Strongly generic sets Ludomir Newelski Instytut Matematyczny - - PowerPoint PPT Presentation

Strongly generic sets

Ludomir Newelski

Instytut Matematyczny Uniwersytet Wroc lawski

June 2011

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Set-up

T is a countable complete theory, M is a model of T, G is a group definable in M. We work in a monster model C. Definition A ⊆ P(G) is a G-algebra if A is closed under: Boolean operations left translation by elements of G. Definition (1) U ⊆ G is generic if G = AU for some finite A ⊆ G. (2) U ⊆ G is strongly generic if every non-empty V ∈ A is generic, where A is the G-algebra generated by U.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

Examples

∅ is strongly generic. G is strongly generic. Assume H is a subgroup of finite index in G. Every left coset of H is strongly generic. A finite union of left cosets of H is strongly generic.

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

The stable case

Theorem 1 Assume T is stable. (1) Assume U ⊆ G is definable.Then U is strongly generic iff U is a union of left cosets of a definable subgroup of finite index in G. (2) Definable strongly generic sets U ⊆ G form a G-algebra A. (3) The restriction function r : SG(M) → S(A) is a bijection between the set of generic types and S(A).

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

Motivating example: the circle S1

M = (R, +, ·, <, . . . ) o-minimal. G = S1, definable in M. Left arcs: (a, b] and right arcs: [a, b) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S1. Finite unions of left arcs form a G-algebra Aℓ. Finite unions of right arcs form a G-algebra Ar. Aℓ ∼ = Ar as G-algebras. r : GenG(M) → S(Aℓ) is a homeomorphism. r : GenG(M) → S(Ar) also. This is not accidental...

Newelski Strongly generic sets

S1 in the saturated setting

Let M∗ ≻ M be ℵ0-saturated. Bad news The only strongly generic definable sets U ⊆ S1(M∗) are ∅ and S1(M∗). Solution Externally definable sets.

Newelski Strongly generic sets

S1 in the saturated setting

Let M∗ ≻ M be ℵ0-saturated. Bad news The only strongly generic definable sets U ⊆ S1(M∗) are ∅ and S1(M∗). Solution Externally definable sets.

Newelski Strongly generic sets

S1 in the saturated setting

Let M∗ ≻ M be ℵ0-saturated. Bad news The only strongly generic definable sets U ⊆ S1(M∗) are ∅ and S1(M∗). Solution Externally definable sets.

Newelski Strongly generic sets

S1 in the saturated setting

Let M∗ ≻ M be ℵ0-saturated. Bad news The only strongly generic definable sets U ⊆ S1(M∗) are ∅ and S1(M∗). Solution Externally definable sets.

Newelski Strongly generic sets

S1 in the saturated setting

Let M∗ ≻ M be ℵ0-saturated. Bad news The only strongly generic definable sets U ⊆ S1(M∗) are ∅ and S1(M∗). Solution Externally definable sets.

Newelski Strongly generic sets

Back to the general case

Let Defext,G(M) = the G-algebra of externally definable subsets of G. Let SGenext,G(M) = {U ⊆ext G : U is strongly generic}. SGenext,G(M) need not be an algebra of sets (see G = S1).

Newelski Strongly generic sets

Back to the general case

Let Defext,G(M) = the G-algebra of externally definable subsets of G. Let SGenext,G(M) = {U ⊆ext G : U is strongly generic}. SGenext,G(M) need not be an algebra of sets (see G = S1).

Newelski Strongly generic sets

Back to the general case

Let Defext,G(M) = the G-algebra of externally definable subsets of G. Let SGenext,G(M) = {U ⊆ext G : U is strongly generic}. SGenext,G(M) need not be an algebra of sets (see G = S1).

Newelski Strongly generic sets

Back to the general case

Let Defext,G(M) = the G-algebra of externally definable subsets of G. Let SGenext,G(M) = {U ⊆ext G : U is strongly generic}. SGenext,G(M) need not be an algebra of sets (see G = S1).

