A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy

Stone duality, more duality, and dynamics in βω

Will Brian

May 22, 2014

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Table of contents

1 A quick look at βω

2 Filters and families

3 Ultrafilters on families

4 A Filter Dichotomy

Will Brian Stone duality, more duality, and dynamics in βω βω If A ⊆ ω, then Aˆ = A is the set of all ultrafilters containing A. n o If F is any filter on ω, then Fˆ = T Aˆ : A ∈ F is a closed subset of βω. Conversely, if C is a closed subset of βω then there is a unique filter F such that Fˆ = C, namely n o F = A ⊆ ω : C ⊆ Aˆ

The above correspondence is called Stone duality. It represents a special case of a famous theorem proved by Marshal Stone in 1936.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω

βω is the space of all ultrafilters on ω.

Will Brian Stone duality, more duality, and dynamics in βω n o If F is any filter on ω, then Fˆ = T Aˆ : A ∈ F is a closed subset of βω. Conversely, if C is a closed subset of βω then there is a unique filter F such that Fˆ = C, namely n o F = A ⊆ ω : C ⊆ Aˆ

The above correspondence is called Stone duality. It represents a special case of a famous theorem proved by Marshal Stone in 1936.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω

βω is the space of all ultrafilters on ω. βω If A ⊆ ω, then Aˆ = A is the set of all ultrafilters containing A.

Will Brian Stone duality, more duality, and dynamics in βω The above correspondence is called Stone duality. It represents a special case of a famous theorem proved by Marshal Stone in 1936.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω

βω is the space of all ultrafilters on ω. βω If A ⊆ ω, then Aˆ = A is the set of all ultrafilters containing A. n o If F is any filter on ω, then Fˆ = T Aˆ : A ∈ F is a closed subset of βω. Conversely, if C is a closed subset of βω then there is a unique filter F such that Fˆ = C, namely n o F = A ⊆ ω : C ⊆ Aˆ

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Stone duality for βω

βω is the space of all ultrafilters on ω. βω If A ⊆ ω, then Aˆ = A is the set of all ultrafilters containing A. n o If F is any filter on ω, then Fˆ = T Aˆ : A ∈ F is a closed subset of βω. Conversely, if C is a closed subset of βω then there is a unique filter F such that Fˆ = C, namely n o F = A ⊆ ω : C ⊆ Aˆ

The above correspondence is called Stone duality. It represents a special case of a famous theorem proved by Marshal Stone in 1936.

Will Brian Stone duality, more duality, and dynamics in βω Define the shift map σ : βω → βω by

σ(p) = ↑{A + 1: A ∈ p} ,

where A + 1 = {n + 1: n ∈ A} and ↑B is the set of all supersets of elements of B. This map is continuous, and almost a surjection. σ is the unique continuous extension to βω of the map on ω given by n 7→ n + 1.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system

By a dynamical system we mean a compact space X together with a map f : X → X .

Will Brian Stone duality, more duality, and dynamics in βω This map is continuous, and almost a surjection. σ is the unique continuous extension to βω of the map on ω given by n 7→ n + 1.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system

By a dynamical system we mean a compact space X together with a map f : X → X . Define the shift map σ : βω → βω by

σ(p) = ↑{A + 1: A ∈ p} ,

where A + 1 = {n + 1: n ∈ A} and ↑B is the set of all supersets of elements of B.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a dynamical system

By a dynamical system we mean a compact space X together with a map f : X → X . Define the shift map σ : βω → βω by

σ(p) = ↑{A + 1: A ∈ p} ,

where A + 1 = {n + 1: n ∈ A} and ↑B is the set of all supersets of elements of B. This map is continuous, and almost a surjection. σ is the unique continuous extension to βω of the map on ω given by n 7→ n + 1.

Will Brian Stone duality, more duality, and dynamics in βω In particular, we can extend the function n 7→ n + m to βω and thus define p + m for any m ∈ ω and p ∈ βω. We can then extend the function n 7→ p + n to βω and thus define p + q for any p, q ∈ βω.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup

Recall that every continuous function f : ω → βω extends to a continuous function βf : βω → βω: ω βω

f βf

βω

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup

Recall that every continuous function f : ω → βω extends to a continuous function βf : βω → βω: ω βω

f βf

βω

In particular, we can extend the function n 7→ n + m to βω and thus define p + m for any m ∈ ω and p ∈ βω. We can then extend the function n 7→ p + n to βω and thus define p + q for any p, q ∈ βω.

Will Brian Stone duality, more duality, and dynamics in βω This makes βω into a left-topological semigroup. If we consider that an ultrafilter p is a (non-σ-additive) measure in which every set has measure 0 or 1, then the ultrafilter p + q simply represents the convolution of the measures p and q.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup

Equivalently, we can write

p + q = {A ⊆ ω : {n:(A − n) ∈ p} ∈ q} .

