Choice-free Stone duality Wesley H. Holliday University of - - PowerPoint PPT Presentation

Choice-free Stone duality

Wesley H. Holliday

University of California, Berkeley Joint work with

Nick Bezhanishvili

University of Amsterdam BLAST 2018 August 7, 2018

What does “choice-free” mean?

We give a choice-free topological duality for Boolean algebras.

What does “choice-free” mean?

We give a choice-free topological duality for Boolean algebras. We work in what Schechter (Handbook of Analysis and Its Foundations) calls quasiconstructive mathematics: “mathematics that permits conventional rules of reasoning plus ZF + DC, but no stronger forms of Choice” (p. 404).

What does “choice-free” mean?

We give a choice-free topological duality for Boolean algebras. We work in what Schechter (Handbook of Analysis and Its Foundations) calls quasiconstructive mathematics: “mathematics that permits conventional rules of reasoning plus ZF + DC, but no stronger forms of Choice” (p. 404). Note: only our applications, not the duality itself, uses DC.

What does “choice-free” mean?

We give a choice-free topological duality for Boolean algebras. We work in what Schechter (Handbook of Analysis and Its Foundations) calls quasiconstructive mathematics: “mathematics that permits conventional rules of reasoning plus ZF + DC, but no stronger forms of Choice” (p. 404). Note: only our applications, not the duality itself, uses DC. Note: of course we won’t prove that every BA is isomorphic to a field of sets, since this implies the Boolean Prime Filter Theorem.

Slogans

Two slogans describing our duality:

Slogans

Two slogans describing our duality:

1

“a mix of Stone and Tarski, connected by Vietoris”;

Slogans

Two slogans describing our duality:

1

“a mix of Stone and Tarski, connected by Vietoris”;

2

“a hyperspace approach, in contrast to a pointfree approach.”

Stone Representation of BAs

Theorem (Stone 1936). Every Boolean algebra is isomorphic to the BA of clopen sets of some topological (Stone) space. Marshall Stone (1903 – 1989)

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra. Let XA be the space of all prime filters.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra. Let XA be the space of all prime filters. The topology is generated by sets a = {x ∈ XA | a ∈ x} for a ∈ A.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra. Let XA be the space of all prime filters. The topology is generated by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a compact Hausdorff space with a clopen basis.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra. Let XA be the space of all prime filters. The topology is generated by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a compact Hausdorff space with a clopen basis. A is isomorphic to the Boolean algebra Clop(XA) of clopen sets.

Stone Representation of BAs

Stone representation uses the Prime Filter Theorem. Let A be a Boolean algebra. Let XA be the space of all prime filters. The topology is generated by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a compact Hausdorff space with a clopen basis. A is isomorphic to the Boolean algebra Clop(XA) of clopen sets. The isomorphism ϕ : A → Clop(XA) is given by ϕ(a) = a.

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective.

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective. Suppose a b. Then ↑a ∩ ↓b = ∅.

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective. Suppose a b. Then ↑a ∩ ↓b = ∅. By the Prime Filter Theorem, there is a prime filter F such that ↑a ⊆ F and F ∩ ↓b = ∅.

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective. Suppose a b. Then ↑a ∩ ↓b = ∅. By the Prime Filter Theorem, there is a prime filter F such that ↑a ⊆ F and F ∩ ↓b = ∅. So F ∈ ϕ(a) and F / ∈ ϕ(b), yielding ϕ(a) ⊆ ϕ(b).

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective. Suppose a b. Then ↑a ∩ ↓b = ∅. By the Prime Filter Theorem, there is a prime filter F such that ↑a ⊆ F and F ∩ ↓b = ∅. So F ∈ ϕ(a) and F / ∈ ϕ(b), yielding ϕ(a) ⊆ ϕ(b). Our aim is to obtain Stone-like representation of Boolean algebras choice free.

Prime Filter Theorem

The Prime Filter Theorem is used for showing that ϕ is injective. Suppose a b. Then ↑a ∩ ↓b = ∅. By the Prime Filter Theorem, there is a prime filter F such that ↑a ⊆ F and F ∩ ↓b = ∅. So F ∈ ϕ(a) and F / ∈ ϕ(b), yielding ϕ(a) ⊆ ϕ(b). Our aim is to obtain Stone-like representation of Boolean algebras choice free. This will resemble Stone’s representation of distributive lattices.

