Introduction
Oligomorphic structures
The proof
A new proof of the existence of cores of
oligomorphic structures
Libor Barto, Michael Kompatscher, Mirek Olšak, Trung Van
Pham, Michael Pinsker
[email protected]
Theory and Logic group
TU Wien
AAA93 Bern, 12/02/2017
The result
Introduction
Oligomorphic structures
Outline
1
Homomorphic equivalence and cores
2
Oligomorphic groups
3
The proof
4
Summary
The proof
The result
Introduction
Oligomorphic structures
The proof
Homomorphic equivalence
Definition
Two structures A and B are homomorphically equivalent, if there
are homomorphisms h : A → B and h0 : B → A.
Examples (finite graphs)
The result
Introduction
Oligomorphic structures
The proof
Homomorphic equivalence
Definition
Two structures A and B are homomorphically equivalent, if there
are homomorphisms h : A → B and h0 : B → A.
Examples (finite graphs)
The result
Introduction
Oligomorphic structures
The proof
Homomorphic equivalence
Definition
Two structures A and B are homomorphically equivalent, if there
are homomorphisms h : A → B and h0 : B → A.
Examples (finite graphs)
The result
Introduction
Oligomorphic structures
Cores of finite structures
Definition
For finite A, we say B is a core of A, if
B is homomorphic equivalent to A
End(B) = Aut(B).
The proof
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Definition
For finite A, we say B is a core of A, if
B is homomorphic equivalent to A
End(B) = Aut(B).
Observation
Every finite structure is hom. equiv. to a unique core.
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Definition
For finite A, we say B is a core of A, if
B is homomorphic equivalent to A
End(B) = Aut(B).
Observation
Every finite structure is hom. equiv. to a unique core.
Proof: take endomorphism with minimal range
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Note that we did not work on the structural level at all:
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Note that we did not work on the structural level at all:
Let M be a transformation monoid on a finite set A.
Let ξ ∈ M have minimal range B = ξ(A).
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Note that we did not work on the structural level at all:
Let M be a transformation monoid on a finite set A.
Let ξ ∈ M have minimal range B = ξ(A).
The monoid
M̃ = {f ∈ B B |∃m ∈ M : f = m|B }
is equal to the group of its invertibles.
The result
Introduction
Oligomorphic structures
The proof
Cores of finite structures
Note that we did not work on the structural level at all:
Let M be a transformation monoid on a finite set A.
Let ξ ∈ M have minimal range B = ξ(A).
The monoid
M̃ = {f ∈ B B |∃m ∈ M : f = m|B }
is equal to the group of its invertibles.
We say M̃ is a core.
The result
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
For infinite structures we adjust the definition of core:
The result
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
For infinite structures we adjust the definition of core:
Definition
B is a (model-complete) core of A, if
The result
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
For infinite structures we adjust the definition of core:
Definition
B is a (model-complete) core of A, if
B is hom. equiv. to A
The result
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
For infinite structures we adjust the definition of core:
Definition
B is a (model-complete) core of A, if
B is hom. equiv. to A
Aut(B) is dense in End(B):
∀f ∈ End(B)∀F ⊂fin B ∃g ∈ Aut(B) g |F = f |F .
The result
Introduction
Oligomorphic structures
The proof
The result
Cores of infinite structures
For infinite structures we adjust the definition of core:
Definition
B is a (model-complete) core of A, if
B is hom. equiv. to A
Aut(B) is dense in End(B):
∀f ∈ End(B)∀F ⊂fin B ∃g ∈ Aut(B) g |F = f |F .
A closed transformation monoid M on a countable set A is a core,
if the group of its invertibles is dense in M.
Introduction
Oligomorphic structures
The proof
The result
Cores of infinite structures
For infinite structures we adjust the definition of core:
Definition
B is a (model-complete) core of A, if
B is hom. equiv. to A
Aut(B) is dense in End(B):
∀f ∈ End(B)∀F ⊂fin B ∃g ∈ Aut(B) g |F = f |F .
