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 ≤ℵ1 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

• f 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

• f ZFC:

The Continuum Hypothesis implies there are 2ℵ1 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

• f ZFC:

The Continuum Hypothesis implies there are 2ℵ1 automorphisms of P(ω)/fin. The number of trivial automorphisms is only 2ℵ0, 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

• f ZFC:

The Continuum Hypothesis implies there are 2ℵ1 automorphisms of P(ω)/fin. The number of trivial automorphisms is only 2ℵ0, 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. We say that α embeds in β, and we write α ֒ → β, if there is an 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. We say that α embeds in β, and we write α ֒ → β, if there is an embedding e : A → B such that e ◦ α = β ◦ e. A A B B β α

e 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. We say that α embeds in β, and we write α ֒ → β, if there is an embedding e : A → B such that e ◦ α = β ◦ e. A A B B β α

e e

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 ≤ℵ1 and α : A → A is an automorphism, then following are equivalent:

1 α ֒

→ f ↑.

2 α↾C ֒

→ f ↑ for every countable, α-invariant subalgebra C of A.

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

• bstruction.

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

• bstruction. In fact, one may show that this proposition provides

the only possible finite obstruction to embedding in the shift map: Theorem Let α be an automorphism of a Boolean algebra A with |A| ≤ ℵ1. Then α embeds in the shift map s↑ if and only if α(x) ≤ x whenever 0 = x = 1.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a permutation without any small obstructions

Let t denote a permutation of ω that consists of infinitely many Z-like orbits:

t

. . . . . . . . . . . . . . . . . . . . . . . . . . .

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a permutation without any small obstructions

Let t denote a permutation of ω that consists of infinitely many Z-like orbits:

t

. . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem There are no finite obstructions to embedding in t↑. In fact, every automorphism of every countable Boolean algebra embeds in t↑.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a permutation without any small obstructions

Let t denote a permutation of ω that consists of infinitely many Z-like orbits:

t

. . . . . . . . . . . . . . . . . . . . . . . . . . . Theorem There are no finite obstructions to embedding in t↑. In fact, every automorphism of every countable Boolean algebra embeds in t↑. Consequently (applying the “main theorem”), every automorphism

• f a Boolean algebra of size ≤ℵ1 embeds in t↑.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a corollary

Corollary Assuming the Continuum Hypothesis, every automorphism of P(ω)/fin embeds in t↑.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a corollary

Corollary Assuming the Continuum Hypothesis, every automorphism of P(ω)/fin embeds in t↑. Let us say that an automorphism of P(ω)/fin is universal if every

• ther automorphism of P(ω)/fin embeds in it. Thus, according to

the corollary above, CH implies there is a universal automorphism

• f P(ω)/fin.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a corollary

Corollary Assuming the Continuum Hypothesis, every automorphism of P(ω)/fin embeds in t↑. Let us say that an automorphism of P(ω)/fin is universal if every

• ther automorphism of P(ω)/fin embeds in it. Thus, according to

the corollary above, CH implies there is a universal automorphism

• f P(ω)/fin.

Theorem Assuming the Continuum Hypothesis, t↑ embeds in 2ℵ1 distinct automorphisms of P(ω)/fin. Because a composition of embeddings is an embedding, CH implies that there are 2ℵ1 distinct universal automorhpisms 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

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

• We already mentioned that every automorphism
• f a countable Boolean algebra embeds in t↑,

so fix an embedding e0 : A0 → P(ω)/fin such that e0 ◦ α0 = t↑ ◦ e0.

e0

t↑

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

• We already mentioned that every automorphism
• f a countable Boolean algebra embeds in t↑,

so fix an embedding e0 : A0 → P(ω)/fin such that e0 ◦ α0 = t↑ ◦ e0.

e0

t↑

• Lift e0 to an embedding e1 of A1 ⊇ A0

into P(ω)/fin such that e1 ◦ α1 = t↑ ◦ e1.

e1

t↑

=

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

• We already mentioned that every automorphism
• f a countable Boolean algebra embeds in t↑,

so fix an embedding e0 : A0 → P(ω)/fin such that e0 ◦ α0 = t↑ ◦ e0.

e0

t↑

• Lift e0 to an embedding e1 of A1 ⊇ A0

into P(ω)/fin such that e1 ◦ α1 = t↑ ◦ e1.

e1

t↑

=

• Continue this up through all the αξ,

taking unions at limit stages.

e2

t↑

= =

. . .

