topological dynamics of stable groupslogica.dmi.unisa.it/ravello/download/newelski.pdf ·...
TRANSCRIPT
![Page 1: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/1.jpg)
Topological dynamics of stable groups
Ludomir Newelski
Instytut MatematycznyUniwersytet Wroc lawski
June 2013
Newelski Topological dynamics of stable groups
![Page 2: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/2.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 3: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/3.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 4: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/4.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 5: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/5.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 6: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/6.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 7: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/7.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 8: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/8.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 9: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/9.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 10: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/10.jpg)
Set-up
T is a stable theory in language LC is a monster model of TM ≺ CG is a group definable in MSG (M) = {tp(a/M) : a ∈ GC}
Definition
Let p, q ∈ SG (M)p ∗ q = tp(a · b/M), where a |= p, b |= q and a |̂ Mb
(SG (M), ∗) is a semi-group
Gen := {p ∈ SG (M) : p is generic} is a maximal subgroup ofSG (M),a minimal left ideal in SG (M).
Gen ∼= GC/(G 0)C
Newelski Topological dynamics of stable groups
![Page 11: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/11.jpg)
A result
Theorem 1
(SG (M), ∗) is an inverse limit of a definable system oftype-definable semigroups (in Meq).
The proof uses:
the definability lemma in local stability theory (the fullversion, Pillay)
topological dynamics, particularly the functionalrepresentation of G -types.
Newelski Topological dynamics of stable groups
![Page 12: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/12.jpg)
A result
Theorem 1
(SG (M), ∗) is an inverse limit of a definable system oftype-definable semigroups (in Meq).
The proof uses:
the definability lemma in local stability theory (the fullversion, Pillay)
topological dynamics, particularly the functionalrepresentation of G -types.
Newelski Topological dynamics of stable groups
![Page 13: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/13.jpg)
A result
Theorem 1
(SG (M), ∗) is an inverse limit of a definable system oftype-definable semigroups (in Meq).
The proof uses:
the definability lemma in local stability theory (the fullversion, Pillay)
topological dynamics, particularly the functionalrepresentation of G -types.
Newelski Topological dynamics of stable groups
![Page 14: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/14.jpg)
A result
Theorem 1
(SG (M), ∗) is an inverse limit of a definable system oftype-definable semigroups (in Meq).
The proof uses:
the definability lemma in local stability theory (the fullversion, Pillay)
topological dynamics, particularly the functionalrepresentation of G -types.
Newelski Topological dynamics of stable groups
![Page 15: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/15.jpg)
Topological dynamics
Definition
(1) X is a G -flow if
X is a compact topological space
G acts on X by homeomorphisms
(2) X is point-transitive if there is a dense G -orbit ⊆ X .(3) Y ⊆ X is a G -subflow of X if Y is closed and G -closed.
Example
Let X be a G -flow and p ∈ X . Then cl(Gp) is a subflow of Xgenerated by p.
Newelski Topological dynamics of stable groups
![Page 16: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/16.jpg)
Topological dynamics
Definition
(1) X is a G -flow if
X is a compact topological space
G acts on X by homeomorphisms
(2) X is point-transitive if there is a dense G -orbit ⊆ X .(3) Y ⊆ X is a G -subflow of X if Y is closed and G -closed.
Example
Let X be a G -flow and p ∈ X . Then cl(Gp) is a subflow of Xgenerated by p.
Newelski Topological dynamics of stable groups
![Page 17: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/17.jpg)
Topological dynamics
Definition
(1) X is a G -flow if
X is a compact topological space
G acts on X by homeomorphisms
(2) X is point-transitive if there is a dense G -orbit ⊆ X .(3) Y ⊆ X is a G -subflow of X if Y is closed and G -closed.
Example
Let X be a G -flow and p ∈ X . Then cl(Gp) is a subflow of Xgenerated by p.
