Invariant descriptive set theory
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
I
Vector spaces on some fixed field are isomorphic iff they have the
same dimension
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
I
Vector spaces on some fixed field are isomorphic iff they have the
same dimension
I
Compact orientable surfaces are homeomorphic iff they have the
same genus
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
I
Vector spaces on some fixed field are isomorphic iff they have the
same dimension
I
Compact orientable surfaces are homeomorphic iff they have the
same genus
I
Complex square matrices are similar iff they have the same Jordan
normal form
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
I
Vector spaces on some fixed field are isomorphic iff they have the
same dimension
I
Compact orientable surfaces are homeomorphic iff they have the
same genus
I
Complex square matrices are similar iff they have the same Jordan
normal form
I
(Ornstein) Bernoulli shifts are isomorphic if they have the same
entropy
Invariant descriptive set theory: an introduction
Invariant descriptive set theory is the study of the complexity of
equivalence relations on standard Borel spaces.
Motivation. In mathematics one often looks for complete invariants to
assign to some notion of equivalence:
I
Vector spaces on some fixed field are isomorphic iff they have the
same dimension
I
Compact orientable surfaces are homeomorphic iff they have the
same genus
I
Complex square matrices are similar iff they have the same Jordan
normal form
I
(Ornstein) Bernoulli shifts are isomorphic if they have the same
entropy
I
...
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
I
Commutative AF -algebras are isomorphic iff their dimension groups
are isomorphic
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
I
Commutative AF -algebras are isomorphic iff their dimension groups
are isomorphic
I
...
In order for these assignment to be of some use, they should be
sufficiently effective.
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
I
Commutative AF -algebras are isomorphic iff their dimension groups
are isomorphic
I
...
In order for these assignment to be of some use, they should be
sufficiently effective.
The right notion of effectiveness turns out to be that of a Borel function,
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
I
Commutative AF -algebras are isomorphic iff their dimension groups
are isomorphic
I
...
In order for these assignment to be of some use, they should be
sufficiently effective.
The right notion of effectiveness turns out to be that of a Borel function,
provided the collections of objects to be classified form a standard Borel
space —
Invariant descriptive set theory: an introduction
I
(Baer) To each countable torsion free Abelian group G of rank 1 it
is possible to assign a sequence hG of natural numbers s.t. G ∼ G 0
iff hG , hG 0 are eventually equal
I
Commutative AF -algebras are isomorphic iff their dimension groups
are isomorphic
I
...
In order for these assignment to be of some use, they should be
sufficiently effective.
The right notion of effectiveness turns out to be that of a Borel function,
provided the collections of objects to be classified form a standard Borel
space — which is often the case.
Borel reducibility
Borel reducibility
Definition
Let E , F be equivalence relations on standard Borel spaces X , Y , resp.
Borel reducibility
Definition
Let E , F be equivalence relations on standard Borel spaces X , Y , resp.
Then E is Borel reducible to F , denoted
E ≤B F ,
Borel reducibility
Definition
Let E , F be equivalence relations on standard Borel spaces X , Y , resp.
Then E is Borel reducible to F , denoted
E ≤B F ,
iff there is f : X → Y Borel s.t.
∀x, x 0 ∈ X (xEx 0 ⇔ f (x)Ff (x 0 ))
When E ≤B F , any classification of objects in Y up to F -equivalence can
be translated to a classification of objects of X up to E -equivalence, by
applying function f .
Borel reducibility
Definition
Let E , F be equivalence relations on standard Borel spaces X , Y , resp.
Then E is Borel reducible to F , denoted
E ≤B F ,
iff there is f : X → Y Borel s.t.
∀x, x 0 ∈ X (xEx 0 ⇔ f (x)Ff (x 0 ))
When E ≤B F , any classification of objects in Y up to F -equivalence can
be translated to a classification of objects of X up to E -equivalence, by
applying function f . In this sense, the complexity of E is less than or
equal to the complexity of F .
Borel reducibility
Definition
Let E , F be equivalence relations on standard Borel spaces X , Y , resp.
Then E is Borel reducible to F , denoted
E ≤B F ,
iff there is f : X → Y Borel s.t.
∀x, x 0 ∈ X (xEx 0 ⇔ f (x)Ff (x 0 ))
When E ≤B F , any classification of objects in Y up to F -equivalence can
be translated to a classification of objects of X up to E -equivalence, by
applying function f . In this sense, the complexity of E is less than or
equal to the complexity of F .
If E ≤B F ≤B E , one writes E ∼B F : the equivalence relations E , F are
Borel bireducible.
