Divisibility in N and N Boris Sobot Faculty of Sciences, - - PowerPoint PPT Presentation

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Divisibility in N and N Boris Sobot Faculty of Sciences, - - PowerPoint PPT Presentation

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Divisibility in N and N Boris Sobot Faculty of Sciences, University of Novi Sad SetTop 2018 Boris Divisibility in N and Sobot (Novi Sad) N July 3rd 2018 1 / 29 The motivation N - discrete topological space on the set of

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Divisibility in βN and ∗N

Boris ˇ Sobot

Faculty of Sciences, University of Novi Sad

SetTop 2018

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 1 / 29

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The motivation

N - discrete topological space on the set of natural numbers βN - the set of ultrafilters on N Principal ultrafilters {A ⊆ N : n ∈ A} are identified with respective elements n ∈ N Idea: extend the divisibility relation | to βN to get results in number theory

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 2 / 29

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The motivation

N - discrete topological space on the set of natural numbers βN - the set of ultrafilters on N Principal ultrafilters {A ⊆ N : n ∈ A} are identified with respective elements n ∈ N Idea: extend the divisibility relation | to βN to get results in number theory

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 2 / 29

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The motivation

N - discrete topological space on the set of natural numbers βN - the set of ultrafilters on N Principal ultrafilters {A ⊆ N : n ∈ A} are identified with respective elements n ∈ N Idea: extend the divisibility relation | to βN to get results in number theory

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 2 / 29

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The motivation

N - discrete topological space on the set of natural numbers βN - the set of ultrafilters on N Principal ultrafilters {A ⊆ N : n ∈ A} are identified with respective elements n ∈ N Idea: extend the divisibility relation | to βN to get results in number theory

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 2 / 29

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  • | -divisibility

U = {S ⊆ N : S is upward closed for |} Definition For F, G ∈ βN F | G iff F ∩ U ⊆ G The restriction of | to N2 is the usual |

  • | is reflexive and transitive, but not antisymmetric. Hence it is an
  • rder on βN/ ∼, where

F ∼ G ⇔ F | G ∧ G | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 3 / 29

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  • | -divisibility

U = {S ⊆ N : S is upward closed for |} Definition For F, G ∈ βN F | G iff F ∩ U ⊆ G The restriction of | to N2 is the usual |

  • | is reflexive and transitive, but not antisymmetric. Hence it is an
  • rder on βN/ ∼, where

F ∼ G ⇔ F | G ∧ G | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 3 / 29

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  • | -divisibility

U = {S ⊆ N : S is upward closed for |} Definition For F, G ∈ βN F | G iff F ∩ U ⊆ G The restriction of | to N2 is the usual |

  • | is reflexive and transitive, but not antisymmetric. Hence it is an
  • rder on βN/ ∼, where

F ∼ G ⇔ F | G ∧ G | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 3 / 29

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  • | -divisibility

U = {S ⊆ N : S is upward closed for |} Definition For F, G ∈ βN F | G iff F ∩ U ⊆ G The restriction of | to N2 is the usual |

  • | is reflexive and transitive, but not antisymmetric. Hence it is an
  • rder on βN/ ∼, where

F ∼ G ⇔ F | G ∧ G | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 3 / 29

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Prime ultrafilters

P ⊆ N - the set of primes Prime ultrafilters: P ∈ βN \ {1} divisible only by 1 and themselves Lemma P ∈ βN is prime iff P ∈ P. Lemma There are 2c prime ultrafilters. Lemma For every F ∈ βN \ {1} there is prime P such that P | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 4 / 29

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Prime ultrafilters

P ⊆ N - the set of primes Prime ultrafilters: P ∈ βN \ {1} divisible only by 1 and themselves Lemma P ∈ βN is prime iff P ∈ P. Lemma There are 2c prime ultrafilters. Lemma For every F ∈ βN \ {1} there is prime P such that P | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 4 / 29

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Prime ultrafilters

P ⊆ N - the set of primes Prime ultrafilters: P ∈ βN \ {1} divisible only by 1 and themselves Lemma P ∈ βN is prime iff P ∈ P. Lemma There are 2c prime ultrafilters. Lemma For every F ∈ βN \ {1} there is prime P such that P | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 4 / 29