=

eξ

t↑

=

. . .

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

• We already mentioned that every automorphism
• f a countable Boolean algebra embeds in t↑,

so fix an embedding e0 : A0 → P(ω)/fin such that e0 ◦ α0 = t↑ ◦ e0.

e0

t↑

• Lift e0 to an embedding e1 of A1 ⊇ A0

into P(ω)/fin such that e1 ◦ α1 = t↑ ◦ e1.

e1

t↑

=

• Continue this up through all the αξ,

taking unions at limit stages.

e2

t↑

= =

. . .

=

eξ

t↑

=

. . .

• In the end, e =

ξ<ω1 eξ embeds α in t↑.

e

t↑

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: what won’t work

Perhaps the most obvious strategy for proving the universality of t↑ is as follows:

• Begin with an automorphism α : A → A

where |A| ≤ ℵ1. α where |A| ≤ ℵ1. Write A as an increasing union of countable, α-invariant subalgebras

• ξ<ω1 Aξ and let αξ = α↾Aξ for all ξ < ω1.

α0

⊂

α1

⊂

α2

⊂

. . .

⊂

αξ

⊂

. . .

• We already mentioned that every automorphism
• f a countable Boolean algebra embeds in t↑,

so fix an embedding e0 : A0 → P(ω)/fin such that e0 ◦ α0 = t↑ ◦ e0.

e0

t↑

• Lift e0 to an embedding e1 of A1 ⊇ A0

into P(ω)/fin such that e1 ◦ α1 = t↑ ◦ e1.

e1

t↑

=

• Continue this up through all the αξ,

taking unions at limit stages.

e2

t↑

= =

. . .

=

eξ

t↑

=

. . .

• In the end, e =

ξ<ω1 eξ embeds α in t↑.

e

t↑

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a sketch of the proof: how to fix it

This strategy does not work as stated, but it can be made to work by choosing the Aξ more carefully. Specifically:

• Fix a continuous chain Mξ : ξ < ω1 of

countable elementary submodels of a suitable fragment of the set-theoretic universe. α

• For each ξ < ω1, let Aξ = A ∩ Mξ and

define αξ = α↾Aξ as before. α0

≺

α1

≺

α2

≺

. . .

≺

αξ

≺

. . .

• The elementarity between the models makes

αξ behave nicely with respect to αξ+1, and makes it possible for any eξ : Aξ → P(ω)/fin embedding αξ into t↑ to be lifted to some eξ+1 : Aξ+1 → P(ω)/fin embedding αξ+1 into t↑.

e0

t↑

• Then the argument outlined on the previous slide

can succeed.

e1

t↑

=

e2

t↑

= =

. . .

=

eξ

t↑

=

. . .

e

t↑

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what if CH fails?

Question Is ¬CH consistent with the existence of universal automorphisms of P(ω)/fin? Might ZFC even imply the existence of such automorphisms?

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what if CH fails?

Question Is ¬CH consistent with the existence of universal automorphisms of P(ω)/fin? Might ZFC even imply the existence of such automorphisms? I have no idea.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what if CH fails?

Question Is ¬CH consistent with the existence of universal automorphisms of P(ω)/fin? Might ZFC even imply the existence of such automorphisms? I have no idea. A more tractable question might be: Question Does OCA + MA imply the existence of universal 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

why OCA + MA?

OCA + MA seems to decide most questions about P(ω)/fin, and work of Farah (and others) seems to indicate that it is something of an optimal hypothesis for ensuring P(ω)/fin has as few self-maps as possible.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

why OCA + MA?

OCA + MA seems to decide most questions about P(ω)/fin, and work of Farah (and others) seems to indicate that it is something of an optimal hypothesis for ensuring P(ω)/fin has as few self-maps as possible. For example, OCA + MA implies that all automorphisms of P(ω)/fin are trivial.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

why OCA + MA?

OCA + MA seems to decide most questions about P(ω)/fin, and work of Farah (and others) seems to indicate that it is something of an optimal hypothesis for ensuring P(ω)/fin has as few self-maps as possible. For example, OCA + MA implies that all automorphisms of P(ω)/fin are trivial. While ZFC implies the existence of nontrivial self-embeddings

• f P(ω)/fin (by recent work of Dow), OCA + MA restricts the

form of these embeddings, and ensures they are “close” to trivial.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

why OCA + MA?