A closed transformation monoid M on a countable set A is a core,
if the group of its invertibles is dense in M.
Example
The rational order (Q; <) is a core.
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
Example
The order (Q ∪ {∞}; <) is not a core, since automorphisms
preserve ∞. But it is hom. equiv. to the core (Q; <).
The result
Introduction
Oligomorphic structures
The proof
Cores of infinite structures
Example
The order (Q ∪ {∞}; <) is not a core, since automorphisms
preserve ∞. But it is hom. equiv. to the core (Q; <).
Infinite structures do not necessarily have a core:
The result
Introduction
Oligomorphic structures
The proof
The result
Cores of infinite structures
Example
The order (Q ∪ {∞}; <) is not a core, since automorphisms
preserve ∞. But it is hom. equiv. to the core (Q; <).
Infinite structures do not necessarily have a core:
Example
Let (Q; <, (In )n∈N ) be the rational order with In = (n, ∞) for every
n ∈ N. There are endomorphisms
(Q; <, (In )n∈N ) → (I0 ; <, (In )n∈N ) → (I1 ; <, (In )n∈N ) → · · ·
but there is no “limit”.
Introduction
Oligomorphic structures
The proof
The result
Cores of infinite structures
Example
The order (Q ∪ {∞}; <) is not a core, since automorphisms
preserve ∞. But it is hom. equiv. to the core (Q; <).
Infinite structures do not necessarily have a core:
Example
Let (Q; <, (In )n∈N ) be the rational order with In = (n, ∞) for every
n ∈ N. There are endomorphisms
(Q; <, (In )n∈N ) → (I0 ; <, (In )n∈N ) → (I1 ; <, (In )n∈N ) → · · ·
but there is no “limit”.
→ we need compactness to obtain a core!
Introduction
Oligomorphic structures
The proof
The result
Manuel’s result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Introduction
Oligomorphic structures
The proof
The result
Manuel’s result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Bodirsky ’05
Every structure A with oligomorphic Aut(A) is homomorphically
equivalent to a unique core.
Introduction
Oligomorphic structures
The end
The proof
The result
Introduction
Oligomorphic structures
The proof
The result
Manuels result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Theorem (Bodirsky ’05)
Every structure A with oligomorphic Aut(A) is homomorphically
equivalent to a unique core.
Reasons to be unhappy...
Introduction
Oligomorphic structures
The proof
The result
Manuels result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Theorem (Bodirsky ’05)
Every structure A with oligomorphic Aut(A) is homomorphically
equivalent to a unique core.
Reasons to be unhappy...
proof does not talk about monoids, as in the finite case
Introduction
Oligomorphic structures
The proof
The result
Manuels result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Theorem (Bodirsky ’05)
Every structure A with oligomorphic Aut(A) is homomorphically
equivalent to a unique core.
Reasons to be unhappy...
proof does not talk about monoids, as in the finite case
proof uses concepts from model theory
Introduction
Oligomorphic structures
The proof
The result
Manuels result
A permutation group G is oligomorphic, if the action G y An has
only finitely many orbits for every n ∈ N.
Theorem (Bodirsky ’05)
Every structure A with oligomorphic Aut(A) is homomorphically
equivalent to a unique core.
Reasons to be unhappy...
proof does not talk about monoids, as in the finite case
proof uses concepts from model theory
condition on Aut(A), not End(A)
Introduction
Oligomorphic structures
The proof
The result
Our Plan
For a finite transformation monoid M on A we found B ⊆ A such
that the monoid
M̃ = {g ∈ B B | ∃m ∈ M : g = m|B }
is a core.
Introduction
Oligomorphic structures
The proof
The result
Our Plan
For a finite transformation monoid M on A we found B ⊆ A such
that the monoid
M̃ = {g ∈ B B | ∃m ∈ M : g = m|B }
is a core.
