TAM

ENESS

John

T.Bald

win

Depart

ment

ofM

ath

em

atics,

Sta

tist

ics

and

Com

pute

r

Scie

nce

Univ

ers

ity

ofIllinois

at

Chic

ago

ww

w.m

ath

.uic

.edu/

jbald

win

August

21,2005

TO

PIC

S

1.

Som

ehisto

ry

2.

backgro

und

on

tam

eness

3.

Anew

suffi

cie

nt

conditio

nand

application

4.

Non-t

am

eness

HIS

TO

RY

AND

CO

NT

EXT

In1970

logic

was

ass

imilating

strikin

gnew

meth

ods:

recurs

ion

theory

:priority

meth

od

set

theory

:fo

rcin

g,dia

monds,

desc

riptive

set

theory

modelth

eory

:M

orley’s

theore

m

pro

ofth

eory

:?

FO

RCIN

GFEST

IVAL

1973?

East

Coast

ModelT

heory

(Yale

):sm

all

models;fo

cus

on

firs

tord

er

theories

of‘real’

stru

ctu

res;

definability

inin

di-

vid

ualm

odels

tools:

com

pactn

ess

,quantifier

elim

ination

innatu

ralla

n-

guages

West

Coast

ModelT

heory

(Berk

ele

y,M

adison,

and

Cor-

nell):

Models

of

arb

itra

rycard

inality

;arb

itra

ryth

eories:

firs

tord

erand

infinitary

tools:

com

pactn

ess

,Erd

os-

Rado,Ehre

nfe

ucht-

Most

ow

ski;

quantifierelim

ination

by

fiat;

com

bin

ato

rialand

som

etim

es

axio

matic

set

theory

;M

orley

rank;in

discern

ible

s

SHELAH’S

REVO

LUT

ION

1.

Clsss

ification

ofth

eories

(not

models

1974):

(a)

frees

firs

tord

er

model

theory

from

axio

matic

set

theory

(b)

syst

em

atizes

modelth

eore

tic

invest

igations

by

pro

-

vid

ing

agenera

lto

ol

2.

monst

er

model–

univ

ers

aldom

ain

3.

dependence

rela

tion

(fork

ing)

4.

analy

ze

types

locally

(a)

geom

etr

yofty

pes

(b)

nonort

hogonality

,p-r

egula

rty

pes,

inte

rnality

,

5.

decom

pose

all

models

into

models

ofsize|L|

6.

uncounta

ble

languages

Marx

ist

Synth

esis

Pilla

y(2

000):

‘There

isonly

one

modelth

eory

’.

LET

MANY

FLO

WERS

GRO

W

Models

ofArith

metic,

finite

modelth

eory

,

univ

ers

alalg

ebra

,

modelth

eory

ofaltern

ative

logic

s,

abst

ract

modelth

eory

,

adm

issible

modelth

eory

IDEO

LO

GY

Can

the

study

ofla

rge

(>2ℵ 0

)st

ructu

res

seriously

impact

the

unders

tandin

gofst

ructu

resdiscovere

din

the

19th

cen-

tury

?

IDEO

LO

GY

Can

the

study

ofla

rge

(>2ℵ 0

)st

ructu

res

seriously

impact

the

unders

tandin

gofst

ructu

resdiscovere

din

the

19th

cen-

tury

?

MET

HO

DO

LO

GIC

ALLY:T

he

answ

eris

decisiv

ely

yes.

The

tools

of

cla

ssifyin

gth

eories

(sta

bility,

supers

tability)

or-

thogonality

,p-r

egula

rdecom

positions,

etc

.,in

vente

dby

Shela

hto

study

the

uncounta

ble

spectr

um

ofarb

itra

ryfirs

t

ord

er

theories

(in

poss

ibly

uncounta

ble

languages)

,were

convert

ed

thro

ugh

geom

etr

icst

ability

theory

asfu

ndam

en-

talto

ols

use

din

the

modern

modelth

eore

tic

invest

igations

ofalg

ebra

icst

ructu

res.

Pro

perly

axio

matized

cate

goricity

implies

quantifier

elim

i-

nation.

Can

the

connection

be

stro

nger?

