The Wadge ordering over the Borel subsets of the Scott domain is - - PowerPoint PPT Presentation

The Wadge ordering over the Borel subsets of the Scott domain is not wqo

Workshop on Wadge Theory and Automata II, June 8th, 2018, Torino Louis Vuilleumier

University of Lausanne and University Paris Diderot

June 8th, 2018, Torino

Table of contents

• 1. Introduction
• 2. Quasi-Polish spaces, a good generalization
• 3. Wadge Theory on the Scott domain
• 4. New results

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Introduction

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish

Metrizable Non-Metrizable

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish

Metrizable Non-Metrizable Polish A Polish space is a separable completely metrizable space.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish

Metrizable Non-Metrizable Polish DCPO A DCPO is a poset in which every directed subset has a supremum.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish

Metrizable Non-Metrizable Polish ω-cont. domain DCPO An ω-continuous domain is a DCPO that has a countable domain theoretic basis.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Generalization of Polish

Metrizable Non-Metrizable Polish ω-cont. domain DCPO Quasi- Polish A quasi-Polish space is a countably based completely quasi-metrizable space.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Quasi-Polish spaces, a good generalization

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces

ωω, 2ω, R, C, Rn, Rω, Iω, countable sets with the discrete topology, separable Banach spaces.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces

ωω, 2ω, R, C, Rn, Rω, Iω, countable sets with the discrete topology, separable Banach spaces.

Examples of quasi-Polish spaces

• 1. The Sierpinski space S = {0, 1} with the topology
• ∅, S, {1}
• .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces

ωω, 2ω, R, C, Rn, Rω, Iω, countable sets with the discrete topology, separable Banach spaces.

Examples of quasi-Polish spaces

• 1. The Sierpinski space S = {0, 1} with the topology
• ∅, S, {1}
• .
• 2. Consider P(ω) with the topology induced by the

quasi-metric d(x, y) = sup {2−n : n ∈ x \ y} . This is the Scott domain, and it is a quasi-Polish space.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces

ωω, 2ω, R, C, Rn, Rω, Iω, countable sets with the discrete topology, separable Banach spaces.

Examples of quasi-Polish spaces

• 1. The Sierpinski space S = {0, 1} with the topology
• ∅, S, {1}
• .
• 2. Consider P(ω) with the topology induced by the

quasi-metric d(x, y) = sup {2−n : n ∈ x \ y} . This is the Scott domain, and it is a quasi-Polish space.

Theorem (de Brecht)

◮ Every Polish space is quasi-Polish; ◮ Every ω-continuous domain is quasi-Polish;

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A generalization Examples of Polish spaces

ωω, 2ω, R, C, Rn, Rω, Iω, countable sets with the discrete topology, separable Banach spaces.

Examples of quasi-Polish spaces

• 1. The Sierpinski space S = {0, 1} with the topology
• ∅, S, {1}
• .
• 2. Consider P(ω) with the topology induced by the

quasi-metric d(x, y) = sup {2−n : n ∈ x \ y} . This is the Scott domain, and it is a quasi-Polish space.

Theorem (de Brecht)

◮ Every Polish space is quasi-Polish; ◮ Every ω-continuous domain is quasi-Polish; ◮ A metrizable space is quasi-Polish iff it is Polish.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem

The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem

The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets.

Example

In the Sierpinski space S, the open set {1} is not Σ0

2.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in quasi-Polish spaces Problem

The Borel classes on quasi-Polish spaces do not give a well behaved hierarchy, i.e. a nested sequence of collection of subsets.

Example

In the Sierpinski space S, the open set {1} is not Σ0

2.

Solution (Selivanov)

Slight modification of the definition of the Borel hierar- chy for non-metrizable sets. Let 2 α < ω1, Σ0

α(X) = n∈ω

(An \ A′

n) : An, A′ n ∈ Σ0 βn, βn < α

• .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Borel hierarchy in all topological spaces

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem

Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem

Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem

Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse.

Theorem (Hausdorff-Kuratowski)

If X is a quasi-Polish space and 1 θ < ω1, then ∆0

θ+1(X) =

• 1α<ω1

Dα(Σ0

θ(X)).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Hierarchies in quasi-Polish spaces Theorem

Let X be an uncountable quasi-Polish space, then the Borel hierarchy on X does not collapse.

Theorem (Hausdorff-Kuratowski)

If X is a quasi-Polish space and 1 θ < ω1, then ∆0

θ+1(X) =

• 1α<ω1

Dα(Σ0

θ(X)).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski)

Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X, Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π0

2(X) with A ⊆ G and a contin-

uous extension g : G → Y of f .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski)

Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X, Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π0

2(X) with A ⊆ G and a contin-

uous extension g : G → Y of f .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski)

Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X, Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π0

2(X) with A ⊆ G and a contin-

uous extension g : G → Y of f .