Newelski Strongly generic sets

Back to G = S1

U ⊆ext S1(M∗) is strongly generic iff U = G 00(M∗) · V for some strongly generic V ⊆def S1(M). Let A∗

ℓ = {G 00(M∗)V : V ∈ Aℓ}

A∗

r = {G 00(M∗)V : V ∈ Ar}

These are G-algebras isomorphic to Aℓ and Ar r : GenG(M∗) ≈ → S(A∗

ℓ) and

r : GenG(M∗) ≈ → S(A∗

r )

Newelski Strongly generic sets

Back to G = S1

U ⊆ext S1(M∗) is strongly generic iff U = G 00(M∗) · V for some strongly generic V ⊆def S1(M). Let A∗

ℓ = {G 00(M∗)V : V ∈ Aℓ}

A∗

r = {G 00(M∗)V : V ∈ Ar}

These are G-algebras isomorphic to Aℓ and Ar r : GenG(M∗) ≈ → S(A∗

ℓ) and

r : GenG(M∗) ≈ → S(A∗

r )

Newelski Strongly generic sets

Back to G = S1

U ⊆ext S1(M∗) is strongly generic iff U = G 00(M∗) · V for some strongly generic V ⊆def S1(M). Let A∗

ℓ = {G 00(M∗)V : V ∈ Aℓ}

A∗

r = {G 00(M∗)V : V ∈ Ar}

These are G-algebras isomorphic to Aℓ and Ar r : GenG(M∗) ≈ → S(A∗

ℓ) and

r : GenG(M∗) ≈ → S(A∗

r )

Newelski Strongly generic sets

Back to G = S1

U ⊆ext S1(M∗) is strongly generic iff U = G 00(M∗) · V for some strongly generic V ⊆def S1(M). Let A∗

ℓ = {G 00(M∗)V : V ∈ Aℓ}

A∗

r = {G 00(M∗)V : V ∈ Ar}

These are G-algebras isomorphic to Aℓ and Ar r : GenG(M∗) ≈ → S(A∗

ℓ) and

r : GenG(M∗) ≈ → S(A∗

r )

Newelski Strongly generic sets

Back to G = S1

U ⊆ext S1(M∗) is strongly generic iff U = G 00(M∗) · V for some strongly generic V ⊆def S1(M). Let A∗

ℓ = {G 00(M∗)V : V ∈ Aℓ}

A∗

r = {G 00(M∗)V : V ∈ Ar}

These are G-algebras isomorphic to Aℓ and Ar r : GenG(M∗) ≈ → S(A∗

ℓ) and

r : GenG(M∗) ≈ → S(A∗

r )

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

The general setting

Definition A maximal G-algebra A ⊆ SGenext,G(M) is called an image algebra. Example Aℓ, Ar, A∗

ℓ, A∗ r are image algebras, in the respective models.

Theorem 2 (1) Image algebras are all G-isomorphic and SGenext,G(M) is a union of them. (2) If A is an image algebra, then there is a G-epimorphism Defext,G(M) → A. This G-epimorphism is a general counterpart of a G-invariant Keisler measure.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

Let Sext,G(M) = S(Defext,G(M)) This is a G-flow. For p ∈ Sext,G(M) and U ∈ Defext,G(M) let dpU = {g ∈ G : g−1U ∈ p}. dpU ∈ Defext,G(M) dp : Defext,G(M) → Defext,G(M) is a G-endomorphism. The function d : Sext,G(M) → End(Defext,G(M)) mapping p to dp is a bijection and induces on Sext,G(M) a semigroup operation ∗.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Topological dynamics

For p, q ∈ Sext,G(M) and U ∈ Defext,G(M) we have: U ∈ p ∗ q ⇐ ⇒ dqU ∈ p. Sext,G(M) is isomorphic to its Ellis semigroup. minimal subflows of Sext,G(M) = minimal left ideals I ⊳ Sext,G(M). For p ∈ Sext,G(M): Im(dp) ⊆ Defext,G(M) is a G-subalgebra, Ker(dp) ⊆ Defext,G(M) is a G-ideal.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

If p ∈ Sext,G(M) is almost periodic, then Im(dp) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ext G arises this way, i.e. SGenext,G(M) =

• {Im(dp) : p ∈ Sext,G(M) is almost periodic}.