Ina

very

rough

sense

,

Morley’s

theore

msa

ys

no.

Atle

ast

forcate

goricity,

infirs

t

ord

er

logic

the

story

isover

atℵ 1

.

Isth

eansw

er

diff

ere

nt

for

infinitary

logic

?

ST

RO

NG

ER

IDEO

LO

GY

For

Shela

hand

pro

bably

Tars

ki,

this

isnot

an

inte

rest

ing

quest

ion.

Their

goalis

tounders

tand

math

em

aticsnotju

st

the

math

em

atics

that

has

already

been

long

invest

igate

d.

ABST

RACT

ELEM

ENTARY

CLASSES

AEC’s

are

‘theories’

not

logic

s.

The

stru

ctu

res

are

‘ense

mblist

e’;

Tars

kiStr

uctu

res.

Meth

odolo

gicalPro

ble

mUnders

tand

the

exact

connec-

tion

betw

een

AEC

and

the

cats

-positive

bounded-c

ontinuous

logic

configura

tion.

MO

NST

ER

MO

DEL

Definitio

n.

The

model

Mis

κ-(

K,¹

K)-

hom

ogeneous

if

A≤

M,A

≤B∈

Kand|B|≤

κim

plies

there

exists

B′ ≤

M

such

that

B∼ =

AB′ .

Sta

ndard

arg

um

ents

show

:

Theore

m1

Ifa

cla

ss(K

0,¹

K)

has

the

am

alg

am

ation

pro

pert

yand

the

join

tem

beddin

gpro

pert

y,fo

revery

κ,

there

isa

stru

ctu

reM

κw

hic

his

κ-(

K,¹

K)-

hom

ogeneous.

For

suffi

cie

ntly

larg

eκ,th

isis

the

Monste

rm

odel.

GALO

IST

YPES

Define:

(M,a

,N)∼ =

(M,a′ ,

N′ )

ifth

ere

exists

N′′

and

stro

ng

em

beddin

gs

f,f′ t

akin

gN

,N′

into

N′′

whic

hagre

eon

Mand

with

f(a

)=

f′ (

a′ )

.

‘The

Galo

isty

pe

of

aover

Min

N’

isth

esa

me

as

‘the

Galo

isty

pe

of

a′ o

ver

Min

N′ ’

if(M

,a,N

)and

(M,a′ ,

N′ )

are

inth

esa

me

cla

ssof

the

equiv

ale

nce

rela

tion

genera

ted

by∼ =

.

AM

ALG

AM

AT

ION

PRO

PERT

Y

If(K

,¹K

)has

the

am

alg

am

ation

pro

pert

y:

1.∼ =

isan

equiv

ale

nce

rela

tion.

2.

(K,¹

K)

has

am

onst

er

model.

S(M

)denote

sth

ese

tofG

alo

isty

pes

over

M.

GALO

IST

YPES

INT

HE

MO

NST

ER

If(K

,¹K

)has

am

onst

er

modelM

,

ga−

tp(a

/M

)=

ga−

tp(a′ /M

)

ifa

and

a′are

conju

gate

under

an

auto

morp

hism

ofM

fixin

gM

.

SYNTACT

ICT

YPES

Fix

alo

gicL.

(E.g

.L

ω,ω

,Lω1,ω

,Lω1,ω

(Q).

Then

tp(a

/M,N

)=

tp(a′ /

M,N

′ )

ifa

satisfi

es

the

sam

eL-

form

ula

sw

ith

para

mete

rsfrom

M

inN

as

a′ s

atisfi

es

inN′ .

KEY

PRO

PERT

IES

Com

pactn

ess

:If〈p

i:

i<

δ〉

isan

incre

asing

sequence

of

synta

ctic

types

⋃ i<δpiis

asy

nta

ctic

type.

Locality

:If

M=

⋃ i<δM

i,p6=

q∈

S(M

)th

en

there

exists

an

isu

ch

that

p|M

i6=

q|M

i.

Tam

eness

:If

p,q∈

S(M

)and

p|N

=p|N

for

every

small

subm

odel

N,th

en

p=

q.