Theorem

A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π0

2(X).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski)

Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X, Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π0

2(X) with A ⊆ G and a contin-

uous extension g : G → Y of f .

Theorem

A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π0

2(X).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Subspaces of quasi-Polish spaces Theorem (Kuratowski)

Let X be a quasi-metrizable space, Y be completely quasi-metrizable, X, Y be countably based, A ⊆ X with f : A → Y continuous. Then there exists G ∈ Π0

2(X) with A ⊆ G and a contin-

uous extension g : G → Y of f .

Theorem

A subspace Y ⊆ X of a quasi-Polish space is quasi- Polish if and only if Y ∈ Π0

2(X).

Theorem (de Brecht)

A space X is a quasi-Polish space if and only if X ∈ Π0

2(P(ω)).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results

As in the Polish case, there exists (de Brecht):

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results

As in the Polish case, there exists (de Brecht):

◮ A game theoretical characterization of quasi-Polish

spaces;

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results

As in the Polish case, there exists (de Brecht):

◮ A game theoretical characterization of quasi-Polish

spaces;

◮ The possibility of turning any Borel set into an open

sets of a quasi-Polish topology without changing the class of Borel sets;

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results

As in the Polish case, there exists (de Brecht):

◮ A game theoretical characterization of quasi-Polish

spaces;

◮ The possibility of turning any Borel set into an open

sets of a quasi-Polish topology without changing the class of Borel sets;

◮ A nice characterization of the Borel sets, known as

Souslin’s theorem (B(X) = ∆1

1(X));

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Further extensions of classical results

As in the Polish case, there exists (de Brecht):

◮ A game theoretical characterization of quasi-Polish

spaces;

◮ The possibility of turning any Borel set into an open

sets of a quasi-Polish topology without changing the class of Borel sets;

◮ A nice characterization of the Borel sets, known as

Souslin’s theorem (B(X) = ∆1

1(X)); ◮ . . .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

◮ It is a measure of complexity that refines the Dif-

ference hierarchy;

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

◮ It is a measure of complexity that refines the Dif-

ference hierarchy;

◮ It induces a quasi-order on P(X);

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

◮ It is a measure of complexity that refines the Dif-

ference hierarchy;

◮ It induces a quasi-order on P(X); ◮ It induces a quasi-order on the Wadge degrees (writ-

ten [A]w) called the Wadge ordering and denoted by (P(X)d, d

w).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory

Let X be a topological space, and let A, B be subsets of

• X. We say that A is Wadge reducible to B, and we write

A w B, if there exists a continuous function f : X → X such that f −1(B) = A.

◮ It is a measure of complexity that refines the Dif-

ference hierarchy;

◮ It induces a quasi-order on P(X); ◮ It induces a quasi-order on the Wadge degrees (writ-

ten [A]w) called the Wadge ordering and denoted by (P(X)d, d

w).

Problem:

Understanding the shape of the Wadge ordering.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Wadge Theory on the Scott domain

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Problem Topological question

What does the Wadge ordering over the Borel subsets

• f the Scott domain look like?

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

The Scott domain vs the Cantor space

There exists a natural bijection via the characteristic function: χ : P(ω) → 2ω A → χA.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

The Scott domain vs the Cantor space

There exists a natural bijection via the characteristic function: χ : P(ω) → 2ω A → χA. The Cantor topology is the topology of positive and neg- ative information.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

The Scott domain vs the Cantor space

There exists a natural bijection via the characteristic function: χ : P(ω) → 2ω A → χA. The Cantor topology is the topology of positive and neg- ative information. The Scott topology is the topology of positive informa- tion.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Order theoretic viewpoint of the Scott domain

◮ The Scott domain is the set P(ω) endowed with the

topology induced by the basis {OF : F ⊆ ω is finite}, where OF = {x ⊆ ω : F ⊆ x}.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Order theoretic viewpoint of the Scott domain

◮ The Scott domain is the set P(ω) endowed with the

topology induced by the basis {OF : F ⊆ ω is finite}, where OF = {x ⊆ ω : F ⊆ x}.

◮ A function f : P(ω) → P(ω) is continuous iff

• 1. if x ⊆ y, then f (x) ⊆ f (y);
• 2. if D ⊆ P(ω) directed, then f

x∈D x

• =

x∈D f (x).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Order theoretic viewpoint of the Scott domain

◮ The Scott domain is the set P(ω) endowed with the

topology induced by the basis {OF : F ⊆ ω is finite}, where OF = {x ⊆ ω : F ⊆ x}.

