Roland Walker (UIC) Distality Rank 2020 0 / 49 Distality Rank - PowerPoint PPT Presentation

2-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 2-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each pair from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 ← → Reset Roland Walker (UIC) Distality Rank 2020 12 / 49

2-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 2-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each pair from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 then all of A inserts indiscernibly... ← → Reset Roland Walker (UIC) Distality Rank 2020 12 / 49

2-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 2-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each pair from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 then all of A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 ← → Reset Roland Walker (UIC) Distality Rank 2020 12 / 49

3-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 3-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each triple from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 ← → Reset Roland Walker (UIC) Distality Rank 2020 13 / 49

3-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 3-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each triple from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 then all of A inserts indiscernibly... ← → Reset Roland Walker (UIC) Distality Rank 2020 13 / 49

3-Distality in Pictures... A Dedekind partition I = I 0 + I 1 + · · · + I 4 is 3-distal iff: for all A = ( a 0 , a 1 , a 2 , a 3 ), if each triple from A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 then all of A inserts indiscernibly... a 0 a 1 a 2 a 3 I 0 I 1 I 2 I 3 I 4 ← → Reset Roland Walker (UIC) Distality Rank 2020 13 / 49

m -Distality Let n > m > 0. Definition We say a Dedekind partition I = I 0 + · · · + I n is m -distal iff: for all sets A = ( a 0 , . . . , a n − 1 ) ⊆ U , if A does not insert indiscernibly into I , then some m -element subset of A does not insert indiscernibly into I . ← → Roland Walker (UIC) Distality Rank 2020 14 / 49

m -Distality for EM-types Let n > m > 0. Definition A complete EM-type Γ is ( n , m ) -distal iff: every Dedekind partition = EM Γ is m -distal. I 0 + · · · + I n | Lemma If Γ is ( m + 1 , m ) -distal, then Γ is ( n , m ) -distal for all n > m. Proof: Induction on n . � Definition A complete EM-type Γ is m -distal iff: it is ( m + 1 , m )-distal. ← → Roland Walker (UIC) Distality Rank 2020 15 / 49

Distality Rank for EM-Types Observation: If a complete EM-type Γ is m -distal, then it is also n -distal for all n > m . Definition The distality rank of a complete EM-type Γ, written DR(Γ), is the least m ≥ 1 such that Γ is m -distal. If no such finite m exists, we say the distality rank of Γ is ω . ← → Roland Walker (UIC) Distality Rank 2020 16 / 49

Skeletons Let n > m > 0. Let I = I 0 + · · · + I n where I 1 = ω ∗ + ω, I n − 1 = ω ∗ + ω, I n = ω ∗ , I 0 = ω, . . . and ω ∗ is ω in reverse order. Definition If I ⊆ U is a sequence indexed by I = I 0 + · · · + I n , we call the corresponding partition I = I 0 + · · · + I n an n -skeleton . Notice that an n -skeleton is a Dedekind partition with n cuts. Proposition A complete EM -type Γ is m-distal if and only if there is an n-skeleton = EM Γ which is m-distal. I 0 + · · · + I n | ← → Proof Roland Walker (UIC) Distality Rank 2020 17 / 49

Distality Rank for Theories Let m > 0. Definition A theory T , not necessarily complete, is m -distal iff: for all completions of T and all tuple sizes κ , every Γ ∈ S EM ( κ · ω ) is m -distal. In the existing literature, a theory is called distal if and only if it is 1-distal. Definition The distality rank of a theory T , written DR( T ), is the least m ≥ 1 such that T is m -distal. If no such finite m exists, we say the distality rank of T is ω . ← → Roland Walker (UIC) Distality Rank 2020 18 / 49

Proposition If T is an L -theory with quantifier elimination and L contains no atomic formula with more than m free variables, then DR( T ) ≤ m. Proof: Let I = I 0 + · · · + I m +1 be Dedekind and A = ( a 0 , . . . , a m ). Suppose all proper subsets of A insert indiscernibly into I . Given φ ∈ L ( x 0 , . . . , x n − 1 ) , there is a T -equivalent formula � � � � θ ij x σ ij (0) , . . . , x σ ij ( m − 1) i j where each θ ij is basic and each σ ij : m → n is a function. Let ( b 0 , . . . , b n − 1 ) ⊆ I and ( d 0 , . . . , d n − 1 ) ⊆ I ∪ A both be increasing. Since all m -sized subsets of A insert indiscernibly into I , then U | = θ ij ( b σ ij (0) , . . . , b σ ij ( m − 1) ) ↔ θ ij ( d σ ij (0) , . . . , d σ ij ( m − 1) ) . � ← → Roland Walker (UIC) Distality Rank 2020 19 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random graph has distality rank 2. a 0 a 1 Edge I ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random graph has distality rank 2. The theory of the random 3-hypergraph has distality rank 3. a 0 a 1 a 2 I ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random graph has distality rank 2. The theory of the random 3-hypergraph has distality rank 3. This generalizes, so... The theory of the random m -(hyper)graph has distality rank m . ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random m -(hyper)graph has distality rank m . ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random m -(hyper)graph has distality rank m . The theories of ( N , σ, 0) and ( Z , σ ), where σ : x �→ x + 1, have distality rank 2. a σ ( a ) I ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random m -(hyper)graph has distality rank m . The theories of ( N , σ, 0) and ( Z , σ ), where σ : x �→ x + 1, have distality rank 2. ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