TAM

ENESS

PARAM

ET

ERS

Iffo

revery

model

Mofcard

inality

µ,

ifp|N

=q|N

for

every

subm

odel

Nw

ith|N|≤

κim

plies

p=

q;

we

say

(K,¹

K)

is(µ

,κ)-

tam

e.

No

ass

um

ption

ofam

alg

am

ation.

HANF

NUM

BERS

Nota

tion

21.

Let

H(λ

,κ)

be

the

least

card

inal

µsu

ch

that

ifa

firs

tord

er

theory

Tw

ith|T|=

λhas

models

ofevery

card

inalle

ssth

an

µw

hic

hom

iteach

ofa

set

Γofty

pes,

with|Γ|=

κ,th

en

there

are

arb

itra

rily

larg

e

models

of

Tw

hic

hom

itΓ.

2.

Write

H(κ

)fo

rH

(κ,κ

).

3.

For

asim

ilarity

type

τ,

H(τ

)m

eans

H(|τ|).

4.

H1

=H

(τ);

H2

=H

(H(τ

)).

PRESENTAT

ION

THEO

REM

Theore

m3

(T

he

Pre

senta

tion

Theore

m)

IfK

isan

AEC

with

Lowenheim

num

ber

LS(K

)(in

avocabula

ryτ

with

|τ|≤

LS(K

)),

there

isa

vocabula

ryτ′

with

card

inality

|LS(K

)|,a

firs

tord

er

τ′ -t

heory

T′and

ase

tΓ

of

2LS(K

)

types

such

that:

K={M

′ |τ:M′ |=

T′ a

nd

M′ o

mits

Γ}.

More

over,

ifM′

isa

τ′ -s

ubst

ructu

reof

N′

where

M′ ,

N′

satisf

yT′ a

nd

om

itΓ

then

M′ |τ¹ K

N′ |τ

.

By

Shela

h’s

pre

senta

tion

theore

m:

Coro

llary

4If

Kis

an

AEC

with

Lowenheim

-Skole

mnum

-

ber

κand

Khas

am

odelof

card

inality

at

least

H(κ

,2κ)

(whic

his

atm

ost

i (2

κ)+

),th

en

Khasarb

itra

rily

larg

em

od-

els.

When

LS(K

)=|τ

(K)|

=κ,H

(κ)=

H1is

the

Hanfnum

ber

for

all

AEC

with

the

sam

eLowenheim

num

ber.

Sam

eHanfnum

ber

forom

itting

Galo

isty

pes.

TW

OT

HEM

ES

What

isth

eeventu

albehavio

rofcate

goricity?

So

ass

um

e

there

exist

arb

itra

rily

larg

em

odels?

Doescate

goricity

insm

all

card

inals

(belo

wi ω

1orbett

erℵ ω

imply

existe

nce

and

cate

goricity

eventu

ally?

Yes,

for

excellence.

Excellence

yie

lds

upward

Lowenheim

-

Skole

m.

CO

NSEQ

UENCES

I

Theore

m(S

hela

h)

If(K

,¹K

)has

am

alg

am

ation

and

is

cate

goricalin

λ+

>H

2and

Kis

then

(K,¹

K)

iscate

gor-

icalon

the

inte

rval[H

2,λ

+).

This

use

sth

ederivation

ofta

meness

from

cate

goricity

de-

scribed

above.

CO

NSEQ

UENCES

II

Gro

ssberg

and

Vandie

ren

began

the

invest

igation

of

sta-

bility

spectr

um

for

tam

eAEC

with:

Theore

m5

Kis

stable

inµ

>H

1and

(∞,χ

)-ta

me

for

som

eχ

<H

1,th

en

Kis

stable

inλ

ifλ

µ=

λ.

Bald

win

,K

ueker,

VanD

iere

npro

ved

the

follow

ing:

Theore

m6

Let

Kbe

an

AEC

with

Lowenheim

-Skole

m

num

ber≤

κ.

Ass

um

eth

at

Ksa

tisfi

es

the

am

alg

am

a-

tion

pro

pert

yand

isκ-w

eakly

tam

eand

Galo

is-s

table

in

κ.

Then,

Kis

Galo

is-s

table

inκ+

nfo

rall

n<

ω.

With

one

furt

her

hypoth

esis

we

get

avery

stro

ng

conclu

-

sion

inth

ecounta

ble

case

.