◮ A function f : P(ω) → P(ω) is continuous iff

• 1. if x ⊆ y, then f (x) ⊆ f (y);
• 2. if D ⊆ P(ω) directed, then f

x∈D x

• =

x∈D f (x).

◮ For any ⊆-increasing function f : P<ω(ω) → P(ω),

there exists a unique continuous extension: ˆ f : P(ω) → P(ω) x →

• F⊆x finite

f (x).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

∆0

2 subsets in the Scott domain

Definition (Selivanov)

A ⊆ P(ω) is approximable if, for all x ∈ A, there exists F ∈ P<ω(ω) such that F ⊆ x and [F, x] = {y ∈ P(ω) | F ⊆ y ⊆ x} ⊆ A.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

∆0

2 subsets in the Scott domain

Definition (Selivanov)

A ⊆ P(ω) is approximable if, for all x ∈ A, there exists F ∈ P<ω(ω) such that F ⊆ x and [F, x] = {y ∈ P(ω) | F ⊆ y ⊆ x} ⊆ A.

Theorem (Selivanov)

A ∈ ∆0

2(P(ω)) if and only if both A and A∁ are approx-

imable.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A first picture of Dα

• Σ0

1(P(ω))

• Theorem (Selivanov)

For any n ∈ ω, Dn(Σ0

1) \ ˇ

Dn(Σ0

1) forms a Wadge degree.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A first picture of Dα

• Σ0

1(P(ω))

• Theorem (Selivanov)

For any n ∈ ω, Dn(Σ0

1) \ ˇ

Dn(Σ0

1) forms a Wadge degree.

Theorem (Becher, Grigorieff, Selivanov)

Let ω α < ω1. Cα

Dα

• Σ0

1(P(ω))

• Yα

Zα

?

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

New results

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A quasi-order on colored posets

Let P = (P, ⊑p, colp) and Q = (Q, ⊑q, colq) be any colored

• posets. P is reducible to Q , written P Q , if there exists

a morphism ϕ : P → Q, i.e. if there exists a mapping ϕ : P → Q such that:

• 1. for all p, p′ ∈ P, we have p ⊑p p′ implies

ϕ(p) ⊑q ϕ(p′);

• 2. for all p ∈ P, we have colp(p) = colq(ϕ(p)).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

An order theoretic viewpoint Recall

Any A ∈ ∆0

2 is uniquely determined by its finite subsets;

Any continuous function is uniquely determined by the behavior over the finite subsets.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

An order theoretic viewpoint Recall

Any A ∈ ∆0

2 is uniquely determined by its finite subsets;

Any continuous function is uniquely determined by the behavior over the finite subsets.

Theorem

Let Papp be the class of all colored approximable posets

• f the form (P<ω(ω), ⊆, colp). We have an isomorphism:
• ∆0

2(P(ω)), w

• ≈ (Papp, ).

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Problem

Previous formulation:

Topological question

What does the Wadge ordering over the ∆0

2 subsets of

the Scott domain look like?

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Problem

Previous formulation:

Topological question

What does the Wadge ordering over the ∆0

2 subsets of

the Scott domain look like? New formulation:

Order theoretic question

What does the -ordering over the colored approximable posets of the form (P<ω(ω), ⊆, colp) look like?

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Difficulty

The quasi-order (P<ω(ω), ⊆) is a mess, it is difficult to really realize what does a set look like because we need to color the whole poset...

http://philippe-fournier-viger.com/www/powersets/powerset_of_abcdef.png

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Difficulty

The quasi-order (P<ω(ω), ⊆) is a mess, it is difficult to really realize what does a set look like because we need to color the whole poset...

http://philippe-fournier-viger.com/www/powersets/powerset_of_abcdef.png

Idea

Define a subclass of colored posets which has a simpler structure; which we can represent; and on which we can define a good notion of games.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Definition of the class Preg

A regular colored poset P = (P, ⊑p, colp) is a subposet

• f (P<ω(ω), ⊆) such that:

i) ∅ ∈ P; ii) (P, ⊑p) is a well founded poset; iii) all ⊑p-maximal elements have color 1; iv) for all S ⊆ P, if S admits a supremum in (P, ⊑p), then it is unique; v) for all p ∈ P such that colp(p) = 1 and p = ∅, there exists a unique immediate predecessor p− ∈ P and at most one immediate successor p+ ∈ P. More-

• ver, if they exist, they are such that colp(p−) =

colp(p+) = 0.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

An order embedding into (∆0

2(P(ω))d, d w)

Theorem

There exists an order embedding: H : (Preg/≈, ≈) → (∆0

2(P(ω))d, d w)

[P]≈ →

• AP
• w

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

An order embedding into (∆0

2(P(ω))d, d w)

Theorem

There exists an order embedding: H : (Preg/≈, ≈) → (∆0

2(P(ω))d, d w)

[P]≈ →

• AP
• w

Conclusion

Study (Preg/≈, ≈)!