Stone Representation of DLs

Theorem (Stone 1937). Every distributive lattice is isomorphic to the distributive lattice of compact open sets of some topological (spectral) space. Marshall Stone (1903 – 1989)

Boolean algebra of regular open sets

Theorem (Tarski 1937). For every topological space X, the set RO(X) of regular open subsets of X forms a (complete) BA. Alfred Tarski (1901 – 1983)

Regular open sets

A set U is regular open if Int(Cl(U)) = U.

Regular open sets

A set U is regular open if Int(Cl(U)) = U. For U, V ∈ RO(X) we put U ∧ V = U ∩ V, U ∨ V = Int(Cl(U ∪ V)), ¬U = Int(X \ U).

Choice-free Stone duality

Let A be a Boolean algebra.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

Good formulation of sober for us: every completely prime filter in Ω(X) is Ω(x) = {U ∈ Ω(X) | x ∈ U} for some x ∈ X.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology. Let Int be the interior operation associated with Up(XA): Int(U) = {x ∈ X | ∀x′ x x′ ∈ U}.

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology. Let Int be the interior operation associated with Up(XA): Int(U) = {x ∈ X | ∀x′ x x′ ∈ U}. U ⊆ X is -regular open if it is regular open in Up(XA)

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology. Let Int be the interior operation associated with Up(XA): Int(U) = {x ∈ X | ∀x′ x x′ ∈ U}. U ⊆ X is -regular open if it is regular open in Up(XA): an upset s.th. if y ∈ U, then ∃z y: z ∈ Int(XA \ U).

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology. Let Int be the interior operation associated with Up(XA): Int(U) = {x ∈ X | ∀x′ x x′ ∈ U}. U ⊆ X is -regular open if it is regular open in Up(XA): an upset s.th. if y ∈ U, then ∃z y: z ∈ Int(XA \ U). (XA, ) is a separative poset, i.e., every principal upset is regular open

Choice-free Stone duality

Let A be a Boolean algebra. Let XA be the space of all proper filters of A. We generate a topology by sets a = {x ∈ XA | a ∈ x} for a ∈ A. Then XA is a spectral space, i.e., compact, T0, coherent (compact

• pen sets are closed under intersection and form a basis), sober.

The specialization order is the inclusion ⊆ on proper filters. Second topology to consider: Up(XA), the -upset topology. Let Int be the interior operation associated with Up(XA): Int(U) = {x ∈ X | ∀x′ x x′ ∈ U}. U ⊆ X is -regular open if it is regular open in Up(XA): an upset s.th. if y ∈ U, then ∃z y: z ∈ Int(XA \ U). (XA, ) is a separative poset, i.e., every principal upset is regular open: if y x, then ∃z y: z ∈ Int(XA \ ↑x).

Choice-free Stone duality

CO(X) = {compact open subsets of X}.

Choice-free Stone duality

CO(X) = {compact open subsets of X}. Let CORO(XA) be the set of compact open -regular open sets.

Choice-free Stone duality

CO(X) = {compact open subsets of X}. Let CORO(XA) be the set of compact open -regular open sets. If U ∈ CORO(XA), then Int(X \ U) ∈ CORO(XA).

Choice-free Stone duality

CO(X) = {compact open subsets of X}. Let CORO(XA) be the set of compact open -regular open sets. If U ∈ CORO(XA), then Int(X \ U) ∈ CORO(XA). Then CORO(XA) is a Boolean algebra, where U ∧ V = U ∩ V, ¬U = Int(X \ U).

Choice-free Stone duality

Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CORO(XA).

Choice-free Stone duality

Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CORO(XA). The isomorphism ϕ : A → CORO(XA) is given by ϕ(a) = a.

Choice-free Stone duality

Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CORO(XA). The isomorphism ϕ : A → CORO(XA) is given by ϕ(a) = a. To show that ϕ is injective we do not need the PFT.

Choice-free Stone duality

Theorem (Choice-free representation of BAs). Each Boolean algebra A is isomorphic to the Boolean algebra CORO(XA). The isomorphism ϕ : A → CORO(XA) is given by ϕ(a) = a. To show that ϕ is injective we do not need the PFT. What kind of space is XA?

UV-spaces

A UV-space is a T0 space X such that:

UV-spaces

A UV-space is a T0 space X such that:

1

CORO(X) is closed under ∩ and Int(X \ ·);

UV-spaces

A UV-space is a T0 space X such that:

1

CORO(X) is closed under ∩ and Int(X \ ·);

2

x y ⇒ there is a U ∈ CORO(X) s.t. x ∈ U and y / ∈ U;

UV-spaces

A UV-space is a T0 space X such that:

1

CORO(X) is closed under ∩ and Int(X \ ·);

2

x y ⇒ there is a U ∈ CORO(X) s.t. x ∈ U and y / ∈ U;

3

every proper filter in CORO(X) is CORO(x) for some x ∈ X.