Coro

llary

7Let

Kbe

an

AEC

that

satisfi

es

the

am

alg

a-

mation

pro

pert

yw

ith

Lowenheim

-Skole

mnum

berℵ 0

that

isω-localandℵ 0

-tam

e.

IfK

isℵ 0

-Galo

is-s

table

then

Kis

Galo

is-s

table

inall

card

inalities.

CO

NSEQ

UENCES

III

Theore

m(G

ross

berg

-VanD

iere

n)

If(K

,¹K

)is

cate

gor-

ical

inλ+

>LS(K

)+and

Kis

(∞,L

S(K

)-ta

me,

then

(K,¹

K)

iscate

goricalon

the

inte

rval[λ

+,∞

).

Theore

m(L

ess

mann)

If(K

,¹K

)is

cate

goricalin

λ+

>

LS(K

)=ℵ 0

and

Kis

(∞,L

S(K

)-ta

me,th

en

(K,¹

K)

is

cate

goricalon

the

inte

rval[λ

+,∞

).

SUFFIC

IENT

CO

ND

ITIO

NS

I

If(K

,¹K

)is

quasim

inim

alexcellent

inth

ese

nse

ofZilber

then

Galo

isty

pes

=sy

nta

ctic

types.

(Coro

llary

topro

ofofcate

goricity

transf

er.)

SUFFIC

IENT

CO

ND

ITIO

NS

II

Theore

m8

IfK

isλ-c

ate

goricalfo

rλ

>H

1,th

en

forany

µ<

λ,

Kis

(µ,H

1)-

weakly

tam

e.

(Weakly

means

the

larg

em

odel

isass

um

ed

tobe

satu

-

rate

d.)

SUFFIC

IENT

CO

ND

ITIO

NS

III

The

Hru

shovsk

iconst

ruction

(even

with

infinitary

input)

giv

es

rise

toan

abst

ract

ele

menta

rycla

ss;

(K,¹

K,d

)

dis

dim

ension

function

whic

him

pose

sa

com

bin

ato

rialge-

om

etr

yand

anotion

of

stro

ng

subm

odelw

hic

hgiv

es

an

abst

ract

ele

menta

rycla

ss.

GEO

MET

RY

Assum

ption

91.

δand

sod

map

into

the

inte

gers

.

2.

dN

satisfi

es

dN(X

)≤|X|.

3.

IfX⊆

Y,

dN(X

)≤

dN(Y

).

Definitio

n10

Let

a∈

cl N

(X)

ifdN(a

/X

)=

0.

Much

more

rest

rictive

hypoth

ese

sth

an

genera

lAEC

or

Gro

ssberg

-Kole

snik

ov

(who

earlie

rdeduced

tam

eness

from

sim

ilar

hypoth

ese

sass

um

ing

the

existe

nce

of

an

abst

ract

independence

rela

tion).

For

each

ain

N,th

ere

isa

finite

ma

with

d(a

/M)=

d(a

/m

a).

Lem

ma

11

Let

M¹ K

N1,N

2w

ith|N

j−

M|=

κ.

There

are

N′ 1,N

′ 2,M

′ with

M′ ⊂

Nj

and|N

j−

M|=

|M′ |

=κ

such

that

each

N′ jis

independent

from

Mover

M′ .

Theore

m12

(B,V

illa

veces,Zam

bra

no)

If(K

,¹K

,d)

ina

counta

ble

language

satisfi

es

1.

free

am

alg

am

ation

2.

free

exte

nsion

over

independent

pairs

3.

weak

3-a

malg

am

ation.

then

(K,¹

K,d

)is

(∞,ℵ

0)-

tam

e.

Note

hypoth

ese

sare

about

arb

itra

rily

larg

em

odels.

UNIV

ERSAL

CO

VERS

When

isth

eexact

sequence:

0→

Z→

V→

A→

0.

(1)

cate

goricalw

here

Vis

aQ

vecto

rsp

ace

and

Ais

ase

mi-

abelian

variety

.

