On the bitopological nature of Stone
Duality
Achim Jung
University of Birmingham, UK
joint work with
M. Andrew Moshier
Chapman University, Orange, CA
June 3, 2010
AAA 80, Bedlewo, June 3, 2010
I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
AAA 80, Bedlewo, June 3, 2010
1
Road map
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2
Representation of Boolean algebras
(Stone 1936)
Question: Is every Boolean algebra a subalgebra of a powerset?
Stone’s construction:
ϕ: B
−→ P(B̂)
where
a 7−→ {F | a ∈ F }
B̂ := pFilt(B),
the prime filters of B
Lemma. [Prime Filter Theorem]
The function ϕ is injective.
Theorem. [Stone Representation Theorem]
Every Boolean algebra is a subalgebra of a powerset.
AAA 80, Bedlewo, June 3, 2010
3
Representation of Boolean algebras
(Stone 1936)
Question: Is every Boolean algebra a subalgebra of a powerset?
Stone’s construction:
ϕ: B
−→ P(B̂)
where
a 7−→ {F | a ∈ F }
B̂ := pFilt(B),
the prime filters of B
Observation.
The function ϕ is a Boolean algebra homomorphism.
Lemma. [Prime Filter Theorem]
The function ϕ is injective.
Theorem. [Stone Representation Theorem]
Every Boolean algebra is a subalgebra of a powerset.
AAA 80, Bedlewo, June 3, 2010
3
Representation of Boolean algebras
(Stone 1936)
Question: Is every Boolean algebra a subalgebra of a powerset?
Stone’s construction:
ϕ: B
−→ P(B̂)
where
a 7−→ {F | a ∈ F }
B̂ := pFilt(B),
the prime filters of B
Observation.
The function ϕ is a Boolean algebra homomorphism.
Lemma. [Prime Filter Theorem]
The function ϕ is injective.
Theorem. [Stone Representation Theorem]
Every Boolean algebra is a subalgebra of a powerset.
AAA 80, Bedlewo, June 3, 2010
3
Representation of Boolean algebras
(Stone 1936)
Question: Is every Boolean algebra a subalgebra of a powerset?
Stone’s construction:
ϕ: B
−→ P(B̂)
where
a 7−→ {F | a ∈ F }
B̂ := pFilt(B),
the prime filters of B
Observation.
The function ϕ is a Boolean algebra homomorphism.
Lemma. [Prime Filter Theorem]
The function ϕ is injective.
Theorem. [Stone Representation Theorem]
Every Boolean algebra is a subalgebra of a powerset.
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Representing morphisms
B
h
B̂
↑
7−→
↓
B′
ĥ(F ′) := h−1(F )
ĥ
B̂ ′
Lemma.
The map ĥ is continuous if B̂ is equipped with the topology
generated by the sets
ϕ(a) = {F | a ∈ F }
a∈B
“Stone topology”
and similarly for B̂.
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Representing morphisms
B
h
B̂
↑
7−→
ĥ(F ′) := h−1(F )
ĥ
↓
B′
B̂ ′
Observation.
This is a contravariant functor.
Lemma.
The map ĥ is continuous if B̂ is equipped with the topology
generated by the sets
ϕ(a) = {F | a ∈ F }
a∈B
“Stone topology”
and similarly for B̂.
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Representing morphisms
B
h
B̂
↑
7−→
ĥ(F ′) := h−1(F )
ĥ
↓
B′
B̂ ′
Observation.
This is a contravariant functor.
Lemma.
The map ĥ is continuous if B̂ is equipped with the topology
generated by the sets
ϕ(a) = {F | a ∈ F }
a∈B
“Stone topology”
and similarly for B̂.
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Be wise, topologize!
Theorem. B ≈ clopen(B̂)
Theorem. The assignment
Hom(B, B ′) −→ Cont(B̂ ′, B̂)
h 7−→ ĥ
is a bijection.
Theorem. The spaces B̂ are exactly the completely
disconnected compact Hausdorff spaces. (The Stone spaces)
Theorem. [Duality Theorem]
The categories Bool and Stone are dually equivalent.
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Bool ≈ Stone
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Representation of bounded distributive lattices
(Stone 1937)
NB. The Prime Filter Theorem does not rely on negation, so
we can try to keep everything the same.
