Yevgeniy Vasilyev Memorial University of Newfoundland Grenfell …eleftheriou/tame/Vasilyev.pdf ·...

G -structures and other dense/codense expansions

Yevgeniy Vasilyev

Memorial University of NewfoundlandGrenfell Campus, Corner Brook, NL, Canada

Joint work with Alexander Berenstein

Workshop on Tame Expansions of O-minimal StructuresKonstanz, October 1-4, 2018

Outline

I geometric theories

I dense/codense expansions

I axiomatization

I Q-independent sets and back-and-forth

I basic properties of dense/codense expansions

I “Q-bases”

I effect on acl

I effect on forking and SU-rank (SU-rank 1 case)

I effect on one-basedness (SU-rank 1 case)

I effect on NIP-like conditions

I application: linearity

I separating “geometry” and “random noise”

Geometric theories

Definition

A first order theory T is called geometric if

I in any model of T , acl satisfies the exchange property

I T eliminates quantifier ∃∞(i.e. for any φ(x , y) there is n ∈ ω such thatwhenever |φ(M, a)| > n, φ(M, a) is infinite)

By geometric structures we mean models of geometric theories.

In any geometric structure, acl induces a pregeometry, with thenatural notion of independence (denoted | ) and dimension.

We can also define dim(φ(x , b)) = max{dim(a/b) | |= φ(a, b)}.

A “map” of geometric structures





Unary expansions

Add a new unary predicate symbol Q to the language L = L(T )and consider models (M,Q) in the expanded languageLQ = L ∪ {Q}, where M |= T .

Unary expansions: examples

I elementary pairs

I belles paires (Poizat 1983) - stable caseI dense pairs (van den Dries 1998) - o-minimal caseI generic/lovely pairs (V. 2001; Ben Yaacov, Pillay, V. 2003) -

supersimple SU-rank 1 and simple casesI lovely pairs of geometric structures (Berenstein, V., 2010)

I generic predicate (Chatzidakis, Pillay 1998)

I indiscernible sequence (Baldwin, Benedikt 2000)

I multiplicative subgroup of a field (Gunaydin, van de Dries2005)

I independent dense subsets of o-minimal structures (Dolich,Miller, Steinhorn 2016)

I independent dense subsets of geometric structures(Berenstein, V. 2016)

Dense/codense subsets

Definition

Let T be a geometric theory, M |= T .A unary expansion (M,Q) of M is dense/codense, if anynonalgebraic 1-type p(x ,A) (in T ) over a finite-dimensionalA ⊂ M has realizations in

I Q(M) ( ”density” property)

I M\ acl(A ∪ Q(M)) (”codensity” or extension property).

Dense/codense in the geometric setting

Lovely pairs vs. H-structures

Definition

Given a geometric T , M |= T and a dense/codense expansion(M,Q) of M,

I if Q(M) is algebraically closed, we call (M,Q) a lovely pair(in this case Q(M) � M);

I if Q(M) is algebraically independent, we call (M,Q) anH-structure

Notation: we use (M,P) for lovely pairs and (M,H) forH-structures.

Lovely pairs vs. H-structures

In fact, for any H-structure (M,H),(M, acl(H(M))) is a lovely pair (in particular, acl(H(M)) � M).

The field case: G -structures

We can consider an ”intermediate” construction between lovelypairs and H-structures:

I Let T be the theory of an algebraically closed or real closedfield (K ,+, ·, 0, 1), and let (K ,H) be its H-structure.

I Let G (K ) = the multiplicative group generated by H(K ).

I (K ,G ) is dense-codense (we call it a G -structure), andH(K ) ⊂ G (K ) ⊂ acl(H(K )).

I G (K ) is a free abelian group.

I G (K ) is linearly independent in K .This is an example of a group with the Mann property.Expansions of fields with multiplicative subgroups with theMann property were studied by van den Dries and Gunaydin.

Axiomatizing density/codensity

Let T be a geometric theory (assume QE for convenience).Then a sufficiently saturated model of the following axioms in thelangauge LQ = L ∪ {Q} is a dense/codense expansion of T :

I T

I density: for any L-formula φ(x , y),

∀y (∃∞x φ(x , y)→ ∃x ∈ Q φ(x , y))

I extension (codensity): for any L-formulas φ(x , y) andψ(x , y , w) where ψ witnesses x ∈ acl(y , w),

∀y (∃∞x φ(x , y)→ ∃x (φ(x , y) ∧ ∀w ∈ Q ¬ψ(x , y , w)))

Theories TP , TH

Theories TP and TH can be obtained by adding axioms saying

I acl(P(M)) = P(M) (for TP)

I H(M) is acl-independent (for TH)

Sufficiently saturated models of TP/TH are again lovely pairs/H-structures.

Note: extension (codensity) axioms are not needed for TH if T is“strongly non-trivial”.

Theory TG

What does a sufficiently saturated model (K ∗,G ) ofTh(K ,+, ·, 0, 1,G ) look like, for a G -structure generated by anH-structure?

Note that (K ∗,G ) is no longer generated by an H-structure:G (K ∗) will have divisible elements.

But G (K ∗) is still linearly independent, dense and codense.

Theory TG

From the work of Gunaydin and van den Dries on fields withmultiplicative subgroups having the Mann property, one gets thatTh(K ,+, ·, 0, 1,G ) can be axiomatized as follows:

For ACF:

I K is an algebraically closed field (of fixed characteristic)

I G (K ) is a subgroup of K×

I G (K ) is linearly independent over Q and satisfies the theoryof free abelian groups (of infinite rank)

Theory TG

For RCF:

I K is a real closed field

I G (K ) is a subgroup of K>0

I G (K ) is dense in K>0

I for any n > 1, G [n] (subgroup of nth powers) has infiniteindex in G

I G (K ) is linearly independent over Q

Theory TG

Thus, in both cases we get a complete theory TG .

From now on, by a G -structure we will mean a sufficientlysaturated model of TG .

Q-independence

For any unary expansion (M,Q) of a geometric structure M and asubset A ⊂ M, we say that A is Q-independent, ifA |

Q(A)Q(M), i.e. for any finite a in A

dim(a/Q(M)) = dim(a/Q(A)).

Note: Any A ⊂ M can be extended to an Q-independent set byadding a subset of Q(M).

Existence of H-structures (lovely pairs, G -structures)

Any structure M |= T with an independent subset H(M) can beextended to an H-structure (N,H) of T in such a way that M isH-independent in (N,H).


Given (M,H) ...


Given (M,H), take a saturated extension of M:


Choose two independent sets of independent realizations of allnon-algebraic 1-types over M:


Include one of them in H:


Iterate ω times:


Take the union:

QE for Q-independent tuples

Given any two H-structures of T , (M,H) and (N,H), andH-independent tuples a ∈ M, b ∈ N, we have

tp(a,H(a)) = tp(b,H(b))⇒ tpH(a) = tpH(b).

Same is true for P-independent tuples in lovely pairs.

In the case of G -structures, we need to add equality of (ordered)group types of G -parts of the tuples.

Back-and-forth for H-independent tuples

Given c ∈ acl(a) ...


... find d ∈ acl(b), bd ≡L ac, with c ∈ H ⇐⇒ d ∈ H.

Tuples are still H-independent.


Given c ∈ H(M) non-algebraic over a ...


... find d ∈ H(N) such that bd ≡L ac (using density).



Given c ∈ acl(aH(M))\H(M) ...


... find h ∈ H(M) such that c ∈ acl(ah) ....


... then find h′ ∈ H(N) such that bh′ ≡L ah...


... then take d ∈ N such that bh′d ≡L ahc.

d ∈ acl(bh′) ⇒ d 6∈ H(N). Tuples are still H-independent.


Given c ∈ M\ acl(aH(M)) ....


... find d ∈ N\ acl(bH(N)) with bd ≡L ac (using extension).


Theory TQ

As a consequence of QE for Q-independent tuples, we get:

I all lovely pairs / H-structures / G -structures are elementarilyequivalent

I this gives rise to the complete theories

I TP (lovely pairs)I TH (H-structures)I or (in the cases of ACF or RCF) TG (G -structures)

I each of TQ has an explicit axiomatization

Quantifier elimination

TQ has QE down to boolean combination of formulas of the form∃y ∈ Q φ(x , y), where φ is an L-formula.

Small and Large

We work in a sufficiently saturated (M,Q) |= TQ .

I Small closure of A ⊂ M is given by scl(A) = acl(A ∪ Q(M)).

I An LQ-definable subset X of M is small if X ⊂ scl(a) for afinite tuple a ∈ M.

I Otherwise, we call X large.

I For any LQ-definable set X ⊂ M, there is an L-definable setY ⊂ M such that X∆Y is small.

Some properties of TQ (TP1, TH

2, TG )

I When passing from T to TQ the following properties arepreserved:

I stability (superstability, except for TG )I simplicity (supersimplicity, except for TG )I NIP

I In SU-rank 1 case:

I one gets a reasonable description of forking in TQ (in terms offorking over Q and forking of ”Q-bases”)

I The SU-rank of TQ reflects the ”geometric complexity” of T .

1A. Berenstein, E. Vassiliev, On lovely pairs of geometric structures, Ann.Pure Appl. Logic, 161 (7), 2010, 866-878

2A. Berenstein, E. Vassiliev, Geometric structures with a dense independentsubset, Selecta Mathematica - N.S., 22(1), 2016, 191-225

“Q-bases”

Suppose C ⊂ M is Q-independent, a ∈ M a tuple.

We can split a into a′ and a′′ where a is independent overC ∪ Q(M) and a′′ ∈ acl(a′C Q(M)).

We can find a finite b ∈ Q(M) such that a′′ ∈ acl(a′bC ).Thus, abC is Q-independent.

Question: can we choose a minimal such b “canonically”?

“Q-bases”

H-structures: There is a unique minimal b ∈ H(M), call it theH-basis of a over C : b = HB(a/C ).

G-structures: There is no unique minimal b ∈ G (K ), but allminimal b are interdefinable, in the group language, over G (C ).We call dclgr (b) the G-basis of a over C , GB(a/C ).

Lovely pairs: if M is supersimple of SU-rank 1 with EI/WEI/ GEI(e.g. ACF) we can take Cb(a/C ).

What happens to acl?

Three closure operators in (M,Q): acl, aclQ and scl.

Clearly, acl(A) ⊂ aclQ(A).

Any Q-independent acl-closed set is aclQ closed. Thus, we have:

I aclQ(A) ⊂ acl(A ∪ Q(M)) = scl(A)

I aclQ(A) =⋂{B|A ⊂ B, B = acl(B) and is Q − independent}

Question: when aclQ = acl?

What happens to acl: H-structures

I HB(a) ∈ aclH(a)

I aclH(a) = acl(aHB(a))

I aclH -closed sets are always H-independent.

I If a ∈ acl(H(M)), then a is interalgebraic with HB(a) (in thesense of aclH).

I aclH = acl ⇐⇒ acl is disintegrated, i.e.acl(A) =

⋃a∈A acl(a).

What happens to acl: G -structures

Similar to H-structures:

aclG (a) = acl(aGB(a))

aclG -closed sets are G -independent.

What happens to acl: lovely pair case

I In a pair (V ,P) of vector spaces, any acl-closed set (subspaceof V ) is P-independent (by modularity), hence, aclP = acl.

I In a pair (K ,P) of algebraically closed fields, aclP 6= acl:

Take a, b, c ∈ K algebraically independent, so thatb, c ∈ P(K ) and a 6∈ P(K ). Let d = ab + c . ThenaclP(a, d) = acl(a, b, c) 6= acl(a, d).

I acl = aclP iff T is linear... More on this later in the talk.

Forking in (M ,Q) in supersimple SU-rank 1 case

Let C ⊂ B ⊂ M, a ∈ M.

Then a | QBC ⇐⇒

a |B Q(M)

C and Q − base of a C |Q−base ofC

Q−base of B.

Lovely pairs: a | PBC ⇐⇒

a |B P(M)

C and Cb(a C/P(M)) |Cb(C/P(M))

Cb(B/P(M)).

H-structures: ( B,C H-independent) a | HBC ⇐⇒

a |B H(M)

C and HB(a C ) |HB(C)

HB(B)

G-structures: ( B,C G -independent) a | QBC ⇐⇒

a |B Q(M)

C and GB(a C ) |GB(C)

GB(B)

Properties of TH : SU-rank 1 case

Description of forking in TH for T supersimple of SU-rank 1:

Let C ⊂ D be aclH -closed, then tp(a/D) forks over C iff

I a ∈ D\C (becoming algebraic), or

I a ∈ acl(D ∪ H)\ acl(C ∪ H) (becoming small), or

I HB(a/D) ( HB(a/C ) (reduction of H-basis)

It follows that H(x) has SU-rank 1.


SU-rank of 1-types in TH :Suppose C = aclH(C ), a a single element.

I T trivial:

I SU(a/C ) = 0 ⇐⇒ a ∈ CI a ∈ scl(C )\C ⇒ SU(a/C ) = 1I a 6∈ acl(C )⇒ SU(a/C ) = 1

I T nontrivial:

I SU(a/C ) = 0 ⇐⇒ a ∈ CI a ∈ scl(C )\C ⇒ SU(a/C ) = |HB(a/C )|I a 6∈ scl(C )⇒ SU(a/C ) = ω (unless a is a “trivial” element)


Proposition

Let T be supersimple of SU-rank 1. Then TH is supersimple

and has SU− rank =

{1, T is trivialω, T nontrivial


Canonical bases in TH :

I T SU-rank 1, B = aclH(B) ⇒ CbH(a/B) is interalgebraicwith Cb(aHB(a/B)/B).

I In particular, we have geometric elimination of imaginaries in(TH)eq down to T eq.


Recall: A theory is 1-based if Cb(a/B) ∈ acleq(a).

1-basedness is not preserved when passing to TH .


I Let T be the theory of infinite vector spaces over F2. Let(V ,H) be an H-structure of T . Take v ∈ H(V ),u ∈ V \H(V ), t = u + v .

I Then CbH(t/u) is interalgebraic withCb(tHB(t/u)/u) = Cb(tv/u) = u. However,u 6∈ aclH(t) = span(t) = {0, t}.

I But two independent realizations of tpH(t/u) are enough:u ∈ aclH(t, t ′). Thus TH is ”2-based” (true in general).


A recent result of Carmona:

I for n ≥ 2: T n-ample ⇐⇒ TH n-ample

I in particular:T 1-based (not 1-ample) ⇒ TH is CM-trivial (not 2-ample)

Properties of TP : SU-rank 1 case

Proposition (V., 2001)

Let T be supersimple of SU-rank 1. Then TP is supersimple

and has SU− rank =

1, T is trivial2, T one− based, nontrivialω, T non− one− based

(generalizing Buechler’s 1991 result for s.m. structures)

If T is one-based then so is TP (Ben Yaacov, Pillay, V. 2003).

Preservation of NIP in dense/codense expansions

I Berenstein, Dolich, Onshuus (2011):T is (strongly) dependent ⇒ TP is (strongly) dependent

I T is (strongly) dependent ⇒ TH is (strongly) dependentIdea of the proof of (T NIP ⇒ TH NIP):

I Since TH has QE down to H-bounded formulas, by aChernikov-Simon’s result, it suffices to show NIP over H;

I Suppose TH has IP over H: there is an LH -formula φ(x , y),a ∈ M and an indiscernible sequence (bi : i < ω) in H(M)such that

|= φ(a, bi ) ⇐⇒ i is even.

I We can replace φ(a,H(M)n) with ψ(a′,H(M)n) where ψ is anL-formula. Then ψ witnesses IP in T , a contradiction.

I RCFG is dependent but not strongly dependent

Application of lovely pairs: a notion of linearity

What does it mean to be a linear geometric theory?

Linearity is well-defined and understood in:

I Strongly minimal theories: linearity = local modularity

I Supersimple theories of SU-rank 1: linearity = one-basedness(weaker than local modularity)

I o-minimal theories: CF-property, non-interpretability of aninfinite field

Comparing strongly minimal and o-minimal settings

strongly minimal:

I linearity = local modularity

I linearity ⇐⇒ 1-basedness (a ≡B a′, a |Ba′ ⇒ a |

a′B)

I linearity ⇐⇒ no interpretable pseudoplane

I linear+nontrivial ⇒ interpretability of infinite vector spaces

I non-local modularity 6⇒ intepretability of an infinite field

o-minimal:

I linearity = no interpretable infinite field

I local modularity ⇒ linearity

I linearity 6⇒ local modularity

I linear+nontrivial ⇒ interpretability of infinite vector spaces

In both settings, linearity ⇐⇒ any normal definable families of”plane curves” has dimension ≤ 1.

More on linearity: families of curves

Plane curve (with parameters a) in a geometric structure M is adefinable one-dimensional subset of M2:

Ca = {(x , y)|M |= φ(x , y , a1, . . . , an)} and for any (u, v) ∈ Ca,dim(uv/a) ≤ 1.

As we vary the parameters a over a definable subset of Mn we geta definable family of plane curves.

Nonlinear example

For example, y = ax2 + bx + c , where a 6= 0, is a definable familyof plane curves in (R,+, ·, 0, 1, <).

Since we are using 3 parameters (which can be chosen algebraicallyindependent), this family has dimension 3.

Nonlinear example

Different choices of (a, b, c) give different parabolas. Differentparabolas can intersect in at most 2 different points. Such familyis called normal: different curves have finite intersection.

Another nonlinear example

A simpler example: y = ax + b, a normal family of dimension 2(two-parameter family).

Linear example

In (R,+, 0, <) we can only form normal families of dimension ≤ 1(one-parameter families): e.g. y = (x + x) + a.

Any two-parameter family that we can create, such as

x = a ∨ y = b

ory = (x + x) + a ∨ y = (x + x) + b

will not be normal.

In search of general notion of linearity

A notion of linearity should:

I have a definition in terms of combinatorial pregeometry (someform of modularity)

I have a definition in terms of definable sets (families of planecurves)

I be equivalent to non-(type)-definability of certain complicatedstructures and/or configurations (e.g. infinite fields,pseudoplanes, quasidesigns)

I be closed under reducts

I in the nontrivial case, imply certain connection with projectivegeometries over division rings, definability of infinite groups(vector spaces)

I should have a natural extension to non-geometric context(e.g. one-basedness in stable theories)

Main challenges in the general geometric case

I doing forking calculus without canonical bases (a strong toolin strongly minimal and SU-rank 1 cases)

I no definable topology (a strong tool in o-minimal orC-minimal cases)

Generic linearity

Call a definable family of plane curves almost normal, if eachcurve has infinite intersection with only finitely many other curves.

A geometric structure M is generically linear, if any almostnormal family of plane curves in M has dimension ≤ 1.

In the strongly minimal and o-minimal cases: generic linearity =linearity .

Theorem3

The following are equivalent for any geometric theory T :

1. T is generically linear(any almost normal definable family of plane curves has dim ≤ 1)

2. T is weakly locally modular(for any a, b,C such that a ∈ acl(bC ), there exist D | abC andc ∈ acl(CD) such that a ∈ acl(bcD))

3. T is weakly 1-based(for any a,B there is a′ ≡B a with a |

Ba′ and a |

a′B)

4. T has no complete type definable almost quasidesign.(a pseudoplane-like configuration)

5. aclP = acl in any (M,P) |= TP

6. scl is modular in any (M,P) |= TP

(a ∈ scl(bC ) ⇒ there exists c ∈ scl(C ) such that a ∈ scl(bc)).

3A. Berenstein, E. Vassiliev, Weakly one-based geometric theories, J. Symb.Logic, 77, No. 2, June 2012

Connection with “classical” lionearity

Generic linearity (weak local modularity, weak 1-basedness) isequivalent to

I local modularity, in the strongly minimal case

I one-basedness, in the SU-rank 1 case

I linearity (CF-property), in the o-minimal case

I linearity as defined by F. Maalouf, in the geometric C-minimalcase

It is also closed under reducts.

Geometry of the small closure

Moreover, for a generically linear T we have:

I the geometry of acl(− ∪ P(M)) is either trivial or splits in adisjoint union of projective geometries over division rings;

I the geometry of acl is a disjoint union of subgeometries ofprojective geometries over division rings;

I if T is ω-categorical (only one countable model), nontrivialand generically linear, then TP interprets an infinite vectorspace over a finite field.

Structure induced on H : generic trivialization4

It turns out that the structure induced on H(M) keeps the“random noise” while “forgetting” the geometry of M.

Given a sufficiently saturated H-structure (M,H), consider H(M)together with traces of definable sets of M (without parameters).

Denote such structure by H∗(M), and its theory by T ∗

(generic trivialization of T ).

4A. Berenstein, E. Vassiliev, Generic trivializations of geometric theories,Math. Logic Q., 60, No. 4-5, 289-303 (2014)

Structure induced on H : generic trivialization

I T ∗ is a trivial (acl(A) = A) geometric theory, with QE

I As T ∗ is a reduct of TH and is trivial, we can expectT ”nice”⇒ T ∗ ”nice”.

I More interestingly, we often have T ∗ ”nice”⇒ T ”nice”.

I To show this we need H∗(M) to somehow ”control” M.

I Can be done by working in acl(H(M)) which is a sufficientlysaturated model of T .

I Easier when acl = dcl: any set definable over dcl(H(M)) isalso definable over H(M).

Moving parameters into H(M): when acl 6= dcl

The main tool that allows to move parameters into H(M):

Proposition

Let (M,H) be an H-structure of a geometric theory T . LetD ⊂ M be a set L-definable over acl(H(M)). Then there existsD ′ ⊂ D L-definable over H(M) such that D\D ′ is finite.

Idea of the Proof

I Let D = φ(M, a, h), where h ∈ H(M) and a ∈ acl(h),witnessed by an L-formula ψ(y , h).

I Consider all the conjugates of D over h (there are finitelymany).

I Each conjugate is cut into disjoint pieces by booleancombinations with other conjugates.

I In each infinite disjoint piece pick an element of H, say c .Then the piece is L-definable over hc by∀y(ψ(y , h)→ (φ(x , y , h)↔ φ(c , y , a))).

Properties of T ∗: strongly minimal case

Proposition

T is strongly minimal ⇐⇒ T ∗ is strongly minimal(and, thus, is the theory of equality)

Proof:⇒ clear⇐ Suppose there is an infinite co-infinite D ⊂ M definable overacl(H(M)). Choose D ′ ⊂ D, definable over H(M), with D\D ′finite. Then D ′ ∩ H(M) is definable in H∗(M) and is infinite andco-infinite.

Properties of T ∗: SU-rank 1 case

Proposition

T is supersimple SU-rank 1 ⇐⇒ T ∗ is supersimple SU-rank 1

Proof:⇒ follows from TH being supersimple and H(x) having SU-rank 1in (M,H).⇐ Assume T is not supersimple of SU-rank 1. Work overacl(H(M)). Assume φ(x , a) is a non-algebraic formula thatk-divides over ∅, witnessed by an indiscernible sequence(ai : i < |T |+).For every i < |T |+ we can find ψi (x , hi ) with hi ∈ H(M) defininga co-finite subset of φ(M, ai ). We may assume that ψi = ψ are thesame for each i and hi form an indiscernible sequence. Thenψ(x ,~h0) defines an infinite subset of H∗(M) that k-divides over ∅in T ∗, a contradiction.

Properties of T ∗: NIP case

Proposition

T is NIP ⇐⇒ T ∗ is NIP

Idea of the proof:⇒ follows from TH being NIP.⇐ Suppose T has IP witnessed by φ(x , y) and an indiscerniblesequence I = (bi : i ∈ ω) and a ∈ M (non-algebraic over I ) suchthat

|= φ(a, bi ) ⇐⇒ i even.

As in the SU-rank 1 case, we can ”pull” I into H(M).

Some questions

I If T is linear (i.e. weakly 1-based), is there a way to “recover”M from the geometry of scl and H∗(M)?

I For a geometric T , does TH have elimination of ∃∞?(true for formulas φ(x , y) that imply y ∈ H)

I Imaginaries in TH?(Dolich, Miller, Steinhorn: EI holds in o-minimal case;we also have GEI in SU-rank 1 case)

I if T is nontrivial and linear, can we interpret an infinite groupin T , or, at least, TP?

I structure of weakly 1-based groups

I weak 1-basedness beyond geometric theories?(progress by Boxall, Bradley-Williams, Kestner, Omar Aziz,Penazzi, NDJFL 2013)

THANK YOU!

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