PERSPECT

IVES

ON

EXPANSIO

NS

John

T.Bald

win

Depart

ment

ofM

ath

em

atics,

Sta

tist

ics

and

Com

pute

rScie

nce

Univ

ers

ity

ofIllinois

at

Chic

ago

ww

w.m

ath

.uic

.edu/

jbald

win

/NO

rsli05.p

df

July

18,2005

SET

TIN

G

Mis

ast

ructu

refo

ra

language

L,

Ais

a

subse

tof

M.

L∗=

L(P

)is

the

expansion

of

Lby

one

unary

pre

dic

ate

and

(M,A

)is

the

L∗ -

stru

ctu

rew

here

Pis

inte

rpre

ted

by

A.

When

does

(M,A

)have

the

sam

est

ability

cla

ssas

M?

PERSPECT

IVES

I.Analy

sis

ofArb

itra

ryexpansions

II.Const

ructing

Expansions:

What

hath

Hru

shovsk

iw

rought?

TW

OFACT

ORS

What

stru

ctu

redoes

M‘induce’on

A?

How

does

A‘sit

in’

M?

IND

UCE

The

basic

form

ula

sin

duced

on

Acan

be:

L∗ :

the

traceson

Aofpara

mete

rfree

L-form

ula

s

(induced

stru

ctu

re);

L#

:th

etr

aces

on

Aof

para

mete

rfree

L(P

)-

form

ula

s(L

#-induced

stru

ctu

re,

A#

);

EXAM

PLES

Form

ast

ructu

reM

with

atw

oso

rted

uni-

vers

e:

1.

The

com

ple

xnum

bers

.

2.

Afibering

over

the

com

ple

xnum

bers

.

Let

Nexte

nd

Mby

putt

ing

one

new

poin

t

inth

efiber

over

aif

and

only

ifa

isa

real

num

ber.

Now

Mand

Nare

isom

orp

hic

and

are

ω-

stable

nfc

p.

But

the

stru

ctu

re(N

,M)

isun-

stable

.

The∗-

induced

stru

ctu

reon

Mis

stable

since

infa

ct

no

new

sets

are

definable

.

Inth

e#

-induced

stru

ctu

re

(∃x)E

(x,y

)∧

x6∈

P

defines

the

reals

soth

e#

-induced

stru

ctu

re

isunst

able

.

SIT

S

Definitio

n1

Mis

ω-s

atu

rate

dover

A,(A

is

small

inM

),if

for

every

a∈

M−

A,

every

L-t

ype

p∈

S(a

A)

isre

alized

inM

.

Definitio

n2

1.

The

set

Ais

weakly

benig

n

inM

iffo

revery

α,β∈

Mif:

stp(α

/A)=

stp(β

/A)

implies

tp∗(

α/A

)=

tp∗(

β/A

).

2.

(M,A

)is

uniform

lyweakly

benig

nif

every

(N,B

)w

hic

his

L(P

)-ele

menta

rily

equiv

ale

nt

to(M

,A)

isweakly

benig

n.

SUFFIC

IENT

CO

ND

ITIO

NS

Bald

win

-Benedik

t

Theore

m3

IfM

isst

able

and

Iis

ase

tof

indiscern

ible

sso

that

(M,I

)is

small,

then

(M,A

)is

stable

.

Expla

inin

gBald

win

-Benedik

t,and

linkin

gw

ith

Poizat:

Casa

novas-

Zie

gle

rpro

ve:

Theore

m4

If(M

,A)

has

the

nfc

p(o

ver

A),

issm

all,

and

the∗-

induced

theory

on

Ais

stable

then

(M,A

)is

stable

.

Exte

ndin

gCasa

novas-

Zie

gle

r,

Baizhanov-B

ald

win

pro

ve:

Theore

m5

If(M

,A)is

uniform

lyweakly

be-

nig

nand

the

#-induced

theory

on

Ais

stable

then

(M,A

)is

stable

.

Theore

m6

impliesT

heore

m5

by

Polk

ow

ska’s

rem

ark

:

Rem

ark

6If

(M,A

)is

small

and

Mhas

nfc

p

then

the∗-

induced

and

#-induced

theories

on

Aare

the

sam

e.

Bousc

are

nsh

owed

(in

our

language):

Theore

m7

IfN

issu

pers

table

and

M≺

N,

then

(N,M

)is

weakly

benig

n.

Baizhanov,Bald

win

,Shela

hsh

owed:

Theore

m8

IfM

issu

pers

table

(M,A

)is

uni-

form

lyweakly

benig

nfo

rany

A.

Question

9If

Mis

stable

must

(M,A

)be

uniform

lyweakly

benig

nfo

rany

A?

Question

10

Isth

ere

ast

able

stru

ctu

reM

and

an

infinite

set

of

indiscern

ible

sI

such

I

isnot

indiscern

ible

in(M

,I)?

Polo

wsk

agave

conditio

ns

‘bounded

PAC’on

(M,A

)so

that

Th∗(

A)

issim

ple

and

(M,A

)is

sim

ple

.