L −→ P(L̂)
L̂ := pFilt(L)
Theorem. [Representation Theorem]
Every bounded distributive lattice is isomorphic to a sublattice
of a powerset.
Morphisms:
AAA 80, Bedlewo, June 3, 2010
L
h↓
L′
7−→
L̂
↑ ĥ
L̂′
ĥ(F ′) := h−1(F ′)
7
Representation of bounded distributive lattices
(Stone 1937)
NB. The Prime Filter Theorem does not rely on negation, so
we can try to keep everything the same.
L −→ P(L̂)
L̂ := pFilt(L)
Theorem. [Representation Theorem]
Every bounded distributive lattice is isomorphic to a sublattice
of a powerset.
Morphisms:
AAA 80, Bedlewo, June 3, 2010
L
h↓
L′
7−→
L̂
↑ ĥ
L̂′
ĥ(F ′) := h−1(F ′)
7
Representation of bounded distributive lattices
(Stone 1937)
NB. The Prime Filter Theorem does not rely on negation, so
we can try to keep everything the same.
L −→ P(L̂)
L̂ := pFilt(L)
Theorem. [Representation Theorem]
Every bounded distributive lattice is isomorphic to a sublattice
of a powerset.
Morphisms:
AAA 80, Bedlewo, June 3, 2010
L
h↓
L′
7−→
L̂
↑ ĥ
L̂′
ĥ(F ′) := h−1(F ′)
7
Be wise . . .
. . . topologize with the Stone topology generated by
ϕ(a) = {F | a ∈ F }, a ∈ L.
Theorem. L is isomorphic to the compact-open subsets of L̂.
Theorem. The spaces L̂ are characterized as those that are
•
•
•
•
T0
compact
locally compact
stably compact (i.e., finite intersections of compact saturated
sets are compact)
• well-filtered
• zero-dimensional
(Now called spectral spaces.)
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Morphisms
Observation. The maps ĥ are precisely those that are
continuous and inverse images of compact saturated sets are
compact.
Theorem 1.
The categories DL and Spec are dually equivalent.
NB. If L is a Boolean algebra (i.e., every element is
complemented) then everything specializes to the previous case.
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I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
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DL ≈ Spec
Bool ≈ Stone
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From topological spaces to algebras
Idea: Read the duality theorems “backwards”, not as providing
representations of algebraic structures but as giving an algebraic
substitute for topological spaces.
Problem 1. Very few spaces are representable in this fashion,
e.g. not the real line, not the unit interval.
Problem 2. Only special morphisms are representable in the
spectral space case.
Solution [Bénabou ’59] Use all open sets in the
representation.
(τ, ⊆) ←−[ (X, τ )
f −1 ←−[ f
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Frames
Algebraically,
F we have complete lattices on the left hand side,
but choose
and ⊓ as operations, as these are the ones that
are preserved by f −1.
Definition. A frame is a complete lattice in which
a⊓
G
B=
G
a⊓b
b∈B
holds. A frame homomorphism is a map preserving
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F
and ⊓.
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Frame duality
From frames to topological spaces:
L 7−→ cpFilt(L)
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completely prime filters
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Frame duality
From frames to topological spaces:
L 7−→ cpFilt(L)
completely prime filters
Disappointment: There is no “Completely Prime Filter
Theorem.”
Theorem. The categories Frm and Top are dual to each
other (but not dually equivalent).
Theorem. The duality between Frm and Top cuts down to a
dual equivalence between spatial frames and sober spaces.
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Frm ∼ Top
DL ≈ Spec
Bool ≈ Stone
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Assessment: commonalities
• algebraic structures vs. topological spaces
• homomorphisms vs. (certain) continuous maps
• the dualities are concrete and induced by a schizophrenic
object, e.g.:
Frm ←
↑
→ Top
↑
⊢
↓
Set
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⊢
↓
Set
L̂ = Frm(L, 2)
X̂ = Top(X, S)
2 = two-element lattice
S = ({0, 1}, {∅, {1}, {0, 1}})
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Assessment: differences
• For Stone and Spec the dualizing object is the discrete
two-element space, for Top it is S.
• For Spec one considers spectral maps, in the other cases just
continuous ones.
• Frames are not algebraic structures in the usual sense, while
the other two are.