UV-spaces

A UV-space is a T0 space X such that:

1

CORO(X) is closed under ∩ and Int(X \ ·);

2

x y ⇒ there is a U ∈ CORO(X) s.t. x ∈ U and y / ∈ U;

3

every proper filter in CORO(X) is CORO(x) for some x ∈ X.

• Proposition. Every UV-space is a spectral space.

Choice-free representation of BAs

Theorem (Choice-free representation of BAs). For each BA A there is a UV-space X such that A is isomorphic to CORO(X).

Choice-free representation of BAs

Theorem (Choice-free representation of BAs). For each BA A there is a UV-space X such that A is isomorphic to CORO(X). We extend this correspondence to a full duality of the corresponding categories.

Vietoris space of a Stone space

Theorem (Vietoris 1922, Stone version). For every Stone space X its Vietoris space, i.e., the space of nonempty closed sets equipped with the hit-and-miss topology, is again a Stone space. Leopold Vietoris (1891 – 2002)

Vietoris space of a Stone space

Let X be a Stone space.

Vietoris space of a Stone space

Let X be a Stone space. Let F(X) be the set of all nonempty closed subsets of X.

Vietoris space of a Stone space

Let X be a Stone space. Let F(X) be the set of all nonempty closed subsets of X. The upper Vietoris topology has the basis U = {F ∈ F(X) | F ⊆ U}, U ∈ Ω(X). The lower Vietoris topology has the subbasis ♦V = {F ∈ F(X) | F ∩ V = ∅}, V ∈ Ω(X).

Vietoris space of a Stone space

Let X be a Stone space. Let F(X) be the set of all nonempty closed subsets of X. The upper Vietoris topology has the basis U = {F ∈ F(X) | F ⊆ U}, U ∈ Ω(X). The lower Vietoris topology has the subbasis ♦V = {F ∈ F(X) | F ∩ V = ∅}, V ∈ Ω(X). The Vietoris topology is the join of the upper and lower Vietoris topologies.

Examples of UV-spaces

Let X be a Stone space and UV(X) its upper Vietoris space.

Examples of UV-spaces

Let X be a Stone space and UV(X) its upper Vietoris space. Then UV(X) is a UV-space.

Examples of UV-spaces

Let X be a Stone space and UV(X) its upper Vietoris space. Then UV(X) is a UV-space. Assuming the PFT, every UV-space is homeomorphic to UV(X) for a Stone space X.

Examples of UV-spaces

Let X be a Stone space and UV(X) its upper Vietoris space. Then UV(X) is a UV-space. Assuming the PFT, every UV-space is homeomorphic to UV(X) for a Stone space X. We can prove a localic version of this result choice free.

UV-maps

Let X and X′ be spectral spaces.

UV-maps

Let X and X′ be spectral spaces. A map f : X → Y is called spectral if f −1[U] is compact open for each compact open U.

UV-maps

Let X and X′ be spectral spaces. A map f : X → Y is called spectral if f −1[U] is compact open for each compact open U. A UV-map between UV-spaces X and X′ is a spectral map f : X → X′

UV-maps

Let X and X′ be spectral spaces. A map f : X → Y is called spectral if f −1[U] is compact open for each compact open U. A UV-map between UV-spaces X and X′ is a spectral map f : X → X′ that is also a p-morphism: if f(x) ′ y′, then ∃y : x y and f(y) = y′. x f(x) y′ x ∃y f(x) y′ ⇒

Duality

Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.

Comparison

Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.

Comparison

Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Let’s further compare these with a localic Stone duality.

Stone locales

A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L: a ∧

• S =
• {a ∧ s | s ∈ S}.

For any space X, Ω(X) ordered by ⊆ is a locale.

Stone locales

A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L: a ∧

• S =
• {a ∧ s | s ∈ S}.

For any space X, Ω(X) ordered by ⊆ is a locale. A Stone locale is a locale that is: compact – S = 1 implies T = 1 for a finite T ⊆ S. zero-dimensional – every element is a join of complemented elements.

Stone locales

A locale is a complete lattice L satisfying the join-infinite distributive law for each a ∈ L and S ⊆ L: a ∧

• S =
• {a ∧ s | s ∈ S}.