Can

be

vie

wed

as

an

expansion

of

Vand

there

isa

com

bi-

nato

rialgeom

etr

ygiv

en

by:

cl(

X)=

ln(a

cl(exp(X

)))

More

overby

directbasic

alg

ebra

icarg

um

entone

can

show

the

thre

ehypoth

ese

s.Just

using

linear

disjo

intn

ess

.T

his

isfree

am

alg

am

ation

soth

ere

are

arb

itra

rily

larg

em

odels.

By

Less

mannℵ 1

cate

goricity

impliescate

goricity

inall

pow

-

ers

.

Zilber

had

pro

ved:

cate

goricity

inall

uncounta

ble

powers

iff‘a

rith

metic’pro

pert

ies

on

sem

iabelian

variety

V’.

We

impro

ve

this

to:

cate

goricity

inℵ 1

iff‘a

rith

metic’pro

pert

ies

on

sem

iabelian

variety

V

GO

AL

The

goalofth

isline

ofre

searc

his

toid

entify

those

pro

per-

ties

of

the

Hru

shovsk

iconst

ruction

whic

hare

‘auto

matic’

and

those

whic

hm

ust

be

verified

inin

div

idualcase

s.

Next

task

:Check

the

statu

sof

the

hypoth

ese

sof

Theo-

rem

12

for

pse

udoexponentiation.

NO

NTAM

ENESS

We

want

tofind

am

onst

er

modelM

and

asu

bm

odel

N

with

ele

ments

a,b

such

that

for

every

small

N0¹ K

N,

there

isan

auto

morp

hism

ofM

fixin

gN

0and

takin

ga

to

bbut

there

isno

such

auto

morp

hism

whic

hfixes

N.

We

find

exam

ple

sby

translating

pro

ble

ms

from

Abelian

gro

up

theory

.

Hart

-Shela

h

There

isan

AEC

(K,¹

K)

whic

his

1.ℵ 1

andℵ 0

cate

gorical.

2.

Ap

fails

inℵ 0

.

3.

Many

models

inall

card

inals≥

λfo

rso

me

λ<

2ℵ 1

.

The

arg

um

ent

that

(K,¹

K)

isnot

(ℵ1,ℵ

0)-

tam

eneeds

som

ework

.It

may

be

thatone

hasto

go

toa

larg

erexam

-

ple

.E.g

.th

eHart

-Shela

hexam

ple

whic

hisℵ 3

-cate

gorical.

WHIT

EHEAD

GRO

UPS

Ais

aW

hitehead

gro

up

(W-g

roup)

ifE

xt(

Z,A

)=

0.

That

is,every

short

exact

sequence:

0→

Z→

H→

A→

0.

(2)

splits

.

BACKG

RO

UND

J.H

.C.W

hitehead

conje

ctu

red

thatevery

Whitehead

gro

up

ofcard

inalityℵ 1

isfree.

Shela

hpro

ved:

1)

(ZFC)

There

isanℵ 1

-fre

e(e

very

counta

ble

subgro

up

is

free)

Abelian

gro

up

ofcard

inalityℵ 1

.

2)

The

Whitehead

Conje

ctu

reis

independent

ofZFC.

THE

EXAM

PLE

Let

Kbe

the

cla

ssof

stru

ctu

res

M=〈G

,Z,I

,H〉,

where

each

of

the

list

ed

sets

isth

eso

lution

set

of

one

of

the

unary

pre

dic

ate

s(G

,Z,I

,H).

Gis

ato

rsio

n-fre

eAbelian

Gro

up.

Zis

acopy

of(Z

,+).

I

isan

index

set

and

His

afa

mily

ofin

finite

gro

ups.

The

vocabula

ryalso

inclu

des

function

sym

bols

F,k

and

π,

nam

ing

functions

F,k

,and

π.

There

isa

bin

ary

+on

G

and

onZ

and

ate

rnary

+.

Fm

aps

Honto

Iand

for

s∈

I,+

(,

,s)

isa

gro

up

oper-

ation

on

Hs=

F−1

(s).

Fin

ally,

πm

aps

Honto

Gso

that

πs=

π|H

sis

apro

jection

from

Hs

onto

G.

The

kern

elofeach

πs

isisom

orp

hic

toZ

via

am

ap

k(

,s)

where

k:Z×

I7→

H.