Smooth equivalence relations
Smooth equivalence relations
Among the simplest equivalence relations are those whose classes can be
characterised by a single real number:
Smooth equivalence relations
Among the simplest equivalence relations are those whose classes can be
characterised by a single real number:
Definition
An equivalence relation E on a standard Borel X is smooth if E ≤B =,
Smooth equivalence relations
Among the simplest equivalence relations are those whose classes can be
characterised by a single real number:
Definition
An equivalence relation E on a standard Borel X is smooth if E ≤B =,
i.e., there exists f : X → R Borel s.t.
∀x, x 0 ∈ X (xEx 0 ⇔ f (x) = f (x 0 ))
Smooth equivalence relations
Among the simplest equivalence relations are those whose classes can be
characterised by a single real number:
Definition
An equivalence relation E on a standard Borel X is smooth if E ≤B =,
i.e., there exists f : X → R Borel s.t.
∀x, x 0 ∈ X (xEx 0 ⇔ f (x) = f (x 0 ))
Fact
Given two uncountable standard Borel spaces Y , Z there is always a
Borel isomorphism g : Y → Z .
Smooth equivalence relations
So an equivalence relation is smooth if Borel reduces to equality on some
uncountable standard Borel spaces
Smooth equivalence relations
So an equivalence relation is smooth if Borel reduces to equality on some
uncountable standard Borel spaces
Using this, the following equivalence relations are smooth:
I
Homeomorphism on compact orientable surfaces
I
Similarity on complex square matrices
I
I
Isomorphism on Bernoulli shifts
(Gromov) Isometry on compact metric spaces
I
...
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Examples
I
Vitali equivalence.
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Examples
I
Vitali equivalence. On R let xEV y ⇔ x − y ∈ Q.
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Examples
I
Vitali equivalence. On R let xEV y ⇔ x − y ∈ Q.
I
Eventual equality on NN .
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Examples
I
Vitali equivalence. On R let xEV y ⇔ x − y ∈ Q.
I
Eventual equality on NN . Let xE0 y ⇔ ∀∞ n x(n) = y (n).
I
Isomorphism 'TFA1 for torsion free Abelian groups of rank 1.
Non-smooth equivalence relations
Not all equivalence relations are smooth:
Examples
I
Vitali equivalence. On R let xEV y ⇔ x − y ∈ Q.
I
Eventual equality on NN . Let xE0 y ⇔ ∀∞ n x(n) = y (n).
I
Isomorphism 'TFA1 for torsion free Abelian groups of rank 1.
It turns out that =R <B EV ∼B E0 ∼B 'TFA1 .
Spaces of countable structures
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp.
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
I R A ⊆ Nn i ,
i
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
ni
I R A ⊆ Nni , i.e. R A ∈ 2N , for each i ∈ I
i
i
I
fjA : Nmj → N,
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
ni
I R A ⊆ Nni , i.e. R A ∈ 2N , for each i ∈ I
i
i
mj
I
fjA : Nmj → N, i.e. fjA ∈ NN , for each j ∈ J
I
ckA ∈ N, for each k ∈ K
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
ni
I R A ⊆ Nni , i.e. R A ∈ 2N , for each i ∈ I
i
i
mj
I
fjA : Nmj → N, i.e. fjA ∈ NN , for each j ∈ J
I
ckA ∈ N, for each k ∈ K
So
Y mj
Y ni
A ∈ ( 2N ) × ( NN ) × NK
i∈I
j∈J
Spaces of countable structures
Let
L = {Ri , fj , ck | i ∈ I , j ∈ J, k ∈ K }
be a countable first-order language, where ni , mj are the arities of Ri , fj ,
resp. A countable structure A — say with universe N — is defined by
fixing interpretations
ni
I R A ⊆ Nni , i.e. R A ∈ 2N , for each i ∈ I
i
i
mj
I
fjA : Nmj → N, i.e. fjA ∈ NN , for each j ∈ J
I
ckA ∈ N, for each k ∈ K
So
Y mj
Y ni
A ∈ ( 2N ) × ( NN ) × NK
i∈I
and
j∈J
Y ni
Y mj
XL = ( 2N ) × ( NN ) × NK
i∈I
j∈J
is the Polish space of countable L-structures (with universe N)
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N.
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN :
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN : for every g ∈ NN , one has g ∈ S∞ iff
1. g is surjective: ∀m ∃n g (n) = m
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN : for every g ∈ NN , one has g ∈ S∞ iff
1. g is surjective: ∀m ∃n g (n) = m
2. g is injective: ∀n, m (n 6= m ⇒ g (n) 6= g (m))
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN : for every g ∈ NN , one has g ∈ S∞ iff
1. g is surjective: ∀m ∃n g (n) = m
2. g is injective: ∀n, m (n 6= m ⇒ g (n) 6= g (m))
So S∞ is a Polish space.