Newelski Topological dynamics of stable groups
![Page 18: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/18.jpg)
Topological dynamics
Definition
(1) X is a G -flow if
X is a compact topological space
G acts on X by homeomorphisms
(2) X is point-transitive if there is a dense G -orbit ⊆ X .(3) Y ⊆ X is a G -subflow of X if Y is closed and G -closed.
Example
Let X be a G -flow and p ∈ X . Then cl(Gp) is a subflow of Xgenerated by p.
Newelski Topological dynamics of stable groups
![Page 19: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/19.jpg)
Topological dynamics
Definition
(1) X is a G -flow if
X is a compact topological space
G acts on X by homeomorphisms
(2) X is point-transitive if there is a dense G -orbit ⊆ X .(3) Y ⊆ X is a G -subflow of X if Y is closed and G -closed.
Example
Let X be a G -flow and p ∈ X . Then cl(Gp) is a subflow of Xgenerated by p.
Newelski Topological dynamics of stable groups
![Page 20: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/20.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 21: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/21.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 22: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/22.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 23: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/23.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 24: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/24.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 25: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/25.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 26: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/26.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 27: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/27.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 28: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/28.jpg)
Topological dynamics
Definition continued
Assume X is a G -flow and p ∈ X .(4) p is almost periodic if cl(Gp) is a minimal subflow of X .(5) U ⊆ X is generic if (∃A ⊆fin G )AU = X .(6) U ⊆ X is weakly generic if (∃V ⊆ X )U ∪V is generic and V isnon-generic.(7) p is [weakly] generic if every open U 3 p is.
Assume X is a G -flow.WGen(X ) = {p ∈ X : p is weakly generic}Gen(X ) = {p ∈ X : p is generic}APer(X ) = {p ∈ X : p is almost periodic}
Newelski Topological dynamics of stable groups
![Page 29: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/29.jpg)
Topological dynamics
Fact
APer(X ) =⋃{minimal subflows of X}
APer(X ) 6= ∅WGen(X ) = cl(APer(X ))
If Gen(X ) 6= ∅, then Gen(X ) = WGen(X ) = APer(X )
Gen(X ) 6= ∅ iff there is just one minimal subflow of X .
Newelski Topological dynamics of stable groups
![Page 30: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/30.jpg)
Topological dynamics
Fact
APer(X ) =⋃{minimal subflows of X}
APer(X ) 6= ∅WGen(X ) = cl(APer(X ))
If Gen(X ) 6= ∅, then Gen(X ) = WGen(X ) = APer(X )
Gen(X ) 6= ∅ iff there is just one minimal subflow of X .
Newelski Topological dynamics of stable groups
![Page 31: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/31.jpg)
Topological dynamics
Fact
APer(X ) =⋃{minimal subflows of X}
APer(X ) 6= ∅WGen(X ) = cl(APer(X ))
If Gen(X ) 6= ∅, then Gen(X ) = WGen(X ) = APer(X )
Gen(X ) 6= ∅ iff there is just one minimal subflow of X .
Newelski Topological dynamics of stable groups
![Page 32: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/32.jpg)
Topological dynamics
Fact
APer(X ) =⋃{minimal subflows of X}
APer(X ) 6= ∅WGen(X ) = cl(APer(X ))
If Gen(X ) 6= ∅, then Gen(X ) = WGen(X ) = APer(X )
Gen(X ) 6= ∅ iff there is just one minimal subflow of X .
Newelski Topological dynamics of stable groups
![Page 33: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/33.jpg)
Topological dynamics
Fact
APer(X ) =⋃{minimal subflows of X}
APer(X ) 6= ∅WGen(X ) = cl(APer(X ))
If Gen(X ) 6= ∅, then Gen(X ) = WGen(X ) = APer(X )
Gen(X ) 6= ∅ iff there is just one minimal subflow of X .
Newelski Topological dynamics of stable groups
![Page 34: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/34.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 35: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/35.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 36: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/36.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 37: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/37.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 38: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/38.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 39: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/39.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 40: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/40.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 41: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/41.jpg)
Model theory
Let DefG (M) = {definable subsets of G}.