The function q 7→ p + q is continuous for every p, but the function p 7→ p + q is continuous only when q is principal.

Will Brian Stone duality, more duality, and dynamics in βω If we consider that an ultrafilter p is a (non-σ-additive) measure in which every set has measure 0 or 1, then the ultrafilter p + q simply represents the convolution of the measures p and q.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup

Equivalently, we can write

p + q = {A ⊆ ω : {n:(A − n) ∈ p} ∈ q} .

The function q 7→ p + q is continuous for every p, but the function p 7→ p + q is continuous only when q is principal. This makes βω into a left-topological semigroup.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy βω is a semigroup

Equivalently, we can write

p + q = {A ⊆ ω : {n:(A − n) ∈ p} ∈ q} .

The function q 7→ p + q is continuous for every p, but the function p 7→ p + q is continuous only when q is principal. This makes βω into a left-topological semigroup. If we consider that an ultrafilter p is a (non-σ-additive) measure in which every set has measure 0 or 1, then the ultrafilter p + q simply represents the convolution of the measures p and q.

Will Brian Stone duality, more duality, and dynamics in βω Proof. If R is a right ideal then R + βN ⊆ R and, in particular, if p ∈ R then p + 1 = σ(p) ∈ R. Thus every right ideal is σ-invariant, and if closed is a subsystem of (βN, σ). Conversely, if X is closed and σ-invariant, then p + βN = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of (βN, σ) is a right ideal. For the second assertion, we need only prove that every minimal right ideal is closed. If R is a minimal right ideal then p + βN = R. Since x 7→ p + x is a continuous function, R is compact.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra

Theorem (Bergelson) The subsystems of (βN, σ) are the closed right ideals of (βN, +), and the minimal subsystems are the minimal right ideals.

Will Brian Stone duality, more duality, and dynamics in βω Conversely, if X is closed and σ-invariant, then p + βN = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of (βN, σ) is a right ideal. For the second assertion, we need only prove that every minimal right ideal is closed. If R is a minimal right ideal then p + βN = R. Since x 7→ p + x is a continuous function, R is compact.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra

Theorem (Bergelson) The subsystems of (βN, σ) are the closed right ideals of (βN, +), and the minimal subsystems are the minimal right ideals.

Proof. If R is a right ideal then R + βN ⊆ R and, in particular, if p ∈ R then p + 1 = σ(p) ∈ R. Thus every right ideal is σ-invariant, and if closed is a subsystem of (βN, σ).

Will Brian Stone duality, more duality, and dynamics in βω For the second assertion, we need only prove that every minimal right ideal is closed. If R is a minimal right ideal then p + βN = R. Since x 7→ p + x is a continuous function, R is compact.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra

Theorem (Bergelson) The subsystems of (βN, σ) are the closed right ideals of (βN, +), and the minimal subsystems are the minimal right ideals.

Proof. If R is a right ideal then R + βN ⊆ R and, in particular, if p ∈ R then p + 1 = σ(p) ∈ R. Thus every right ideal is σ-invariant, and if closed is a subsystem of (βN, σ). Conversely, if X is closed and σ-invariant, then p + βN = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of (βN, σ) is a right ideal.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy dynamics and algebra

Theorem (Bergelson) The subsystems of (βN, σ) are the closed right ideals of (βN, +), and the minimal subsystems are the minimal right ideals.

Proof. If R is a right ideal then R + βN ⊆ R and, in particular, if p ∈ R then p + 1 = σ(p) ∈ R. Thus every right ideal is σ-invariant, and if closed is a subsystem of (βN, σ). Conversely, if X is closed and σ-invariant, then p + βN = p + N = p + N ⊆ X = X for every p ∈ X . Thus every subsystem of (βN, σ) is a right ideal. For the second assertion, we need only prove that every minimal right ideal is closed. If R is a minimal right ideal then p + βN = R. Since x 7→ p + x is a continuous function, R is compact.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition

A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ ∈/ F and ω ∈ F. 2 Upwards heredity: if A ∈ F and A ⊆ B, then B ∈ F. 3 Finite intersection property: if A, B ∈ F then A ∩ B ∈ F.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition

A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ ∈/ F and ω ∈ F. 2 Upwards heredity: if A ∈ F and A ⊆ B, then B ∈ F. 3 Finite intersection property: if A, B ∈ F then A ∩ B ∈ F.

If we omit (2) then we get the definition of a filter base.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A familiar definition

A filter F on ω is a set of subsets of ω satisfying: 1 Nontriviality: ∅ ∈/ F and ω ∈ F. 2 Upwards heredity: if A ∈ F and A ⊆ B, then B ∈ F. 3 Finite intersection property: if A, B ∈ F then A ∩ B ∈ F.