Finding Examples... Corollary Suppose L is a language where all function symbols are unary and all relation symbols have arity at most m ≥ 2 . If T is an L -theory with quantifier elimination, then DR( T ) ≤ m. This corollary helps us find examples by putting an upper bound on distality rank: The theory of the random m -(hyper)graph has distality rank m . The theories of ( N , σ, 0) and ( Z , σ ), where σ : x �→ x + 1, have distality rank 2. We can not apply the corollary to groups... ← → Roland Walker (UIC) Distality Rank 2020 20 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : Let I a 0 · · · a m − 1 be an algebraically independent set. a m − 1 a 0 a 1 I ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : Let I a 0 · · · a m − 1 be an algebraically independent set. Let a m = a 0 + · · · + a m − 1 , and let A = ( a 0 , . . . , a m ). a m − 1 a 0 a 1 a m I ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : Let I a 0 · · · a m − 1 be an algebraically independent set. Let a m = a 0 + · · · + a m − 1 , and let A = ( a 0 , . . . , a m ). Now we can insert any m elements of A without breaking indiscernibility... a m − 1 a 0 a 1 a m I ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : Let I a 0 · · · a m − 1 be an algebraically independent set. Let a m = a 0 + · · · + a m − 1 , and let A = ( a 0 , . . . , a m ). Now we can insert any m elements of A without breaking indiscernibility... a m − 1 a 0 a 1 a m I However, inserting all of A breaks indiscernibility... ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

For example, if T is the complete theory of a strongly minimal group , then DR( T ) = ω : Let I a 0 · · · a m − 1 be an algebraically independent set. Let a m = a 0 + · · · + a m − 1 , and let A = ( a 0 , . . . , a m ). Now we can insert any m elements of A without breaking indiscernibility... a m − 1 a 0 a 1 a m I However, inserting all of A breaks indiscernibility... a m − 1 a 0 a 1 a m I ← → Reset Roland Walker (UIC) Distality Rank 2020 21 / 49

Distal and non-distal NIP theories (Simon 2013) Simon worked strictly in the context of NIP theories and proved several structural results concerning distality: • Distality is invariant under base change; i.e., T B is distal ⇐ ⇒ T is distal . • Distality can be characterized by the orthogonality of commuting global invariant types; i.e., if p ( x ) and q ( y ) are global invariant types that commute, then p ( x ) ∪ q ( y ) ⊢ p ⊗ q . • It’s sufficient to check one-dimensional sequences I ⊂ U 1 . ← → Roland Walker (UIC) Distality Rank 2020 22 / 49

Base Change Adding named parameters does not increase distality rank... Proposition If T is a complete theory and B ⊆ U is a small set of parameters, then DR( T B ) ≤ DR( T ) . Proof: Let I = ( b i : i ∈ I ) be indiscernible over B . Given m > 0, suppose there is a Dedekind partion I 0 + . . . + I m +1 of I and a set A = ( a 0 , . . . , a m ) witnessing that T B is not m -distal. It follows that I ′ = ( b i + B : i ∈ I ) and A ′ = ( a 0 + B , . . . , a m + B ) witness that T is not m -distal. � ← → Roland Walker (UIC) Distality Rank 2020 23 / 49

Base Change If T is NIP, adding named parameters does not change distality rank... Base Change Theorem If T is NIP and B ⊆ U is a small set of parameters, then DR( T B ) = DR( T ). Proof of Theorem: DR( T B ) ≤ DR( T ) by the previous proposition. We need to show that T B is m -distal ⇒ T is m -distal. But first, we need more background... ← → Roland Walker (UIC) Distality Rank 2020 24 / 49

