Properties of the automorphism group and a probabilistic construction of a class of countable labeled structures Dragan Maˇ sulovi´ c and Igor Dolinka Department of Mathematics and Informatics University of Novi Sad, Serbia AAA 83 Novi Sad, 17 Mar 2012 Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 1 / 23
Automorphism groups of Fra¨ ıss´ e limits Questions: Probabilistic construction Simplicity of the automorphism group Small index property Bergman property Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 2 / 23
Automorphism groups of Fra¨ ıss´ e limits Questions: Probabilistic construction Simplicity of the automorphism group Small index property Bergman property A helpful assumption: Aut ( F ) is oligomorphic Contrast: rational Urysohn space Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 2 / 23
Automorphism groups of Fra¨ ıss´ e limits We would like to consider some of these questions but in the setting where Aut ( F ) is not oligomorphic. Our starting point: labeled graphs The requirement that Aut ( F ) be oligomorphic will be replaced by other types finiteness requirements. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 3 / 23
Labeled structures L = { R i : i ∈ N } — countable relational language Ar ( L ) = { ar ( R ) : R ∈ L } A relational language L has bounded arity if there is an n ∈ N such that Ar ( L ) ⊆ { 1 , 2 , . . . , n } . Definition. An L -structure A is labeled if for every n ∈ Ar ( L ) and every a ∈ A n there exists exactly one R ∈ L n such that A | = R ( a ) . An L -structure A is partially labeled if for every n ∈ Ar ( L ) and every a ∈ A n there exists at most one R ∈ L n such that A | = R ( a ) . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 4 / 23
Labeled structures Definition. A labeled L -structure A ∗ is a filling of a partially labeled L -structure A if they have the same base set and A � A ∗ . A class A of partially labeled L -structures has uniform fillings in a class B of labeled L-structures if there is a mapping ( · ) ∗ : A → B such that for all A , B ∈ A : A ∗ is a filling of A , and if f is an isomorphism from A onto B , then f is also an isomorphism from A ∗ onto B ∗ . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 5 / 23
Labeled structures Our labeled structures may implement certain restrictions expressed by means of special Horn clauses over L ∪ {� = } . A Horn restriction over L is a Horn clause of the form Φ = ¬ ( R 1 ( v 1 ) ∧ . . . ∧ R n ( v n )) where R 1 , . . . , R n ∈ L ∪ {� = } and R i ∈ L for at least one i . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 6 / 23
Labeled structures – The setup Condition (A) L — a countable relational language of bounded arity Σ — a set of Horn restrictions over L Σ | L 0 is finite for every finite L 0 ⊆ L P Σ — the class of all finite partially lbld L -structures satisfying Σ K Σ — the class of all labeled structures in P Σ for all A , B ∈ K Σ : A ⊔ B ∈ P Σ for all A , B , C ∈ K Σ : B ⊔ A C belongs to P Σ ; P Σ has uniform fillings in K Σ . Fact. K Σ is a Fra¨ ıss´ e class. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 7 / 23
Example: Graphs L = { R 0 ( · , · ) , R 1 ( · , · ) } Σ : ¬ R 1 ( x , x ) ¬ ( R 1 ( x , y ) ∧ R 0 ( y , x ) ∧ x � = y ) Uniform fillings: if ( a , a ) is not labeled in G , label it by R 0 ; if R 1 ( a , b ) but ( b , a ) is unlabeled, label ( b , a ) by R 1 ; if neither ( a , b ) nor ( b , a ) are labeled, label both by R 0 . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 8 / 23
Example: K m -free graphs L = { R 0 ( · , · ) , R 1 ( · , · ) } Σ : ¬ R 1 ( x , x ) ¬ ( R 1 ( x , y ) ∧ R 0 ( y , x ) ∧ x � = y ) �� �� � ¬ x i � = x j ∧ R 1 ( x i , x j ) ∧ R 1 ( x j , x i ) 1 � i < j � m Uniform fillings: if ( a , a ) is not labeled in G , label it by R 0 ; if R 1 ( a , b ) but ( b , a ) is unlabeled, label ( b , a ) by R 1 ; if neither ( a , b ) nor ( b , a ) are labeled, label both by R 0 . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 9 / 23
Example: Edge-colored graphs I — a nonempty countable set 0 ∈ I — arbitrary but fixed L I = { R b ( · , · ) : b ∈ I } Σ I : ¬ R b ( x , x ) for all b ∈ I \ { 0 } ¬ ( R b ( x , y ) ∧ R c ( y , x ) ∧ x � = y ) for all b , c ∈ I s. t. b � = c Uniform fillings: symmetrize, label every unlabeled tuple by R 0 . Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 10 / 23
