An axiomatic approach to free amalgamation

An axiomatic approach to free amalgamation

Gabriel ConantUIC

July 15, 2015Neostability Theory Workshop

Casa Matematica Oaxaca

Gabriel Conant (UIC) Free Amalgamation July 15, 2015 1 / 22

Main Theme

Goal: Understand and classify unstable theories without the strictorder property (SOP).

Examples/Categories of NSOP theories• simple unstable theories

random graph, ACFA, pseudo-finite fields

• non-simple NSOP3 theoriesω-free PAC fields, T ∗feq,∞-dim. vector spaces with bilinear formChernikov-Ramsey (2015): these examples are NSOP1

• SOP3 theoriesgeneric Kn-free graph, rational Urysohn space, universal(Kn + K3)-free graph


Questions

• Stable forking conjecture (forking in simple theories witnessed bystable formulas)

• Elimination of hyperimaginaries for simple theories (and beyond)Adler (2007): no SOP theory has e.h.i.Casanovas-Wagner (2004): there is an NSOP theory without e.h.i.- rational Urysohn sphere (has strong order property)

• Equivalence of forking and þ-forking for simple theories

• Are SOP1 and SOP3 the same?

• Is every NSOP NTP2 theory simple?

• Classify the NSOP theories for which simplicity, NTP2, and NSOP3coincide.


Stationary Independence RelationsMany NSOP theories of “generic structures” can be equipped with aternary relation | satisfying nice properties:

• invariance, monotonicity, symmetry, transitivity• full existence: for all A,B,C there is A′ ≡C A such that A′ |

CB.

• stationarity: for all A,A′,B,C, if A ≡C A′, A |C

B, and A′ |C

B, thenA ≡BC A′.

Examples• random graph, generic Kn-free graph: free amalgamation of graphs• rational Urysohn space: free amalgamation of metric spaces

Stationary independence relations have been used to studyautomorphism groups of countable structures.

Tent-Ziegler (2013): The isometry group of the rational Urysohn spaceis simple modulo bounded isometries.


Stationary Independence Relations

Suppose T is a theory in which the monster model M can be equippedwith a stationary independence relation | (i.e., satisfyinginvariance, monotonicity, symmetry, transitivity, full existence,stationarity).

Note: Very close to Adler’s “mock stability”.

What are the model theoretic consequences?

• C.-Terry (2014): | implies | f , and so every set is an extensionbase for nonforking.• Is there an unstable NIP theory with an SIR?

(C.) No totally ordered theory can have an SIR.• Is there an SOP theory with an SIR?

Guess: Fraısse limit of atomless boolean algebras


Free Amalgamation

DefinitionSuppose L is a relational language andM is an L-structure. Define| fa on subsets ofM as follows:

A | faC

B if and only if:

(i) A ∩ B ⊆ C;(ii) for all R ∈ L and x ∈ ABC, if R(x) holds then x ∈ AC or x ∈ BC.

Suppose K is a Fraısse class, in a finite relational language L, which isclosed under free amalgamation of L-structures. LetM be the Fraısselimit of K. Then | fa is a stationary independence relation on finitesubsets ofM.In this case, | fa also satisfies:freedom: for all A,B,C,D, if A |

CB and C ∩ AB ⊆ D ⊆ C, then

A |D

B.


Axiomatic Free Amalgamation

DefinitionSuppose T is a complete first-order theory, with monster model M.Then T is a free amalgamation theory if there is a ternary relation | ,defined on small algebraically closed subsets of M, which satisfies:• invariance, monotonicity, symmetry, full existence, and transitivity –

where each axiom is modified for closed sets;• stationarity: for all closed A,A′,B,C, if C ⊆ A ∩ B, A′ ≡C A, A |

CB,

and A′ |C

B, then A′ ≡B A;

• freedom: for all closed A,B,C,D, if A |C

B and C ∩ AB ⊆ D ⊆ Cthen A |

DB;

• closure: for all closed A,B,C, if C ⊆ A ∩ B and A |C

B then AB isclosed.

In this case, ( | ,acl) is a free amalgamation scheme.


