff b = (a ∨ p′) ∧ p

Note that:if a ∧ p′ = 0, then (a ∨ p′) ∧ p is an atom.

if b = (a ∨ p′) ∧ p, then b ∧ p = b.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

PL→ DQF

Let L = (L,≤, (·)′) be a Piron lattice.Σ = atoms of LL = {atoms[p] | p ∈ L}

ap?−→ b iff b = (a ∨ p′) ∧ p

Note that:if a ∧ p′ = 0, then (a ∨ p′) ∧ p is an atom.

if b = (a ∨ p′) ∧ p, then b ∧ p = b.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

PL→ DQF

Let L = (L,≤, (·)′) be a Piron lattice.Σ = atoms of LL = {atoms[p] | p ∈ L}

ap?−→ b iff b = (a ∨ p′) ∧ p

Note that:if a ∧ p′ = 0, then (a ∨ p′) ∧ p is an atom.if b = (a ∨ p′) ∧ p, then b ∧ p = b.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 15 / 19

DQF-→ PL-Morphisms

PL-morphismh(∧

S) =∧

p∈S h(s) for all S ⊆ L2.h(p′) = h(p)∗

for all atoms b ∈ L1 there exists an atom a ∈ L2 such that

b ≤ h(a).

Let f : Σ1 → Σ2 be a DQF-morphism.

f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF-→ PL-Morphisms

PL-morphismh(∧

S) =∧

p∈S h(s) for all S ⊆ L2.h(p′) = h(p)∗

for all atoms b ∈ L1 there exists an atom a ∈ L2 such that

b ≤ h(a).

Let f : Σ1 → Σ2 be a DQF-morphism.

f−1 preserves⋂

=∧,⋃

and ¬.

We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF-→ PL-Morphisms

PL-morphismh(∧

S) =∧

p∈S h(s) for all S ⊆ L2.h(p′) = h(p)∗

for all atoms b ∈ L1 there exists an atom a ∈ L2 such that

b ≤ h(a).

Let f : Σ1 → Σ2 be a DQF-morphism.

f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).

Remains to show: f−1 preserves ∼.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

DQF-→ PL-Morphisms

PL-morphismh(∧

S) =∧

p∈S h(s) for all S ⊆ L2.h(p′) = h(p)∗

for all atoms b ∈ L1 there exists an atom a ∈ L2 such that

b ≤ h(a).

Let f : Σ1 → Σ2 be a DQF-morphism.

f−1 preserves⋂

=∧,⋃

and ¬.We have s ∈ f−1[{t}], for t = f (s).Remains to show: f−1 preserves ∼.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 16 / 19

Preservation of ∼

Let �A = {s ∈ Σ | t ∈ A whenever s → t}

Theoremf is a bounded morphism iff f−1 preserves �.

∼A = � ◦ ¬A

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

Preservation of ∼

Let �A = {s ∈ Σ | t ∈ A whenever s → t}

Theoremf is a bounded morphism iff f−1 preserves �.

∼A = � ◦ ¬A

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

Preservation of ∼

Let �A = {s ∈ Σ | t ∈ A whenever s → t}

Theoremf is a bounded morphism iff f−1 preserves �.

∼A = � ◦ ¬A

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 17 / 19

PL-→ DQF-morphismLet h : L1 → L2 be a PL-morphism. Define ` by

`(x) =∧

x≤h(y)

y .

` is a left adjoin of h:

`(x) ≤ y ⇔ x ≤ h(y).

` sends atoms to atoms.

f = ` � (atoms of L1)

f−1 preserves �, because

� = ∼ ◦ ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL-→ DQF-morphismLet h : L1 → L2 be a PL-morphism. Define ` by

`(x) =∧

x≤h(y)

y .

` is a left adjoin of h:

`(x) ≤ y ⇔ x ≤ h(y).

` sends atoms to atoms.

f = ` � (atoms of L1)

f−1 preserves �, because

� = ∼ ◦ ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL-→ DQF-morphismLet h : L1 → L2 be a PL-morphism. Define ` by

`(x) =∧

x≤h(y)

y .

` is a left adjoin of h:

`(x) ≤ y ⇔ x ≤ h(y).

` sends atoms to atoms.

f = ` � (atoms of L1)

f−1 preserves �, because

� = ∼ ◦ ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

PL-→ DQF-morphismLet h : L1 → L2 be a PL-morphism. Define ` by

`(x) =∧

x≤h(y)

y .

` is a left adjoin of h:

`(x) ≤ y ⇔ x ≤ h(y).

` sends atoms to atoms.

f = ` � (atoms of L1)

f−1 preserves �, because

� = ∼ ◦ ¬.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 18 / 19

Thank you for your attention.

J.M. Bergfeld (ILLC) Duality for the logic of quantum actions 30 November ’13 19 / 19