Alternation Rank Let φ ∈ L U ( x ) and I = ( b i : i ∈ I ) ⊆ U | x | be an infinite indiscernible sequence. Definition We use alt( φ, I ) to denote the number of alternations of φ on I , i.e.,     �  n < ω : ∃ i 0 < · · · < i n ∈ I U | ¬ [ φ ( b i j ) ↔ φ ( b i j +1 )] sup =  . j < n Definition We use alt( φ ) to denote the alternation rank of φ , i.e., alt( φ, J ) : J ⊆ U | x | is an infinite indiscernible sequence � � sup . ← → Roland Walker (UIC) Distality Rank 2020 25 / 49

IP and NIP Definition A formula φ ∈ L ( x , y ) is IP iff: there is a d ∈ U | y | such that alt( φ ( x , d )) = ∞ . Definition The theory T is IP iff: there is a φ ∈ L U ( x ) with alt( φ ) = ∞ . In both cases, we use NIP to denote the, often more desirable, condition of not being IP. ← → Roland Walker (UIC) Distality Rank 2020 26 / 49

Limit Types Let ( I , < ) be a linear order and let I = ( b i : i ∈ I ) ⊆ U be a sequence of tuples. Definition Given A ⊆ U , if the partial type { φ ∈ L A ( x ) : ∃ i ∈ I ∀ j ≥ i U | = φ ( b j ) } is complete, we call it the limit type of I over A , written limtp A ( I ). Moreover, if it exists, we call limtp U ( I ) the global limit type of I and may simply write lim( I ). If I is indiscernible, then limtp I ( I ) exists. If T is NIP and I is indiscernible, the global limit type lim( I ) exists. ← → Roland Walker (UIC) Distality Rank 2020 27 / 49

In order to prove the Base Change Theorem, we need the following lemma... Let m > 0. Base Change Lemma Suppose T is NIP. If I = I 0 + · · · + I m +1 is a Dedekind partition, A = ( a 0 , . . . , a m ) is a set of parameters such that every proper subset inserts indiscernibly into I , and D ⊆ U is a small set of parameters, then there is a set A ′ = ( a ′ m ) such that A ′ ≡ I A and for each 0 , . . . , a ′ σ : m → m + 1 increasing, we have � � a ′ σ (0) · · · a ′ c − σ (0) , . . . , c − σ ( m − 1) | = limtp D . σ ( m − 1) ← → Proof Roland Walker (UIC) Distality Rank 2020 28 / 49

Now we can prove the Base Change Theorem... Base Change Theorem If T is NIP and B ⊆ U is a small set of parameters, then DR( T B ) = DR( T ). Proof of Theorem (continued): It remains to show that T B is m -distal ⇒ T is m -distal. Suppose Γ ∈ S EM is not m -distal. = EM Γ be a skeleton which is indiscernible over B . Let I 0 + · · · + I m +1 | There exists a set A = ( a 0 , . . . , a m ) such that every proper subset inserts indiscernibly over ∅ but A does not. Applying the lemma with D = B ∪ I yields a set A ′ such that every proper subset inserts indiscernibly over B but A ′ does not. � ← → Roland Walker (UIC) Distality Rank 2020 29 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | Proof: ( ⇐ ) Suppose Γ ∈ S EM is not m -distal. Let J | = Γ with index Q × ( m + 1). Q Q Q Q J ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | Proof: ( ⇐ ) Suppose Γ ∈ S EM is not m -distal. Let J | = Γ with index Q × ( m + 1). Let K ⊆ J with index Z ≥ 0 + Z + · · · + Z + Z ≤ 0 . Q Q Q Q K ⊆ J Z ≥ 0 Z ≤ 0 Z Z ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | Proof: ( ⇐ ) Suppose Γ ∈ S EM is not m -distal. Let J | = Γ with index Q × ( m + 1). Let K ⊆ J with index Z ≥ 0 + Z + · · · + Z + Z ≤ 0 . a 0 a 1 a 2 Q Q Q Q ¯ ¯ ¯ ¯ b 0 b 1 b 2 b 3 K ⊆ J Z ≥ 0 Z ≤ 0 Z Z Since K is a skeleton, there is ( φ, A , B ) witnessing that K is not m -distal, ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | Proof: ( ⇐ ) Suppose Γ ∈ S EM is not m -distal. Let J | = Γ with index Q × ( m + 1). Let K ⊆ J with index Z ≥ 0 + Z + · · · + Z + Z ≤ 0 . a 0 a 1 a 2 Q Q Q Q ¯ ¯ ¯ ¯ b 0 b 1 b 2 b 3 K ⊆ J Z ≥ 0 Z ≤ 0 Z Z Since K is a skeleton, there is ( φ, A , B ) witnessing that K is not m -distal, so by the Base Change Lemma, there is A ′ ≡ K A such that for each σ , we have � � a ′ σ (0) · · · a ′ c − σ (0) , . . . , c − σ ( m − 1) | = limtp J . σ ( m − 1) ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Theorem Suppose T is NIP. A complete EM-type Γ is m-distal if and only if there is = EM Γ . an m-distal Dedekind partition I 0 + · · · + I m +1 | Proof: ( ⇐ ) Suppose Γ ∈ S EM is not m -distal. Let J | = Γ with index Q × ( m + 1). Let K ⊆ J with index Z ≥ 0 + Z + · · · + Z + Z ≤ 0 . a 0 a 1 a 2 Q Q Q Q ¯ ¯ ¯ ¯ b 0 b 1 b 2 b 3 K ⊆ J Z ≥ 0 Z ≤ 0 Z Z Since K is a skeleton, there is ( φ, A , B ) witnessing that K is not m -distal, so by the Base Change Lemma, there is A ′ ≡ K A such that for each σ , we have � � a ′ σ (0) · · · a ′ c − σ (0) , . . . , c − σ ( m − 1) | = limtp J . σ ( m − 1) It follows that ( φ, A ′ , B ) witnesses that J is not m -distal. � ← → Roland Walker (UIC) Distality Rank 2020 30 / 49