DefG (M) is an algebra od sets, closed under left translation inG
SG (M) = S(DefG (M))
G acts on SG (M) by left translation:
g · p = {ϕ(g−1 · x) : ϕ(x) ∈ p}
SG (M) is a point-transitive G -flow,the set {tp(g/M : g ∈ G} is a dense orbit.
p ∈ SG (M) is a generic type iff p is a generic point in theG -flow SG (M).
Newelski Topological dynamics of stable groups
![Page 42: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/42.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 43: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/43.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 44: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/44.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 45: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/45.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 46: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/46.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 47: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/47.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 48: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/48.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 49: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/49.jpg)
Topological dynamics
Let X be a point-transitive G -flow.G 3 g πg : X
≈→ X , πg (x) = g · x ,
E (X ) = cl({πg : g ∈ G}) ⊆ XX
cl is the topological closure w.r. to pointwise convergencetopology in XX
E (X ) is the Ellis (enveloping) semigroup of X
E (X ) is a point-transitive G -flow:1. for f ∈ E (X ) and g ∈ G , g ∗ f = πg ◦ f2. {πg : g ∈ G} is a dense G -orbit.
◦ is continuous on E (X ), in the first coordinate.
Newelski Topological dynamics of stable groups
![Page 50: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/50.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 51: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/51.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 52: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/52.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 53: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/53.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 54: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/54.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 55: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/55.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 56: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/56.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 57: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/57.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 58: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/58.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 59: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/59.jpg)
Ellis semigroup
Definition
1. I ⊆ E (X ) is an ideal if I 6= ∅ and fI ⊆ I for every f ∈ E (X ).2. j ∈ E (X ) is an idempotent if j2 = j .
Properties of E (X )
Minimal subflows of E (X ) = minimal ideals in E (X ).
Let I ⊆ E (X ) be a minimal ideal and j ∈ I be an idempotent.Then jI ⊆ I is a group (with identity j), called an idealsubgroup of E (X ) andI is a union of its ideal subgroups.
The ideal subgroups of E (X ) are isomorphic.
E (X ) explains the structure of X .
Sometimes X ∼= E (X ), as G -flows. For example,E (SG (M)) ∼= SG (M), as G -flows and semigroups.
Newelski Topological dynamics of stable groups
![Page 60: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/60.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 61: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/61.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 62: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/62.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 63: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/63.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 64: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/64.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 65: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/65.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 66: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/66.jpg)
Functional representation
Let A ⊆ P(G ) be a G -algebra of sets(i.e. closed under left translation in G ).Then S(A) is a G -flow.For p ∈ S(A) we define dp : A → P(G ) by:
dp(U) = {g ∈ G : g−1U ∈ p}
Definition
A is d-closed if A is closed under dp for every p ∈ S(A).
Example
A = DefG (M) is d-closed, because every p ∈ SG (M) is definable.
Newelski Topological dynamics of stable groups
![Page 67: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/67.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 68: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/68.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 69: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/69.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 70: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/70.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 71: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/71.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 72: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/72.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 73: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/73.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 74: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/74.jpg)
Functional representation
Assume A is d-closed.
For p ∈ S(A), dp ∈ End(A) := {G -endomorphisms of A}.Let d : S(A)→ End(A) map p to dp. Then d is a bijection.
d induces ∗ on S(A) so that
d : (S(A), ∗)∼=→ (End(A), ◦)
Theorem 2
(E (S(A)), ◦) ∼=1 (S(A), ∗) ∼=2 (End(A), ◦)
Proof
1. For p ∈ S(A) let lp(q) = p ∗ q.Then lp ∈ E (S(A)) and p 7→ lp gives ∼=1.2. This is d .
Newelski Topological dynamics of stable groups
![Page 75: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/75.jpg)
Functional representation
Example
If A = DefG (M) then A is d-closed and ∗ on SG (M) = S(A) fromTheorem 2 is just the free multiplication of G -types.
