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Introduction Classes of structures Characterizing homogenizable

Almost amalgamation classes generating homogenizable structures

Ove Ahlman, Uppsala University

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Table of Contents

Introduction Classes of structures Characterizing homogenizable

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

We only consider finite relational languages.

Definition

For a structure M and a substructure A ⊆ M, M is called A−homogeneous if for each embedding f0 : A → M, there is an automorphism f : M → M such that f extends f0 i.e. ∀a ∈ A, f0(a) = f(a).

M

A f

M is homogeneous if it is A−homogeneous for each finite A ⊆ M.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Let K be a class of structures.

◮ K has the hereditary property (HP) if for each A ∈ K

and B ⊆ A, B ∈ K.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Let K be a class of structures.

◮ K has the hereditary property (HP) if for each A ∈ K

and B ⊆ A, B ∈ K.

◮ A ∈ K is an amalgamation base for K if for each

B, C ∈ K and f0 : A → B, g0 : A → C there is D ∈ K and f1 : B → D, g1 : C → D such that for each a ∈ A, f1(f0(a)) = g1(g0(a)). B

f1

• A

f0

• g0

C

g1

D

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Let K be a class of structures.

◮ K has the hereditary property (HP) if for each A ∈ K

and B ⊆ A, B ∈ K.

◮ A ∈ K is an amalgamation base for K if for each

B, C ∈ K and f0 : A → B, g0 : A → C there is D ∈ K and f1 : B → D, g1 : C → D such that for each a ∈ A, f1(f0(a)) = g1(g0(a)). B

f1

• A

f0

• g0

C

g1

D

◮ K satisfies the amalgamation property (AP) if each

A ∈ K is an amalgamation base.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Let K be a class of structures.

◮ K has the hereditary property (HP) if for each A ∈ K

and B ⊆ A, B ∈ K.

◮ A ∈ K is an amalgamation base for K if for each

B, C ∈ K and f0 : A → B, g0 : A → C there is D ∈ K and f1 : B → D, g1 : C → D such that for each a ∈ A, f1(f0(a)) = g1(g0(a)). B

f1

• A

f0

• g0

C

g1

D

◮ K satisfies the amalgamation property (AP) if each

A ∈ K is an amalgamation base.

◮ K satisfies the joint embedding property (JEP) if for

each A, B ∈ K there is C ∈ K such that both A and B embeds into C

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Age(M) = {A : A ֒ → M, A is finite}

Theorem (Fra¨ ıss´ e 1953)

Let K be a class of finite structures closed under isomorphism satisfying HP, JEP and AP. Then there is a unique countable homogeneous structure M such that Age(M) = K. In the relational context JEP can be excluded

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Age(M) = {A : A ֒ → M, A is finite}

Theorem (Fra¨ ıss´ e 1953)

Let K be a class of finite structures closed under isomorphism satisfying HP, JEP and AP. Then there is a unique countable homogeneous structure M such that Age(M) = K. In the relational context JEP can be excluded If C is a set of structures let Forb(C) = {A : ∀C ∈ C, C ֒ → A} If Forb(C) satisfies AP call the unique homogeneous structure M such that Age(M) = Forb(C) the generic C−free structure.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is called homogenizable if there are a finite number of ∅−definable relations R1, . . . , Rn in M such that if we enrich the language of M with symbols for R1, . . . , Rn then this new structure M′ is homogeneous.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is called homogenizable if there are a finite number of ∅−definable relations R1, . . . , Rn in M such that if we enrich the language of M with symbols for R1, . . . , Rn then this new structure M′ is homogeneous. Note: Adding a finite number of relations means that Homogenizable ⇒ ω − categorical

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is called homogenizable if there are a finite number of ∅−definable relations R1, . . . , Rn in M such that if we enrich the language of M with symbols for R1, . . . , Rn then this new structure M′ is homogeneous. Note: Adding a finite number of relations means that Homogenizable ⇒ ω − categorical Trivial Example 1:

• Using the formula P(x) ≡ ∃y∃z(E(x, z) ∧ E(x, y))

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The random bipartite graph N: ...

• ...

...

• ...

Using the formula R(x, y) ≡ ∃z(E(x, z) ∧ E(y, z)).

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The random bipartite graph N: ...

• ...

...

• ...

Using the formula R(x, y) ≡ ∃z(E(x, z) ∧ E(y, z)). Age(N) = {A : A is bipartite}. If we want to write Age(N) = Forb(C) then C will be an infinite set.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The random bipartite graph N: ...

• ...

...

• ...

Using the formula R(x, y) ≡ ∃z(E(x, z) ∧ E(y, z)). Age(N) = {A : A is bipartite}. If we want to write Age(N) = Forb(C) then C will be an infinite set.

Theorem (Fra¨ ıss´ e 1953)

Let K be a class of finite structures closed under isomorphism satisfying HP, JEP and AP. Then there is a unique countable homogeneous structure M such that Age(M) = K. Is there a similar theorem where we replace homogeneous with homogenizable?