OCA + MA seems to decide most questions about P(ω)/fin, and work of Farah (and others) seems to indicate that it is something of an optimal hypothesis for ensuring P(ω)/fin has as few self-maps as possible. For example, OCA + MA implies that all automorphisms of P(ω)/fin are trivial. While ZFC implies the existence of nontrivial self-embeddings

• f P(ω)/fin (by recent work of Dow), OCA + MA restricts the

form of these embeddings, and ensures they are “close” to trivial. For example, OCA + MA implies that the shift map does not embed into its inverse, and vice versa.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

two more permutations of ω

Let r denote a permutation of ω that consists of infinitely many finite cycles, one of size n! for every n:

r

. . . Let t ∨ r denote the permutation of ω obtained by putting a copy

• f t next to a copy of r:

. . . . . . . . . . . . . . . . . . . . . . . . . . .

t ∨ r

. . .

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a jointly universal pair

Theorem Every trivial automorphism of P(ω)/fin embeds in either t↑ or in (t ∨ r)↑.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a jointly universal pair

Theorem Every trivial automorphism of P(ω)/fin embeds in either t↑ or in (t ∨ r)↑. Hence OCA + MA implies that every automorphism of P(ω)/fin embeds in either t↑ or in (t ∨ r)↑.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

a jointly universal pair

Theorem Every trivial automorphism of P(ω)/fin embeds in either t↑ or in (t ∨ r)↑. Hence OCA + MA implies that every automorphism of P(ω)/fin embeds in either t↑ or in (t ∨ r)↑. More specifically, if f is a mod-finite permutation of ω, then f ↑ embeds in t↑ if it has no “cyclic part” (i.e., if f contains only finitely many finite cycles), and otherwise it embeds in (t ∨ r)↑. Theorem OCA + MA implies that r↑ does not embed in t↑. Thus t↑ is not universal under OCA + MA . . .

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what about (t ∨ r)↑?

. . . but (t ∨ r)↑ might be.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what about (t ∨ r)↑?

. . . but (t ∨ r)↑ might be. Recall that ω∗ denotes the Stone space of P(ω)/fin, and that f ∗ denotes the self-homeomorphism of ω∗ induced by a mod-finite permutation f of ω; i.e., f ∗ = Stone(f ↑).

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what about (t ∨ r)↑?

. . . but (t ∨ r)↑ might be. Recall that ω∗ denotes the Stone space of P(ω)/fin, and that f ∗ denotes the self-homeomorphism of ω∗ induced by a mod-finite permutation f of ω; i.e., f ∗ = Stone(f ↑). Theorem Assuming there are no nontrivial automorphisms of P(ω)/fin (e.g., under OCA + MA), then (t ∨ r)↑ is universal if and only if there is a continuous function q : ω∗ → ω∗ such that q ◦ r∗ = s∗ ◦ q.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

what about (t ∨ r)↑?

. . . but (t ∨ r)↑ might be. Recall that ω∗ denotes the Stone space of P(ω)/fin, and that f ∗ denotes the self-homeomorphism of ω∗ induced by a mod-finite permutation f of ω; i.e., f ∗ = Stone(f ↑). Theorem Assuming there are no nontrivial automorphisms of P(ω)/fin (e.g., under OCA + MA), then (t ∨ r)↑ is universal if and only if there is a continuous function q : ω∗ → ω∗ such that q ◦ r∗ = s∗ ◦ q. OCA + MA implies that any such function q must be nontrivial; i.e., it cannot be induced by a function ω → βω.

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

• pen questions

Question Is (t ∨ r)↑ a universal automorphism of P(ω)/fin under OCA + MA?

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

• pen questions

Question Is (t ∨ r)↑ a universal automorphism of P(ω)/fin under OCA + MA? Question Does every automorphism of P(ω)/fin embed in a trivial automorphism?

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

• pen questions

Question Is (t ∨ r)↑ a universal automorphism of P(ω)/fin under OCA + MA? Question Does every automorphism of P(ω)/fin embed in a trivial automorphism? Question Does CH imply that the shift map is conjugate/isomorphic to its inverse?

Will Brian Universal automorphisms of P(ω)/fin

automorphisms of P(ω)/fin universal automorphisms with CH universal automorphisms without CH

The end

Thank you for listening

Will Brian Universal automorphisms of P(ω)/fin