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN : for every g ∈ NN , one has g ∈ S∞ iff
1. g is surjective: ∀m ∃n g (n) = m
2. g is injective: ∀n, m (n 6= m ⇒ g (n) 6= g (m))
So S∞ is a Polish space. Moreover, the operations of composition
(g , h) 7→ gh and inversion g 7→ g −1 are continuous in S∞ ,
The logic action
To simplify notation, from now on I will assume that L is relational, i.e.,
Y ni
XL =
2N
i∈I
Elements of XL will be usually denoted by letters like x, y , . . ..
Let S∞ = Sym(N) be the group of permutations of N. This is a Gδ
subset of NN : for every g ∈ NN , one has g ∈ S∞ iff
1. g is surjective: ∀m ∃n g (n) = m
2. g is injective: ∀n, m (n 6= m ⇒ g (n) 6= g (m))
So S∞ is a Polish space. Moreover, the operations of composition
(g , h) 7→ gh and inversion g 7→ g −1 are continuous in S∞ , so it is a
Polish group.
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
i.e., gx = y iff g : N → N is an isomorphism between the L-structures
x, y .
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
i.e., gx = y iff g : N → N is an isomorphism between the L-structures
x, y . The orbit equivalence relation is isomorphism:
x ' y ⇔ ∃g ∈ S∞ gx = y .
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
i.e., gx = y iff g : N → N is an isomorphism between the L-structures
x, y . The orbit equivalence relation is isomorphism:
x ' y ⇔ ∃g ∈ S∞ gx = y .
Definition
A subset A of a standard Borel space X is analytic if it is the Borel image
of a Borel subset of a standard Borel space:
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
i.e., gx = y iff g : N → N is an isomorphism between the L-structures
x, y . The orbit equivalence relation is isomorphism:
x ' y ⇔ ∃g ∈ S∞ gx = y .
Definition
A subset A of a standard Borel space X is analytic if it is the Borel image
of a Borel subset of a standard Borel space:
A ∈ Σ11 (X ) ⇔ ∃f : Y → X Borel ∃B ∈ B(Y ) f (B) = A
The logic action
The action of S∞ on N induces an action of S∞ on XL , the logic action:
gx = y ⇔ ∀i yi (h1 , . . . , hn ) = xi (g −1 (h1 ), . . . , g −1 (hn ))
i.e., gx = y iff g : N → N is an isomorphism between the L-structures
x, y . The orbit equivalence relation is isomorphism:
x ' y ⇔ ∃g ∈ S∞ gx = y .
Definition
A subset A of a standard Borel space X is analytic if it is the Borel image
of a Borel subset of a standard Borel space:
A ∈ Σ11 (X ) ⇔ ∃f : Y → X Borel ∃B ∈ B(Y ) f (B) = A
So the isomorphism relation ' on XL is analytic, as a subset of XL2 (but
in general not Borel).
Borel sets in XL
However, the following holds:
Theorem (Miller)
Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borel
action of G on X .
Borel sets in XL
However, the following holds:
Theorem (Miller)
Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borel
action of G on X . Then every orbit {gx}g ∈G is Borel.
Borel sets in XL
However, the following holds:
Theorem (Miller)
Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borel
action of G on X . Then every orbit {gx}g ∈G is Borel.
Definition
Lω1 ω is the extension of L obtained by allowing countable disjunctions
and conjunctions
∨n ϕn , ∧n ϕn
Borel sets in XL
However, the following holds:
Theorem (Miller)
Let G be a Polish group, X a standard Borel space, (g , x) 7→ gx a Borel
action of G on X . Then every orbit {gx}g ∈G is Borel.
Definition
Lω1 ω is the extension of L obtained by allowing countable disjunctions
and conjunctions
∨n ϕn , ∧n ϕn
where each ϕn has free variables between v0 , . . . , vk−1 for some k
independent of n (so each formula has finitely many free variables).
Borel sets in XL
Proposition
Let ϕ(v0 , . . . , vk−1 ) be a formula of Lω1 ω .
Borel sets in XL
Proposition
Let ϕ(v0 , . . . , vk−1 ) be a formula of Lω1 ω . Then the set Aϕ,k ⊆ XL × Nk
defined by
(x, a0 , . . . , ak−1 ) ∈ Ak ⇔ x |= ϕ(a0 , . . . , ak−1 )
Borel sets in XL
Proposition
Let ϕ(v0 , . . . , vk−1 ) be a formula of Lω1 ω . Then the set Aϕ,k ⊆ XL × Nk
defined by
(x, a0 , . . . , ak−1 ) ∈ Ak ⇔ x |= ϕ(a0 , . . . , ak−1 )
is Borel.