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I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong
proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
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Frm ∼ Top
DL ≈ Spec
Bool ≈ Stone
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A finitary approach to the representation of
topological spaces
[Smyth ’92] [J. & Sünderhauf ’96]
Motivation. Given a topological space, consider
(L, ≺)
←−[
(X, τ )
where
L := {(U, K) ∈ τ × κ | U ⊆ K}
and
(U, K) ≺ (U ′, K ′) :⇔ K ⊆ U ′
(κ is the set of compact saturated subsets)
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Strong proximity lattices
Definition. A strong proximity lattice is a distributive lattice
(L; ∧, ∨, tt, ff ) equipped with a binary relation ≺ satisfying
ff ≺ a ≺ tt
a ≺ b, a ≺ b′
a ≺ b, a′ ≺ b
⇐⇒
⇐⇒
a ≺ b ∧ b′
a ∨ a′ ≺ b
a∧x≺b
=⇒
∃x′ ≻ x. a ∧ x′ ≺ b
a≺y∨b
=⇒
∃y ′ ≺ y. a ≺ y ′ ∨ b
Observation. It follows that ≺ ◦ ≺
=
≺.
Observation. ≺ ⊆ ≤ is not required, but a distributive
lattice with ≺:=≤ is an example of a strong proximity lattice.
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Spectra of strong proximity lattices
To a strong proximity lattice assign
L̂ := rpFilt(L)
the set of round prime filters, i.e. those for which
a∈F
=⇒
∃a′ ≺ a. a′ ∈ F
Equip L̂ with the Stone topology generated by
ϕ(a) = {F | a ∈ F },
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a ∈ L.
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Theorem. The spaces that arise in this way are characterized
as those which satisfy
•
•
•
•
T0
compact
locally compact
stably compact (i.e., finite intersections of compact saturated
sets are compact)
• well-filtered
known as stably compact spaces.
NB. The only difference to spectral spaces is that
zero-dimensionality has been dropped. This allows classical
spaces, such as the unit interval, to be represented.
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The duality
Lemma. [Round Prime Filter Theorem] For a 6≺ b in a
strong proximity lattice there exists a round prime filter F such
that a ∈ F and b 6∈ F .
Continuous functions between stably compact spaces are
captured by certain relations between strong proximity lattices.
Theorem. The categories SPL and SCS are dually equivalent.
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Assessment
⊕ Almost generalizes Stone’s classical dualities.
⊕ Uses a finitary algebraic (albeit relational) structure.
⊕ Is a dual equivalence, not just a duality.
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Assessment
⊕ Almost generalizes Stone’s classical dualities.
⊕ Uses a finitary algebraic (albeit relational) structure.
⊕ Is a dual equivalence, not just a duality.
⊖ Does not employ homomorphisms on the algebra side (and
what should those be?).
⊖ Difficult to see this as a representation theorem of an
interesting algebraic structure.
⊖ Is not concrete, i.e., not induced by a schizophrenic object.
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I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
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Inspiration: Priestley duality
[Priestley ’70]
Boolean algebras and distributive lattices give rise to two
topologies on the spectrum:
τ+ generated by {F | a ∈ F },
a∈L
τ− generated by {F | a 6∈ F },
a∈L
In Priestley duality, the spectrum L̂ is equipped with τ+ ∨ τ−
(and an order relation).
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More inspiration . . .
Theorem. The following categories are equivalent:
• spectral spaces (L̂, τ+) and spectral maps;
• Priestley spaces (L̂, τ+ ∨ τ−) and monotone continuous maps;
• compact regular bitopological spaces (L̂, τ+, τ−) and
bicontinuous maps.
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Frm
SPL
DL
Bool
Frm
SPL
DL
Bool
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D-frames
bitopological spaces: (X, τ+, τ−) — no further axioms!
d-frames: (L+, L−, con, tot), where con, tot ⊆ L+ × L−
Intuition:
dFrm ←− biTop
(τ+, τ−, con, tot) ←−[ (X, τ+, τ−)
where
(O, U ) ∈ con
(O, U ) ∈ tot
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:⇔
:⇔
O∩U =∅
O∪U =X
“consistent”
“total”
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Axioms for d-frames
• L+, L− are frames (with operations
F
and ⊓)
• (a′, b′) ⊑ (a, b) ∈ con
=⇒
(a′, b′) ∈ con
• (a′, b′) ⊒ (a, b) ∈ tot
=⇒
(a′, b′) ∈ tot
• (a, b), (a′, b′) ∈ con
=⇒
(a⊓a′, b⊔b′), (a⊔a′, b⊓b′) ∈ con
• (a, b), (a′, b′) ∈ tot
=⇒
(a ⊓ a′, b ⊔ b′), (a ⊔ a′, b ⊓ b′) ∈ tot
• (0, 1), (1, 0) ∈ con and (0, 1), (1, 0) ∈ tot
F↑
• con is Scott-closed, i.e., closed under
• (a, b) ∈ con and (a, b′) ∈ tot imply b ⊑ b′
Any others?