Example: Metric spaces with rational distances L met = { D q : q ∈ Q � 0 } Σ met : ¬ ( x � = y ∧ D 0 ( x , y )) ¬ D q ( x , x ) for every q ∈ Q s. t. q > 0 ¬ ( D p ( x , y ) ∧ D q ( y , x )) for all p , q ∈ Q � 0 s. t. p � = q ¬ ( D q 1 ( u 1 , v 1 ) ∧ . . . ∧ D q n ( u n , v n ) ∧ D q 0 ( u 0 , v 0 )) , for all q 0 , q 1 , . . . , q n ∈ Q s. t. q 0 , q 1 , . . . , q n > 0 and q 0 > q 1 + . . . + q n , and all possible choices ( u i , v i ) ∈ { ( x i − 1 , x i ) , ( x i , x i − 1 ) } where 1 ≤ i ≤ n and ( u 0 , v 0 ) ∈ { ( x 0 , x n ) , ( x n , x 0 ) } Uniform fillings: nontrivial, but obvious Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 11 / 23
A negative example ( Q , < ) [up to 1-dim bi-interpretability] Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 12 / 23
The small index property G = Aut ( F ) has the small index property if, for every H � G : H is open if and only if ( G : H ) < 2 ω Theorem. Assume that (A) holds and let K Σ be the Fra¨ ıss´ e limit of K Σ . Then K Σ has ample generic automorphisms, and therefore it has the small index property. cf. A. S. Kechris, C. Rosendal: Turbulence, amalgamation, and generic automorphisms of homogeneous structures, Proc. London Math. Soc. (3) 94 (2007) 302–350. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 13 / 23
The small index property Consequently, the following Fra¨ ıss´ e limits have the small index property: the random graph R (proved by W. Hodges, I. Hodkinson, D. Lascar, S. Shelah 1993), the Henson graph H m , m � 3 (proved by Herwig 1998), the edge-colored random graph over a countable set of colors I (if I is finite, the strong small index property was proved by Cameron and Tarzi), the random deterministic transition system over a countable set of transitions I , the random I -fuzzy graph, where I is a countable bounded meet-semilattice, the rational Urysohn space, and the Urysohn sphere of radius 1 (follows from the results of Kechris and Rosendal 2007, Solecki 2005). Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 14 / 23
The Bergman property An infinite group G has the Bergman property if for any generating subset E ⊆ G such that 1 ∈ E = E − 1 we have G = E k for some positive integer k . Droste, G¨ obel 2005: strong uncountable cofinality ⇒ Bergman property Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 15 / 23
The Bergman property Condition (A+) Assume that (A) holds, and that there is a uniform filling ( · ) ∗ : P Σ → K Σ such that: for all A , B , C , D ∈ K Σ such that C ∩ D = ∅ , if f : C ֒ → A and → B , then f ∪ g : ( C ⊔ D ) ∗ ֒ g : D ֒ → ( A ⊔ B ) ∗ ; and for all pairwise disjoint A , B , C ∈ K Σ we have (( A ⊔ B ) ∗ ⊔ C ) ∗ = ( A ⊔ ( B ⊔ C ) ∗ ) ∗ . Example. Metric spaces with rational distances do not fulfill (A+), but if the distances are bounded by 1, then (A+) holds. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 16 / 23
The Bergman property ıss´ Theorem. Assume that (A+) holds and let K Σ be the Fra¨ e limit of K Σ . Then Aut ( K Σ ) has [strong uncountable cofinality, and consequently] the Bergman property. cf. C. Rosendal: A topological version of the Bergman property, Forum Math. 21 (2009) 299–332. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 17 / 23
The Bergman property The automorphism groups of the following Fra¨ ıss´ e limits have the Bergman property: the random graph R (Kechris and Rosendal 2007), the Henson graph H m , m � 3 (Kechris and Rosendal 2007), the edge-colored random graph over a cntbl set of colors I , the random deterministic transition system over a countable set of transitions I , the random I -fuzzy graph, where I is a countable bounded meet-semilattice, and the Urysohn sphere of radius 1 (Rosendal 2009). Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 18 / 23
A general probabilistic construction Recall: K Σ is a Fra¨ ıss´ e class, so let K Σ be its Fra¨ ıss´ e limit. µ n ( · ) — prob measure on L n s. t. µ n ( R ) > 0 for all R ∈ L n We start with Φ 0 = ∅ ∈ P Σ Given a labeled L -structure Φ n ∈ P Σ with the base set { a 1 , . . . , a n } we construct Φ n + 1 ∈ K Σ ⊆ P Σ with the base set { a 1 , . . . , a n , a n + 1 } as follows. Dragan Maˇ sulovi´ c and Igor Dolinka (UNS) Labeled structures 17 Mar 2012 19 / 23
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