Newelski Topological dynamics of stable groups
![Page 76: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/76.jpg)
Functional representation
Example
If A = DefG (M) then A is d-closed and ∗ on SG (M) = S(A) fromTheorem 2 is just the free multiplication of G -types.
Newelski Topological dynamics of stable groups
![Page 77: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/77.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 78: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/78.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 79: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/79.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 80: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/80.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 81: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/81.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 82: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/82.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 83: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/83.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 84: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/84.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 85: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/85.jpg)
(SG (M), ∗) in the definable realm
Definition
1. ∆ ⊆ L is invariant if the family of relatively ∆-definable subsetsof G is closed under left and right translation in G .2. Let Inv = {∆ ⊆fin L : ∆ is invariant}.
Fact
Inv is cofinal in [L]<ω.
Let ∆ ∈ Inv .
Notation
DefG ,∆ = {relatively ∆-definable subsets of G}
SG ,∆ = S(DefG ,∆(M)),
the space of complete ∆-types of G over M.
Newelski Topological dynamics of stable groups
![Page 86: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/86.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 87: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/87.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 88: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/88.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 89: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/89.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 90: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/90.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 91: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/91.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 92: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/92.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 93: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/93.jpg)
(SG (M), ∗) in the definable realm
1 DefG ,∆(M) is a d-closed G -algebra of sets.(this relies on the full definability lemma in local stabilitytheory)
2 (SG ,∆(M), ∗) ∼= (E (SG ,∆(M)), ◦) ∼= (End(DefG ,∆(M)), ◦)(this is by Theorem 2)
3
DefG (M) =⋃
∆∈Inv
DefG ,∆(M)
4 〈SG ,∆(M),∆ ∈ Inv〉 is an inverse system of G - flows andsemi-groups(the connecting functions are restrictions)
5 SG (M) = invlim∆∈InvSG ,∆(M)(as G -flows and semigroups)
Newelski Topological dynamics of stable groups
![Page 94: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/94.jpg)
Commuting diagram
(SG (M), ∗) r //
d ∼=��
(SG ,∆(M), ∗)
d ∼=��
End(DefG (M), ◦) r // End(DefG ,∆(M), ◦)
The horizontal arrows are restrictions.All arrows are semigroup homomorphisms.
Newelski Topological dynamics of stable groups
![Page 95: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/95.jpg)
Commuting diagram
(SG (M), ∗) r //
d ∼=��
(SG ,∆(M), ∗)
d ∼=��
End(DefG (M), ◦) r // End(DefG ,∆(M), ◦)
The horizontal arrows are restrictions.All arrows are semigroup homomorphisms.
Newelski Topological dynamics of stable groups
![Page 96: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/96.jpg)
Commuting diagram
(SG (M), ∗) r //
d ∼=��
(SG ,∆(M), ∗)
d ∼=��
End(DefG (M), ◦) r // End(DefG ,∆(M), ◦)
The horizontal arrows are restrictions.All arrows are semigroup homomorphisms.
Newelski Topological dynamics of stable groups
![Page 97: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/97.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 98: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/98.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 99: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/99.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 100: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/100.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 101: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/101.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 102: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/102.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 103: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/103.jpg)
(SG (M), ∗) in the definable realm
Proposition
(SG ,∆(M), ∗) is a type-definable semigroup (in Meq).
Proof.
SG ,∆(M) is a type-definable set in Meq
(identify p ∈ SG ,∆(M) with the tuple of canonicalϕ-definitions of p, ϕ ∈ ∆)
∗ is relatively definable on SG ,∆(M).(Use d : SG ,∆(M) ∼= End(DefG ,∆(M)), the full definabilitylemma and compactness.DefG ,∆(M) is ind-definable in Meq.)
Newelski Topological dynamics of stable groups
![Page 104: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/104.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 105: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/105.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 106: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/106.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 107: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/107.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 108: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/108.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 109: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/109.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 110: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/110.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 111: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/111.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Assume f ∈ End(A).