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Axioms for d-frames
• L+, L− are frames (with operations
F
and ⊓)
• (a′, b′) ⊑ (a, b) ∈ con
=⇒
(a′, b′) ∈ con
• (a′, b′) ⊒ (a, b) ∈ tot
=⇒
(a′, b′) ∈ tot
• (a, b), (a′, b′) ∈ con
=⇒
(a⊓a′, b⊔b′), (a⊔a′, b⊓b′) ∈ con
• (a, b), (a′, b′) ∈ tot
=⇒
(a ⊓ a′, b ⊔ b′), (a ⊔ a′, b ⊓ b′) ∈ tot
• (0, 1), (1, 0) ∈ con and (0, 1), (1, 0) ∈ tot
F↑
• con is Scott-closed, i.e., closed under
• (a, b) ∈ con and (a, b′) ∈ tot imply b ⊑ b′
Any others?
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Morphisms
Just pairs of frame homomorphisms which together preserve con
and tot.
The assignment
(τ+, τ−, con, tot)
↑
(f −1, f −1)
′
(τ+′ , τ−
, con′, tot′)
(X, τ+, τ−)
←−[
f
↓
′
(X ′, τ+′ , τ−
)
indeed gives rise to a functor from biTop to dFrm.
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The points of d-frames
Again, we take our cue from the theory of frames. (Abstract)
points are pairs (F+, F−) of completely prime filters in L+,
respectively, L−, such that
(a, b) ∈ con
=⇒
a 6∈ F+ or b 6∈ F−
(a, b) ∈ tot
=⇒
a ∈ F+ or b ∈ F−
As before, denote the set of points of a d-frame
L = (L+, L−, con, tot) by L̂.
Equip L̂ with two topologies
ϕ+(a) := {(F+, F−) ∈ L̂ | a ∈ F+},
ϕ−(b) := {(F+, F−) ∈ L̂ | b ∈ F−},
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a ∈ L+
b ∈ L−
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Morphisms
−1
′
′
(h−1
+ (F+ ), h− (F− )) ∈ L̂
↑
L = (L+, L−, con, tot)
(h+, h−)
↓
L′ = (L′+, L′−, con′, tot′)
7−→
(F+′ , F−′ ) ∈ L̂′
Together this constitutes a functor from dFrm to biTop.
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Theorem. The categories dFrm and biTop are dual to each
other.
The duality is concrete and the schizophrenic object (as an
element of dFrm) is (2, 2) with minimal con and tot.
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Theorem. The categories dFrm and biTop are dual to each
other.
The duality is concrete and the schizophrenic object (as an
element of dFrm) is (2, 2) with minimal con and tot.
NB. Of course, there is no Completely Prime Filter Theorem,
so this is point-free, just like the duality between Frm and Top.
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Theorem. The categories dFrm and biTop are dual to each
other.
The duality is concrete and the schizophrenic object (as an
element of dFrm) is (2, 2) with minimal con and tot.
NB. Of course, there is no Completely Prime Filter Theorem,
so this is point-free, just like the duality between Frm and Top.
NB. Notions of “spatial” and “sober” can be defined which
are much like those for frames and topological spaces.
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Theorem. The categories dFrm and biTop are dual to each
other.
The duality is concrete and the schizophrenic object (as an
element of dFrm) is (2, 2) with minimal con and tot.
NB. Of course, there is no Completely Prime Filter Theorem,
so this is point-free, just like the duality between Frm and Top.
NB. Notions of “spatial” and “sober” can be defined which
are much like those for frames and topological spaces.
NB. D-frames are different from the “biframes” of
Banaschewski, Brümmer, and Hardie (1983).
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dFrm ∼ biTop
Frm ∼ Top
SPL ≈ SCS
DL ≈ Spec
Bool ≈ Stone
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A reformulation of d-frames: stage 1
Replace (L+, L−, con, tot) by (L+ × L−, (1, 0), (0, 1), con, tot, ).