If we omit (2) then we get the definition of a filter base.

If we omit (3) then we get the definition of a Furstenberg family, or simply a family.

Will Brian Stone duality, more duality, and dynamics in βω If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family.

Will Brian Stone duality, more duality, and dynamics in βω The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F.

Will Brian Stone duality, more duality, and dynamics in βω A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets.

Will Brian Stone duality, more duality, and dynamics in βω A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ.

Will Brian Stone duality, more duality, and dynamics in βω If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ.

Will Brian Stone duality, more duality, and dynamics in βω The sets in Θ#Σ are called piecewise syndetic.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Examples of families

Every filter is a family. If F is a family, then so is kF, the dual of F, defined as the set of all subsets of ω that meet every element of F. Notice that kkF = F for every family F. The set of all infinite subsets of ω is a family. Its dual is the filter of cofinite sets. A is thick iff A contains arbitrarily long intervals. We denote the family of thick sets by Θ. A is syndetic iff it has “bounded gaps” in ω. The family of syndetic sets is denoted by Σ. Note that Σ = kΘ. If F and G are families, then F#G = {A ∩ B : A ∈ F and B ∈ G} (sometimes denoted F·G) is also a family. The sets in Θ#Σ are called piecewise syndetic. Will Brian Stone duality, more duality, and dynamics in βω Theorem (Glasner, 1980) A family F has the Ramsey property if and only if kF is a filter.

Theorem (Furstenberg, 1981) For any family F, F#kF has the Ramsey property.

Families with the Ramsey property: any ultrafilter F, the infinite sets, the sets containing arbitrarily long arithmetic sequences (van der Waerden), the IP sets (Hindman) Families without the Ramsey property: non-maximal filters, Θ, Σ, dense sets with respect to the topology on Q

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Connections to Ramsey theory

A family F has the Ramsey property if whenever A ∈ F and A = B1 ∪ · · · ∪ Bn, then there is some i ≤ n such that Bi ∈ F.

Will Brian Stone duality, more duality, and dynamics in βω Theorem (Furstenberg, 1981) For any family F, F#kF has the Ramsey property.

Families with the Ramsey property: any ultrafilter F, the infinite sets, the sets containing arbitrarily long arithmetic sequences (van der Waerden), the IP sets (Hindman) Families without the Ramsey property: non-maximal filters, Θ, Σ, dense sets with respect to the topology on Q

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Connections to Ramsey theory

A family F has the Ramsey property if whenever A ∈ F and A = B1 ∪ · · · ∪ Bn, then there is some i ≤ n such that Bi ∈ F. Theorem (Glasner, 1980) A family F has the Ramsey property if and only if kF is a filter.

Will Brian Stone duality, more duality, and dynamics in βω Families with the Ramsey property: any ultrafilter F, the infinite sets, the sets containing arbitrarily long arithmetic sequences (van der Waerden), the IP sets (Hindman) Families without the Ramsey property: non-maximal filters, Θ, Σ, dense sets with respect to the topology on Q

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Connections to Ramsey theory

A family F has the Ramsey property if whenever A ∈ F and A = B1 ∪ · · · ∪ Bn, then there is some i ≤ n such that Bi ∈ F. Theorem (Glasner, 1980) A family F has the Ramsey property if and only if kF is a filter.

Theorem (Furstenberg, 1981) For any family F, F#kF has the Ramsey property.

Will Brian Stone duality, more duality, and dynamics in βω Families without the Ramsey property: non-maximal filters, Θ, Σ, dense sets with respect to the topology on Q

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Connections to Ramsey theory

A family F has the Ramsey property if whenever A ∈ F and A = B1 ∪ · · · ∪ Bn, then there is some i ≤ n such that Bi ∈ F. Theorem (Glasner, 1980) A family F has the Ramsey property if and only if kF is a filter.

Theorem (Furstenberg, 1981) For any family F, F#kF has the Ramsey property.

Families with the Ramsey property: any ultrafilter F, the infinite sets, the sets containing arbitrarily long arithmetic sequences (van der Waerden), the IP sets (Hindman)

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Connections to Ramsey theory

A family F has the Ramsey property if whenever A ∈ F and A = B1 ∪ · · · ∪ Bn, then there is some i ≤ n such that Bi ∈ F. Theorem (Glasner, 1980) A family F has the Ramsey property if and only if kF is a filter.

Theorem (Furstenberg, 1981) For any family F, F#kF has the Ramsey property.