M0¹ K

M1

if

M0

isa

subst

ructu

reof

M,

but

ZM

0=

ZM

and

GM

0is

apure

subgro

up

of

GM

1.

FACT

S

Definitio

n13

We

say

the

AEC

(K,¹

K)

adm

its

clo

sure

s

iffo

revery

X⊆

M∈

K,th

ere

isa

min

imalclo

sure

of

Xin

M.

That

is,

M|⋂ {

N:X⊆

N¹ K

M}=

cl M

(X)¹ K

M.

The

cla

ss(K

,¹K

)is

an

abst

ract

ele

menta

rycla

ssth

at

adm

its

clo

sure

sand

has

the

am

alg

am

ation

pro

pert

y.

NO

TLO

CAL

Lem

ma

14

(K,¹

K)

isnot

(ℵ1,ℵ

1)-

local.

That

is,th

ere

isan

M0∈

Kofcard

inalityℵ 1

and

acontinuous

incre

asing

chain

ofm

odels

M0 i

for

i<ℵ 1

and

two

distinctty

pes

p,q∈

S(M

0)

with

p|M

0 i=

q|M

ifo

reach

i.

Let

Gbe

an

Abelian

gro

up

ofcard

inalityℵ 1

whic

hisℵ 1

-fre

e

but

not

aW

hitehead

gro

up.

There

isan

Hsu

ch

that,

0→Z→

H→

G→

0

isexact

but

does

not

split.

WHY?

Let

M0

=〈G

,Z,a

,G⊕

Z〉

M1

=〈G

,Z,{

a,t

1},{G

⊕Z

,H}〉

M2

=〈G

,Z,{

a,t

2},{G

⊕Z

,G⊕

Z}〉

Let

p=

tp(t

1/M

0,M

1)

and

q=

tp(t

2/M

0,M

2).

Sin

ce

the

exactse

quence

for

HM

2sp

lits

and

thatfo

rH

M1

does

not,

p6=

q.

NO

Tℵ 1

-LO

CAL

But

forany

counta

ble

M′ 0¹ K

M0,

p|M

′ 0=

q|M

′ 0,as

0→

Z→

H′ i→

G′ →

0.

(3)

splits

.

G′ =

G(M

′ 0),

H′ =

π−1

(ti,

G′ ).

NO

Tℵ 0

-TAM

E

Itis

easy

tose

eth

at

if(K

,¹K

)is

(ℵ1,ℵ

0)-

tam

eth

en

itis

(ℵ1,ℵ

1)-

local,

so

(K,¹

K)

isnot

(ℵ1,ℵ

0)-

tam

e.

So

infa

ct,

(K,¹

K)

isnot

(χ,ℵ

0)-

tam

efo

rany

χ.

NO

Tκ-T

AM

E

With

som

euse

ofdia

monds,

foreach

success

orcard

inalκ,

there

isa

κ-fre

ebut

not

free

gro

up

ofcard

inality

κw

hic

h

isnot

Whitehead.

This

show

sth

at,

consist

ently,

For

arb

itra

rily

larg

eκ,

(K,¹

K)

isnot

(κ+

,κ)-

tam

efo

rany

κ.

BECO

MIN

GTAM

E

Gro

ssberg

and

Van

Die

ren

ask

ed

for(K

,¹K

),and

µ1

<µ2

soth

at

(K,¹

K)

isnot

(≥µ1,µ

1)-

tam

ebut

is(≥

µ2,µ

2)-

tam

e.

This

requires

findin

ga

way

tobound

the

card

inality

of

G

without

imposing

too

much

stru

ctu

reon

G.

Even

more

inte

rest

ing.

Isit

poss

ible

tofind

cate

gorical

exam

ple

sw

hic

honly

becom

eta

me

nearth

eHanfnum

ber.

LO

CALIT

YAND

CO

MPACT

NESS

The

existe

nce

of

asu

pfo

ran

incre

asing

chain

of

Galo

is

types

isalso

pro

ble

matic.

Every

ω-c

hain

has

asu

p.

Locality

up

toκ

implies

com

pactn

ess

at

κ+

.

There

are

NO

know

nsu

fficie

nt

conditio

ns

forlo

cality

.