Abstractly: (L, tt, ff , con, tot) where
• L is a frame
• tt and ff are a complemented pair.
Proposition. With the obvious homomorphisms, this yields a
category equivalent to dFrm.
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The logical operations
On L+ × L− define operations
(a, b) ∧ (a′, b′) := (a ⊓ a′, b ⊔ b′)
(a, b) ∨ (a′, b′) := (a ⊔ a′, b ⊓ b′)
Proposition. (L+ × L−, ∧, ∨, tt, ff ) is a distributive lattice.
Note that the order derived from the logical operations is at
“90 degrees” to the order derived from the frame operations.
(Belnap ’77)
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A reformulation of d-frames: stage 2
Replace L+ × L− by con. According to the axioms of d-frames,
con carries the following structure:
information
F↑ order: ⊥ := (0, 0), ⊓,
over .
F↑
with ⊓ distributing
logical order: ∧, ∨, tt, and ff .
In order to capture tot, we also consider ≺ given by
(a, b) ≺ (a′, b′) :⇔ (a′, b) ∈ tot
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Skew-frames
Abstractly, a skew-frame is a structure
G
(L, ⊥, ⊓, ↑, ∧, ∨, tt, ff , ≺)
subject to the following additional axioms:
≺◦≺ ⊆ ≺ ⊆ ≤
ff ≺ α ≺ tt
α ≺ β, α ≺ β ′
α ≺ β, α′ ≺ β
⇐⇒
⇐⇒
α ≺ β ∧ β′
α ∨ α′ ≺ β
α ⊓ β = (α ∧ ⊥) ∨ (β ∧ ⊥) ∨ (α ∧ β)
α ⊑ α′ , β ⊑ β ′ , α ≺ β
=⇒
α′ ≺ β ′
There is some justification for calling ≺ a strong entailment.
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Skew-frames
Abstractly, a skew-frame is a structure
G
(L, ⊥, ⊓, ↑, ∧, ∨, tt, ff , ≺)
subject to the following additional axioms:
≺◦≺ ⊆ ≺ ⊆ ≤
ff ≺ α ≺ tt
α ≺ β, α ≺ β ′
α ≺ β, α′ ≺ β
⇐⇒
⇐⇒
α ≺ β ∧ β′
α ∨ α′ ≺ β
α ⊓ β = (α ∧ ⊥) ∨ (β ∧ ⊥) ∨ (α ∧ β)
α ⊑ α′ , β ⊑ β ′ , α ≺ β
=⇒
α′ ≺ β ′
There is some justification for calling ≺ a strong entailment.
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A reformulation of d-frames: final stage
With the obvious structure-preserving morphisms, we get a
category skew-Frm.
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A reformulation of d-frames: final stage
With the obvious structure-preserving morphisms, we get a
category skew-Frm.
Theorem. The categories dFrm and skew-Frm are
equivalent.
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dFrm ∼ biTop
Frm ∼ Top
SPL ≈ SCS
DL ≈ Spec
Bool ≈ Stone
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Compactness
For d-frames say that (L+, L−, con, tot) is compact if tot is
Scott-open.
For skew-frames this amounts to the axiom
G
↑
A≺
G
↑
B
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=⇒
α≺β
for some α ∈ A, β ∈ B
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Regularity
For d-frames say that a′ ⊳ a in L+ if there is r ∈ L− such that
(a′, r) ∈ con and (a, r) ∈ tot.
(L+, L−, con, tot) is regular if for all a ∈ L+,
a=
G
↑
{a′ | a′ ⊳ a}
and likewise for all b ∈ L−.
For skew-frames require
γ=
G
↑
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{α ⊓ β | α ≺ γ ≺ β}
for all γ
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Compact regular skew-frames
Observation.
For a compact regular skew-frame
F↑
(L, ⊥, ⊓, , ∧, ∨, tt, ff , ≺) the reduct (L, ∧, ∨, tt, ff , ≺) is a
strong proximity lattice.
Furthermore, with the right definition of morphism (“adjoint
pairs of continuous consequence relations”):
Theorem. The categories SPL and crdFrm are equivalent.
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Compact regular skew-frames
Observation.
