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Strongly generic sets Ludomir Newelski Instytut Matematyczny Uniwersytet Wroc lawski June 2011 Newelski Strongly generic sets Set-up T is a countable complete theory, M is a model of T , G is a group definable in M . We work in a monster

  1. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  2. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  3. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  4. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  5. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  6. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  7. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  8. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  9. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  10. Motivating example: the circle S 1 M = ( R , + , · , <, . . . ) o-minimal. G = S 1 , definable in M . Left arcs: ( a , b ] and right arcs: [ a , b ) Finite unions of left [right] arcs are strongly generic. These are all definable strongly generic sets U ⊆ S 1 . Finite unions of left arcs form a G -algebra A ℓ . Finite unions of right arcs form a G -algebra A r . A ℓ ∼ = A r as G -algebras. r : Gen G ( M ) → S ( A ℓ ) is a homeomorphism. r : Gen G ( M ) → S ( A r ) also. This is not accidental... Newelski Strongly generic sets

  11. S 1 in the saturated setting Let M ∗ ≻ M be ℵ 0 -saturated. Bad news The only strongly generic definable sets U ⊆ S 1 ( M ∗ ) are ∅ and S 1 ( M ∗ ). Solution Externally definable sets. Newelski Strongly generic sets

  12. S 1 in the saturated setting Let M ∗ ≻ M be ℵ 0 -saturated. Bad news The only strongly generic definable sets U ⊆ S 1 ( M ∗ ) are ∅ and S 1 ( M ∗ ). Solution Externally definable sets. Newelski Strongly generic sets

  13. S 1 in the saturated setting Let M ∗ ≻ M be ℵ 0 -saturated. Bad news The only strongly generic definable sets U ⊆ S 1 ( M ∗ ) are ∅ and S 1 ( M ∗ ). Solution Externally definable sets. Newelski Strongly generic sets

  14. S 1 in the saturated setting Let M ∗ ≻ M be ℵ 0 -saturated. Bad news The only strongly generic definable sets U ⊆ S 1 ( M ∗ ) are ∅ and S 1 ( M ∗ ). Solution Externally definable sets. Newelski Strongly generic sets

  15. S 1 in the saturated setting Let M ∗ ≻ M be ℵ 0 -saturated. Bad news The only strongly generic definable sets U ⊆ S 1 ( M ∗ ) are ∅ and S 1 ( M ∗ ). Solution Externally definable sets. Newelski Strongly generic sets

  16. Back to the general case Let Def ext , G ( M ) = the G -algebra of externally definable subsets of G . Let SGen ext , G ( M ) = { U ⊆ ext G : U is strongly generic } . SGen ext , G ( M ) need not be an algebra of sets (see G = S 1 ). Newelski Strongly generic sets

  17. Back to the general case Let Def ext , G ( M ) = the G -algebra of externally definable subsets of G . Let SGen ext , G ( M ) = { U ⊆ ext G : U is strongly generic } . SGen ext , G ( M ) need not be an algebra of sets (see G = S 1 ). Newelski Strongly generic sets

  18. Back to the general case Let Def ext , G ( M ) = the G -algebra of externally definable subsets of G . Let SGen ext , G ( M ) = { U ⊆ ext G : U is strongly generic } . SGen ext , G ( M ) need not be an algebra of sets (see G = S 1 ). Newelski Strongly generic sets

  19. Back to the general case Let Def ext , G ( M ) = the G -algebra of externally definable subsets of G . Let SGen ext , G ( M ) = { U ⊆ ext G : U is strongly generic } . SGen ext , G ( M ) need not be an algebra of sets (see G = S 1 ). Newelski Strongly generic sets

  20. Back to G = S 1 U ⊆ ext S 1 ( M ∗ ) is strongly generic iff U = G 00 ( M ∗ ) · V for some strongly generic V ⊆ def S 1 ( M ). Let A ∗ ℓ = { G 00 ( M ∗ ) V : V ∈ A ℓ } A ∗ r = { G 00 ( M ∗ ) V : V ∈ A r } These are G -algebras isomorphic to A ℓ and A r r : Gen G ( M ∗ ) ≈ → S ( A ∗ ℓ ) and r : Gen G ( M ∗ ) ≈ → S ( A ∗ r ) Newelski Strongly generic sets

  21. Back to G = S 1 U ⊆ ext S 1 ( M ∗ ) is strongly generic iff U = G 00 ( M ∗ ) · V for some strongly generic V ⊆ def S 1 ( M ). Let A ∗ ℓ = { G 00 ( M ∗ ) V : V ∈ A ℓ } A ∗ r = { G 00 ( M ∗ ) V : V ∈ A r } These are G -algebras isomorphic to A ℓ and A r r : Gen G ( M ∗ ) ≈ → S ( A ∗ ℓ ) and r : Gen G ( M ∗ ) ≈ → S ( A ∗ r ) Newelski Strongly generic sets

  22. Back to G = S 1 U ⊆ ext S 1 ( M ∗ ) is strongly generic iff U = G 00 ( M ∗ ) · V for some strongly generic V ⊆ def S 1 ( M ). Let A ∗ ℓ = { G 00 ( M ∗ ) V : V ∈ A ℓ } A ∗ r = { G 00 ( M ∗ ) V : V ∈ A r } These are G -algebras isomorphic to A ℓ and A r r : Gen G ( M ∗ ) ≈ → S ( A ∗ ℓ ) and r : Gen G ( M ∗ ) ≈ → S ( A ∗ r ) Newelski Strongly generic sets