Ker(f ) = {U ∈ A : f (U) = ∅}
Im(f ) = {f (U) : U ∈ A}
Ker(f ) is a G -ideal in A.
Im(f ) is a G -subalgebra of A.
Crucial point
Assume f , g ∈ End(A). Then
Ker(f ◦ g) ⊇ Ker(g) and Im(f ◦ g) ⊆ Im(f )
Newelski Topological dynamics of stable groups
![Page 112: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/112.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 113: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/113.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 114: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/114.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 115: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/115.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 116: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/116.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 117: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/117.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 118: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/118.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 119: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/119.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 120: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/120.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 121: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/121.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 122: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/122.jpg)
Subgroups of E (S(A)) ∼= S(A) ∼= End(A)
Proposition
Assume that S is a subgroup of End(A).Then all f ∈ S have acommon kernel K = KS and common image R = RS .Let SK ,R ={f ∈ End(A) : Ker(f ) = K , Im(f ) = R and f |R permutes R}.Then SK ,R is a maximal subgroup of End(A) containing S .
Let I = {ideal subgroups of S(A)}K = { kernels of ideal subgroups of S(A) ∼= End(A)}R = { images of ideal subgroups of S(A) ∼= End(A)}
Fact
The mapping I 3 S 7→ 〈KS ,RS〉 ∈ K ×R is a bijectionI → K ×R.The fibers of the surjective mappingAPer(S(A)) 3 p 7→ Ker(dp) ∈ K are precisely the minimalsubflows of S(A).
Newelski Topological dynamics of stable groups
![Page 123: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/123.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 124: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/124.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 125: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/125.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 126: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/126.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 127: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/127.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 128: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/128.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 129: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/129.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 130: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/130.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 131: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/131.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 132: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/132.jpg)
Maximal subgroups of SG (M) and SG ,∆(M)
Example
Let H < G be ∆-definable, ∆-connected (i.e. Mlt∆(H) = 1)(So: ∃!pH ∈ SG ,∆(M) generic of H.)Let N = NG (H) < G and SpH
= {n · pH : n ∈ N} ⊆ SG ,∆(M).Then SpH
is a maximal subgroup of SG ,∆(M) and SpH∼=def N/H.
Theorem 3
1. All maximal subgroups of SG ,∆(M) are of this form.2. If S ⊆ SG (M) is a maximal subgroup, then S = invlim∆∈InvS∆
for some maximal subgroups S∆ ⊆ SG ,∆(M).
Newelski Topological dynamics of stable groups
![Page 133: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/133.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 134: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/134.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 135: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/135.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 136: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/136.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 137: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/137.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 138: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/138.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 139: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/139.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 140: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/140.jpg)
Types as functions
SG (M) 3 p dp : DefG (M)→ DefG (M)
SG ,∆(M) 3 p dp : DefG ,∆(M)→ DefG ,∆(M)
dp Ker(dp), Im(dp)
Ker(dp) = {U ∈ DefG ,∆(M) : [U] ∩ cl(Gp) = ∅}
Idea
The larger the type p ∈ SG (M), p ∈ SG ,∆(M)
The smaller the flow cl(Gp).
The larger the kernel Ker(dp).
The smaller the image Im(dp).
The larger the (local) Morley rank of p.
Newelski Topological dynamics of stable groups
![Page 141: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/141.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 142: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/142.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 143: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/143.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 144: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/144.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 145: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/145.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 146: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/146.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 147: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/147.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 148: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/148.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 149: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/149.jpg)
Types as functions
Ker(dp), Im(dp): measures of the size of p.Let p ∈ SG (M) (or p ∈ SG ,∆(M)...)Let p∗n = p ∗ · · · ∗ p︸ ︷︷ ︸
n
.So dp∗n = dp ◦ · · · ◦ dp︸ ︷︷ ︸n
.
Let R(p) = 〈RM∆(p) : ∆ ∈ Inv〉.
