Universal automorphisms of P ( ) / fin Will Brian University of - PowerPoint PPT Presentation

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH Universal automorphisms of P ( ω ) / fin Will Brian University of North Carolina at Charlotte BLAST 2018 University of Denver Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Every function f : ω → ω induces a function f ↑ : P ( ω ) / fin → P ( ω ) / fin . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Every function f : ω → ω induces a function f ↑ : P ( ω ) / fin → P ( ω ) / fin . If f is a mod-finite permutation of ω , then the map f ↑ induced by f is an automorphism of P ( ω ) / fin . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Every function f : ω → ω induces a function f ↑ : P ( ω ) / fin → P ( ω ) / fin . If f is a mod-finite permutation of ω , then the map f ↑ induced by f is an automorphism of P ( ω ) / fin . Automorphisms of this kind are called trivial . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Every function f : ω → ω induces a function f ↑ : P ( ω ) / fin → P ( ω ) / fin . If f is a mod-finite permutation of ω , then the map f ↑ induced by f is an automorphism of P ( ω ) / fin . Automorphisms of this kind are called trivial . A “mod-finite permutation” of ω means a bijection A → B , where both A and B are co-finite subsets of ω . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH P ( ω ) / fin and its trivial self-maps P ( ω ) / fin is the Boolean algebra of all subsets of ω modulo the ideal of finite sets. Its Stone space is ω ∗ = βω \ ω . Every function f : ω → ω induces a function f ↑ : P ( ω ) / fin → P ( ω ) / fin . If f is a mod-finite permutation of ω , then the map f ↑ induced by f is an automorphism of P ( ω ) / fin . Automorphisms of this kind are called trivial . A “mod-finite permutation” of ω means a bijection A → B , where both A and B are co-finite subsets of ω . Theorem (Parovičenko, 1963) Every Boolean algebra of size ≤ℵ 1 embeds in P ( ω ) / fin . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH an example The successor map on s : ω → ω is an example of a mod-finite permutation: . . . . Its lifting to P ( ω ) / fin , namely s ↑ ([ A ]) = [ A + 1 ] is called the shift map . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH non-trivial autohomeomorphisms Whether every automorphism of P ( ω ) / fin is trivial is independent of ZFC: Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH non-trivial autohomeomorphisms Whether every automorphism of P ( ω ) / fin is trivial is independent of ZFC: The Continuum Hypothesis implies there are 2 ℵ 1 automorphisms of P ( ω ) / fin . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH non-trivial autohomeomorphisms Whether every automorphism of P ( ω ) / fin is trivial is independent of ZFC: The Continuum Hypothesis implies there are 2 ℵ 1 automorphisms of P ( ω ) / fin . The number of trivial automorphisms is only 2 ℵ 0 , so CH implies that “most” automorphisms are nontrivial. Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH non-trivial autohomeomorphisms Whether every automorphism of P ( ω ) / fin is trivial is independent of ZFC: The Continuum Hypothesis implies there are 2 ℵ 1 automorphisms of P ( ω ) / fin . The number of trivial automorphisms is only 2 ℵ 0 , so CH implies that “most” automorphisms are nontrivial. On the other hand, Shelah proved it is consistent with ZFC that every automorphism of P ( ω ) / fin is trivial. Shelah and Stepr¯ ans later showed that this is a consequence of PFA, and Veličković ultimately weakened the assumption to OCA+MA. Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH Mapping one automorphism into another Suppose A and B are Boolean algebras, and that α and β are automorphisms of A and B , respectively. Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH Mapping one automorphism into another Suppose A and B are Boolean algebras, and that α and β are automorphisms of A and B , respectively. → β , if there is an We say that α embeds in β , and we write α ֒ embedding e : A → B such that e ◦ α = β ◦ e . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH Mapping one automorphism into another Suppose A and B are Boolean algebras, and that α and β are automorphisms of A and B , respectively. → β , if there is an We say that α embeds in β , and we write α ֒ embedding e : A → B such that e ◦ α = β ◦ e . β B B e e A A α Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH Mapping one automorphism into another Suppose A and B are Boolean algebras, and that α and β are automorphisms of A and B , respectively. → β , if there is an We say that α embeds in β , and we write α ֒ embedding e : A → B such that e ◦ α = β ◦ e . β B B e e A A α Equivalently, α ֒ → β if there is a subalgebra C of A such that ( B , β ) is isomorphic to ( C , α ↾ C ) . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH the main theorem The main result of this talk is an analogue for automorphisms of Parovičenko’s result for algebras: Main Theorem Let f be a mod-finite permutation of ω . If A is a Boolean algebra of size ≤ℵ 1 and α : A → A is an automorphism, then following are equivalent: 1 α ֒ → f ↑ . → f ↑ for every countable, α -invariant subalgebra C of A . 2 α ↾ C ֒ 3 there is no “finite obstruction” to embedding α in f ↑ . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH an example of a finite obstruction Recall that s denotes the successor map n �→ n + 1. Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH an example of a finite obstruction Recall that s denotes the successor map n �→ n + 1. Proposition If x ∈ P ( ω ) / fin with [ ∅ ] � = x � = [ ω ] , then s ↑ ( x ) �≤ x . Will Brian Universal automorphisms of P ( ω ) / fin

automorphisms of P ( ω ) / fin universal automorphisms with CH universal automorphisms without CH an example of a finite obstruction Recall that s denotes the successor map n �→ n + 1. Proposition If x ∈ P ( ω ) / fin with [ ∅ ] � = x � = [ ω ] , then s ↑ ( x ) �≤ x . So, for example, if α : A → A has nontrivial fixed points, then α does not embed in s ↑ , because this proposition provides an obstruction. Will Brian Universal automorphisms of P ( ω ) / fin