Families with the Ramsey property: any ultrafilter F, the infinite sets, the sets containing arbitrarily long arithmetic sequences (van der Waerden), the IP sets (Hindman) Families without the Ramsey property: non-maximal filters, Θ, Σ, dense sets with respect to the topology on Q

Will Brian Stone duality, more duality, and dynamics in βω Explicitly, G is an (ultra)filter on F iff G is a (maximal) upwards hereditary subset of F such that if A, B ∈ G then A ∩ B ∈ G. For example: An ultrafilter on ω is just an ultrafilter on the family of nonempty sets. A free ultrafilter on ω is just an ultrafilter on the family of infinite sets. If F is a filter then F itself is the unique ultrafilter on F.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Ultrafilters on families

For any family F, an F-ultrafilter is an ultrafilter on (F, ⊆).

Will Brian Stone duality, more duality, and dynamics in βω For example: An ultrafilter on ω is just an ultrafilter on the family of nonempty sets. A free ultrafilter on ω is just an ultrafilter on the family of infinite sets. If F is a filter then F itself is the unique ultrafilter on F.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Ultrafilters on families

For any family F, an F-ultrafilter is an ultrafilter on (F, ⊆). Explicitly, G is an (ultra)filter on F iff G is a (maximal) upwards hereditary subset of F such that if A, B ∈ G then A ∩ B ∈ G.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Ultrafilters on families

For any family F, an F-ultrafilter is an ultrafilter on (F, ⊆). Explicitly, G is an (ultra)filter on F iff G is a (maximal) upwards hereditary subset of F such that if A, B ∈ G then A ∩ B ∈ G. For example: An ultrafilter on ω is just an ultrafilter on the family of nonempty sets. A free ultrafilter on ω is just an ultrafilter on the family of infinite sets. If F is a filter then F itself is the unique ultrafilter on F.

Will Brian Stone duality, more duality, and dynamics in βω Proposition n o Let F be any family. The closure in βω of S Gˆ: G ∈ U(F) is equal to (kF#F)ˆ.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results I

Let U(F) denote the set of all ultrafilters on a family F. Proposition F is a Boolean algebra if and only if F has the Ramsey property. In this case U(F) is naturally a Stone space, and in fact S(U(F)) = kˆF.

Proposition F has the Ramsey property if and only if U(F) ⊆ βω.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results I

Let U(F) denote the set of all ultrafilters on a family F. Proposition F is a Boolean algebra if and only if F has the Ramsey property. In this case U(F) is naturally a Stone space, and in fact S(U(F)) = kˆF.

Proposition F has the Ramsey property if and only if U(F) ⊆ βω.

Proposition n o Let F be any family. The closure in βω of S Gˆ: G ∈ U(F) is equal to (kF#F)ˆ.

Will Brian Stone duality, more duality, and dynamics in βω If the closures of these sets are always equal, must the sets themselves also be equal? No! Theorem n o Suppose S Gˆ: G ∈ U(F) is not closed in βω. Then n o n o S Gˆ: G ∈ U(F) 6= S Gˆ: G ∈ U(kF) .

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results II

Corollary n o n o The closures of S Gˆ: G ∈ U(F) and S Gˆ: G ∈ U(kF) are equal for any family F.

Will Brian Stone duality, more duality, and dynamics in βω No! Theorem n o Suppose S Gˆ: G ∈ U(F) is not closed in βω. Then n o n o S Gˆ: G ∈ U(F) 6= S Gˆ: G ∈ U(kF) .

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results II

Corollary n o n o The closures of S Gˆ: G ∈ U(F) and S Gˆ: G ∈ U(kF) are equal for any family F.

If the closures of these sets are always equal, must the sets themselves also be equal?

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results II

Corollary n o n o The closures of S Gˆ: G ∈ U(F) and S Gˆ: G ∈ U(kF) are equal for any family F.

If the closures of these sets are always equal, must the sets themselves also be equal? No! Theorem n o Suppose S Gˆ: G ∈ U(F) is not closed in βω. Then n o n o S Gˆ: G ∈ U(F) 6= S Gˆ: G ∈ U(kF) .

Will Brian Stone duality, more duality, and dynamics in βω It may be true that every nontrivial pair of dual families (not just Θ and Σ) is a counterexample too.

Question n o n o Are S Gˆ: G ∈ U(F) and S Gˆ: G ∈ U(kF) equal only when F or kF is a filter?

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results III

n o If we set F = Θ then, as we will see shortly, S Gˆ: G ∈ U(F) has algebraic significance in (βω, +). In particular, this set is the smallest two-sided ideal of this semigroup. It is known that this ideal is not closed, so this theorem provides a negative answer to the preceding question.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Some basic results III

n o If we set F = Θ then, as we will see shortly, S Gˆ: G ∈ U(F) has algebraic significance in (βω, +). In particular, this set is the smallest two-sided ideal of this semigroup. It is known that this ideal is not closed, so this theorem provides a negative answer to the preceding question.