Type Determinacy Let n > m > 0. Definition Given p ∈ S A ( x 0 , . . . , x n − 1 ), we say that the n -type p is m -determined iff: it is completely determined by the m -types � q ∈ S A ( x i 0 , . . . , x i m − 1 ) : q ⊆ p and i 0 < · · · < i m − 1 < n � it contains. ← → Roland Walker (UIC) Distality Rank 2020 32 / 49

Type Determinacy Let n > m > 0. Definition Given p ∈ S A ( x 0 , . . . , x n − 1 ), we say that the n -type p is m -determined iff: it is completely determined by the m -types � q ∈ S A ( x i 0 , . . . , x i m − 1 ) : q ⊆ p and i 0 < · · · < i m − 1 < n � it contains. Theorem If T is m-distal, then for any n global invariant types p 0 ( x 0 ) , . . . , p n − 1 ( x n − 1 ) which commute pairwise, their product p 0 ⊗ · · · ⊗ p n − 1 is m-determined. Furthermore , if T is NIP, the converse holds as well. ← → Roland Walker (UIC) Distality Rank 2020 32 / 49

Fix m ≥ 2. Let L = { R , <, P 0 , . . . , P m − 1 } . Let T be the complete theory of an m -partite m -uniform ordered random (hyper)graph; i.e. the theory axiomatized by the following: 1 All models are linearly ordered by < . 2 The ordering is partitioned by P 0 < · · · < P m − 1 where each part has no endpoints. 3 All models are m -partite m -uniform (hyper)graphs, with parts P 0 , . . . , P m − 1 and edge relation R . 4 For each s , t < ω and each j < m , we have the following axiom: � ∀ distinct X 0 , . . . , X s − 1 , Y 0 , . . . , Y t − 1 ∈ ∀ z 0 < z 1 ∈ P j ∃ z ∈ P j P i i � = j � � � � z 0 < z < z 1 ∧ X r Rz ∧ Y r � Rz r < s r < t ← → Roland Walker (UIC) Distality Rank 2020 34 / 49

Fix m = 3. If Γ ∈ S EM (1 · ω ), then DR(Γ) = 1... a 0 a 1 a 2 P i ← → Reset Roland Walker (UIC) Distality Rank 2020 35 / 49

Fix m = 3. If Γ ∈ S EM (1 · ω ), then DR(Γ) = 1... a 0 a 1 a 2 P i However, DR ( T ) = 3... a 0 a 1 a 2 P 0 P 1 P 2 ← → Reset Roland Walker (UIC) Distality Rank 2020 35 / 49

Strong m -Distality ← → Roland Walker (UIC) Distality Rank 2020 36 / 49

Suppose a Dedekind partition I = I 0 + I 1 is strongly 1-distal . If I is indiscernible over D 0 I 0 I 1 D 0 ← → Reset Roland Walker (UIC) Distality Rank 2020 37 / 49

Suppose a Dedekind partition I = I 0 + I 1 is strongly 1-distal . If I is indiscernible over D 0 and a ∈ U inserts indiscernibly... a I 0 I 1 D 0 ← → Reset Roland Walker (UIC) Distality Rank 2020 37 / 49

Suppose a Dedekind partition I = I 0 + I 1 is strongly 1-distal . If I is indiscernible over D 0 and a ∈ U inserts indiscernibly... a I 0 I 1 D 0 then a inserts indiscernibly over D 0 ... ← → Reset Roland Walker (UIC) Distality Rank 2020 37 / 49

Roland Walker (UIC) Distality Rank 2020 0 / 49 Distality Rank - PowerPoint PPT Presentation

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