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Let P ∈ Preg

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Fix a numbering on P

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Define AP

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A small additional information Definition

Let ⊤ be a new symbol and P ∈ Preg be a colored regular

• poset. We define ⊑⊤

p as a quasi-order on P ∪ {⊤} which

extends ⊑p:

• 1. if p, p′ ∈ P, then p ⊑⊤

p p′ iff p ⊑ p′;

• 2. if p ∈ P, then p ⊑⊤

p ⊤ and ⊤ ⊑⊤ p p;

• 3. ⊤ ⊑⊤

p ⊤.

Moreover, we extend the coloring colp with col⊤

p by set-

ting col⊤

p (⊤) = 0.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

The game GPreg(P, Q )

I II p0 q0 p1 q1 p2 q2 . . . . . . p q where pn ∈ P ∪ {⊤}, qn ∈ Q ∪ {⊤}, there exists n0 ∈ ω such that, for all n n0, pn = pn0.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

The game GPreg(P, Q )

I II p0 q0 p1 q1 p2 q2 . . . . . . p q where pn ∈ P ∪ {⊤}, qn ∈ Q ∪ {⊤}, there exists n0 ∈ ω such that, for all n n0, pn = pn0. We say that II wins if and only if the two following con- ditions are satisfied:

• 1. pn ⊑⊤

p pm → qn ⊑⊤ q qm for all n, m ∈ ω;

• 2. col⊤

p (pn) = col⊤ q (qn) for all n ∈ ω.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• Q
• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• Q

1

• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 2
• Q

1

• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 2
• Q

1

• 2
• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 3
• 2
• Q

1

• 2
• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 3
• 2
• Q

1

• 3

2

• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 3
• 2
• 4
• Q

1

• 3

2

• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

Example of a run of the game GPreg(P, Q )

P 1

• 3
• 2
• 4
• Q

1

• 3

2

• 4
• On the Wadge ordering on the Scott domain

Unil, Paris-VII June 8th, 2018, Torino

A characterization of P Q Theorem

Let P, Q ∈ Preg. II has a winning strategy in GPreg(P, Q ) if and only if P Q .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

P2 P1 but P1 P2

P1 P2

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Ill foundedness of the Wadge ordering

Easy to generalize from P1 and P2 to Pn for all n ∈ N\{0}.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Ill foundedness of the Wadge ordering

Easy to generalize from P1 and P2 to Pn for all n ∈ N\{0}.

Theorem

There exists an infinite -strictly decreasing chain of colored regular posets, namely: P1 ≻ P2 ≻ P3 ≻ P4 ≻ . . .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Ill foundedness of the Wadge ordering

Easy to generalize from P1 and P2 to Pn for all n ∈ N\{0}.

Theorem

There exists an infinite -strictly decreasing chain of colored regular posets, namely: P1 ≻ P2 ≻ P3 ≻ P4 ≻ . . .

Corollary

The Wadge ordering on the ∆0

2 subsets of the Scott do-

main is ill founded.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Q1 Q2 and Q2 Q1

Q1 Q2

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Infinite antichains in the Wadge ordering

Easy to generalize from Q1 and Q2 to Qn for all n ∈ N\{0}.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Infinite antichains in the Wadge ordering

Easy to generalize from Q1 and Q2 to Qn for all n ∈ N\{0}.

Theorem

There exists an infinite sequence of colored regular posets that are pairwise -incomparable, namely: if n = m, n, m ∈ N \ {0}, then Qn ⊥ Qm.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

Infinite antichains in the Wadge ordering

Easy to generalize from Q1 and Q2 to Qn for all n ∈ N\{0}.

Theorem

There exists an infinite sequence of colored regular posets that are pairwise -incomparable, namely: if n = m, n, m ∈ N \ {0}, then Qn ⊥ Qm.

Corollary

The Wadge ordering on the ∆0

2 subsets of the Scott do-

main contains infinite antichains.

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino

A new picture of Dω

• Σ0

1(P(ω))

• Cω

Dω(Σ0

1(P(ω)))

Yω Zω · · · · · · . . .

On the Wadge ordering on the Scott domain Unil, Paris-VII June 8th, 2018, Torino