Question

11

If(M

,A)

issm

all

and

Mhas

nfc

pand

Th∗(

A)

issim

ple

must

(M,A

)be

sim

ple

?

THE

HRUSHO

VSK

IM

ACHIN

E

INPUT

:K

0,δ

,µ

DESIR

ED

OUT

PUT

:Nic

eth

eory

ACT

UAL

OUT

PUT

:

Nic

est

ructu

re:

the

generic

An

infinitarily

defined

cla

ss

What

isa

‘firs

tord

er

stru

ctu

re’?

Poss

ible

meanin

g:

The

cla

ssof

existe

ntially

clo

sed

models

(in

appro

priate

language)

isfirs

tord

er.

PRO

BLEM

Inth

eab

initio

or

free

am

alg

am

ation

case

s

the

generic

isω-s

atu

rate

dand

its

theory

is

the

answ

er.

Even

so,

the

cla

ssof

existe

ntially

clo

sed

models

may

not

be

ele

menta

ry(in

giv

en

language).

Inm

ore

com

plicate

dsitu

ations,

the

generic

may

not

be

satu

rate

d.

FRAM

EW

ORK

S

Robin

son

Theory

CAT

S

AEC/Q

uasim

inim

alexcellence

MAK

ING

EC

MO

DELS

ELEM

ENTARY

I.Axio

matize

more

care

fully.

II.Adju

stth

ela

nguage.

QUANT

IFIE

RCO

MPLEXIT

Y

Definitio

n12

Tis

modelcom

ple

teif

every

form

ula

φ(x

)is

equiv

ale

nt

toan

existe

ntial

form

ula

.

Definitio

n13

Tis

nearly

modelcom

ple

teif

every

form

ula

φ(x

)is

equiv

ale

ntto

aBoole

an

com

bin

ation

ofexiste

ntialfo

rmula

s.

LIN

DST

RO

MS’S

LIT

TLE

THEO

REM

Theore

m14

IfT

isπ2-a

xio

matizable

and

cat-

egoricalin

som

ein

finite

card

inality

then

Tis

modelcom

ple

te.

Hru

shovsk

i-like

Const

ructions

theory

axio

ms

com

ple

xity

smse

t/fu

sions

π2

m.c

.

rank

ωbic

olfield

π2

n.m

.c.

rank

2bic

olfield

π2

m.c

.

Spencer-

Shela

hπ2

n.m

.c.

The

origin

al

axio

matizations

of

the

smse

t

and

offu

sions

(Hru

shovsk

i)and

ofth

en−α

-

random

gra

ph

(Bald

win

-Shela

h)were

π3.

The

pro

ofs

ofth

era

nk

2bic

olo

red

field

(Bald

win

-

Holland)

did

not

find

axio

ms

explicit

axio

ms.

But

Holland

did

.Axio

ms

were

explicitly

pro

-

vid

ed

forbic

olo

red

field

sby

Baudisch,M

art

in-

Pizzaro

,and

Zie

gle

r.

Holland,Lask

ow

skiand

Bald

win

-Holland

(re-

spectively

)m

ade

the

impro

vem

ents

.

Definitio

n15

Afo

rmula

φ(x

1,.

..,x

k)hasex-

actly

rank

k−

1if

forevery

generic

(overth

e

para

mete

rsof

φ)

solu

tion

a=〈a

1,.

..,a

k〉

of

φ(x

1,.

..,x

k),

RM

(a)=

k−

1and

any

pro

per

subse

quence

of

ais

independent.

Assum

ption

16

The

underlyin

gth

eory

Tfsa

t-

isfies

the

follow

ing

conditio

n:

Ifψ

i(x,y

)fo

r

i<

mare

afinite

set

of

k+

1-a

ryfo

rmula

s

such

thatfo

rso

me

g,fo

reach

i<

m,

ψi(

x,g

)

has

rank

at

most

k−

1,

there

isa

form

ula

φ(x

,y)su

ch

that

φ(x

,g)hasexactly

rank

k−

1

and

for

each

i<

m,

RM

(φ(x

,g)∧

ψi(

x,g

))<

k−

1.

Theore

m17

IfT

fis

stro

ngly

min

imalw

ith

elim

ination

ofim

agin

aries

and

dm

pand

sat-

isfies

Ass

um

ption

16

then

Tµ k

ism

odelcom

-

ple

te.

Inpart

icula

r,th

era

nk

kbic

olo

red

field

s

are

modelcom

ple

te.

Exam

ple

Ifw

hen

tryin

gto

collapse

the

bic

olo

red

field

(ora

fusion),

the

function

µis

chose

nbadly,

the

generic

isnot

satu

rate

d

and

its

theory

isnot

even

nearly

modelcom

-

ple

te.

Ifth

egeneric

isnot

satu

rate

d,can

we

expla

ina

sense

inw

hic

hit

isnic

e.

Perh

aps

get

afa

llback-t

heore

m?