  23. Back to G = S 1 U ⊆ ext S 1 ( M ∗ ) is strongly generic iff U = G 00 ( M ∗ ) · V for some strongly generic V ⊆ def S 1 ( M ). Let A ∗ ℓ = { G 00 ( M ∗ ) V : V ∈ A ℓ } A ∗ r = { G 00 ( M ∗ ) V : V ∈ A r } These are G -algebras isomorphic to A ℓ and A r r : Gen G ( M ∗ ) ≈ → S ( A ∗ ℓ ) and r : Gen G ( M ∗ ) ≈ → S ( A ∗ r ) Newelski Strongly generic sets

  24. Back to G = S 1 U ⊆ ext S 1 ( M ∗ ) is strongly generic iff U = G 00 ( M ∗ ) · V for some strongly generic V ⊆ def S 1 ( M ). Let A ∗ ℓ = { G 00 ( M ∗ ) V : V ∈ A ℓ } A ∗ r = { G 00 ( M ∗ ) V : V ∈ A r } These are G -algebras isomorphic to A ℓ and A r r : Gen G ( M ∗ ) ≈ → S ( A ∗ ℓ ) and r : Gen G ( M ∗ ) ≈ → S ( A ∗ r ) Newelski Strongly generic sets

  25. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  26. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  27. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  28. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  29. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  30. The general setting Definition A maximal G -algebra A ⊆ SGen ext , G ( M ) is called an image algebra. Example A ℓ , A r , A ∗ ℓ , A ∗ r are image algebras, in the respective models. Theorem 2 (1) Image algebras are all G -isomorphic and SGen ext , G ( M ) is a union of them. (2) If A is an image algebra, then there is a G -epimorphism Def ext , G ( M ) → A . This G -epimorphism is a general counterpart of a G -invariant Keisler measure. Newelski Strongly generic sets

  31. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  32. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  33. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  34. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  35. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  36. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  37. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  38. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  39. Topological dynamics Let S ext , G ( M ) = S ( Def ext , G ( M )) This is a G -flow. For p ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) let d p U = { g ∈ G : g − 1 U ∈ p } . d p U ∈ Def ext , G ( M ) d p : Def ext , G ( M ) → Def ext , G ( M ) is a G -endomorphism. The function d : S ext , G ( M ) → End ( Def ext , G ( M )) mapping p to d p is a bijection and induces on S ext , G ( M ) a semigroup operation ∗ . Newelski Strongly generic sets

  40. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  41. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  42. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  43. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  44. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  45. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  46. Topological dynamics For p , q ∈ S ext , G ( M ) and U ∈ Def ext , G ( M ) we have: U ∈ p ∗ q ⇐ ⇒ d q U ∈ p . S ext , G ( M ) is isomorphic to its Ellis semigroup. minimal subflows of S ext , G ( M ) = minimal left ideals I ⊳ S ext , G ( M ). For p ∈ S ext , G ( M ): Im ( d p ) ⊆ Def ext , G ( M ) is a G -subalgebra, Ker ( d p ) ⊆ Def ext , G ( M ) is a G -ideal. Newelski Strongly generic sets

  47. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  48. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  49. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  50. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  51. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  52. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  53. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  54. Strongly generic sets explained If p ∈ S ext , G ( M ) is almost periodic, then Im ( d p ) consists of strongly generic sets and is an image algebra. Every strongly generic set U ⊆ ext G arises this way, i.e. � SGen ext , G ( M ) = { Im ( d p ) : p ∈ S ext , G ( M ) is almost periodic } . Every image algebra arises this way, i.e. Assume A ⊆ SGen ext , G ( M ) is an image algebra and I ⊳ S ext , G ( M ). Then A = Im ( d p ) for some p ∈ I . Newelski Strongly generic sets

  55. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  56. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  57. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  58. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  59. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  60. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  61. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  62. Strongly generic sets explained Assume I ⊳ S ext , G ( M ). All d p , p ∈ I , have common kernel K I ⊆ Def ext , G ( M ). Im ( d p ) ∼ = Def ext , G ( M ) / K I , so all image algebras are G -isomorphic. Let A = Im ( d p ) be an image algebra. Then d p : Def ext , G ( M ) → A is a G -epimorphism. Also S ( A ) ≈ I . So we have proved Theorem 2. Newelski Strongly generic sets

  63. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  64. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  65. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  66. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  67. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  68. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  69. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  70. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  71. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  72. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  73. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  74. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  75. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

  76. Ideal groups Definition (1) u ∈ S ext , G ( M ) is an idempotent iff u 2 = u . (2) J = { u ∈ S ext , G ( M ) : u 2 = u } . (3) For I ⊳ S ext , G ( M ) let J ( I ) = J ∩ I . J ( I ) � = ∅ . Let u ∈ J ( I ). Then uI ⊆ I is a group. Groups uI are called ideal groups. All ideal groups are isomorphic and I is the disjoint union of them.(well known from topological dynamics). p , q ∈ I are in the same uI iff Im ( d p ) = Im ( d q ). If u ∈ J ( I ) and A = Im ( d u ), then d u : Def ext , G ( M ) → A is a retraction. Newelski Strongly generic sets

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