For any space X, Ω(X) ordered by ⊆ is a locale. A Stone locale is a locale that is: compact – S = 1 implies T = 1 for a finite T ⊆ S. zero-dimensional – every element is a join of complemented elements. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.

Comparison

Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.

Comparison

Theorem (Choice-free UV-duality for BAs). The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Stone duality). Assuming the PFT, the category of Stone spaces with continuous maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Theorem (Choice-free localic Stone duality). The category of Stone locales with localic maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms. Let’s now relate the first and third approaches.

Vietoris space of a Stone locale

The Vietoris space of X may be defined using the open sets instead of the closed sets, which led Johnstone to define for any Stone locale L the Vietoris space of L.

Vietoris space of a Stone locale

The Vietoris space of X may be defined using the open sets instead of the closed sets, which led Johnstone to define for any Stone locale L the Vietoris space of L. Similarly, we can define the upper Vietoris space of L.

Upper Vietoris space of a Stone locale

The upper Vietoris space of L has as its set of points L− = {a ∈ L | a = 1} with the topology generated by the sets a = {b ∈ L− | a ∨ b = 1}, a ∈ L.

Upper Vietoris space of a Stone locale

The upper Vietoris space of L has as its set of points L− = {a ∈ L | a = 1} with the topology generated by the sets a = {b ∈ L− | a ∨ b = 1}, a ∈ L. Idea: switch from nonempty closed sets to their open complements.

Upper Vietoris space of a Stone locale

The upper Vietoris space of L has as its set of points L− = {a ∈ L | a = 1} with the topology generated by the sets a = {b ∈ L− | a ∨ b = 1}, a ∈ L. Idea: switch from nonempty closed sets to their open

• complements. Since the basic opens of upper Vietoris are

U = {F ∈ F(X) | F ⊆ U} for U ∈ Ω(X), when we switch to open complements we look at U = {V ∈ Ω(X)\{X} | Vc ⊆ U} = {V ∈ Ω(X)\{X} | U ∪V = X}.

Upper Vietoris space of a Stone locale

The upper Vietoris space of L has as its set of points L− = {a ∈ L | a = 1} with the topology generated by the sets a = {b ∈ L− | a ∨ b = 1}, a ∈ L. Idea: switch from nonempty closed sets to their open

• complements. Since the basic opens of upper Vietoris are

U = {F ∈ F(X) | F ⊆ U} for U ∈ Ω(X), when we switch to open complements we look at U = {V ∈ Ω(X)\{X} | Vc ⊆ U} = {V ∈ Ω(X)\{X} | U ∪V = X}. Theorem (Representation of UV-spaces). X is a UV-space iff X is homeomorphic to the upper Vietoris space of a Stone locale.

Upper Vietoris locale of a Stone locale

Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L.

Upper Vietoris locale of a Stone locale

Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L. Similarly, one can define the upper Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the upper Vietoris space of L.

Upper Vietoris locale of a Stone locale

Johnstone also defined the notion of the Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the Vietoris space of L. Similarly, one can define the upper Vietoris locale of a Stone locale L such that its space of points is homeomorphic to the upper Vietoris space of L.

• Theorem. X is a UV-space iff X is homeomorphic to the space of

points of the upper Vietoris locale of a Stone locale.

Hyperspace & pointfree approaches related

upper Vietoris locale of L upper Vietoris space of X pt pt locale L of filters Stone space X upper Vietoris space of L BA A UV(A) locales: spaces:

Duality dictionary

BA UV Stone BA UV-space Stone space homomorphism UV-map continuous map filter ↑x, x ∈ X closed set ideal U ∈ ORO(X)

• pen set

principal filter U ∈ CORO(X) clopen set principal ideal U ∈ CORO(X) clopen set maximal filter {x}, x ∈ Max(X) {x}, x ∈ X maximal ideal X \ ↓x, x ∈ Max(X) X \ {x}, x ∈ X relativization subspace U ∈ CORO(X) subspace U ∈ Clop(X) complete algebra complete UV-space ED Stone space atom isolated point isolated point atomic algebra Cl(Xiso) = X Cl(Xiso) = X atomless algebra Xiso = ∅ Xiso = ∅ homomorphic image subspace induced by ↑x, x ∈ X closed set subalgebra image under UV-map image under continuous map direct product UV-sum disjoint union canonical completion RO(X) ℘(X) MacNeille completion RO({x ∈ X | ↑x ∈ CORO(X)}) RO(X)

Table: Dictionary for BA, UV, and Stone.