Borel sets in XL
Proposition
Let ϕ(v0 , . . . , vk−1 ) be a formula of Lω1 ω . Then the set Aϕ,k ⊆ XL × Nk
defined by
(x, a0 , . . . , ak−1 ) ∈ Ak ⇔ x |= ϕ(a0 , . . . , ak−1 )
is Borel.
Proof.
By induction on the construction of ϕ.
Borel sets in XL
In particular, if σ is a sentence of Lω1 ω , i.e., k = 0, then
Aσ = Aσ0 = {x | x |= σ} is Borel
Borel sets in XL
In particular, if σ is a sentence of Lω1 ω , i.e., k = 0, then
Aσ = Aσ0 = {x | x |= σ} is Borel and invariant under '.
Borel sets in XL
In particular, if σ is a sentence of Lω1 ω , i.e., k = 0, then
Aσ = Aσ0 = {x | x |= σ} is Borel and invariant under '.
The converse is true as well:
Theorem (Lopez-Escobar)
The invariant Borel subsets of XL are exactly those of the form Aσ , for σ
a sentence of Lω1 ω .
Borel sets in XL
In particular, if σ is a sentence of Lω1 ω , i.e., k = 0, then
Aσ = Aσ0 = {x | x |= σ} is Borel and invariant under '.
The converse is true as well:
Theorem (Lopez-Escobar)
The invariant Borel subsets of XL are exactly those of the form Aσ , for σ
a sentence of Lω1 ω .
Corollary (Scott)
For every countable L-structure A there is a sentence σA of Lω1 ω such
that for any countable L-structure B, one has
B ' A iff B |= σA
Borel sets in XL
In particular, if σ is a sentence of Lω1 ω , i.e., k = 0, then
Aσ = Aσ0 = {x | x |= σ} is Borel and invariant under '.
The converse is true as well:
Theorem (Lopez-Escobar)
The invariant Borel subsets of XL are exactly those of the form Aσ , for σ
a sentence of Lω1 ω .
Corollary (Scott)
For every countable L-structure A there is a sentence σA of Lω1 ω such
that for any countable L-structure B, one has
B ' A iff B |= σA
(such a sentence is called a Scott sentence of A)
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
In particular, the usual classes of countable structures are Borel and
invariant in XL (where L is a suitable language), so they form standard
Borel spaces:
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
In particular, the usual classes of countable structures are Borel and
invariant in XL (where L is a suitable language), so they form standard
Borel spaces:
I
graphs: L = {R}, where R is a binary relation symbol
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
In particular, the usual classes of countable structures are Borel and
invariant in XL (where L is a suitable language), so they form standard
Borel spaces:
I
graphs: L = {R}, where R is a binary relation symbol
I
groups: L = {·, −1 , 1}, where · is a binary function symbol,
unary function symbol, 1 is a constant symbol
−1
is a
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
In particular, the usual classes of countable structures are Borel and
invariant in XL (where L is a suitable language), so they form standard
Borel spaces:
I
graphs: L = {R}, where R is a binary relation symbol
I
groups: L = {·, −1 , 1}, where · is a binary function symbol,
unary function symbol, 1 is a constant symbol
I
fields: L = . . .
−1
is a
Borel sets in XL
Fact
A Borel subset of a standard Borel space, with the induced σ-algebra, is
itself standard Borel.
In particular, the usual classes of countable structures are Borel and
invariant in XL (where L is a suitable language), so they form standard
Borel spaces:
I
graphs: L = {R}, where R is a binary relation symbol
I
groups: L = {·, −1 , 1}, where · is a binary function symbol,
unary function symbol, 1 is a constant symbol
I
fields: L = . . .
I
linear orders: L = {<}, where < is a binary relation symbol
I
...
−1
is a
Classification by countable structures
Definition
An equivalence relation E is classifiable by countable structures if there is
an invariant Borel class Z of countable structures such that E is Borel
reducible to isomorphism on Z :
E ≤B 'Z
Classification by countable structures
Definition
An equivalence relation E is classifiable by countable structures if there is
an invariant Borel class Z of countable structures such that E is Borel
reducible to isomorphism on Z :
E ≤B 'Z
Examples.
I
Isomorphism on commutative AF algebras
Classification by countable structures
Definition
An equivalence relation E is classifiable by countable structures if there is
an invariant Borel class Z of countable structures such that E is Borel
reducible to isomorphism on Z :
E ≤B 'Z
Examples.