Example: Fraısse limits with free amalgamation

• Th(M), whereM is the Fraısse limit, in a finite relational languageL, of a Fraısse class closed under free amalgamation ofL-structures.

Examples- the random graph,- generic Kn-free graph,- generic K r

n -free r -hypergraph, where n > r and K rn is the

complete r -hypergraph on n vertices.


Example: Urysohn space with spectrum {0,1,2,3}

• Let U3 be the Urysohn space with spectrum {0,1,2,3}, i.e. Fraısselimit of finite metric spaces with distances in {0,1,2,3}.Also constructed by Casanovas-Wagner as the free third root of thecomplete graph.

Let T = Th(U3) in language with binary relations d(x , y) = r forr ∈ {1,2,3}.Define the ternary relation:

A | 2C

B ⇔ d(a,b) = 2 for all a ∈ A\C, b ∈ B\C

Note that | 2 is not the same as free amalgamation of metricspaces. In general, free amalgamation of metric spaces fails thefreedom axiom.


Example: universal (Kn + K3)-free graph

• Th(Gn), where n ≥ 3 and Gn is the countable, universal, existentiallyclosed (Kn + K3)-free graph.

The age of Gn is not closed under free amalgamation. Gn is not evenhomogeneous as a graph. However, it is pseudo-homogeneous inthe following sense:

Patel (2006): The class of algebraically closed subsets of Gn isclosed under free amalgamation.


Example: certain Hrushovski constructions

• Let L be a finite relational language and suppose (Kf ,≤) is a classof finite L-structures where:- ≤ is a notion of strong substructure defined from a predimension,- f is a control function, with Kf closed under free amalgamation of strong

substructures.

LetMf be the ℵ0-categorical Hrushovski generic of (Kf ,≤).

Assume: ( | fa,acl) satisfies closure, i.e. for all closed A,B,C, if

C ⊆ A ∩ B and A | faC

B then AB is closed.

Are there any interesting examples of this?


Model Theory of Free Amalgamation TheoriesIf T is a free amalgamation theory then T is NSOP.

Theorem (C. (strengthening Patel (2006)))

Any free amalgamation theory is NSOP4.

This was already known for the previous examples.

• Shelah (1980)?: random graph (simple)• Shelah (1996): generic Kn-free graph• Hrushovski (2002): generic K r

n -free r -hypergraph, where r > 2(simple)• Patel (2006): Fraısse limits with free amalgamation• Patel (2006): universal (Kn + K3)-free graph• Evans-Wong (2009): Hrushovski genericsMf

• C.-Terry (2014): Urysohn space with spectrum {0,1,2,3}


Proof of NSOP4

Let ( | ,acl) be a free amalgamation scheme. Fix an indiscerniblesequence (ai)i<ω, and let p(x , y) = tp(a0,a1). Want to show:

p(x1, x2) ∪ p(x2, x3) ∪ p(x3, x4) ∪ p(x4, x1) is consistent.

Without loss of generality, ai is closed. Let c be the commonintersection of (ai)i<ω, i.e., c = (a0,j : a0,j = a1,j).• full existence: there is b0 ≡a1 a0 such that b0 | a1

a2.• freedom: since b0 ∩ a1 = c = a1 ∩ a2, we have b0 | c

a2

Claim: b0a2 ≡c a2b0Proof: b0 ≡c a2 so there is b′ such that b0a2 ≡c a2b′.

• invariance, symmetry: b′ |c

a2

• stationarity: a2b′ ≡c a2b0

By the claim, there is b1 such that b0a2a1 ≡c a2b0b1.Gabriel Conant (UIC) Free Amalgamation July 15, 2015 13 / 22

Proof of NSOP4

Let ( | ,acl) be a free amalgamation scheme. Fix an indiscerniblesequence (ai)i<ω, and let p(x , y) = tp(a0,a1). Want to show:

p(x1, x2) ∪ p(x2, x3) ∪ p(x3, x4) ∪ p(x4, x1) is consistent.