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The rational numbers ◗ are homogeneous. ...

...

◗ ◗ ◗

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The rational numbers ◗ are homogeneous. ...

...

The non-negative rational nunmbers ◗+ ˙ ∪{0} are homogenizable.

...

But Age(◗) = Age(◗ ˙ ∪{0}).

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The rational numbers ◗ are homogeneous. ...

...

The non-negative rational nunmbers ◗+ ˙ ∪{0} are homogenizable.

...

But Age(◗) = Age(◗ ˙ ∪{0}). The random bipartite graph M is homogenizable ...

• ...

...

• ...

But M ˙ ∪M is also homogenizable and Age(M) = Age(M ˙ ∪M)

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

The rational numbers ◗ are homogeneous. ...

...

The non-negative rational nunmbers ◗+ ˙ ∪{0} are homogenizable.

...

But Age(◗) = Age(◗ ˙ ∪{0}). The random bipartite graph M is homogenizable ...

• ...

...

• ...

But M ˙ ∪M is also homogenizable and Age(M) = Age(M ˙ ∪M) One could demand the structures to be primitive, but this is not a good limitation for homogenizable structures (M is not primitive).

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is model-complete if each formula ϕ(¯ x) is equivalent to some ∃−formula ψϕ(¯ x) over Th(M).

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is model-complete if each formula ϕ(¯ x) is equivalent to some ∃−formula ψϕ(¯ x) over Th(M).

Fact

If M is homogeneous then it is model-complete.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Definition

A structure M is model-complete if each formula ϕ(¯ x) is equivalent to some ∃−formula ψϕ(¯ x) over Th(M).

Fact

If M is homogeneous then it is model-complete.

Theorem (Saracino 1973)

If M is ω−categorical, then there is a unique model-complete structure N such that Age(M) = Age(N). Thus for a class satisfying HP and AP, the unique homogeneous structure is also the unique model-complete structure.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Generalize Fra¨ ıss´ e’s theorem to homogenizable classes using model-completeness.

Question

How do we characterize the classes of structures, who are ages

• f model-complete homogenizable structures?

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Generalize Fra¨ ıss´ e’s theorem to homogenizable classes using model-completeness.

Question

How do we characterize the classes of structures, who are ages

• f model-complete homogenizable structures?

Clearly if M is model-complete and homogenizable then Age(M) does not satisfy AP.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Generalize Fra¨ ıss´ e’s theorem to homogenizable classes using model-completeness.

Question

How do we characterize the classes of structures, who are ages

• f model-complete homogenizable structures?

Clearly if M is model-complete and homogenizable then Age(M) does not satisfy AP.

Note

A common problem is that one usually describe a class of finite structures by saying Forb(C), but this does not say anything about wheter the class satisfy AP.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Theorem (Covington 1989)

Forb(N) is the age of a homogenizable model-complete structure. The graph N is the path on 4 vertices.

• Ove Ahlman,

Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Theorem (Covington 1989)

Forb(N) is the age of a homogenizable model-complete structure. The graph N is the path on 4 vertices.

• The proof is quite explicit showing exactly which formula is

needed and why it is homogenizing.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Theorem (Covington 1990)

All classes of structures satisfying LFA are ages of a homogenizable model-complete structure. LFA means that there are diagrams without amalgams. B1 . . . Bn A1

• C1

. . . An

• Cn

Such that for any A → B and A → C without an amalgam, there is an i such that Bi

B

Ai

• Ci
• A
• C

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Theorem (Covington 1990)

All classes of structures satisfying LFA are ages of a homogenizable model-complete structure. LFA means that there are diagrams without amalgams. B1 . . . Bn A1

• C1

. . . An

• Cn

Such that for any A → B and A → C without an amalgam, there is an i such that Bi

B

Ai

• Ci
• A
• C

This was a generalization of her previous result, although the paper is much less constructive.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

A class of structures F is homomorphism closed if for each A ∈ F, if B is such that f : A → B through a bijective homomorphism then B ∈ F.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

A class of structures F is homomorphism closed if for each A ∈ F, if B is such that f : A → B through a bijective homomorphism then B ∈ F.

Theorem (Hartman, Hubicka, Nesetril 2015)

Let F be a set of finite connected homomorphism closed structures such that there exists a model-complete structure M such that age(M) = Forb(F). If the size of the largest minimal g−separating g−cut in F is finite then M is homogenizable.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

A class of structures F is homomorphism closed if for each A ∈ F, if B is such that f : A → B through a bijective homomorphism then B ∈ F.