For a compact regular skew-frame
F↑
(L, ⊥, ⊓, , ∧, ∨, tt, ff , ≺) the reduct (L, ∧, ∨, tt, ff , ≺) is a
strong proximity lattice.
Furthermore, with the right definition of morphism (“adjoint
pairs of continuous consequence relations”):
Theorem. The categories SPL and crdFrm are equivalent.
and
Theorem. The corresponding maps between compact regular
bitopological spaces (= stably compact spaces) are precisely the
bicontinuous ones.
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I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
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46
dFrm ∼ biTop
Frm ∼ Top
SPL ≈ SCS
DL ≈ Spec
Bool ≈ Stone
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Spectral skew-frames
Say that there is a dense set of reflexive elements in a compact
regular skew-frame (or that the skew-frame is spectral) if
α≺β
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=⇒
∃γ. α ≺ γ ≺ γ ≺ β
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Spectral skew-frames
Say that there is a dense set of reflexive elements in a compact
regular skew-frame (or that the skew-frame is spectral) if
α≺β
=⇒
∃γ. α ≺ γ ≺ γ ≺ β
Theorem. The category of spectral skew-frames is equivalent
to the category of distributive lattices and lattice
homomorphisms.
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I. The classical dualities of Stone
II. Frame duality
III. Stably compact spaces and strong proximity lattices
IV. The bitopological view
V. Compactness and regularity
VI. Negation
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dFrm ∼ biTop
Frm ∼ Top
SPL ≈ SCS
DL ≈ Spec
Bool ≈ Stone
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Symmetry
Say that a d-frame is symmetric if it is equipped with an
isomorphism i : L− → L+ subject to the conditions
(a, b) ∈ con
(a, b) ∈ tot
AAA 80, Bedlewo, June 3, 2010
⇐⇒
⇐⇒
a ⊓ i(b) = 0 in L+
a ⊔ i(b) = 1 in L+
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Symmetry
Say that a d-frame is symmetric if it is equipped with an
isomorphism i : L− → L+ subject to the conditions
(a, b) ∈ con
(a, b) ∈ tot
⇐⇒
⇐⇒
a ⊓ i(b) = 0 in L+
a ⊔ i(b) = 1 in L+
Proposition. The category of symmetric d-frames with
symmetry-preserving morphisms is equivalent to Frm.
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Negation
On the skew-frame corresponding to a d-frame define
¬α = ¬(a, b) := (i(b), i−1(a))
This operation satisfies the rules
α⊑β
=⇒
¬◦¬
=
¬α ⊑ ¬β
Belnap
¬
α∧γ ≺β
⇐⇒
α ≺ β ∨ ¬γ
α ⊓ ¬α′ = ⊥
⇐⇒
{α, α′} is bounded
sequent calculus
Proposition. The categories of symmetric d-frames and
skew-frames with negation are equivalent (and equivalent
to Frm).
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Moshier skew-frames
Proposition. A regular d-frame can have at most one
symmetry; a regular skew-frame at most one negation. It is
already characterised by
¬◦¬
α∧γ ≺β
=
⇐⇒
¬
α ≺ β ∨ ¬γ
Definition. A compact regular skew-frame with negation is
called (from today!) a Moshier skew-frame.
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Moshier’s theorems
Theorem. The category of Moshier skew-frames is dually
equivalent to the category of compact Hausdorff spaces and
continuous maps.
Theorem. A compact regular skew-frame has a negation if
and only if the inverse Gentzen cut-rule is valid:
α ∧ α′ ≺ β ∨ β ′
=⇒
∃γ. α ≺ β ∨ γ and γ ∧ α′ ≺ β ′
(The other direction is always true in a compact regular
skew-frame.)
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Boolean skew-frames
Proposition. If a spectral skew-frame admits a negation, then
its reflexive elements form a Boolean algebra.
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dFrm ∼ biTop
Frm ∼ Top
SPL ≈ SCS
M-skewFrm ≈ compHd
DL ≈ Spec
Bool ≈ Stone
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Papers
• A. Jung and M. A. Moshier. On the bitopological nature of
Stone duality. Technical Report CSR-06-13, School of
Computer Science, The University of Birmingham, 2006. 110
pages
• A. Jung and M. A. Moshier. A Hofmann-Mislove theorem for
bitopological spaces. Journal of Logic and Algebraic
Programming, 76:161–174, 2008
(Available from my home page.)
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