Example applications: antichains of BAs

Example applications: antichains of BAs

By an antichain in a BA, we mean a collection C of elements such that for all x, y ∈ C with x = y, we have x ∧ y = 0.

Example applications: antichains of BAs

By an antichain in a BA, we mean a collection C of elements such that for all x, y ∈ C with x = y, we have x ∧ y = 0.

• Proposition. Every infinite BA contains infinite chains and

infinite antichains.

Example applications: antichains of BAs

By an antichain in a BA, we mean a collection C of elements such that for all x, y ∈ C with x = y, we have x ∧ y = 0.

• Proposition. Every infinite BA contains infinite chains and

infinite antichains. The standard Stone duality proof uses the fact that if X is an infinite set and U ⊆ X, then either U is infinite or X \U is infinite.

Example applications: antichains of BAs

By an antichain in a BA, we mean a collection C of elements such that for all x, y ∈ C with x = y, we have x ∧ y = 0.

• Proposition. Every infinite BA contains infinite chains and

infinite antichains. The standard Stone duality proof uses the fact that if X is an infinite set and U ⊆ X, then either U is infinite or X \U is infinite. Our proof is very similar, but we use the fact that if X is an infinite separative poset and U ∈ RO(X), then either U is infinite

• r ¬U = Int(X \ U) = {x ∈ X | ∀x′ x x′ ∈ U} is infinite.

Example applications: antichains of BAs

• Proposition. Every infinite BA contains infinite chains and

infinite antichains.

Proof.

Example applications: antichains of BAs

• Proposition. Every infinite BA contains infinite chains and

infinite antichains.

• Proof. By duality, it suffices to show that in any infinite UV-space X,

there is an infinite descending chain U0 U1 . . . of sets from CORO(X), as well as an infinite family of pairwise disjoint sets from CORO(X).

Example applications: antichains of BAs

• Proposition. Every infinite BA contains infinite chains and

infinite antichains.

• Proof. By duality, it suffices to show that in any infinite UV-space X,

there is an infinite descending chain U0 U1 . . . of sets from CORO(X), as well as an infinite family of pairwise disjoint sets from CORO(X). For this it suffices to show that (⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n.

Example applications: antichains of BAs

• Proposition. Every infinite BA contains infinite chains and

infinite antichains.

• Proof. By duality, it suffices to show that in any infinite UV-space X,

there is an infinite descending chain U0 U1 . . . of sets from CORO(X), as well as an infinite family of pairwise disjoint sets from CORO(X). For this it suffices to show that (⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. For then by DC, there is an infinite descending chain U0 ⊇ U1 ⊇ . . . of sets from CORO(X) with Ui ∩ ¬Ui+1 = ∅ for each i ∈ N, in which case {U0 ∩ ¬U1, U1 ∩ ¬U2, . . . } is our antichain.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V,

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces;

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V,

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V, and otherwise we claim that Un ∩ ¬V is infinite, in which case we can set Un+1 := Un ∩ ¬V.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V, and otherwise we claim that Un ∩ ¬V is infinite, in which case we can set Un+1 := Un ∩ ¬V. Since Un ∈ RO(X), we may regard Un as a separative poset.

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V, and otherwise we claim that Un ∩ ¬V is infinite, in which case we can set Un+1 := Un ∩ ¬V. Since Un ∈ RO(X), we may regard Un as a separative poset. Given V ∈ RO(X), we have Un ∩ V, Un ∩ ¬V ∈ RO(Un)

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V, and otherwise we claim that Un ∩ ¬V is infinite, in which case we can set Un+1 := Un ∩ ¬V. Since Un ∈ RO(X), we may regard Un as a separative poset. Given V ∈ RO(X), we have Un ∩ V, Un ∩ ¬V ∈ RO(Un) and Un ∩ ¬V = ¬n(Un ∩ V), where ¬n is the negation in RO(Un).