Every image algebra arises this way, i.e. Assume A ⊆ SGenext,G(M) is an image algebra and I ⊳ Sext,G(M). Then A = Im(dp) for some p ∈ I.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Strongly generic sets explained

Assume I ⊳ Sext,G(M). All dp, p ∈ I, have common kernel KI ⊆ Defext,G(M). Im(dp) ∼ = Defext,G(M)/KI, so all image algebras are G-isomorphic. Let A = Im(dp) be an image algebra. Then dp : Defext,G(M) → A is a G-epimorphism. Also S(A) ≈ I. So we have proved Theorem 2.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Ideal groups

Definition (1) u ∈ Sext,G(M) is an idempotent iff u2 = u. (2) J = {u ∈ Sext,G(M) : u2 = u}. (3) For I ⊳ Sext,G(M) let J(I) = J ∩ I. J(I) = ∅. Let u ∈ J(I). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p, q ∈ I are in the same uI iff Im(dp) = Im(dq). If u ∈ J(I) and A = Im(du), then du : Defext,G(M) → A is a retraction.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Image algebras and groups

Let A be an image algebra. Let GA = {U ∈ A : 1 ∈ U}. Let I ⊳ Sext,G(M). Choose u ∈ J(I) with A = Im(du). Proposition 3 GA = Stab(p) for every p ∈ uI.In particular, GA ⊆ G ∞

M .

Here G ∞

M is the invariant-connected component of G C over M.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Topological dynamics vs model theory

minimal flows in SG(M), Sext,G(M). ideal groups in Sext,G(M). almost periodic types. Question Which dynamical properties of G are independent of the particular choice of M | = T ? Ideal groups are related to G/G 00

M .

Question How do ideal groups change when we change M ? Needed: additional assumptions. E.g. existence of a bounded orbit.

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Definition Assume M is κ+-saturated strongly ℵ0-homogeneous and p ∈ Sext,G(M). The orbit of p is bounded if 2|I| ≤ κ, where I = cl(Gp). Proposition 4 Assume p ∈ Sext,G(M) has a bounded orbit. Then: (1) G ∞ exists and equals G 00. (2) The G-orbit of every almost periodic type in Sext,G(M) has size ≤ 2ℵ0. (3) Stab(q) = G 00 ∩ M for every almost periodic q ∈ Sext,G(M). Definition G has a bounded external orbit if there is a type p ∈ Sext,G(M) with a bounded orbit for every M that is κ+-saturated strongly ℵ0-homogeneous, where κ = 4(ℵ0).

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Bounded external orbits

Theorem 5 The following are equivalent: (1) G has a bounded external orbit. (2) There are boundedly many strongly generic externally definable sets. In this case the strongly generic externally definable sets are unions

• f G 00-cosets.

Definition M ≺∗ N ⇐ ⇒ Mext ≺ Next ↾ Lext,M Here Mext is the expansion of M by externally definable sets, in the language Lext,M.

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Ideal groups: changing model

Theorem 6 Assume G has a bounded external orbit, M is sufficiently saturated and M ≺∗ N. r : Sext,G(N) → Sext,G(M) is restriction. (1) Assume s′ ∈ Sext,G(N), s = r(s′) and both s, s′ are almost periodic. Let I = cl(G Ms) and I ′ = cl(G Ns′). Then r : I ′ → I is a bijection respecting the partitions of I and I ′ into ideal groups. (2) Assume I ⊳ Sext,G(M), I ′ ⊳ Sext,G(N), r[I ′] = I, u ∈ J(I), u′ ∈ J(I ′) and r[u′I ′] = uI. Then r : u′I ′ → uI is a twisted group isomorphism. That is, rp ◦ r : u′I ′ → uI is a group isomorphism, where p = r(u′)−1 in uI (group inverse) and rp is the right translation by p in the group uI. That is, for any q0, q1 ∈ u′I ′: r(q0 ∗ q1) ∗ p = r(q0) ∗ p ∗ r(q1) ∗ p

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets

Special cases

Problem Find an example where the twist is non-trivial. Assume G 00 exists , I ⊳ Sext,G(M) and u ∈ J(I). There is a natural group epimorphism F : uI → G/G 00. If G is abelian, then F splits i.e. there is a group embedding i : G/G 00 → uI with F ◦ i = id. F splits also for non-abelian G, if the strongly generic externally definable sets form an algebra of sets (equivalently: every I ⊳ Sext,G(M) is a group). Problem No group is known, where F is not an isomorphism.

Newelski Strongly generic sets