It may be true that every nontrivial pair of dual families (not just Θ and Σ) is a counterexample too.

Question n o n o Are S Gˆ: G ∈ U(F) and S Gˆ: G ∈ U(kF) equal only when F or kF is a filter?

Will Brian Stone duality, more duality, and dynamics in βω Intuitively, this proposition states that dual families are topologically “orthogonal” in βω. Question (The intersection question) Let F be a family, G an ultrafilter on F, and H an ultrafilter on kF. Is it necessarily true that G#H is an ultrafilter on ω?

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A basis result

Theorem Let F be any family and let G be an ultrafilter on F. Then n o Gˆ ∩ Hˆ : H ∈ kF

is a basis for the topology of Gˆ.

Will Brian Stone duality, more duality, and dynamics in βω Question (The intersection question) Let F be a family, G an ultrafilter on F, and H an ultrafilter on kF. Is it necessarily true that G#H is an ultrafilter on ω?

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A basis result

Theorem Let F be any family and let G be an ultrafilter on F. Then n o Gˆ ∩ Hˆ : H ∈ kF

is a basis for the topology of Gˆ.

Intuitively, this proposition states that dual families are topologically “orthogonal” in βω.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A basis result

Theorem Let F be any family and let G be an ultrafilter on F. Then n o Gˆ ∩ Hˆ : H ∈ kF

is a basis for the topology of Gˆ.

Intuitively, this proposition states that dual families are topologically “orthogonal” in βω. Question (The intersection question) Let F be a family, G an ultrafilter on F, and H an ultrafilter on kF. Is it necessarily true that G#H is an ultrafilter on ω?

Will Brian Stone duality, more duality, and dynamics in βω Theorem The following are equivalent for a filter F: F is a Θ-ultrafilter. (Fˆ, σ) is a minimal subsystem of (βω, σ). (Fˆ, +) is a minimal right ideal of (βω, +).

The union of all minimal right ideals of (βω, +), usually denoted M, is the smallest two-sided ideal of (βω, +).

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Θ-ultrafilters

It turns out that Θ-ultrafilters play a special role in the dynamical and algebraic structure of βω.

Will Brian Stone duality, more duality, and dynamics in βω The union of all minimal right ideals of (βω, +), usually denoted M, is the smallest two-sided ideal of (βω, +).

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Θ-ultrafilters

It turns out that Θ-ultrafilters play a special role in the dynamical and algebraic structure of βω. Theorem The following are equivalent for a filter F: F is a Θ-ultrafilter. (Fˆ, σ) is a minimal subsystem of (βω, σ). (Fˆ, +) is a minimal right ideal of (βω, +).

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Θ-ultrafilters

It turns out that Θ-ultrafilters play a special role in the dynamical and algebraic structure of βω. Theorem The following are equivalent for a filter F: F is a Θ-ultrafilter. (Fˆ, σ) is a minimal subsystem of (βω, σ). (Fˆ, +) is a minimal right ideal of (βω, +).

The union of all minimal right ideals of (βω, +), usually denoted M, is the smallest two-sided ideal of (βω, +).

Will Brian Stone duality, more duality, and dynamics in βω Proof. Let G be any Σ-ultrafilter. A positive answer to the intersection question is precisely the (Stone dual of the) assertion that Gˆ has the required property.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Σ-ultrafilters

If the intersection question has a positive answer, then we obtain the following: Proposition There is a closed subset of βω that meets every minimal right ideal in exactly one point.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Σ-ultrafilters

If the intersection question has a positive answer, then we obtain the following: Proposition There is a closed subset of βω that meets every minimal right ideal in exactly one point.

Proof. Let G be any Σ-ultrafilter. A positive answer to the intersection question is precisely the (Stone dual of the) assertion that Gˆ has the required property.

Will Brian Stone duality, more duality, and dynamics in βω We say that a (non-principal) filter F on ω is almost Fr´echet or feeble if there is a finite-to-one map f : ω → ω such that βf (F) is the filter of cofinite sets. We say that a filter F on ω is almost an ultrafilter if there is a finite-to-one map f : ω → ω such that βf (F) is an ultrafilter. The Filter Dichotomy states that every filter on ω is either almost Fr´echetor almost an ultrafilter. FD is consistent with and independent of ZFC: it follows from u < g (in fact, it is equivalent to a slightly modified version of this inequality) and is false under MA.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Finite-to-one maps and the Filter Dichotomy

Recall that every map f : ω → ω induces a map βf : βω → βω.