Variants

ofth

eW

hitehead

const

ruction

should

giv

e(c

on-

sist

ently)

acla

ssth

at

isnot

(ℵ2,ℵ

2)-

com

pact.

TAM

ENESS

AND

AM

ALG

AM

AT

ION

How

clo

sely

are

tam

eness

and

am

alg

am

ation

inte

rtw

ined?

TW

OPRO

PERT

IES

Kis

modelcom

ple

teif

N⊂

Mand

N∈

K,im

plies

N¹ K

M.

Definitio

n15

We

say

the

AEC

(K,¹

K)

adm

its

clo

sure

s

iffo

revery

X⊆

M∈

K,th

ere

isa

min

imalclo

sure

of

Xin

M.

That

is,

M|⋂ {

N:X⊆

N¹ K

M}=

cl M

(X)¹ K

M.

Non-t

am

ewith

A.P

.

Theore

m16

For

any

AEC

(K,¹

K)

whic

hadm

its

clo

-

sure

s,we

can

ass

ign

(K′ ,¹′ K

)w

hic

hhasth

eam

alg

am

ation

pro

pert

yand

and

isno

more

tam

eth

an

K.

IDEA:Expand

the

language

by

addin

ga

new

bin

ary

rela

-

tion.

Form

acom

ple

tegra

ph

on

each

K-s

tructu

re.

K′is

the

smallest

AEC

conta

inin

gth

ese

expansions

such

that

each

finite

com

ple

tegra

ph

isexte

nded

toa

min

imalm

odel

that

expands

toa

mem

berof

K(w

hic

his

also

acom

ple

te

gra

ph).

N′ ¹

KM

ifcl′ N

′(A)=

cl′ M

′(A)

forevery

finite

A.

Getap

by

no

edgesam

alg

am

ation

ofth

eunderlyin

ggra

phs.

Definitio

n17

Let

Kbe

an

AEC

with

are

lationalvocab-

ula

ryτ.

The

vocabula

ryτ′of

K′is

obta

ined

by

addin

g

one

additio

nalbin

ary

rela

tion

R.

We

say

the

dom

ain

ofa

τ′ -s

tructu

reA

isan

R-s

etif

Rdefines

acom

ple

tegra

ph

on

A. 1.

The

cla

ssK′is

those

τ′ -s

tructu

res

Mw

hic

hass

ign

a

τ′ -s

tructu

reM

A=

cl′ M

(A)

toeach

finite

subse

tA

of

M

such

that:

(a)

Ifth

efinite

subse

tA

of

Mis

an

R-s

et

there

isa

τ′ -s

tructu

reM

Asu

ch

that

A⊆

MA⊆

Mw

ith

MA≤

LS(K

),M

Ais

an

R-s

et,

and

MA|τ∈

K.

(b)

IfN⊂

Msa

tisfi

es

the

conditio

ns

on

MA

inre

quire-

ment

1),

then

MA|τ¹ K

N|τ

.

2.

IfM

1⊆

M2

are

each

inK′ ,

then

M1¹′ K

M2

iffo

reach

finite

R-s

et

Ain

M1,cl′ M

1(A

)=

cl′ M

2(A

).

Connections

Motivate

dby

issu

es

indata

base

sBald

win

-Benedik

tin

tro-

duced

the

notion

of‘sm

all’to

study

em

bedded

finite

model

theory

.

Casa

novas

and

Zie

gle

rexte

nded

and

cla

rified

this

work

;

pro

vid

ing

afram

ework

whic

hin

clu

ded

Poizat’s

theory

of

‘belles

paires’.

Bald

win

and

Baizhanov

refined

this

furt

her:

Theore

m18

If(M

,A)

isuniform

lyweakly

benig

nand

the

#-induced

theory

on

Ais

stable

then

(M,A

)is

stable

.

The

beautifu

lwork

ofVan

den

Dries

and

Ghury

ikan

could

have

applied

these

resu

lts.

REFERENCES

My

monogra

ph

athtt

p:/

/w

ww

2.m

ath

.uic

.edu/

jbald

win

/m

odel.htm

l

conta

ins

much

of

this.

The

actu

alpapers

discuss

ed

here

may

not

appear

on

the

web

fora

couple

ofm

onth

s.