Without loss of generality, ai is closed. Let c be the commonintersection of (ai)i<ω, i.e., c = (a0,j : a0,j = a1,j).We have:• b0 ≡a1 a0

• b0a2a1 ≡c a2b0b1

Therefore (b0,a1,a2,b1) realizes the type above:

• b0a1 ≡ a0a1, and so p(b0,a1)

• by definition: p(a1,a2)

• a2b1 ≡ b0a1, and so p(a2,b1)

• b1b0 ≡ a1a2, and so p(b1,b0)


Remarks

(1) This cannot be improved to NSOP3.

• Shelah (1996): generic Kn-free graph is SOP3;• Patel (2006): generic (Kn + K3)-free graph is SOP3;• Evans-Wong (2009): any non-simpleMf is SOP3.

(2) Stationary independence relations are not sufficient.

• C.-Terry (2014): the rational Urysohn space is SOPn for all n.


Toward Rosiness

Recall that a theory T has weak elimination of imaginaries if for anye ∈Meq there is a finite real tuple c ∈M such that e ∈ dcleq(c) andc ∈ acleq(e) (i.e. c is a weak canonical parameter for e).

T is rosy (resp. real rosy) if þ-forking satisfies local character in T eq

(resp. in T ).

simple : forking :: rosy : þ-forking

Ealy-Goldbring (2012): If T has weak elimination of imaginaries then“rosy” and “real rosy” coincide.


Weak Elimination of Imaginaries

Theorem (C.)

If T is a free amalgamation theory, and acl is locally finite, then T hasweak elimination of imaginaries.

Corollary:(a) Suppose T = Th(M), whereM is the Fraısse limit, in a finite

relational language, of a Fraısse class with free amalgamation.Then T is rosy of Uþ-rank 1.

(b) Suppose T = Th(Gn), where Gn is the universal (Kn + K3)-freegraph. Then T is rosy of Uþ-rank 2.

In both cases, acl satisfies “base monotonicity”, and so | þ coincideswith algebraic independence

A | aC

B ⇔ acl(AC) ∩ acl(BC) = acl(C).


Sketch of w.e.i.

• Let a ∈Mn be a tuple and E(x , y) a 0-definable equivalence relationon Mn. We want to find a weak canonical parameter for aE .

• Without loss of generality, we may assume a is closed (uses acllocally finite).

• Let c be a minimal length subtuple of a such that there is anE-related indiscernible sequence (ai)i<ω with common intersection cand a0 = a.

• Without any extra assumptions: c ∈ acleq(aE ).

• Using the free amalgamation scheme: aE ∈ dcleq(c).


Characterizing Simplicity

Theorem (C.)

Suppose T is a free amalgamation theory and algebraic independencesatisfies base monotonicity (i.e. the lattice of algebraically closedsubsets of M is modular). Then the following are equivalent.

(i) T is simple.(ii) T is NTP2.(iii) T is NSOP3.

(iv) | a coincides with | f .

This includes many of the previous examples (e.g. Fraısse limits offree amalgamation classes).

Conjecture: Conditions (i), (ii), and (iii) are equivalent for any freeamalgamation theory.


Key tool

Let ( | ,acl) be a free amalgamation scheme. Suppose (bi)i<ω is asequence of closed tuples and C ⊆ b0 is closed. Then (bi)i<ω is| -independent over C if, for all n < ω, bn ≡C b0 and bn | C

b<n.

LemmaSuppose ( | ,acl) is a free amalgamation scheme for T . Fix closedtuples a,b and let C = a ∩ b. Then, if (bi)i<ω is | -independent overC, with b0 = b, there is a tuple a∗ such that a∗bi ≡C ab for all i < ω.

Slogan: | -independent sequences cannot witness any dividingbeyond failures of algebraic independence.

The lemma can fail if | is only a stationary independence relation(e.g. free amalgamation of metric spaces in the rational Urysohnspace).


Future Directions

• Understand the consequences of stationary independence relations.

• Keep the freedom axiom, but weaken stationarity to one of thevarious versions of amalgamation over algebraically closed sets:

- for all closed A1,A2,B1,B2,C, with C ⊆ B1 ∩ B2, if Ai | CBi , B1 | C

B2,A1 ≡C A2, then there is D such that D ≡B1 A1, D ≡B2 A2, and D |

CB1B2.


thank you


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An axiomatic approach to free amalgamation