Will Brian Stone duality, more duality, and dynamics in βω We say that a filter F on ω is almost an ultrafilter if there is a finite-to-one map f : ω → ω such that βf (F) is an ultrafilter. The Filter Dichotomy states that every filter on ω is either almost Fr´echetor almost an ultrafilter. FD is consistent with and independent of ZFC: it follows from u < g (in fact, it is equivalent to a slightly modified version of this inequality) and is false under MA.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Finite-to-one maps and the Filter Dichotomy

Recall that every map f : ω → ω induces a map βf : βω → βω. We say that a (non-principal) filter F on ω is almost Fr´echet or feeble if there is a finite-to-one map f : ω → ω such that βf (F) is the filter of cofinite sets.

Will Brian Stone duality, more duality, and dynamics in βω The Filter Dichotomy states that every filter on ω is either almost Fr´echetor almost an ultrafilter. FD is consistent with and independent of ZFC: it follows from u < g (in fact, it is equivalent to a slightly modified version of this inequality) and is false under MA.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Finite-to-one maps and the Filter Dichotomy

Recall that every map f : ω → ω induces a map βf : βω → βω. We say that a (non-principal) filter F on ω is almost Fr´echet or feeble if there is a finite-to-one map f : ω → ω such that βf (F) is the filter of cofinite sets. We say that a filter F on ω is almost an ultrafilter if there is a finite-to-one map f : ω → ω such that βf (F) is an ultrafilter.

Will Brian Stone duality, more duality, and dynamics in βω FD is consistent with and independent of ZFC: it follows from u < g (in fact, it is equivalent to a slightly modified version of this inequality) and is false under MA.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Finite-to-one maps and the Filter Dichotomy

Recall that every map f : ω → ω induces a map βf : βω → βω. We say that a (non-principal) filter F on ω is almost Fr´echet or feeble if there is a finite-to-one map f : ω → ω such that βf (F) is the filter of cofinite sets. We say that a filter F on ω is almost an ultrafilter if there is a finite-to-one map f : ω → ω such that βf (F) is an ultrafilter. The Filter Dichotomy states that every filter on ω is either almost Fr´echetor almost an ultrafilter.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Finite-to-one maps and the Filter Dichotomy

Recall that every map f : ω → ω induces a map βf : βω → βω. We say that a (non-principal) filter F on ω is almost Fr´echet or feeble if there is a finite-to-one map f : ω → ω such that βf (F) is the filter of cofinite sets. We say that a filter F on ω is almost an ultrafilter if there is a finite-to-one map f : ω → ω such that βf (F) is an ultrafilter. The Filter Dichotomy states that every filter on ω is either almost Fr´echetor almost an ultrafilter. FD is consistent with and independent of ZFC: it follows from u < g (in fact, it is equivalent to a slightly modified version of this inequality) and is false under MA.

Will Brian Stone duality, more duality, and dynamics in βω Proposition (folklore) If F is a non-principal ultrafilter on ω, then F does not have the property of Baire in 2ω. In fact, if U ⊆ 2ω is open then F ∩ U does not have the property of Baire.

Proof. Suppose F has the property of Baire on U. Then there is some <ω s ∈ 2 such that X = F ∩ [s] is meager (or comeager). Let fn(x) fix the first n terms of x and change those after the first n. If n = length(s), then fn(X ) ∩ X = ∅ and fn(X ) ∪ X = [s]. Thus fn(X ) is comeager (or meager), contradicting the fact that fn is a homeomorphism.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy The property of Baire

Every set of subsets of ω can be identified with a subset of 2ω via characteristic functions.

Will Brian Stone duality, more duality, and dynamics in βω Proof. Suppose F has the property of Baire on U. Then there is some <ω s ∈ 2 such that X = F ∩ [s] is meager (or comeager). Let fn(x) fix the first n terms of x and change those after the first n. If n = length(s), then fn(X ) ∩ X = ∅ and fn(X ) ∪ X = [s]. Thus fn(X ) is comeager (or meager), contradicting the fact that fn is a homeomorphism.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy The property of Baire

Every set of subsets of ω can be identified with a subset of 2ω via characteristic functions. Proposition (folklore) If F is a non-principal ultrafilter on ω, then F does not have the property of Baire in 2ω. In fact, if U ⊆ 2ω is open then F ∩ U does not have the property of Baire.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy The property of Baire

Every set of subsets of ω can be identified with a subset of 2ω via characteristic functions. Proposition (folklore) If F is a non-principal ultrafilter on ω, then F does not have the property of Baire in 2ω. In fact, if U ⊆ 2ω is open then F ∩ U does not have the property of Baire.

Proof. Suppose F has the property of Baire on U. Then there is some <ω s ∈ 2 such that X = F ∩ [s] is meager (or comeager). Let fn(x) fix the first n terms of x and change those after the first n. If n = length(s), then fn(X ) ∩ X = ∅ and fn(X ) ∪ X = [s]. Thus fn(X ) is comeager (or meager), contradicting the fact that fn is a homeomorphism.