Lemma
1. R(p∗n) grow (coordinatewise),Ker(dp∗n ) grow and Im(dp∗n )shrink with n = 1, 2, 3, . . . .2. The growth/shrinking of these three sequences is strictlycorrelated.
Theorem 4
Let p ∈ SG (M). Then p is ”profinitely many steps away” from atranslate of a generic type of a connected type-definable subgroupof G .
Newelski Topological dynamics of stable groups
![Page 150: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/150.jpg)
Proof of Theorem 4
Let ∆ ∈ Inv , n∆ = RM∆(G ), p∆ = p|∆ ∈ SG ,∆(M). Then:
1 p∗n∆∆ ∈ a maximal subgroup S of SG ,∆(M).
2 p∗n∆∆ is a translate of a generic type of a ∆-definable
∆-connected group H < G
3 For every l ≥ n∆, items 1. and 2. hold for p∗l∆ in place ofp∗n∆
∆ , with the same S and H.
Corollary
SG ,∆(M)n∆ =⋃{subgroups of SG ,∆(M)}.
Newelski Topological dynamics of stable groups
![Page 151: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/151.jpg)
Proof of Theorem 4
Let ∆ ∈ Inv , n∆ = RM∆(G ), p∆ = p|∆ ∈ SG ,∆(M). Then:
1 p∗n∆∆ ∈ a maximal subgroup S of SG ,∆(M).
2 p∗n∆∆ is a translate of a generic type of a ∆-definable
∆-connected group H < G
3 For every l ≥ n∆, items 1. and 2. hold for p∗l∆ in place ofp∗n∆
∆ , with the same S and H.
Corollary
SG ,∆(M)n∆ =⋃{subgroups of SG ,∆(M)}.
Newelski Topological dynamics of stable groups
![Page 152: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/152.jpg)
Proof of Theorem 4
Let ∆ ∈ Inv , n∆ = RM∆(G ), p∆ = p|∆ ∈ SG ,∆(M). Then:
1 p∗n∆∆ ∈ a maximal subgroup S of SG ,∆(M).
2 p∗n∆∆ is a translate of a generic type of a ∆-definable
∆-connected group H < G
3 For every l ≥ n∆, items 1. and 2. hold for p∗l∆ in place ofp∗n∆
∆ , with the same S and H.
Corollary
SG ,∆(M)n∆ =⋃{subgroups of SG ,∆(M)}.
Newelski Topological dynamics of stable groups
![Page 153: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/153.jpg)
Proof of Theorem 4
Let ∆ ∈ Inv , n∆ = RM∆(G ), p∆ = p|∆ ∈ SG ,∆(M). Then:
1 p∗n∆∆ ∈ a maximal subgroup S of SG ,∆(M).
2 p∗n∆∆ is a translate of a generic type of a ∆-definable
∆-connected group H < G
3 For every l ≥ n∆, items 1. and 2. hold for p∗l∆ in place ofp∗n∆
∆ , with the same S and H.
Corollary
SG ,∆(M)n∆ =⋃{subgroups of SG ,∆(M)}.
Newelski Topological dynamics of stable groups
![Page 154: Topological dynamics of stable groupslogica.dmi.unisa.it/Ravello/download/newelski.pdf · Topological dynamics of stable groups Ludomir Newelski Instytut Matematyczny Uniwersytet](https://reader033.vdocuments.us/reader033/viewer/2022050219/5f654853d04dff390023c01a/html5/thumbnails/154.jpg)
Proof of Theorem 4
Let ∆ ∈ Inv , n∆ = RM∆(G ), p∆ = p|∆ ∈ SG ,∆(M). Then:
1 p∗n∆∆ ∈ a maximal subgroup S of SG ,∆(M).
2 p∗n∆∆ is a translate of a generic type of a ∆-definable
∆-connected group H < G
3 For every l ≥ n∆, items 1. and 2. hold for p∗l∆ in place ofp∗n∆
∆ , with the same S and H.
Corollary
SG ,∆(M)n∆ =⋃{subgroups of SG ,∆(M)}.
Newelski Topological dynamics of stable groups