ABST

RACT

ELEM

ENTARY

CLASSES

Acla

ssof

L-s

tructu

res,

(K,¹

K),

issa

idto

be

an

abst

ractele

menta

rycla

ss:

AEC

ifboth

K

and

the

bin

ary

rela

tion¹ K

are

clo

sed

under

isom

orp

hism

and

satisf

yth

eJonss

on

condi-

tions

plu

s

IfA

,B,C

∈K

,A¹ K

C,

B¹ K

Cand

A⊆

B

then

A¹ K

B;

More

care

fulfo

rmula

tion

ofunio

ns

ofchain

s;

Existe

nce

ofLowenheim

num

ber.

QUASIM

INIM

ALIT

YI

TrialD

efinitio

nM

is‘q

uasim

inim

al’

ifevery

firs

tord

er

(Lω1,ω

?)

definable

subse

tof

Mis

counta

ble

or

cocounta

ble

.

a∈

acl′ (

X)

ifth

ere

isa

firs

tord

er

form

ula

with

counta

bly

many

solu

tions

over

Xw

hic

h

issa

tisfi

ed

by

a.

Exerc

ise

?If

fta

kes

Xto

Yis

an

ele

men-

tary

isom

orp

hism

,f

exte

ndsto

an

ele

menta

ry

isom

orp

hism

from

acl′ (

X)

toacl′ (

Y).

QUASIM

INIM

AL

EXCELLENCE

Acla

ss(K

,cl)

isquasim

inim

alexcellent

ifit

adm

its

acom

bin

ato

rialgeom

etr

yw

hic

hsa

t-

isfies

on

each

M∈

K

there

isa

uniq

ue

type

ofa

basis,

ate

chnic

alhom

ogeneity

conditio

n:

ℵ 0-h

om

ogeneity

over∅

and

over

models.

Definitio

n18

Let

C⊆

H∈

Kand

let

Xbe

afinite

subse

tof

H.

We

say

tpqf(

X/C)

is

defined

over

the

finite

C0

conta

ined

inC

if:

forevery

G-p

art

ialm

onom

orp

hism

fm

appin

g

Xin

toH′ ,

forevery

G-p

art

ialm

onom

orp

hism

f1

mappin

gC

into

H′ ,

iff∪

(f1|C

0)

isa

G-

part

ial

monom

orp

hism

,f∪

f1

isalso

aG-

part

ialm

onom

orp

hism

.

Inth

efo

llow

ing

definitio

nit

isess

entialth

at

⊂be

unders

tood

as

pro

per

subse

t.

Definitio

n19

1.Forany

Y,cl−

(Y)=

⋃ X⊂Y

cl(

X).

2.W

ecall

C(t

he

unio

nof)

an

n-d

imensional

cl-in

dependent

syst

em

ifC

=cl−

(Z)

and

Zis

an

independent

set

ofcard

inality

n.

[Conditio

nIV

:Q

uasim

inim

alExcellence]Let

G⊆

H,H

′ ∈K

with

Gem

pty

or

inK

.Sup-

pose

Z⊂

H−

Gis

an

n-d

imensionalin

depen-

dent

syst

em

,C

=cl−

(Z),

and

Xis

afinite

subse

tof

cl(

Z).

Then

there

isa

finite

C0

conta

ined

inC

such

that

tpqf(

X/C)

isde-

fined

over

C0.

Excellence

yie

lds:

Lem

ma

20

An

isom

orp

hism

betw

een

inde-

pendent

Xand

Yexte

nds

toan

isom

orp

hism

ofcl(

X)

and

cl(

Y).

This

giv

escate

goricity

inall

uncounta

ble

pow

-

ers

ifth

eclo

sure

offinite

sets

iscounta

ble

.

AEC

Kfrom

Hru

shovsk

iconst

ruction.

dM

(A)=

inf{

δ(B

):A⊆

B⊂ ω

M}

M¹ K

Niff

forevery

finite

A⊂

M

dM

(A)=

dN(A

)

FACT

:(K

,¹K

)is

an

aec.

Quasi

Min

Excell

ofHru

shovsk

iConst

ruction

a∈

cl M

(X)

iffdM

(a/X

)=

0.

Note

clo

sure

isL

ω1,ω

-definable

and

a∈

cl M

(X)im

pliestp

(a/X

0)hasfinitely

many

solu

tions

for

som

efinite

X0⊆

X.

Fact:

(K,c

l M)

isa

quasim

inim

al

excellent

cla

ss.

Application

Forbic

olo

red

field

sand

any

µth

eScott

sen-

tence

of

the

generic

modelis

cate

goricalin

all

card

inalities.

Com

pare

with

work

of

Villa

veces

and

Zam

-

bra

no.

Zilber

Variation

Indefinin

gth

ecla

ssK

for

the

Hru

shovsk

i

const

ruction

allow

sente

nces

of

Lω1,ω

.

Specifi

cally,

fix

(Z,+

).

Itst

illwork

s!

Pse

udo-e

xponentiation

et

al

Connecting

the

two

part

softh

eta

lk.

When

the

second

appro

ach

does

not

yie

lda

‘firs

tord

er

stru

ctu

re’,

study

the

theory

of

the

generic

by

the

firs

t

appro

ach.

Bib

liogra

phy

See

accom

panyin

gfile

.