Example applications: antichains of BAs

(⋆) for any n ∈ N, there is a descending chain U0 ⊇ U1 ⊇ · · · ⊇ Un of infinite sets from CORO(X) such that Ui ∩ ¬Ui+1 = ∅ for i ∈ n. We prove (⋆) by induction. Let U0 = X. For the inductive step: Since Un is infinite and X is T0, there are x, y ∈ Un such that x y. Then by the separation property of UV-spaces, there is a V ∈ CORO(X) such that x ∈ V and y ∈ V, which with y ∈ Un and Un, V ∈ RO(X) implies that there is a z y such that z ∈ Un ∩ ¬V. Since Un, V ∈ CORO(X), we have Un ∩ V, Un ∩ ¬V ∈ CORO(X) by the closure conditions on UV-spaces; and since z ∈ Un ∩ ¬V and x ∈ Un ∩ V, we have z ∈ Un ∩ ¬(Un ∩ V) = ∅ and x ∈ Un ∩ ¬(Un ∩ ¬V) = ∅. Thus, if Un ∩ V is infinite, then we can set Un+1 := Un ∩ V, and otherwise we claim that Un ∩ ¬V is infinite, in which case we can set Un+1 := Un ∩ ¬V. Since Un ∈ RO(X), we may regard Un as a separative poset. Given V ∈ RO(X), we have Un ∩ V, Un ∩ ¬V ∈ RO(Un) and Un ∩ ¬V = ¬n(Un ∩ V), where ¬n is the negation in RO(Un). Then since Un is infinite, either Un ∩ V or ¬n(Un ∩ V) is infinite.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces. Assuming AC, we can do the same for the duality via UV-spaces.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces. Assuming AC, we can do the same for the duality via UV-spaces. Let A be a BA.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces. Assuming AC, we can do the same for the duality via UV-spaces. Let A be a BA. Let XA be the space of all proper filters, with topology generated by a subbasis of sets of the form { a : a ∈ A}, {XA \ a : a ∈ A}.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces. Assuming AC, we can do the same for the duality via UV-spaces. Let A be a BA. Let XA be the space of all proper filters, with topology generated by a subbasis of sets of the form { a : a ∈ A}, {XA \ a : a ∈ A}. Let ≤ be the inclusion of proper filters.

Priestley-like duality

The spectral duality of distributive lattices can be reformulated in terms of Priestley spaces. Assuming AC, we can do the same for the duality via UV-spaces. Let A be a BA. Let XA be the space of all proper filters, with topology generated by a subbasis of sets of the form { a : a ∈ A}, {XA \ a : a ∈ A}. Let ≤ be the inclusion of proper filters. Then (XA, ≤) is a Priestley space.

Priestley-like duality

In addition, if ClopRO(X, ≤) = {clopen ≤-regular open sets}:

1

if U ∈ ClopRO(X, ≤), then Int≤(X \ U) ∈ ClopRO(X, ≤);

Priestley-like duality

In addition, if ClopRO(X, ≤) = {clopen ≤-regular open sets}:

1

if U ∈ ClopRO(X, ≤), then Int≤(X \ U) ∈ ClopRO(X, ≤);

2

x ≤ y ⇒ there is a U ∈ ClopRO(X, ≤) s.t. x ∈ U and y / ∈ U;

Priestley-like duality

In addition, if ClopRO(X, ≤) = {clopen ≤-regular open sets}:

1

if U ∈ ClopRO(X, ≤), then Int≤(X \ U) ∈ ClopRO(X, ≤);

2

x ≤ y ⇒ there is a U ∈ ClopRO(X, ≤) s.t. x ∈ U and y / ∈ U;

3

every proper filter in ClopRO(X, ≤) is ClopRO(x) for some x ∈ X.

Priestley-like duality

Theorem (Priestley-like representation of BAs). Every Boolean algebra A is isomorphic to ClopRO(XA, ≤).

Priestley-like duality

Theorem (Priestley-like representation of BAs). Every Boolean algebra A is isomorphic to ClopRO(XA, ≤). Such ordered spaces are order-homeomorphic to the Vietoris space of a Stone space ordered by ⊆.

Conclusions and further directions

Conclusions and further directions

We developed choice-free topological duality for Boolean algebras.

Conclusions and further directions

We developed choice-free topological duality for Boolean algebras. With choice this can be converted into a Priestley-like

• rder-topological duality.

Conclusions and further directions

We developed choice-free topological duality for Boolean algebras. With choice this can be converted into a Priestley-like

• rder-topological duality.

We also have extensions of this duality to modal algebras (see our presentation at Advances in Modal Logic 2018).

Conclusions and further directions

We developed choice-free topological duality for Boolean algebras. With choice this can be converted into a Priestley-like

• rder-topological duality.

We also have extensions of this duality to modal algebras (see our presentation at Advances in Modal Logic 2018). It should also be possible to give choice-free dualities for Heyting algebras and distributive lattices.

Thank you!