Will Brian Stone duality, more duality, and dynamics in βω For example, Θ and Σ have the property of Baire. If [ Un = {[s]: s contains n consecutive 1’s} ,

[ Vn = {[s]: s contains n consecutive 0’s} , T S ω then Θ = n∈ω Un and Σ = n∈ω(2 \ Vn). Thus Θ is a (comeager) Gδ and Σ is a (meager) Fσ.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Baire pairs

Proposition Let F be a (non-principal) family and suppose F has the property of Baire. Then kF also has the property of Baire. Furthermore, exactly one of F and kF is comeager, and the other is meager.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Baire pairs

Proposition Let F be a (non-principal) family and suppose F has the property of Baire. Then kF also has the property of Baire. Furthermore, exactly one of F and kF is comeager, and the other is meager.

For example, Θ and Σ have the property of Baire. If [ Un = {[s]: s contains n consecutive 1’s} ,

[ Vn = {[s]: s contains n consecutive 0’s} , T S ω then Θ = n∈ω Un and Σ = n∈ω(2 \ Vn). Thus Θ is a (comeager) Gδ and Σ is a (meager) Fσ.

Will Brian Stone duality, more duality, and dynamics in βω Proof. For the first part, recall that F can be extended to an ultrafilter p. Since p is not comeager on any open set, F is not comeager on any open set. Thus F is meager. It is know that every meager filter is almost Fr´echet. For the second part, recall that if F is a filter then U(F) = {F} and U(kF) ⊆ βω.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A trivial instance of the Filter Dichotomy

Proposition If F is a filter with the Baire property, then F is almost Fr´echet. In this case, every member of U(F) is almost Fr´echetand every member of U(kF) is almost an ultrafilter.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A trivial instance of the Filter Dichotomy

Proposition If F is a filter with the Baire property, then F is almost Fr´echet. In this case, every member of U(F) is almost Fr´echetand every member of U(kF) is almost an ultrafilter.

Proof. For the first part, recall that F can be extended to an ultrafilter p. Since p is not comeager on any open set, F is not comeager on any open set. Thus F is meager. It is know that every meager filter is almost Fr´echet. For the second part, recall that if F is a filter then U(F) = {F} and U(kF) ⊆ βω.

Will Brian Stone duality, more duality, and dynamics in βω Proof sketch. Consider the partition of ω into the intervals

 1 1  [0, 0], [1, 2], [3, 5], [6, 9],..., 2 n(n + 1), 2 n(n + 1) + n ,.... This partition works “uniformly” for members of U(Σ). Each member of Θ “splits” over our partition: if A is an infinite subset of the partition and T ∈ Θ, then either S A ∩ T ∈ Θ or (ω − S A) ∩ T ∈ Θ. Using these two facts, we can prove that this partition works for every member of U(Θ) as well.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A nontrivial instance of the Filter Dichotomy

Theorem Every member of U(Θ) is almost an ultrafilter, and every member of U(Σ) is almost Fr´echet.

Will Brian Stone duality, more duality, and dynamics in βω This partition works “uniformly” for members of U(Σ). Each member of Θ “splits” over our partition: if A is an infinite subset of the partition and T ∈ Θ, then either S A ∩ T ∈ Θ or (ω − S A) ∩ T ∈ Θ. Using these two facts, we can prove that this partition works for every member of U(Θ) as well.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A nontrivial instance of the Filter Dichotomy

Theorem Every member of U(Θ) is almost an ultrafilter, and every member of U(Σ) is almost Fr´echet.

Proof sketch. Consider the partition of ω into the intervals

 1 1  [0, 0], [1, 2], [3, 5], [6, 9],..., 2 n(n + 1), 2 n(n + 1) + n ,....

Will Brian Stone duality, more duality, and dynamics in βω Each member of Θ “splits” over our partition: if A is an infinite subset of the partition and T ∈ Θ, then either S A ∩ T ∈ Θ or (ω − S A) ∩ T ∈ Θ. Using these two facts, we can prove that this partition works for every member of U(Θ) as well.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A nontrivial instance of the Filter Dichotomy

Theorem Every member of U(Θ) is almost an ultrafilter, and every member of U(Σ) is almost Fr´echet.

Proof sketch. Consider the partition of ω into the intervals

 1 1  [0, 0], [1, 2], [3, 5], [6, 9],..., 2 n(n + 1), 2 n(n + 1) + n ,.... This partition works “uniformly” for members of U(Σ).

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy A nontrivial instance of the Filter Dichotomy

Theorem Every member of U(Θ) is almost an ultrafilter, and every member of U(Σ) is almost Fr´echet.