For a closed oligomorphic transformation monoid M on A we want
B ⊆ A such that the monoid
M̃ = {g ∈ B B |
is a core.
∃m ∈ M : g
= m|B }
Introduction
Oligomorphic structures
The proof
The result
Our Plan
For a finite transformation monoid M on A we found B ⊆ A such
that the monoid
M̃ = {g ∈ B B | ∃m ∈ M : g = m|B }
is a core.
For a closed oligomorphic transformation monoid M on A we want
B ⊆ A such that the monoid
M̃ = {g ∈ B B | ∀F ⊂fin B ∃m ∈ M : g |F = m|F }
is a core.
Introduction
Oligomorphic structures
The proof
The result
Our Plan
For a finite transformation monoid M on A we found B ⊆ A such
that the monoid
M̃ = {g ∈ B B | ∃m ∈ M : g = m|B }
is a core.
For a closed oligomorphic transformation monoid M on A we want
B ⊆ A such that the monoid
M̃ = {g ∈ B B | ∀F ⊂fin B ∃m ∈ M : g |F = m|F }
is a core.
B needs to reflect the “minimal range” condition from the finite
case.
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
It follows that:
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
It follows that:
Lemma
There is a top. closed set I ⊆ M such that
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
It follows that:
Lemma
There is a top. closed set I ⊆ M such that
I is left ideal: f ∈ I , m ∈ M → mf ∈ I
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
It follows that:
Lemma
There is a top. closed set I ⊆ M such that
I is left ideal: f ∈ I , m ∈ M → mf ∈ I
I is closed under ∼
Introduction
Oligomorphic structures
The proof
The result
A compactness argument
For f , g ∈ M, set f ∼ g if f (ā) and g (ā) are in the same orbit for
all tuples ā in A.
Lemma
Let M be a topologically closed weakly oligomorphic
transformation monoid. Then M/ ∼ is a compact monoid.
It follows that:
Lemma
There is a top. closed set I ⊆ M such that
I is left ideal: f ∈ I , m ∈ M → mf ∈ I
I is closed under ∼
I is minimal with those properties
Introduction
Oligomorphic structures
The proof
Existence of the core
By minimality of I we have
∀F ⊂fin A ∀f , g ∈ I ∃m ∈ M : mf |F = g |F .
The result
Introduction
Oligomorphic structures
The proof
Existence of the core
By minimality of I we have
∀F ⊂fin A ∀f , g ∈ I ∃m ∈ M : mf |F = g |F .
Let B =
S
f ∈I
f (A).
The result
Introduction
Oligomorphic structures
The proof
Existence of the core
By minimality of I we have
∀F ⊂fin A ∀f , g ∈ I ∃m ∈ M : mf |F = g |F .
Let B =
S
f ∈I
f (A).
Then
M̃ = {g ∈ B B | ∀F ⊂fin B ∃m ∈ M : g |F = m|F }
is a core.
The result
Introduction
Oligomorphic structures
The proof
The result
Existence of the core
By minimality of I we have
∀F ⊂fin A ∀f , g ∈ I ∃m ∈ M : mf |F = g |F .
Let B =
S
f ∈I
f (A).
Then
M̃ = {g ∈ B B | ∀F ⊂fin B ∃m ∈ M : g |F = m|F }
is a core.
(...show elements of M̃ locally look like invertibles)
Introduction
Oligomorphic structures
The result
We can derive the result for structures:
The proof
The result
Introduction
Oligomorphic structures
The proof
The result
We can derive the result for structures:
Theorem (Barto, K, Olšak, Pham, Pinsker ’17)
Let A be a structure such that End(A) is weakly oligomorphic.
Then A is homomorphically equivalent to a unique core B.
Moreover Aut(B) is oligomorphic.
The result
Introduction
Oligomorphic structures
The end
Thank you for your attention!
The proof
The result