Almost amalgamation classes generating homogenizable structures Ove - PowerPoint PPT Presentation

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 f 0 : A → M , there is an automorphism f : M → M such that f extends f 0 i.e. ∀ a ∈ A , f 0 ( 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 f 0 : A → B , g 0 : A → C there is D ∈ K and f 1 : B → D , g 1 : C → D such that for each a ∈ A , f 1 ( f 0 ( a )) = g 1 ( g 0 ( a )). B f 0 f 1 � C � D A g 0 g 1 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 f 0 : A → B , g 0 : A → C there is D ∈ K and f 1 : B → D , g 1 : C → D such that for each a ∈ A , f 1 ( f 0 ( a )) = g 1 ( g 0 ( a )). B f 0 f 1 � C � D A g 0 g 1 ◮ 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 f 0 : A → B , g 0 : A → C there is D ∈ K and f 1 : B → D , g 1 : C → D such that for each a ∈ A , f 1 ( f 0 ( a )) = g 1 ( g 0 ( a )). B f 0 f 1 � C � D A g 0 g 1 ◮ 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 R 1 , . . . , R n in M such that if we enrich the language of M with symbols for R 1 , . . . , R n 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 R 1 , . . . , R n in M such that if we enrich the language of M with symbols for R 1 , . . . , R n 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 R 1 , . . . , R n in M such that if we enrich the language of M with symbols for R 1 , . . . , R n 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 ψ ϕ (¯ x ) over Th ( M ). Ove Ahlman, Uppsala University Almost amalgamation classes generating homogenizable