Proof sketch. Consider the partition of ω into the intervals

 1 1  [0, 0], [1, 2], [3, 5], [6, 9],..., 2 n(n + 1), 2 n(n + 1) + n ,.... This partition works “uniformly” for members of U(Σ). Each member of Θ “splits” over our partition: if A is an infinite subset of the partition and T ∈ Θ, then either S A ∩ T ∈ Θ or (ω − S A) ∩ T ∈ Θ. Using these two facts, we can prove that this partition works for every member of U(Θ) as well.

Will Brian Stone duality, more duality, and dynamics in βω However, the “splitting” part does not work in general. Proposition There is a comeager family F on ω such that, for any partition {In : n ∈ ω} of ω into finite intervals, there is some B ∈ F and S S A ⊆ ω such that neither B ∩ ( n∈A In) nor B ∩ (ω − n∈A In) is in F.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Generalizing the proof

The “uniform” part of this proof works in the more general case: Proposition Let F be a meager family. There is a finite-to-one function f : ω → ω such that if G ∈ U(F) then βf (G) is the Fr´echetfilter.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Generalizing the proof

The “uniform” part of this proof works in the more general case: Proposition Let F be a meager family. There is a finite-to-one function f : ω → ω such that if G ∈ U(F) then βf (G) is the Fr´echetfilter.

However, the “splitting” part does not work in general. Proposition There is a comeager family F on ω such that, for any partition {In : n ∈ ω} of ω into finite intervals, there is some B ∈ F and S S A ⊆ ω such that neither B ∩ ( n∈A In) nor B ∩ (ω − n∈A In) is in F.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy The dichotomy question

It is still possible, of course, that the conclusion of our theorem holds in general for families with the property of Baire. Question (A filter dichotomy for ZFC) Let F be a family with the property of Baire. Is it true that every F-ultrafilter is almost Fr´echet(or almost an ultrafilter) and that every kF-ultrafilter is almost an ultrafilter (resp., almost Fr´echet)? In other words, does the Filter Dichotomy hold for ultrafilters on families with the property of Baire?

Will Brian Stone duality, more duality, and dynamics in βω Proposition If F is a comeager family then no F-ultrafilter is countably based. In fact, no F-ultrafilter can have a basis smaller than mCohen (the smallest cardinal at which MA fails for the Cohen order h2<ω, ⊆i).

Proposition Let F be a comeager family. If F splits over any interval partition, then every F-ultrafilter is almost an ultrafilter.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Partial progress

Proposition If F is a comeager family then |U(F)| = 22ℵ0 . In fact, there are 22ℵ0 F-ultrafilters that are almost ultrafilters.

Will Brian Stone duality, more duality, and dynamics in βω Proposition Let F be a comeager family. If F splits over any interval partition, then every F-ultrafilter is almost an ultrafilter.

A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Partial progress

Proposition If F is a comeager family then |U(F)| = 22ℵ0 . In fact, there are 22ℵ0 F-ultrafilters that are almost ultrafilters.

Proposition If F is a comeager family then no F-ultrafilter is countably based. In fact, no F-ultrafilter can have a basis smaller than mCohen (the smallest cardinal at which MA fails for the Cohen order h2<ω, ⊆i).

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Partial progress

Proposition If F is a comeager family then |U(F)| = 22ℵ0 . In fact, there are 22ℵ0 F-ultrafilters that are almost ultrafilters.

Proposition If F is a comeager family then no F-ultrafilter is countably based. In fact, no F-ultrafilter can have a basis smaller than mCohen (the smallest cardinal at which MA fails for the Cohen order h2<ω, ⊆i).

Proposition Let F be a comeager family. If F splits over any interval partition, then every F-ultrafilter is almost an ultrafilter.

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy More questions

Question If F is comeager, is it true that no F-ultrafilter can be almost Fr´echet?

Question Is it possible to have a nonmeager family F such that every F-ultrafilter is almost Fr´echet?

Question What more can be said about (comeager) families that split over some interval partition?

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy Primary sources

families: E. Akin, Recurrence in Topological Dynamics (1997), chapter 2 dynamics, algebra, and combinatorics in βω: V. Bergelson, “Minimal idempotents and ergodic Ramsey theory” (2003) filters and the filter dichotomy: A. Blass, “Combinatorial cardinal characteristics of the continuum” (2003), mostly section 9 algebra in βω: N. Hindman and D. Strauss, Algebra in the Stone-Cechˇ compactification (1998) topology in βω: J. van Mill, “An introduction to βω” (1984) topology in 2ω: J. Oxtoby, Measure and category, second edition (1980)

Will Brian Stone duality, more duality, and dynamics in βω A quick look at βω Filters and families Ultrafilters on families A Filter Dichotomy The End

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Will Brian Stone duality, more duality, and dynamics in βω