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Prime ultrafilters

P ⊆ N - the set of primes Prime ultrafilters: P ∈ βN \ {1} divisible only by 1 and themselves Lemma P ∈ βN is prime iff P ∈ P. Lemma There are 2c prime ultrafilters. Lemma For every F ∈ βN \ {1} there is prime P such that P | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 4 / 29

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Prime ultrafilters

P ⊆ N - the set of primes Prime ultrafilters: P ∈ βN \ {1} divisible only by 1 and themselves Lemma P ∈ βN is prime iff P ∈ P. Lemma There are 2c prime ultrafilters. Lemma For every F ∈ βN \ {1} there is prime P such that P | F.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 4 / 29

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Prime ultrafilters

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 5 / 29

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The second level

A2 = {a2 : a ∈ A} The only ultrafilter above P containing P 2 is P2 generated by {A2 : A ∈ P}

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 6 / 29

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The second level

A2 = {a2 : a ∈ A} The only ultrafilter above P containing P 2 is P2 generated by {A2 : A ∈ P}

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 6 / 29

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The second level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 7 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

where F · G = {A ∈ P(N) : {n ∈ N : {m ∈ N : mn ∈ A} ∈ G} ∈ F}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 8 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

where F · G = {A ∈ P(N) : {n ∈ N : {m ∈ N : mn ∈ A} ∈ G} ∈ F}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 8 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

where F · G = {A ∈ P(N) : {n ∈ N : {m ∈ N : mn ∈ A} ∈ G} ∈ F}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 8 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

where F · G = {A ∈ P(N) : {n ∈ N : {m ∈ N : mn ∈ A} ∈ G} ∈ F}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 8 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

where F · G = {A ∈ P(N) : {n ∈ N : {m ∈ N : mn ∈ A} ∈ G} ∈ F}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 8 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

Lemma There are either finitely many or 2c ultrafilters containing F(P,2).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 9 / 29

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The second level

A(2) = {ab : a, b ∈ A, GCD(a, b) = 1} F(P,2) = {A(2) : A ∈ P, A ⊆ P} Ultrafilters containing F(P,2) are also divisible only by 1, P and themselves

  • Example. P · P ⊇ F(P,2)

Lemma There are either finitely many or 2c ultrafilters containing F(P,2).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 9 / 29

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The second level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 10 / 29

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The second level

Theorem Let P be prime. There is unique ultrafilter F ⊇ F(P,2) if and only if P is Ramsey. Theorem (CH) There is a prime P such that there are 2c ultrafilters F ⊇ F(P,2).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 11 / 29

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The second level

Theorem Let P be prime. There is unique ultrafilter F ⊇ F(P,2) if and only if P is Ramsey. Theorem (CH) There is a prime P such that there are 2c ultrafilters F ⊇ F(P,2).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 11 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Bears similarities to another kind of product of filters F × G = {X ∈ P(N2) : (∃A ∈ F)(∃B ∈ G)A × B ⊆ X}.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 12 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Lemma There are either finitely many or 2c ultrafilters containing F(P,1),(Q,1).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 13 / 29

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The second level

AB = {ab : a ∈ A, b ∈ B, GCD(a, b) = 1} F(P,1),(Q,1) = {AB : A ∈ P, B ∈ Q, A, B ⊆ P are disjoint} Ultrafilters containing F(P,1),(Q,1) are divisible only by 1, P, Q and themselves They are exactly ultrafilters containing AB for some disjoint A, B ⊆ P

  • Example. P · Q, Q · P ⊇ F(P,1),(Q,1)

Lemma There are either finitely many or 2c ultrafilters containing F(P,1),(Q,1).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 13 / 29

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The second level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 14 / 29

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The second level

Theorem Let P, Q be primes. If there is unique F ⊇ F(P,1),(Q,1) then both P and Q are P-points. Theorem For every prime P there is a prime Q such that there are 2c ultrafilters F ⊇ F(P,1),(Q,1).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 15 / 29

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The second level

Theorem Let P, Q be primes. If there is unique F ⊇ F(P,1),(Q,1) then both P and Q are P-points. Theorem For every prime P there is a prime Q such that there are 2c ultrafilters F ⊇ F(P,1),(Q,1).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 15 / 29

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The third level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 16 / 29

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The third level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 17 / 29

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The third level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 18 / 29

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The third level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 19 / 29

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The third level

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 20 / 29

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Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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SLIDE 49

Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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Nonstandard arithmetic

A superstructure over X: V0(X) = X, Vn+1(X) = Vn(X) ∪ P(Vn(X)), V (X) =

n∈ω Vn(X).

V (Y ) is a nonstandard extension of V (X) if X ⊂ Y and there is a rank-preserving function ∗ : V (X) → V (Y ) such that ∗X = Y and satisfying: The Transfer Principle. For every bounded formula ϕ and every a1, a2, . . . , an ∈ V (X), ϕ(a1, a2, . . . , an) holds in V (X) if and only if ϕ(∗a1, ∗a2, . . . , ∗an) holds in V (Y ).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 21 / 29

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Nonstandard arithmetic

By Transfer, for x, y ∈ ∗N: x∗|y iff (∃k ∈ ∗N)y = kx. Each element n ∈ N is identified with ∗n. In every nonstandard extension V (∗N) of V (N) holds a generalization

  • f the Fundamental Theorem of Arithmetic. (Here p is the unique

increasing function from N to P.) Theorem (a) For every z ∈ ∗N and every internal sequence h(n) : n ≤ z there is unique x ∈ ∗N such that ∗p(n)h(n) ∗ x for n ≤ z and ∗p(n) ∗∤ x for n > z; we denote such element by

n≤z ∗p(n)h(n).

(b) Every x ∈ ∗N can be uniquely represented as

n≤z ∗p(n)h(n) for

some z ∈ ∗N and some internal sequence h(n) : n ≤ z such that h(z) > 0.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 22 / 29

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Nonstandard arithmetic

By Transfer, for x, y ∈ ∗N: x∗|y iff (∃k ∈ ∗N)y = kx. Each element n ∈ N is identified with ∗n. In every nonstandard extension V (∗N) of V (N) holds a generalization

  • f the Fundamental Theorem of Arithmetic. (Here p is the unique

increasing function from N to P.) Theorem (a) For every z ∈ ∗N and every internal sequence h(n) : n ≤ z there is unique x ∈ ∗N such that ∗p(n)h(n) ∗ x for n ≤ z and ∗p(n) ∗∤ x for n > z; we denote such element by

n≤z ∗p(n)h(n).

(b) Every x ∈ ∗N can be uniquely represented as

n≤z ∗p(n)h(n) for

some z ∈ ∗N and some internal sequence h(n) : n ≤ z such that h(z) > 0.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 22 / 29

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Nonstandard arithmetic

By Transfer, for x, y ∈ ∗N: x∗|y iff (∃k ∈ ∗N)y = kx. Each element n ∈ N is identified with ∗n. In every nonstandard extension V (∗N) of V (N) holds a generalization

  • f the Fundamental Theorem of Arithmetic. (Here p is the unique

increasing function from N to P.) Theorem (a) For every z ∈ ∗N and every internal sequence h(n) : n ≤ z there is unique x ∈ ∗N such that ∗p(n)h(n) ∗ x for n ≤ z and ∗p(n) ∗∤ x for n > z; we denote such element by

n≤z ∗p(n)h(n).

(b) Every x ∈ ∗N can be uniquely represented as

n≤z ∗p(n)h(n) for

some z ∈ ∗N and some internal sequence h(n) : n ≤ z such that h(z) > 0.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 22 / 29

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SLIDE 54

The connection

For every x ∈ ∗N the family {S ⊆ N : x ∈ ∗S} is an ultrafilter; it is denoted by v(x). Thus a function v : ∗N → βN is obtained. v is onto if V (∗N) is an enlargement. µ(F) = v−1[{F}] is the monad of F ∈ βN.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 23 / 29

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SLIDE 55

The connection

For every x ∈ ∗N the family {S ⊆ N : x ∈ ∗S} is an ultrafilter; it is denoted by v(x). Thus a function v : ∗N → βN is obtained. v is onto if V (∗N) is an enlargement. µ(F) = v−1[{F}] is the monad of F ∈ βN.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 23 / 29

slide-56
SLIDE 56

The connection

For every x ∈ ∗N the family {S ⊆ N : x ∈ ∗S} is an ultrafilter; it is denoted by v(x). Thus a function v : ∗N → βN is obtained. v is onto if V (∗N) is an enlargement. µ(F) = v−1[{F}] is the monad of F ∈ βN.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 23 / 29

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SLIDE 57

The connection

Similarities between V (∗N) and βN:

  • for n ∈ N, v(n) = n (the corresponding principal ultrafilter);
  • x ∈ ∗N is prime iff v(x) is a prime ultrafilter;
  • x ∈ ∗N is divisible by n ∈ N iff v(x) is divisible by n...

Theorem The following conditions are equivalent for every two ultrafilters F, G ∈ βN: (i) F | G; (ii) in every enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y; (iii) in some enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y. ((i)⇒(ii) for any nonstandard extension.)

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 24 / 29

slide-58
SLIDE 58

The connection

Similarities between V (∗N) and βN:

  • for n ∈ N, v(n) = n (the corresponding principal ultrafilter);
  • x ∈ ∗N is prime iff v(x) is a prime ultrafilter;
  • x ∈ ∗N is divisible by n ∈ N iff v(x) is divisible by n...

Theorem The following conditions are equivalent for every two ultrafilters F, G ∈ βN: (i) F | G; (ii) in every enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y; (iii) in some enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y. ((i)⇒(ii) for any nonstandard extension.)

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 24 / 29

slide-59
SLIDE 59

The connection

Similarities between V (∗N) and βN:

  • for n ∈ N, v(n) = n (the corresponding principal ultrafilter);
  • x ∈ ∗N is prime iff v(x) is a prime ultrafilter;
  • x ∈ ∗N is divisible by n ∈ N iff v(x) is divisible by n...

Theorem The following conditions are equivalent for every two ultrafilters F, G ∈ βN: (i) F | G; (ii) in every enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y; (iii) in some enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y. ((i)⇒(ii) for any nonstandard extension.)

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 24 / 29

slide-60
SLIDE 60

The connection

Similarities between V (∗N) and βN:

  • for n ∈ N, v(n) = n (the corresponding principal ultrafilter);
  • x ∈ ∗N is prime iff v(x) is a prime ultrafilter;
  • x ∈ ∗N is divisible by n ∈ N iff v(x) is divisible by n...

Theorem The following conditions are equivalent for every two ultrafilters F, G ∈ βN: (i) F | G; (ii) in every enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y; (iii) in some enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y. ((i)⇒(ii) for any nonstandard extension.)

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 24 / 29

slide-61
SLIDE 61

The connection

Similarities between V (∗N) and βN:

  • for n ∈ N, v(n) = n (the corresponding principal ultrafilter);
  • x ∈ ∗N is prime iff v(x) is a prime ultrafilter;
  • x ∈ ∗N is divisible by n ∈ N iff v(x) is divisible by n...

Theorem The following conditions are equivalent for every two ultrafilters F, G ∈ βN: (i) F | G; (ii) in every enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y; (iii) in some enlargement V (∗N), there are x, y ∈ ∗N such that v(x) = F, v(y) = G and x ∗| y. ((i)⇒(ii) for any nonstandard extension.)

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 24 / 29

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SLIDE 62

The connection

Lemma Let V (∗N) be any nonstandard extension. (a) x ∈ ∗N is of the form p2 for some p ∈ ∗P if and only if v(x) = P2 for some prime ultrafilter P. (b) x ∈ ∗N is of the form p · q for two distinct primes p, q such that v(p) = v(q) = P if and only if v(x) ⊇ F(P,2). (c) x ∈ ∗N is of the form p · q for two primes p, q such that v(p) = P, v(q) = Q and P = Q if and only if v(x) ⊇ F(P,1),(Q,1).

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 25 / 29

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SLIDE 63

The connection

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 26 / 29

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SLIDE 64

Above finite levels of the | -hierarchy

Theorem (a) There is the | -greatest class MAX of ultrafilters. (b) Every F ∈ βN \ MAX has an immediate successor in (βN, | ). (c) Every F ∈ βN such that there are p ∈ P and n ∈ N \ {0} so that pn∗F has an immediate predecessor in (βN, | ). (d) Every | -ascending sequence of ultrafilters of length ω has the least upper bound. (e) Every | -descending sequence of ultrafilters of length ω has the greatest lower bound.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 27 / 29

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SLIDE 65

Above finite levels of the | -hierarchy

Theorem (a) There is the | -greatest class MAX of ultrafilters. (b) Every F ∈ βN \ MAX has an immediate successor in (βN, | ). (c) Every F ∈ βN such that there are p ∈ P and n ∈ N \ {0} so that pn∗F has an immediate predecessor in (βN, | ). (d) Every | -ascending sequence of ultrafilters of length ω has the least upper bound. (e) Every | -descending sequence of ultrafilters of length ω has the greatest lower bound.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 27 / 29

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SLIDE 66

Above finite levels of the | -hierarchy

Theorem (a) There is the | -greatest class MAX of ultrafilters. (b) Every F ∈ βN \ MAX has an immediate successor in (βN, | ). (c) Every F ∈ βN such that there are p ∈ P and n ∈ N \ {0} so that pn∗F has an immediate predecessor in (βN, | ). (d) Every | -ascending sequence of ultrafilters of length ω has the least upper bound. (e) Every | -descending sequence of ultrafilters of length ω has the greatest lower bound.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 27 / 29

slide-67
SLIDE 67

Above finite levels of the | -hierarchy

Theorem (a) There is the | -greatest class MAX of ultrafilters. (b) Every F ∈ βN \ MAX has an immediate successor in (βN, | ). (c) Every F ∈ βN such that there are p ∈ P and n ∈ N \ {0} so that pn∗F has an immediate predecessor in (βN, | ). (d) Every | -ascending sequence of ultrafilters of length ω has the least upper bound. (e) Every | -descending sequence of ultrafilters of length ω has the greatest lower bound.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 27 / 29

slide-68
SLIDE 68

Above finite levels of the | -hierarchy

Theorem (a) There is the | -greatest class MAX of ultrafilters. (b) Every F ∈ βN \ MAX has an immediate successor in (βN, | ). (c) Every F ∈ βN such that there are p ∈ P and n ∈ N \ {0} so that pn∗F has an immediate predecessor in (βN, | ). (d) Every | -ascending sequence of ultrafilters of length ω has the least upper bound. (e) Every | -descending sequence of ultrafilters of length ω has the greatest lower bound.

Boris ˇ Sobot (Novi Sad) Divisibility in βN and ∗ N July 3rd 2018 27 / 29

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SLIDE 69

References

[1] M. Di Nasso, M. Forti, Topological and nonstandard extensions,

  • Monatsh. Math. 144 (2005), 89–112.

[2] S.-A. Ng, H. Render, The Puritz order and its relationship to the Rudin-Keisler order, in: Reuniting the antipodes - Constructive and nonstandard views of the continuum, (Schuster P., Berger U., Osswald H., eds.), Kluwer Academic Publishers (2001), 157–166. [3] C. Puritz, Skies, constellations and monads, in: Contributions to non-standard analysis (Luxemburg W. A. J., Robinson A., eds.), North Holland (1972), 215–243. [4] B. ˇ Sobot: Divisibility in the Stone- ˇ Cech compactification, Rep.

  • Math. Logic 50 (2015), 53-66.

[5] B. ˇ Sobot: | -divisibility of ultrafilters, arXiv 1703.05999 [6] B. ˇ Sobot: Divisibility in nonstandard arithmetic, arXiv 1806.06236

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SLIDE 70

Thank you for your attention!

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