Theorem (Hartman, Hubicka, Nesetril 2015)

Let F be a set of finite connected homomorphism closed structures such that there exists a model-complete structure M such that age(M) = Forb(F). If the size of the largest minimal g−separating g−cut in F is finite then M is homogenizable. Minimal g−separating g − cut is a technical term from graph theory which essentially say that how the structures can be split up into different parts. If the graph language is not used then all the terms are considered in the context of the Gaifmann graph.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

A class of structures F is homomorphism closed if for each A ∈ F, if B is such that f : A → B through a bijective homomorphism then B ∈ F.

Theorem (Hartman, Hubicka, Nesetril 2015)

Let F be a set of finite connected homomorphism closed structures such that there exists a model-complete structure M such that age(M) = Forb(F). If the size of the largest minimal g−separating g−cut in F is finite then M is homogenizable. Minimal g−separating g − cut is a technical term from graph theory which essentially say that how the structures can be split up into different parts. If the graph language is not used then all the terms are considered in the context of the Gaifmann graph. This characterization is not explicit but it does indicate how large the formulas which homogenize needs to be.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

For k ∈ ❩+, M is >k−homogeneous if for each A ⊆ M such that |A| >k, M is A−homogeneous.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

For k ∈ ❩+, M is >k−homogeneous if for each A ⊆ M such that |A| >k, M is A−homogeneous. M is cofinitely homogeneous if M is > k−homogeneous for some k. A class of structure has cofinite amalgamation property (CAP) if, for some k, each A ∈ K with |A| > k is an amalgamation base.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

For k ∈ ❩+, M is >k−homogeneous if for each A ⊆ M such that |A| >k, M is A−homogeneous. M is cofinitely homogeneous if M is > k−homogeneous for some k. A class of structure has cofinite amalgamation property (CAP) if, for some k, each A ∈ K with |A| > k is an amalgamation base.

Lemma

If K is a class of structures which satisfies CAP, then K satisfies LFA. The converse is not true, Forb(N) does not satisfy CAP.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Lemma

If K is a class of structures which satisfies CAP, then K satisfies LFA. The converse is not true, Forb(N) does not satisfy CAP.

Corollary

Let K be a class of structures closed under isomorphism satisfying HP, JEP and CAP. Then there exists a model-complete cofinitely homogeneous structure M such that Age(M) = K.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Theorem (A. 2018)

If M is a cofinitely homogeneous countably infinite graph then M (or Mc) is isomorphic to one of the following

◮ A homogeneous graph. ◮ Gn ˙

∪H for some n ∈ ❩+ and finite homogeneous graph H.

◮ Ht,1 or Ht,2 for some t.

• The graph (G1 ˙

∪K1)c

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Question

So how do we characterize the homogenizable classes?

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Question

So how do we characterize the homogenizable classes? A homogenizable structure can be seen as

• 1. ω−categorical structure

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Question

So how do we characterize the homogenizable classes? A homogenizable structure can be seen as

• 1. ω−categorical structure
• 2. Add formulas to homogenize, bounded size, tuples.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

ω−categoricity

Let SBOP be the property that for any A ∈ K there are B1, . . . , Bn such that A → B1, . . . , A → Bn do each pairwise not have an amalgam and if A → C then this has an amalgam together with some A → Bi.

Lemma (Albert, Burris 1988)

Let K be a class of structures satisfying HP, JEP and SBOP. Then there exists a model-complete ω−categorical structure M such that Age(M) = K.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Homogenizability

K satisfies AFP if there is k, t such that whenever A → B, A → C does not have an amalgam there is A0 ⊆ A, B0 ⊇ B, C0 ⊇ C such that |A0| < k, |B0 − B| < t, |C0 − C0| < t and A0 → B0, A0 → C0 does not have an amalgam.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Homogenizability

K satisfies AFP if there is k, t such that whenever A → B, A → C does not have an amalgam there is A0 ⊆ A, B0 ⊇ B, C0 ⊇ C such that |A0| < k, |B0 − B| < t, |C0 − C0| < t and A0 → B0, A0 → C0 does not have an amalgam.

Lemma (A. 2017)

If M is a model-complete ω−categorical structure then Age(M) satisfies AFP if and only if M is homogenizable.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Homogenizability

K satisfies AFP if there is k, t such that whenever A → B, A → C does not have an amalgam there is A0 ⊆ A, B0 ⊇ B, C0 ⊇ C such that |A0| < k, |B0 − B| < t, |C0 − C0| < t and A0 → B0, A0 → C0 does not have an amalgam.

Lemma (A. 2017)

If M is a model-complete ω−categorical structure then Age(M) satisfies AFP if and only if M is homogenizable.

Corollary

K is a class of structures satisfying HP, JEP, SBOP and AFP if and only if there is a model-complete homogenizable structure M such that Age(M) = K.

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Summary

◮ Questions

◮ What can we say about non-model-complete homogenizable

structures? How many are there for a given age?

◮ Is there a homogenizable non-model-complete structure for

which there is no model-complete homogenizable structure with the same age?

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable

Introduction Classes of structures Characterizing homogenizable

Thank you!

Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable