Structures of SU-rank omega with a dense independentsubset of generics

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Structures of SU-rank omega with a dense

independentsubset of generics

Alexander Berenstein, Juan Felipe Carmona, Evgueni Vassiliev

To cite this version:

INDEPENDENT SUBSET OF GENERICS

ALEXANDERBERENSTEIN,JUANFELIPECARMONA

∗

,ANDEVGUENIVASSILIEV

Abstra t. Extendingtheworkdonein[5,9 ℄intheo-minimalandgeometri

settings,westudyexpansionsofmodelsofasupersimpletheoryofSU-rank

ω

witha"dense odense"independent olle tion

H

ofelementofrank

ω

,where density of

H

meansit interse ts any denable set of

SU

-rank omega. We showthatundersomete hni al onditions,the lassofsu hstru turesisrst

order. Weprovethatthe expansionissupersimple and hara terize forking

and anoni albasesoftypesintheexpansion. Wealsoanalyzetheee tthese

expansionshaveonone-basedness andCM-triviality. Inthe one-based ase,

wedes ribeanatural"geometryofgeneri smodulo

H

"asso iatedwithsu h expansionsandshowitismodular.

1. Introdu tion

Thereareseveralpapersthatdealwithexpansionsofsimpletheorieswithanew

unarypredi ate. Forexample,thereistheexpansionwitharandomsubset[8℄that

givesa asewherethenewtheoryisagainsimpleandforkingremainsthesame,in

ontrasttothe aseoflovelypairs[2,15℄,wherethepairisusuallymu hri herand

the omplexityof forking isrelated to thegeometri propertiesof the underlying

theory[15℄.

In[5℄therst andthethird authors studied, in thesettingof geometri

stru -tures,addingapredi ateforanalgebrai allyindependentset

H

whi hisdenseand odenseinamodel

M

(meaningeverynon-algebrai formulainasinglevariablehas arealizationin

H

andarealizationgeneri over

H

anditsparameters). Thepaper generalizedideasdevelopedintheframeworkofo-minimaltheoriesin[9℄. Thekey

toolusedin[5℄wasthatthe losureoperator

acl

hastheex hangepropertyandthus givesamatroidthatintera tswellwiththedenablesubsets. Aspe ial aseunder

onsideration was SU-rank onetheories, where forking independen e agrees with

algebrai independen e. Inthis strongersettingtheauthors hara terizedforking

and gaveades ription of anoni albases in the expansion. As in thelovelypair

ase, the omplexity of forking is relatedto theunderlying geometry of the base

theory

T

.

In this paper we start with a theory

T

that has

SU

-rank omega and we use

the losureoperatorasso iatedto theweightofgeneri types,namelyfor

M |= T

,

a ∈ M

,

A ⊂ M

,wehave

a ∈ cl(A)

if

SU (a/A) < ω

. This losureoperatorhasthe ex hangepropertyandmanyoftheresultsobtainedin[5℄ anbeprovedinthenew

framework: weexpand

M

by a new predi ate onsisting in a

cl

-dense

cl

- odense 2000Mathemati sSubje tClassi ation. 03C45.

Key words and phrases. supersimple theories, SU-rank omega, unarypredi ate expansions,

one-basedness,ampleness,CM-triviality.

∗

supersimpleandwegeta leardes riptionof anoni albases intheexpansion,up

tointeralgebrai ity(seeProposition5.6).

Inthespe ial asewhere thetheory of

M

issuperstablewith auniquetypeof

U

-rank

ω

,thepredi ate

H

isaMorleysequen eofgeneri s;this aseisrelatedto theworkdonein[1℄. OurworkisalsorelatedtoworkofFornasieroonlovelypairs

of losureoperators[10℄.

Of spe ial interest is the ee t of our expansion on the geometri omplexity,

namely theampleness hierar hy. Followingthe ideasof [7℄, weshowthat the

ex-pansionpreservesCM-triviality, but one-basednessispreservedonlyin thetrivial

ase.

We then use this expansion to study the underlying geometry of the losure

operator lo alized in

H

. We show that if

T

is a one-based supersimple theory

of SU-rank

ω

,

(N, H)

a su iently (e.g.

|T |

+

-) saturated

H

-stru ture, then the lo alized losure operator

cl(− ∪ H)

is modular and its asso iated geometry is a disjointunionofproje tivegeometriesoverdivisionringsandtrivialgeometries.

Thispaperisorganizedasfollows. Inse tion2wedene

H

-stru turesasso iated tomodels

M

ofatheory

T

. Weshowthattwo

H

-stru turesasso iatedtothesame theory are elementary equivalent and all

T

ind

this ommon theory. Finally we

provethatthatundersomete hni al onditions(eliminationofthequantier

∃

large

andthe typedenablityofthepredi ates

Q

ϕ,ψ

)thesaturatedmodelsof

T

ind

are

again

H

-stru tures.

In se tion 3 we study four dierent examples of theories of

SU

-rank

ω

: dif-ferentially losed elds, ve torspa es with a generi automorphism,

H

-pairs and lovelypairsofgeometri theories. In ea h aseweshowthe orrespondingtheory

of

H

-stru turesisrstorder.

Inse tion4weanalyzethedenablesetsin theexpansion,weprovethat every

denablesetisaboolean ombinationofoldformulasboundedbyexistential

quan-tiersoverthenewpredi ate. Inse tion5we hara terizeforkingintheexpansion

and hara terize anoni albases.Inse tion6westudythequestionofpreservation

of one-basedness and CM-triviality under ourexpansion. Finally in se tion7 we

studythegeometryof

cl(− ∪ H)

.

2.

H

-stru tures: definition andfirst properties

Let

T

beasimpletheoryof

SU

-rank

ω

. Let

H

beanewunarypredi ateandlet

cl

H

= cl ∪{H}

. Let

T

′

bethe

L

H

-theoryof allstru tures

(M, H)

, where

M |= T

and

H(M )

isanindependentsubsetofgeneri elementsof

M

,thatis,allelements

have

SU

-rank

ω

. Note that saying that

H(M )

is an independent olle tion of

generi sis arst order property, it is simply the onjun tionsof formulasof the

form

¬ϕ(x

1

, . . . , x

n

)

,where

SU (ϕ(x

1

, . . . , x

n

)) < ωn

.

For

M |= T

,

A ⊂ M

and

b ∈ M

,wewrite

b ∈ cl(A)

andsaythat

b

issmall over

A

if

SU (b/A) < ω

. Bytheadditivitypropertiesof

SU

rankwehavethat

L

givesa pregeometryon

M

. Wewrite

dim

cl

(ϕ(x

1

, . . . , x

n

)) = n

andsaythat

ϕ(x

1

, . . . , x

n

)

islarge if

SU (ϕ(x

1

, . . . , x

n

)) = ωn

Wewillassumethat foreveryformula

ϕ(x, ~y)

thereisaformula

ψ(~y)

su h that for any

~a ∈ M ϕ(x, ~a)

is largeif andonly if

ψ(~a)

. We write

∃

large

_{ϕ(x, ~y)}

hange the pregeometry

acl

for the pregeometry

L

and thequantier

∃

∞

for the quantier

∃

large

. Notation2.1. Let

(M, H(M )) |= T

′

andlet

A ⊂ M

. Wewrite

H(A)

for

H(M ) ∩

A

.

Notation 2.2. Throughout this paper independen e means independen e in the

sense of

T

andwe usethe familiar symbol

|

⌣

. Wewrite

tp(~a)

for the

L

-typeof

a

and

dcl

,

acl

for the denable losureand the algebrai losure in the language

L

. Similarly we write

dcl

H

, acl

H

, tp

H

for the denable losure, the algebrai losure andthe typeinthe language

L

H

.

Denition2.3. Wesaythat

(M, H(M ))

isan

H

-stru tureif (1)

(M, H(M )) |= T

′

(2) (Density/ oheirproperty)If

A ⊂ M

isnite and

q ∈ S

1

(A)

is thetypeof ageneri element(of

SU

-rank

ω

),thereis

a ∈ H(M )

su hthat

a |= q

. (3) (Co-density/extensionproperty)If

A ⊂ M

isniteand

q ∈ S

1

(A)

,thereis

a ∈ M

,

a |= q

and

a |

⌣

A

H(M )

.

Lemma 2.4. Let

(M, H(M )) |= T

′

. Then

(M, H(M ))

is an

H

-stru ture if and onlyif:

(2') (Generalizeddensity/ oheirproperty)If

A ⊂ M

isniteand

q ∈ S

n

(A)

has

SU

-rank

ωn

,thenthereis

~a ∈ H(M )

n

su hthat

~a |= q

.

(3') (Generalized o-density/extensionproperty)If

A ⊂ M

isnitedimensional and

q ∈ S

n

(A)

,thenthereis

~a ∈ M

n

realizing

q

su hthat

tp(~a/A ∪ H(M ))

does notfork over

A

.

Proof. We prove(2') and leave(3') to thereader. Let

~b |= q

, we may write

~b =

(b

1

, . . . , b

n

)

. Sin e

(M, H(M ))

isan

H

-stru ture,applyingthedensitypropertywe annd

a

1

∈ H(M )

su hthat

tp(a

1

/A) = tp(b

1

/A)

. Let

q(x, b

1

, A) = tp(b

2

, b

1

, A)

andlet

A

1

= A ∪ {a

1

}

. Finally onsiderthetype

q(x, a

1

, A)

over

A

1

,whi histhe typeofageneri element. Applying thedensity propertywe annd

a

2

∈ H(M )

su hthat

tp(a

2

, a

1

/A) = tp(b

2

, b

1

/A)

. We ontinueindu tivelytondthedesired

tuple

(a

1

, a

2

, . . . , a

n

)

.



Notethat if

(M, H(M ))

isan

H

-stru ture, theextensionpropertyimplies that

M

is

ℵ

0

-saturated.

Denition2.5. Let

A

beasubsetofan

H

-stru ture

(M, H(M ))

. Wesaythat

A

is

H

-independent if

A

isindependentfrom

H(M )

over

H(A)

.

Lemma 2.6. Any model

M

of

T

with a distinguished independent subset

H(M )

anbeembedded inan

H

-stru turein an

H

-independent way.

Proof. Givenanymodel

M

withadistinguishedindependentsubset

H(M )

of gener-i s,we analwaysndanelementaryextension

N

of

M

andaset

H(N )

extending

H(M )

su h that foreverygeneri

1

-type

p(x, acl( ~

m))

(i.e.

SU (p(x)) = ω

), where

~

m ∈ M

,thereis

d ∈ N

su h that

d |= p(x, acl( ~

m))

and

d |

⌣

H(M)

m

~

. Addasimilar

statementfortheextensionproperty. Nowapplya hainargument.



Lemma2.7. Let

(M, H)

and

(N, H)

besu ientlysaturated

H

-stru tures,

~a ∈ M

and

~a

′

_{∈ N H}

-independent tuples su h that

tp(~a, H(~a)) = tp(~a

′

_{, H(~a}

′

₎₎

. Then

tp

H

(~a) = tp

H

(~a

′

)

.

Proof. Write

~a = ~a

0

~a

1

~h

, where

~a

0

is independentover

H(M )

,

~h = H(~a) ∈ H(M)

and

~a

1

∈ L(~a

0

~h)

. Similarlywrite

~a

′

_{= ~a}

′

0

~a

′

1

~h

′

.

It su es to show that for any

b ∈ M

there are

~h

1

∈ H(M )

,

~h

′

1

∈ H(N )

and

b

′

_{∈ N}

su h that

~a~h

1

b

and

~a

′

_~h

′

1

b

′

are ea h

H

-independent,

tp(~a

0

~a

1

~h~h

1

b) =

tp(~a

′

0

~a

′

1

~h

′

~h

1

′

b

′

)

,and

b ∈ H(M )

i

b

′

_{∈ H(N )}

.

Case1:

b ∈ cl(~a) ∩ H(M )

. By

H

-independen eof

~a

,wemusthave

b ∈ cl(~h)

and sin e

H

forms an independentset we musthave

b ∈ ~h

. Let

b

′

_{∈ ~h}

′

be su h that

tp(b

′

_~a

′

_{) = tp(b~a)}

andtheresultfollows. Herewe antake

~h

1

and

~h

′

1

to beempty Case 2:

b ∈ H(M )

and is non small over

~a

. Then

tp(b/~a)

is generi . Bythe densityproperty, we annd

b

′

_{∈ H(N )}

su h that

tp(b

′

_~a

′

_{) = tp(b~a)}

. Here again

we antake

~h

1

and

~h

′

1

to beempty. Case3:

b ∈ cl(~a)

. We laimthat

b |

⌣

~

a

H(M )

. Indeedlet

~h

1

(sayoflength

k

)in

H(M ) \ ~h

. Sin e

~a

is

H

-independent, theelements in

H(M ) \ ~h

areindependent over

~a

andthus

SU (~h

1

/~a) = SU (~h

1

/~h) = ωk

. Ontheotherhand

SU (b/~a) < ω

,so thetypes

tp(b/~a)

,

tp(~h

1

/~a)

areorthogonalandthe laimfollows.

Thusthetuple

~ab

is

H

-independent. Let

p(x, ~a) = tp(b/~a)

. Nowusethe exten-sionpropertytond

b

′

_{∈ N}

′

su hthat

b

′

_{|= p(x, ~a}

′

₎

,

b

′

_|

⌣

~

a

′

H(N )

,sobytransitivity

~a

′

_b

′

is

H

-independent.

Case 4:

b ∈ cl(H(M )~a)

. Add a tuple

~h

1

∈ H(M )

su h that

~ab~h

1

is

H

-independent,anduseCase2andCase3.

Case 5:

b 6∈ cl(H(M )~a)

. Bythe extension property, there is

b

′

_{∈ N}

su h that

b

′

_{6∈ cl(H(N )~a}

′

₎

and

tp(b

′

_~a

′

_{) = tp(b~a)}

. The tuples stay

H

-independent, so again we antake

~h

1

and

~h

′

1

to beempty.



Thepreviousresulthasthefollowing onsequen e:

Corollary 2.8. All

H

-stru turesareelementarily equivalent.

Wewrite

T

ind

forthe ommon ompletetheoryofall

H

-stru turesofmodelsof

T

.

Denition 2.9. We say that

T

ind

is rst order if the

|T |

+

-saturatedmodels of

T

ind

areagain

H

-stru tures. To axiomatize

T

ind

and to showthat

T

ind

is rstoder, wefollowthe ideasof

[15,Prop2.15℄,[3℄and[2℄. Here weuseforthersttimethat

T

eliminates

∃

large

.

Re allthatwhenever

T

eliminates

∃

large

theexpressiontheformula

ϕ(x,~b)

islarge isrstorder.

Wealsoneedthefollowingdenition from[2,Denition2.4℄:

Denition 2.10. Let

ψ(~y, ~z)

and

ϕ(~x, ~y)

be

L

-formulas.

Q

ϕ,ψ

is the predi ate whi hisdenedtoholdofatuple

~c

(in

M

)ifforall

~b

satisfying

ψ(~y, ~c)

,theformula

ϕ(~x,~b)

doesnotdivide over

~c

.

Thefollowingresultfollowswordbywordfromtheproofof[2,Proposition4.5℄,

(1)

Q

ϕ,ψ

istype-denable(inM )for all

L

-formulas

ϕ(~x, ~y)

,

ψ(~y, ~z)

. (2) The extensionproperty isrstorder.

(3) Any

|T |

+

-saturatedmodel of

T

ind

satisesthe extensionproperty.

Corollary 2.12. Let

T

beasimpletheory of

SU

-rankomega that satiseswnf p. Then the extensionproperty isrst order.

Proposition 2.13. Assume

T

eliminates

∃

large

andthat the predi ates

Q

ϕ,ψ

are

L

-type-denablefor all

L

-formulas

ϕ(~x, ~y)

,

ψ(~y, ~z)

. Then

T

ind

isrstorder.

Proof. Thetheory

T

ind

isdes ribedby

T

′

,thedensitypropertyandtheextension

property.

T

′

isarstorderproperty.

Thedensityproperty anbedes ribedin rstorderbythes heme:

For all

L

-formulas

ϕ(x, ~y)

∀~y(ϕ(x, ~y)

large

=⇒ ∃x(ϕ(x, ~y) ∧ x ∈ H))

.

Thusallsaturatedmodelsofthes hemesatisfythedensityproperty. Finallyby

Proposition 2.11any

|T |

+

-saturatedmodelof

T

ind

satisfytheextension property.



Notation 2.14. Let

(M, H(M ))

be an

H

-stru ture and let

A ⊂ M

. We write

cl

H

(A)

for

cl(AH(M ))

andwe allitthe small losureof

A

over

H

.

3. examples

Inthis se tion wegivealist of examplesofsimple theoriesof

SU

-rank

ω

that eliminate

∃

large

andwhere theextension propertyisrstorder. Wealso listsome

examples that eliminate the quantier

∃

large

but where it remains as an open

questioniftheextensionpropertyisrstorder.

3.1. Dierentially losed elds. Let

T = DCF

0

, the theory of dierentially

losedelds. Thistheoryisstableof

U

rank

ω

andalso

RM (DCF

0

) = ω

.

Let

p(x)

betheuniquegeneri typeofthetheory. Thistypeis omplete, station-aryanddenable over

∅

. Let

ϕ(x, ~y)

beaformulaandlet

ψ(~y)

beits

p

-denition. Then for

(K, d) |= DCF

0

,

~a ∈ K

, theformula

ϕ(x, ~a)

is largei

ψ(~a)

. Thus this theoryeliminatesthequantier

∃

large

.

Nowletusstudytheextensionproperty. Re allthat

DCF

0

hasquantier elimi-nation[12,Theorem2.4℄andeliminatesimaginaries[12,Theorem3.7℄.Itisproved

in[12,Theorem2.13℄that

DCF

0

hasuniformbounding(i.e. iteliminates

∃

∞

)and

thus it has nf p. This is also expli itly explained in [12, page 52℄. It followsby

Corollary2.12thattheextensionpropertyisrstorder.

3.2. Free pseudoplane-innite bran hing tree. Let

T

be the theory of the

free pseudoplane, that is, a graph without y les su h that every vertex has

in-nitely many edges. The theory of the free pseudoplane is stable of

U

-rank

ω

and

M R(T ) = ω

. For every

A

,

acl(A) = dcl(A) = A ∪ {x|

therearepoints

a, b ∈

Thegeneri typeis denableover

∅

and thus bydenabilityoftypes

T

eliminates thequantier

∃

large

.

An

H

-stru ture

(M, H)

asso iatedto

T

isaninnite olle tionoftreeswithan innite olle tionof sele tedpoints

H(M )

at innitedistan e onefrom theother andwithinnitemanytreesnot onne tedtothem. If

(N, H) |= T h(M, H)

,then

N

hasinnitelymanysele tedpoints

H(N )

atinnitedistan eonefromtheother. If

(N, H)

is

ℵ

0

-saturated, then by saturation it also has innitely many trees

whi h are not onne ted to the points

H(N )

. We will prove that in this ase

(N, H)

isan

H

-stru ture. Thedensitypropertyis lear. Nowlet

A ⊂ N

benite and assumethat

A = dcl(A)

and let

c ∈ N

. If

U (c/A) = ω

hooseapoint

b

in a treenot onne tedto

A ∪ H

,then

tp(c/A) = tp(b/A)

and

b |

⌣

A

H

. If

U (c/A) = 0

thereisnothingto prove. If

U (c/A) = n > 0

, let

a

bethenearestpointfrom

A

to

c

. Sin ethereisatmostonepointof

H

onne tedto

a

andthetreesareinnitely bran hing,we an hooseapoint

b

with

d(b, a) = n

andsu hthat

d(b, A ∪ H) = n

; then

tp(c/A) = tp(b/A)

and

b |

⌣

A

H

. This provesthat

(N, H)

is an

H

-stru ture andthatthat

T

ind

isrstorder.

3.3. Ve tor spa e with a generi automorphism. Let

T

be the theory of

(innite-dimensional)ve torspa esoveradivisionring

F

,andlet

T

σ

byits(unique)

generi automorphismexpansion.

This theory hasaunique generi ,whi h is denable over

∅

. Bydenability of types,

T

σ

eliminatesthequantier

∃

large

.

Nowweprovethat theextensionpropertyisrstorder.

Let

(M, H)

be an

H

-stru ture asso iated to

T

σ

, let

(N, H) |= T h(M, H)

be

|T |

+

-saturatedandlet

a,~b ∈ N

.

Notethatthetypeoftheelement

a

overatuple

~b

in

T

σ

isdeterminedby

qf tp

−

(σ

Z

_(a)/σ

Z

_(~b)),

wherethesupers ript

−

referstothelanguageof

T

,and

σ

Z

_{(~c) = . . . , σ}

−1

_{(~c), ~c, σ(~c), σ}

2

_{(~c), . . . .}

Therearethreepossiblesituations for

tp(a/~b)

: (1)

a ∈ span(σ

Z

_(~b))

(2)

a, σ(a), . . . , σ

n−1

_(a)

areindependentover

σ

Z

_(~b)

,but

σ

n

(a) ∈ span(a, σ(a), . . . , σ

n−1

(a)σ

Z

_(~b))

(3)

σ

Z

_(a)

isindependentover

σ

Z

_(~b)

For therst ase,wehavethat

a ∈ dcl(~b)

andthus

a |

⌣

~b

H

. Forthese ond ase,assumenowthat

σ

n

_{(a) ∈ span(a, σ(a), . . . , σ}

n−1

_(a)σ

Z

_(~b))

.

Sin e

M

is an

ℵ

0

-saturated, we annd

a

′

_,~b

′

_{∈ M}

su h that

tp(a,~b) = tp(a

′

_,~b

′

₎

and sin e

(M, H)

isan

H

-stru ture wemayassumethat

a

′

_|

⌣

~b

′

H

. Inparti ular, theelements

a

′

_{, σ(a}

′

_{), . . . , σ}

n−1

_(a

′

₎

donotsatisfyanynontriviallinear ombination

with elements in

dcl(~b

′

_{H(M ))}

. Sin e

(N, H) |= T h((M, H))

is

|T |

+

-saturated, we an nd

(a

′′

_{,~b) |= tp(a}

′

_,~b

′

₎

su h that

a

′′

_{, σ(a}

′′

_{), . . . , σ}

n−1

_(a

′′

₎

do not satisfy

anynontriviallinear ombination with elements in

dcl(~bH(N ))

. This shows that

a

′′

_|

For the third ase, sin e

(N, H) |= (M, H)

, we have that

H(N )

is an innite olle tionofindependentgeneri s. Let

a

0

, . . . , a

n

2

−1

∈ H(N )

bedistin tand on-sider

c

0

= a

0

+ · · · + a

n−1

, . . . , c

n−1

= a

n

2

−n

+ · · · + a

n

2

−1

. Then the elements

c

0

, . . . , c

n−1

are independent generi s and neither one an be written as alinear ombination oflessthat

n

elementsin

H

. Sin e

(N, H)

is

|T |

+

-saturated,we an

ndinnitelymanyindependentgeneri sthatareindependentover

H(N )

. If

σ

Z

_(a)

is independentover

σ

Z

_(~b)

we an hoose

a

′

generi independentfrom

~bH(N)

and

thus

a

′

_|

⌣

~b

H

.

3.4. Theories of Morley rank omega with denable Morley rank. Let

T

bea

ω

-stabletheoryofrank

ω

andlet

M |= T

be

|T |

+

-saturated. Assumealsothat

theMorleyrankisdenable,thatis,foreveryformula

ϕ(x, ~y)

withoutparameters andevery

α ∈ {0, 1, . . . , ω}

thereisaformula

ψ

α

(~y)

withoutparameterssu h that for

~a ∈ M

,

M R(ϕ(x, ~a)) ≥ α

ifandonlyif

ψ

α

(~a)

. Tosimplify thenotation,wewill write

M R(ϕ(x, ~a)) ≥ α

insteadof

ψ

α

(~a)

. Wewill provethat

T

ind

isrstorder.

Eliminationof

∃

large

. Considerrst

ϕ(x, ~y)

andlet

~b ∈ M

. Then

ϕ(x,~b)

islarge ifandonlyif

M R(ϕ(x,~b)) ≥ ω

,so

T

eliminatesthequantier

∃

large

.

Extensionproperty. Nowassumethat

(M, H)

isan

H

-stru tureandlet

(N, H) |=

T h(M, H)

be

|T |

+

-saturated. Let

a ∈ N

andlet

~b ∈ N

. If

M R(tp(a/~b)) = 0

there isnothingtoprove. Assume thenthat

M R(tp(a/~b)) = n > 0

.

Let

ϕ(x, ~y) ∈ tp(a,~b)

with

M R(ϕ(x,~b)) = n

and

M d(ϕ(x,~b)) = M d(tp(a/~b))

. Let

(a

′

_,~b

′

_{) |= tp(a,~b)}

belong to

M

. Sin e

(M, H)

is an

H

-stru ture, we may

assume that

a

′

_|

⌣

~b

′

H

and thus for every formula

θ(x, ~y, ~z)

and every tuple

~h ∈

H

, if

M R(θ(x,~b

′

_{,~h)) < M R(ϕ(x,~b}

′

_{)) = n}

then

¬θ(x,~b

′

_{,~h) ∈ tp(a}

′

_/~b

′

_H)

. So

(M, H) |= ∀d

′

_{M R(ϕ(x, ~}

_d

′

_{)) ≥ n}

_=⇒

_{∃cϕ(c, ~}

_d

′

_{) ∧ ∀~h ∈ H(M R(θ(x, ~}

_d

′

_{,~h)) <}

n =⇒ ¬θ(c, ~

d

′

_,~h))

. Sin e

(N, H) |= T H(M, H)

is

|T |

+

-saturated,we annd

a

′

su hthat

M R(ϕ(a

′

_{,~b)) ≥}

n

and whenever

~h ∈ H(N)

and

θ(x,~b,~h)

is a formula with Morley rank smaller

than

n

wehave

¬θ(a

′

_,~b,~h)

. Thisshowsthat

M R(a

′

_{/~bH) = M R(a}

′

_{/~b) = M R(a/~b)}

,

M d(a

′

_{/~b) = M d(a/~b)}

, both

a

and

a

′

aregeneri s ofthe formula

ϕ(x,~b)

and thus

tp(a/~b) = tp(a

′

_/~b)

. Finallyby onstru tion

a

′

_|

⌣

~b

H

. It followsthat

T

ind

is rst order.

3.5.

H

-triples. Re allfrom[3℄thatif

T

0

issupersimple

SU

-rankonetheorywhose pregeometry isnot trivial, then

T = T

ind

0

has

SU

-rankomega. Themodelsof

T

are stru tures of the form

(M, H

1

)

, where

M |= T

0

and

H

1

is a

acl

0

-dense and

acl

0

- odensesubsetof

M

. Wewrite

L

0

forthelanguageasso iatedto

T

0

and

L

for thelanguageasso iated to

T

. Similarly, wewrite

acl

0

forthealgebrai losure in thelanguage

L

0

andfor

A ⊂ M |= T

0

,wewrite

S

0

n

(A)

forthespa eof

L

0

-

n

-types over

A

.

In this subse tion we hange our notation and we let

H

2

be a new predi ate symbolthatwillbeinterpretedbyadenseand odense

T

-generi subsetof

(M, H

1

)

. Thestru tures

(M, H

1

, H

2

)

werealreadystudiedin[3℄. Were allthedenitions and themain result. Themain toolfor studying

T

ind

is to takeintoa ountthe

basetheory

T

0

andusetriples.

Denition3.1. Wesaythat

(M, H

1

(M ), H

2

(M ))

isan

H

-triple asso iatedto

T

0

if:

(1)

M |= T

0

,

H

1

(M )

is an

acl

0

-independent subsetof

M

,

H

2

(M )

is an

acl

0

-independentsubsetof

M

over

H

1

.

(2) (Densitypropertyfor

H

1

)If

A ⊂ M

isnite dimensionaland

q ∈ S

0

1

(A)

is non-algebrai ,there is

a ∈ H

1

(M )

su hthat

a |= q

.

(3) (Densitypropertyfor

H

2

/H

1

)If

A ⊂ M

isnitedimensionaland

q ∈ S

0

1

(A)

is non-algebrai , there is

a ∈ H

2

(M )

su h that

a |= q

and

a 6∈ acl

0

(A ∪

H

1

(M ))

.

(4) (Extension property) If

A ⊂ M

is nite dimensional and

q ∈ S

0

1

(A)

is non-algebrai ,there is

a ∈ M

,

a |= q

and

a 6∈ acl

0

(A ∪ H

1

(M ) ∪ H

2

(M ))

. It is observed in [3℄ that if

(M, H

1

(M ), H

2

(M ))

,

(N, H

1

(N ), H

2

(N ))

are

H

-triples, then

T h(M, H

1

(M ), H

2

(M )) = T h(N, H

1

(N ), H

2

(N ))

and we denote the

ommontheoryby

T

tri

0

.

Thefolowingresultisprovedin[3℄forgeometri theories.

Proposition3.2. Let

T

bean

SU

rank onestrongly non-trivial supersimple the-ory,let

M |= T

andlet

H

1

(M ) ⊂ M

,

H

2

(M ) ⊂ M

be distinguishedsubsets. Then

(M, H

1

(M ), H

2

(M ))

isa

H

2

-stru tureasso iatedto

T

ifandonlyif

(M, H

1

(M ), H

2

(M ))

isan

H

-triple.

Thus, to showthat the lass of

H

2

-stru turesasso iated to

T

is rst order,it su estoprovethatthisisthe asefor

H

-triplesasso iatedto

T

0

. Aspointedout in[3℄wehave:

Proposition3.3. The theory

T

tri

isaxiomatizedby:

(1) T.

(2)

M |= T

0

,

H

1

(M )

isan

acl

0

-independent subset of

M

,

H

2

(M )

is an

acl

0

-independent subsetof

M

over

H

1

.

(3) Forall

L

-formulas

ϕ(x, ~y)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃x(ϕ(x, ~y) ∧ x ∈ H

1

))

.

(4) For all

L

-formulas

ϕ(x, ~y)

,

m ∈ ω

, and all

L

-formulas

ψ(x, z

1

, . . . , z

m

, ~y)

su hthat for some

n ∈ ω ∀~z∀~y∃

≤n

_{xψ(x, ~z, ~y)}

(so

ψ(x, ~y, ~z)

is always alge-brai in

x

)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃x(ϕ(x, ~y) ∧ x ∈ H

2

) ∧

∀w

1

. . . ∀w

m

∈ H

1

¬ψ(x, w

1

, . . . , w

m

, ~y))

(5) For all

L

-formulas

ϕ(x, ~y)

,

m ∈ ω

, and all

L

-formulas

ψ(x, z

1

, . . . , z

m

, ~y)

su hthat for some

n ∈ ω ∀~z∀~y∃

≤n

_{xψ(x, ~z, ~y)}

(so

ψ(x, ~y, ~z)

is always alge-brai in

x

)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃xϕ(x, ~y) ∧

Thuswhen

T

0

isastronglynon-trivialsupersimple

SU

-rankonetheory,

T

ind

₌

T

tri

isrstorder.

3.6.

H

stru tures of lovely pairs of

SU

-rankone theories. Let

T

bea

geo-metri theory,

T

P

itslovelypairsexpansion,andlet

cl(−) = acl(− ∪ P (M ))

bethesmall losureoperatorinalovelypair

(M, P )

. Ourgoalistoexpand

T

P

to atheory

T

ind

P

inthelanguage

L

P H

= L

P

∪ {H}

,byaddinga

cl

-independentdense setto amodelof

T

P

.

ThefollowingdenitionisanalogoustoDenition 3.1.

Denition3.4. Wesaythat an

L

P H

-stru ture

(M, P, H)

isa

P H

-stru tureof

T

if

(1)

P (M )

isanelementarysubstru tureof

M

; (2)

H(M )

is

acl

-independentover

P (M )

; (3) foranynon-algebrai type

q ∈ S

T

1

(A)

overanite-dimensionalset

A ⊂ M

,

q

isrealizedin

(densityof

P

over

H

)

P (M )\ acl(H(M )A)

; (densityof

H

over

P

)

H(M )\ acl(P (M )A)

; (extension)

M \ acl(P (M )H(M )A)

.

Remark3.5. (a)Itsu estorequire

P (M )

tobedenseinthe usualsense,i.e.

q

havingarealization in

P (M )

.

(b) We an geta

P H

-stru ture froman

H

-triple

(M, H

1

, H

2

)

(see previous ex-ample),by letting

P (M ) = acl(H

1

)

.

( )Ausualelementary hainargumentshowsthatany

L

P H

stru ture

(M, P, H)

satisfying(1,2)embedsina

P H

-stru ture

(N, P, H)

sothat

H(N ) |

⌣

H(M)

M P (N )

and

P (N ) |

⌣

P (M)

M H(N )

. In parti ular,

P H

-stru turesexist.

(d)Redu ts

(M, P )

and

(M, H)

of

(M, P, H)

arelovely pairs and

H

-stru tures, respe tively.

Whileinlinearexamplesthe

SU

-rankof

T

P

is twoinsteadof

ω

,thema hinery forthispaperstillgoesthroughweour urrentassumptionsfor

cl

.

Denition3.6. Wesaythat

(M, P, H)

isan

cl

-stru tureif (1)

(M, P )

isalovelypairand

H

isan

cl

-independentset

(2) (Density/ oheir property for

cl

) If

A ⊂ M

is nite dimensional and

q ∈

S

P

1

(A)

islarge,there is

a ∈ H(M )

su hthat

a |= q

.

(3) (Extensionproperty)If

A ⊂ M

isnitedimensionaland

q ∈ S

P

1

(A)

islarge, thereis

a ∈ M

,

a |= q

and

a 6∈ cl(A ∪ H(M ))

.

Proposition 3.7.

(M, P, H)

isan

cl

-stru tureif and only if

(M, P, H)

is a

P H

-stru ture.

Proof. Assumerstthat

(M, P, H)

isa

cl

-stru ture. Thenthepair

(M, P )

islovely and thus

(M, P, H)

satises the density axiom for

P

. Now let

A ⊂ M

be nite dimensionalandlet

q ∈ S

1

(A)

benon-algebrai . Let

q ∈ S

ˆ

P

1

(A)

beanextensionof

q

that ontainsnosmallformulawithparametersin

A

. ThenbytheDensity/ oheir property for

cl

it followsthat there is

a ∈ H(M )

su h that

a |= ˆ

q

. In parti ular,

sin e the same

q

ˆ

is not small, there is

c ∈ M

,

c |= ˆ

q

and

c 6∈ cl(A ∪ H(M )) =

acl(A ∪ P (M ) ∪ H(M ))

. Thustheextension propertyholdsaswell.

Now assumethat

(M, P, H)

is an

P H

-stru ture. Then

H

is an

cl

-independent set,and bythe densitypropertyfor

P

andthe extensionpropertyit follows that

(M, P )

isa lovelypair. Now let

A ⊂ M

benite dimensional andlet

q ∈ S

ˆ

P

1

(A)

benon-small. Wemay enlarge

A

and assume that

A

is

P

-independent. Let

q

be

the restri tion of

q

ˆ

to the language

L

. Note that

q

ˆ

is the unique extension of

q

to anon-small type. Bythedensity for

H

over

P

, there is

a ∈ H(M )

su h that

a |= q

,

a 6∈ cl(A)

and thus

a |= ˆ

q

. Finallytheextension propertyfollowsfromthe

extensionpropertyfor

P H

-stru tures.



Wewillnowshowthat the lassof

P H

-stru turesis "rstorder",that is, that there isasetof axiomswhose

|T |

+

-saturatedmodelsarethe

P H

-stru tures. The axiomatizationworksasin

H

-triples.

Proposition3.8. Assume

T

eliminates

∃

∞

. Then the theory

T

P H

isaxiomatized by:

(1)

T

(2) axiomssaying that

P

distinguishes anelementary substru ture. (3) Forall

L

-formulas

ϕ(x, ~y)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃x(ϕ(x, ~y) ∧ x ∈ P ))

.

(4) For all

L

-formulas

ϕ(x, ~y)

,

m ∈ ω

, and all

L

-formulas

ψ(x, z

1

, . . . , z

m

, ~y)

su hthat for some

n ∈ ω ∀~z∀~y∃

≤n

_{xψ(x, ~z, ~y)}

(so

ψ(x, ~y, ~z)

is always alge-brai in

x

)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃x(ϕ(x, ~y) ∧ x ∈ H) ∧

∀w

1

. . . ∀w

m

∈ P ¬ψ(x, w

1

, . . . , w

m

, ~y))

(5) For all

L

-formulas

ϕ(x, ~y)

,

m ∈ ω

, and all

L

-formulas

ψ(x, z

1

, . . . , z

m

, ~y)

su hthat for some

n ∈ ω ∀~z∀~y∃

≤n

_{xψ(x, ~z, ~y)}

(so

ψ(x, ~y, ~z)

is always alge-brai in

x

)

∀~y(ϕ(x, ~y)

nonalgebrai

=⇒ ∃x(ϕ(x, ~y) ∧ x 6∈ P ∧ x 6∈ H) ∧

∀w

1

. . . ∀w

m

∈ P ∪ H¬ψ(x, w

1

, . . . , w

m

, ~y))

Furthermore,if

(M, P, H) |= T

P H

is

|T |

+

-saturated,then

(M, P, H)

isa

P H

-stru ture.

Nowwelistafamilyofstru turesof

SU

-rank

ω

wherewedoknowifthe orre-spondingtheoryof

H

-stru turesisaxiomatizable. Inboth asesitisopenwhether ornottheextensionpropertyisrstorder.

3.7. ACFA. Let

T = ACF A

,(a ompletion)of thetheory ofalgebrai ally losed elds withageneri automorphism. This theory issimpleof

SU

rank

ω

and itis unstable.

Let

p(x)

be thegeneri typeof thetheory, namelythe typeofa transformally independentelement. This typeis omplete,stationary and denableover

∅

. Let

ϕ(x, ~y)

be aformula and let

ψ(~y)

be its

p

-denition. Then for

(K, σ) |= ACF A

,

~a ∈ K

, the formula

ϕ(x, ~a)

is large i

ψ(~a)

. Thus this theory eliminates the quantier

∃

large

.

QuestionDoestheextensionpropertyhold forACFA?Does

T

0

satisfywnf p?

3.8. Hrushovskiamalgamationwithout ollapsing. Inthissubse tionwe

fol-lowthepresentationofHrushovskiamalgamationsfrom[16℄,alltheresultswe

let

C

bethe lassof

L

-stru tureswhere

R

issymmetri andnotreexive. For

A ∈ C

anite stru turewelet

δ(A) = |A| − |R(A)|

and welet

C

0

f in

bethesub lass of

C

onsistingofallnite

L

-stru tures

M

wherefor

A ⊂ M

wehave

δ(A) ≥ 0

. Finally

M

0

standsfortheFraïssélimitofthe lass

C

0

f in

. Let

T

0

bethetheoryof

M

0

,then

M R(T

0

) = ω

and

M d(T

0

) = 1

.

Now let

M |= T

0

and for

A ⊂ M

nite we dene

d(A) = inf{δ(B) : A ⊂ B}

. Then

d

is thedimensionfun tion ofapregeometryand thatforanelement

a

and aset

B

,

d(a/B) = 1

ifandonlyif

M R(a/B) = ω

ifandonlyif

U (a/B) = ω

. Thus the pregeometry studied in [16℄ orrespondsto thepregeometry asso iated to

cl

. Sin e thetheory

T

0

hasauniquegeneri type, by denabilityof typesthetheory

T

0

eliminatesthequantier

∃

large

.

QuestionDoestheextensionpropertyhold for

T

0

? Does

T

0

satisfynf p?

4. Definable setsin

H

-stru tures

Fix

T

a

SU

rank

ω

theoryandlet

(M, H(M )) |= T

ind

. Ournextgoalistoobtain

ades riptionofdenablesubsetsof

M

and

H(M )

in thelanguage

L

H

.

Notation4.1. Let

(M, H(M ))

bean

H

-stru ture. Let

~a

beatuplein

M

. We de-noteby

etp

H

(~a)

the olle tionofformulasoftheform

∃x

1

∈ H . . . ∃x

m

∈ Hϕ(~x, ~y)

, where

ϕ(~x, ~y)

isan

L

-formula su hthatthereexists

~h ∈ H

with

M |= ϕ(~h, ~a)

. Lemma 4.2. Let

(M, H(M ))

,

(N, H(N ))

be

H

-stru tures. Let

~a

,

~b

be tuples of the samearityfrom

M

,

N

respe tively. Then thefollowing areequivalent:

(1)

etp

H

(~a) = etp

H

(~b)

.

(2)

~a

,

~b

havethe same

L

H

-type.

Proof. Clearly(2)implies(1). Assume (1),then

tp(~a) = tp(~b)

. Claim

dim

cl

(~b/H) = dim

cl

(~a/H)

.

Let

~h = (h

1

, . . . , h

l

) ∈ H(M )

besu h

k := dim

cl

(~a/~h) = dim

cl

(~a/H(M ))

. We mayassumethat

~a

1

_{= (a}

1

, . . . , a

k

)

areindependentover

H

and

~a

2

_{= (a}

k+1

, . . . , a

n

) ∈

cl(a

1

, . . . , a

k

, h

1

, . . . , h

l

)

. Choose

ψ(~x, ~y, ~z)

su hthatforany

~b ∈ M

,

~c ∈ M ψ(~b, ~c, ~z)

is alwayssmall in

~z

and

M |= ψ(~h, ~a

1

_{, ~a}

2

₎

. Sin e

etp

H

(~a) = etp

H

(~b)

weget that

dim

cl

(~b/H) ≤ k

. Asimilarargumentshowsthat

dim

cl

(~a/H(M )) ≤ dim

cl

(~b/H(N ))

.

Claim

tp

H

(~b) = tp

H

(~a)

.

As before, let

~h = (h

1

, . . . , h

l

) ∈ H(M )

be su h that

k := dim

cl

(~a/~h) =

dim

cl

(~a/H(M ))

. Then

~a~h

is

H

-independent. Sin e

N

issaturatedasan

L

-stru ture there are

~h

′

_{= (h}

′

1

, . . . , h

′

l

) ∈ H

su h that

tp(~a,~h) = tp(~b,~h

′

₎

. Bythe laim above

~b~h

′

is

H

-independent,sotheresultfollowsfrom Lemma2.7.



Now we are interestedin the

L

H

-denable subsets of

H(M )

. This material is verysimilartotheresultspresentedin [5℄.

Lemma 4.3. Let

(M

0

, H(M

0

))  (M

1

, H(M

1

))

and assumethat

(M

1

, H(M

1

))

is

|M

0

|

-saturated. The

M

0

(seen asasubsetof

M

1

) isa

H

-independentset.

Proof. Assume not. Then there are

a

1

, . . . , a

n

∈ M

0

\ H(M

0

)

su h that

a

n

∈

holds. Sin e

(M

0

, H(M

0

))  (M

1

, H(M

1

))

thereis

~b

′

_{∈ H(M}

0

)

~

z

su hthat

ϕ(a

n

, a

1

, . . . , a

n−1

,~b

′

) ∧ ¬∃

large

xϕ(x, a

1

, . . . , a

n−1

,~b

′

)

holds,so

a

n

∈ acl(a

1

, . . . , a

n−1

, H(M

0

))

,a ontradi tion.



Proposition4.4. Let

(M, H(M ))

bean

H

-stru tureandlet

Y ⊂ H(M )

n

be

L

H

-denable. Then thereis

X ⊂ M

n

_L

-denablesu hthat

Y = X ∩ H(M )

n

.

Proof. Let

(M

1

, H(M

1

))  (M, H(M ))

be

κ

-saturatedwhere

κ > |M | + |L|

andlet

~a,~b ∈ H(M

1

)

n

besu hthat

tp(~a/M ) = tp(~b/M )

. Wewill provethat

tp

H

(~a/M ) =

tp

H

(~b/M )

and theresultwillfollowby ompa tness. Sin e

~a,~b ∈ H(M

1

)

n

,weget

by Lemma 4.3 that

M~a

,

M~b

are

H

-independent sets and thus by Lemma 2.7 we

get

tp

H

(~a/M ) = tp

H

(~b/M )

.



Denition 4.5. Let

(M, H) |= T

ind

be saturated. We say that an

L

H

-formula

ψ(x, ~c)

denes aH-large subset of

M

ifthere is

b |= ψ(x, ~c)

su h that

b 6∈ cl(H~c)

. Thisisequivalentasrequiringthatthereareinnitelymanyrealizationsof

ψ(x, ~c)

thatarenotsmallover

H(M )~c

.

Denition 4.6. Let

(M, H) |= T

ind

be

κ

-saturated and let

A ⊂ M

be smaller

than

κ

. Let

~b ∈ M

be a tuple. We say that

~b

is in the H-small losure of

A

if

~b ∈ cl(AH(M))

. Let

X ⊂ M

n

be

A

-denable. We say that

X

is H-small if

X ⊂ cl(A ∪ H(M ))

.

Proposition 4.7. Let

(M, H(M ))

be an

H

-stru ture. Let

~a = (a

1

, . . . , a

n

) ∈ M

. Then thereisauniquesmallesttuple

~h ∈ H(M)

su hthat

~a |

⌣

~

_h

H

.

Proof. Sin e

T

is supersimple, there is a nite tuple

~h ∈ H

su h that

~a |

⌣

~

_h

H

. Choosesu hatuple sothat

|~h|

(the lengthofthe tuple)isminimal. Wewill now showsu h atuple

~h

isunique(up topermutation).

We an write

~a = (~a

1

, ~a

2

)

so that

~a

1

isindependent tuple of generi swhi h is independent from

H(M )

and

~a

2

∈ cl(~a

1

)

. If

~a

2

= ∅

, then

~h = ∅

and the result follows. Sowemayassumethat

~a

2

6= ∅

.

Then

~a

2

∈ cl(~a

1

,~h)

. Let

~h

′

beanothersu htuple. Let

~h

1

bethelistof ommon elementsin both

~h

and

~h

′

,sowe anwrite

~h = (~h

1

,~h

2

)

and

~h

′

_{= (~h}

1

,~h

′

2

)

. Claim

~h

2

= ~h

′

2

= ∅

.

Assume otherwise. Then thereis

c ∈ ~a

2

su h that

c ∈ cl(~a

1

,~h

1

,~h

2

) \ cl(~a

1

,~h

1

)

. Sin e

~a |

⌣

~

_h

′

H

, we must havethat

c ∈ cl(~a

1

,~h

1

,~h

′

2

) \ cl(~a

1

,~h

1

)

. By theex hange property

dim

cl

(~h

′

2

/~a

1

~h

1

~h

2

) < dim

cl

(~h

′

2

/~a

1

~h

1

)

. Sin e

~a

1

is a tuple of generi el-ementsthat are independent over

H

we get that

dim

cl

(~h

′

2

/~h

1

~h

2

) < dim

cl

(~h

′

2

/~h

1

)

andsin e

H

isindependent,

~h

2

hasa ommonelementwith

~h

′

2

,a ontradi tion.



Remark 4.8. Let

(M, H(M ))

be an

H

-stru ture. Let

~a = (a

1

, . . . , a

n

) ∈ M

and let

C ⊂ M

be su hthat

C

is

H

-independent. As before, thereisaunique smallest tuple

~h ∈ H(M)

su hthat

~a |

⌣

~

hC

H

.

Notation4.9. Let

(M, H(M ))

be an

H

-stru ture. Let

~a = (a

1

, . . . , a

n

) ∈ M

. Let

~h ∈ H(M)

bethe smallesttuple su hthat

~a |

⌣

~

h

H

. We all

~h

the

H

-basisof

~a

and wedenoteitas

HB(~a)

. Given

C ⊂ M

su hthat

C

is

H

-independent,let

~h ∈ H(M)

the smallesttuple su hthat

~a |

denote itas

HB(~a/C)

. Note that

H

-basis is unique upto permutation, therefore we will view the

H

-basis

~h = (h

1

, . . . , h

k

)

either asa nite set

{h

1

, . . . , h

k

}

or as theimaginary representingthis niteset. Ifweviewitasatuple,wewillexpli itly

sayso.

Proposition4.10. Let

(M, H(M ))

be an

H

-stru ture. Let

a

1

, . . . , a

n

, a

n+1

∈ M

andlet

C ⊂ M

besu hthat

C

is

H

-independent. Then

HB(a

1

, . . . , a

n

, a

n+1

/C) =

HB(a

1

, . . . , a

n

/C) ∪ HB(a

n+1

/Ca

1

, . . . , a

n

HB(a

1

, . . . , a

n

/C))

, where all

H

-basis areseenassets.

Proof. Let

~h

1

= HB(a

1

, . . . , a

n

/C)

. Firstnotethatsin e

a

1

, . . . , a

n

|

⌣

C~

h

1

H

,then

theset

a

1

, . . . , a

n

C~h

1

is

H

-independentandwe andene

~h

2

= HB(a

n+1

/Ca

1

, . . . , a

n

~h

1

)

. Finally,let

~h = HB(a

1

, . . . , a

n

, a

n+1

/C)

.

Claim

~h ⊂ ~h

1

~h

2

. Wehave

a

1

, . . . , a

n

|

⌣

C~

h

1

H

and

a

n+1

|

⌣

C~

h

1

~

h

2

a

1

,...,a

n

H

,sobytransitivity,

a

1

, . . . , a

n

a

n+1

_⌣

|

_C~

_h

1

~

h

2

H

and by the minimality of an

H

-basis, we have

~h ⊂

~h

1

~h

2

.

Claim

~h ⊃ ~h

1

~h

2

.

By denition,

a

1

, . . . , a

n

a

n+1

|

⌣

C~

h

H

, so

a

1

, . . . , a

n

|

⌣

C~

h

H

and byminimality wehave

~h

1

⊂ ~h

. Wealsogetbytransitivitythat

a

n+1

|

⌣

Ca

1

,...,a

n

~

h

1

~

h

H

andbythe

minimalityof

H

-basisweget

~h

2

⊂ ~h

asdesired.



Proposition 4.11. Let

(M, H(M ))

be an

H

-stru ture. Let

a

1

, . . . , a

n

∈ M

and let

C ⊂ D ⊂ M

be su h that

C

,

D

are

H

-independent. Assume that there is

h ∈ HB(a

1

, . . . , a

n

/C) \ HB(a

1

, . . . , a

n

/D)

. Then

h ∈ D

.

Proof. Write

h

D

= HB(a

1

, . . . , a

n

/D)

andseeitasaset.Then

a

1

, . . . , a

n

D |

⌣

h

D

H(D)

H

and

a

1

, . . . , a

n

|

⌣

h

D

CH(D)

H

. By minimality of

HB(a

1

, . . . , a

n

/C)

we get that

HB(a

1

, . . . , a

n

/C) ⊂ h

D

H(D)

and thus if

h ∈ HB(a

1

, . . . , a

n

/C) \ h

D

, we must

have

h ∈ H(D)

.



We will now apply the

H

-basis to hara terize denable sets in terms of

L

-denablesets.

Proposition 4.12. Let

(M, H(M ))

be an

H

-stru ture and let

Y ⊂ M

be

L

H

-denable. Then there is

X ⊂ M L

-denablesu hthat

Y △X

is

H

-small, where

△

standsfor aboolean onne tivefor the symmetri dieren e.

Proof. If

Y

is

H

-smallor

H

- osmall,theresultis lear,sowemayassumethatboth

Y

and

M \ Y

are

H

-large. Assumethat

Y

isdenableover

~a

andthat

~a = ~aHB(~a)

. Let

b ∈ Y

besu h that

b 6∈ cl(~aH)

and let

c ∈ M \ Y

be su h that

c 6∈ cl(~aH)

. Then

b~a

,

c~a

are

H

-independentandthusthere is

X

bc

an

L

-denable setsu h that

b ∈ X

bc

and

c 6∈ X

bc

. By ompa tness,wemaygetasingle

L

-denableset

X

su h thatfor

b

′

_{∈ Y}

and

c

′

_{∈ M \ X}

notinthe

H

-small losureof

~a

,wehave

b

′

_{∈ X}

and

c

′

_{∈ M \ Z}

. Thisshowsthat

Y △X

is

H

-small.



Ournextgoalisto hara terizethealgebrai losure in

H

-stru tures. Thekey toolisthefollowingresult:

Lemma4.13. Let

(M, H(M ))

bean

H

-stru ture,andlet

A ⊂ M

be

acl

- losedand

Proof. Suppose

a ∈ M

,

a 6∈ A

. If

a 6∈ cl(AH)

,then

A ∪ {a}

is

H

-independent,and usingtheextensionproperty,we annd

a

i

,

i ∈ ω

,

acl

-independentover

A∪H(M )

, realizing

tp(a/A)

. ByLemma2.7,ea h

a

i

realizes

tp

H

(a/A)

,andthus

a 6∈ acl

H

(A)

. If

a ∈ cl(AH)

, takea minimaltuple

~h ∈ H(M)

su h that

a ∈ acl(A~h)

. Using onjugatesof

~h

over

A

itiseasytosee that

tp(a/A)

has

∞

-manyrealizations.



Corollary 4.14. Let

(M, H(M ))

be an

H

-stru ture, and let

A ⊂ M

. Then

acl

H

(A) = acl(A, HB(A))

.

Proof. ByProposition4.7,itis learthat

HB(A) ∈ acl(A)

,so

acl

H

(A) ⊃ acl(A, HB(A))

.

On the other hand,

A ∪ HB(A)

is

H

- losed, so by the previous Proposition,

acl(A ∪ HB(A)) = acl

H

(A ∪ HB(A))

andthus

acl

H

(A) ⊂ acl(A, HB(A))



5. Supersimpli ity

In this se tion we prove that

T

ind

is supersimple and hara terize forking in

T

ind

.

Theorem5.1. The theory

T

ind

issupersimple.

Proof. Wewillprovethatnon-dividinghaslo al hara ter.

Let

(M, H(M )) |= T

ind

be saturated. Let

C ⊂ D ⊂ M

and assume that

C = acl

H

(C)

and

D = acl

H

(D)

. Note that both

C

and

D

are

H

-independent. Let

~a ∈ M

. We will nd a olle tionof onditions forthe typeof

~a

over

C

that guaranteethat

tp

H

(~a/D)

doesnotdivide over

C

.

Wemaywrite

~a = (~a

1

, ~a

2

) ∈ M

sothat

~a

1

isan independenttuple ofgeneri s over

DH

,

~a

2

isatuplesu h that

~a

2

∈ cl(~a

1

DH)

.

Assumethat thefollowing onditionsholdfor

C

: (1)

HB(~a/D) = HB(~a/C)

.

(2)

SU (~a

2

/C~a

1

H) = SU (~a

2

/D~a

1

H)

Claim

tp

H

(~a/D)

doesnotdivideover

C

.

Let

p(~x, D) = tp(~a

1

, D)

. Let

{D

i

: i ∈ ω}

be an

L

H

-indis ernible sequen e over

C

. Sin e

~a

1

is an independent tuple of generi sover

D

,

tp(~a

1

/D)

does not divide over

C

and

∪

i∈ω

p(~x, D

i

)

is onsistent. We annd

~a

′

1

|= ∪

i∈ω

p(~x, D

i

)

su h that

{~a

′

1

D

i

: i ∈ ω}

is indis ernible and

~a

′

1

is an independent tuple of generi s over

∪

i∈ω

D

i

. By the generalized extension property, we may assume that

~a

′

1

is independent over

∪

i∈ω

D

i

H

. Note that

~a

1

D

is

H

-independent,

~a

′

1

D

i

is also

H

-independent for any

i ∈ ω

. So by Lemma 2.7

tp

H

(~a

1

D) = tp

H

(~a

′

1

D

i

)

for any

i ∈ ω

.

Now let

~h = HB(~a/C)

(viewed as a tuple) and let

q(~y, ~a

1

, D) = tp(~h, ~a

1

, D)

. Notethat

~h

isanindependenttupleofgeneri sover

~a

1

D

(aswellasanindependent tuple over

~a

1

C

). Sin e

{D

i

~a

′

1

: i ∈ ω}

is an

L

-indis ernible sequen e, there is

~h

′

_{|= ∪}

i∈ω

q(~y, ~a

′

1

, D

i

)

. Wemayassumethat

~h

′

is independentfrom

∪

i∈ω

D

i

~a

′

1

and thusitisatupleofgeneri sover

∪

i∈ω

D

i

~a

′

1

. Furthermorewemayassumethatthe sequen e

{D

i

~a

′

1

: i ∈ ω}

isindis ernibleover

~h

′

.

Bythegeneralized oheir/densityproperty, wemayassumethat

~h

′

_{∈ H}

. Note

that sin e ea h

~a

′

1

D

i

is

H

-independent, then

~h

′

_~a

′

1

D

i

is also

H

-independent. On theother hand,

tp(~h, ~a

1

, D) = tp(~h

′

_{, ~a}

′

1

, D

i

)

for ea h

i

, sobyLemma 2.7 we have

tp

H

(~h, ~a

1

, D) = tp

H

(~h

′

, ~a

′

1

, D

i

)

. Thisshowsthat

tp(~a

1

,~h/D)

doesnotdivide over

Now onsider

t(~z, ~a

1

,~h, D) = tp(~a

2

, ~a

1

,~h, D)

. Bytheassumption

SU (~a

2

/C~a

1

~h) =

SU (~a

2

/D~a

1

~h)

, so

tp(~a

2

/~a

1

~hD)

does not divide over

~a

1

~hC

. Sin e

tp(~a

1

~hD) =

tp(~a

′

1

~h

′

D

0

)

,then

t(~z, ~a

′

1

,~h

′

, D

0

)

doesnotdivide over

~a

′

1

~h

′

C

. Sin e

{D

i

: i ∈ ω}

is an

L

-indis ernible sequen e over

C~a

′

1

~h

′

, there is

~a

′

2

|=

∪

i∈ω

t(~z, ~a

′

1

,~h

′

, D

i

)

. Wemayassumeasbeforethat

~a

′

2

_⌣

|

_~

_a

′

1

~

h

′

C

∪

i

D

i

. Bytheextension property,wemayassumethat

~a

′

2

_⌣

|

_~

_a

′

1

~

h

′

∪

i

D

i

H

. Using

transi-tivitywealsohave

~a

′

2

_⌣

|

_~

_a

′

1

~

h

′

C

H ∪

i

D

i

anditfollowsthat

~a

′

2

_⌣

|

_~

_a

′

1

~

h

′

D

i

H

,sowehave

~a

′

1

~a

′

2

_⌣

|

~

_h

′

_D

_i

H

,andthus,also

~a

′

1

~a

′

2

D

i

_⌣

|

~

_h

′

_H(D

_i

₎

H

forea hindex

i

. Sin eboth

~a

1

~a

2

~hD

,

~a

′

1

~a

′

2

~h

′

D

i

are

H

-independentand

tp(~a

′

1

~a

′

2

~h

′

D

i

) = tp(~a

1

~a

2

~hD)

byLemma2.7

tp

H

(~a

′

1

~a

′

2

~h

′

D

i

) = tp

H

(~a

1

~a

2

~hD)

. Thisshowsthat

tp(~a/D)

doesnotdivideover

C

.

Sin e

T

issupersimple,forany

D

and

~a

we analways hooseanitesubset

C

0

of

D

su hthat

C = acl

H

(C

0

)

satisesthe onditions(1)and(2)above. Thisshows that

T

ind

issupersimple.



Proposition 5.2. Let

(M, H) |= T

ind

be saturated, let

C ⊂ D ⊂ M

besu h that

C = acl

H

(C)

,

D = acl

H

(D)

andlet

a ∈ M

. Then

tp(a/D)

forksover

C

i

a ∈ D\C

or

a ∈ cl(HD) \ cl(HC)

or

HB(a/C) ) HB(a/D)

or,

HB(a/C) = HB(a/D)

and

SU (a/CH) 6= SU (a/DH)

.

Proof. In the proof of Theorem 5.1 we showed that if

a ∈ C

or if,

HB(a/C) =

HB(a/D)

and

SU (a/CH) = SU (a/DH)

then

tp(a/D)

does notforkover

C

. So itremainstoshowtheotherdire tion,whi hwedo aseby ase.

Case 1: Assumethat

a ∈ D \ C

,then

a

be omesalgebrai over

D

and

tp(a/D)

forksover

C

.

Case 2: Assume that

a ∈ cl(DH) \ cl(CH)

. Then

SU (tp(a/DH)) < ω

and

SU (tp(a/CH)) = ω

. Wewillprovethat

tp

H

(a/D)

dividesover

C

.

Let

d ∈ D

~

andlet

~c ∈ C

besu hthat

a ∈ cl(~c~

dH)

, so

SU (tp(a/ ~

d~cH)) < ω

. By additivity of Las ar rank, we an hoose

d

~

to be independent generi sover

HC

. Let

~h ∈ H

besu hthat

a ∈ cl(~c ~

d~h)

. Let

p(x, ~y) = tp

H

(a, ~

d/C)

.

Let

{ ~

d

i

: i ∈ ω}

be an

L

-indis ernible sequen e in

tp(~

d/C)

over

C

su h that

{ ~

d

i

: i ∈ ω}

is independent over

C

. By the generalized extension property, we mayassumethat

{ ~

d

i

: i ∈ ω}

is independent over

HC

. Notethat by Lemma 2.7

{ ~

d

i

: i ∈ ω}

isan

L

H

-indis erniblesequen eofgeneri sover

C

. Assume,inorderto geta ontradi tion,thatthereis

a

′

_{|= ∪}

i∈ω

p(x, ~

d

i

)

. Thenthereare

{~h

i

: i ∈ ω}

su h that

a

′

_{∈ cl(~}

_d

i

, ~c,~h

i

)

forevery

i

,that is,

SU (a

′

_{/ ~}

_d

i

, ~c,~h

i

) < ω

. But

a

′

_{6∈ cl(CH)}

, so

~

d

0

_⌣

6 |

_CH

d

~

1

,a ontradi tion.

Case 3: Assumethat

HB(a/D) 6= HB(a/C)

and

a ∈ cl(CH)

. Then

HB(a/D)

is aproper subsetof

HB(a/C)

. Write

~h

C

= HB(a/C)

,

~h

D

= HB(a/D)

and let

~h

E

∈ H

besu hthat

~h

C

= ~h

D

~h

E

. Notethat

~h

E

6= ∅

andthat

~h

E

isanindependent tupleover

C

.

Let

p(x, ~y) = tp

H

(a,~h

E

/C)

. Let

{~h

i

E

: i ∈ ω}

bean

L

-indis erniblesequen e in

tp(~h

E

/C)

su hthat

{~h

i

E

: i ∈ ω}

isindependentover

C

. Then bythegeneralized density property, we may assume that the sequen e

{~h

i

E

: i ∈ ω}

belongs to

H

.

Note that byLemma 2.7, the sequen e

{~h

i

E

: i ∈ ω}

is

L

H

-indis ernible over

C

. Wewillshowthat

∪

i∈ω

p(x,~h

i