I
Isomorphism on commutative AF algebras
I
Isometry on ultrametric Polish spaces
Classification by countable structures
Definition
An equivalence relation E is classifiable by countable structures if there is
an invariant Borel class Z of countable structures such that E is Borel
reducible to isomorphism on Z :
E ≤B 'Z
Examples.
I
Isomorphism on commutative AF algebras
I
Isometry on ultrametric Polish spaces
(Darji, Marcone, C; 2005) Homeomorphism on dendrites (compact,
connected, locally connected metric spaces not containing simple
closed curves)
I
Classification by countable structures
Definition
An equivalence relation E is classifiable by countable structures if there is
an invariant Borel class Z of countable structures such that E is Borel
reducible to isomorphism on Z :
E ≤B 'Z
Examples.
I
Isomorphism on commutative AF algebras
I
Isometry on ultrametric Polish spaces
(Darji, Marcone, C; 2005) Homeomorphism on dendrites (compact,
connected, locally connected metric spaces not containing simple
closed curves)
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I
...
Complete elements
Complete elements
Definition
If C is a class of equivalence relations on standard Borel spaces, an
equivalence relation E is C-complete if:
Complete elements
Definition
If C is a class of equivalence relations on standard Borel spaces, an
equivalence relation E is C-complete if:
1. E ∈ C
2. ∀F ∈ C F ≤B E
Complete elements
Definition
If C is a class of equivalence relations on standard Borel spaces, an
equivalence relation E is C-complete if:
1. E ∈ C
2. ∀F ∈ C F ≤B E
It turns out that several classes C of equivalence relations have a complete
member.
Complete elements
Definition
If C is a class of equivalence relations on standard Borel spaces, an
equivalence relation E is C-complete if:
1. E ∈ C
2. ∀F ∈ C F ≤B E
It turns out that several classes C of equivalence relations have a complete
member. The interest is to find examples of C-complete relations, or of
relations that are not C-complete but not for trivial reasons.
S∞ -complete equivalence relations
If C is the class of equivalence relations classifiable by countable
structures, a C-complete equivalence relation is also called S∞ -complete.
S∞ -complete equivalence relations
If C is the class of equivalence relations classifiable by countable
structures, a C-complete equivalence relation is also called S∞ -complete.
Examples of S∞ -equivalence relations
S∞ -complete equivalence relations
If C is the class of equivalence relations classifiable by countable
structures, a C-complete equivalence relation is also called S∞ -complete.
Examples of S∞ -equivalence relations
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The isomorphism relation on countable graphs, or on countable trees
S∞ -complete equivalence relations
If C is the class of equivalence relations classifiable by countable
structures, a C-complete equivalence relation is also called S∞ -complete.
Examples of S∞ -equivalence relations
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The isomorphism relation on countable graphs, or on countable trees
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(Gao, C.; 2001) The isomorphism relation on countable Boolean
algebras; the homeomorphism relation on zero-dimensional compact
metric spaces; the conjugacy relation on the group of
homeomorphism of Cantor space
S∞ -complete equivalence relations
If C is the class of equivalence relations classifiable by countable
structures, a C-complete equivalence relation is also called S∞ -complete.
Examples of S∞ -equivalence relations
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The isomorphism relation on countable graphs, or on countable trees
I
(Gao, C.; 2001) The isomorphism relation on countable Boolean
algebras; the homeomorphism relation on zero-dimensional compact
metric spaces; the conjugacy relation on the group of
homeomorphism of Cantor space
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(Darji, Marcone, C; 2005) The homeomorphism relation on dendrites
Analytic equivalence relations
The preorder ≤B is defined between all equivalence relations on standard
Borel spaces. However, the class Σ11 of analytic equivalence relations is,
up to date, the biggest class for which some structural properties and
significant examples could be found.
Analytic equivalence relations
The preorder ≤B is defined between all equivalence relations on standard
Borel spaces. However, the class Σ11 of analytic equivalence relations is,
up to date, the biggest class for which some structural properties and
significant examples could be found.
Theorem
There exists a Σ11 -complete equivalence relation.
Analytic equivalence relations
Examples of Σ11 -complete equivalence relations
Analytic equivalence relations
Examples of Σ11 -complete equivalence relations
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(Louveau, Rosendal; 2005) Biembeddability on countable trees
Analytic equivalence relations
Examples of Σ11 -complete equivalence relations
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(Louveau, Rosendal; 2005) Biembeddability on countable trees
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(Ferenczi, Louveau, Rosendal) Isomorphism on separable Banach
spaces