Theories with a finite number of countable models and a small

r

of Canada

du Canada

NAME OF AUTHCWNOM DE L'AUTEUR

TITLE OF THESISITITRE DE LA T H ~ E

ON MICROFICHE

r

s-7 SUR MICROFICHE

WOODROW, Robert Edward

Theories w i t h a f i n i t e number of countable models and a

small language 77-

UNIVERSITY/UNIVERSIT~

SI-

FRASER U N I V E R S I ~ ~

DEGREE FOf( WHICH THESIS WAS PpESENTED/ E GRADE POUR LEWEL CETTE THESE FUT P R ~ S ~ N T ~PhYEAR THIS DEGREE C O N F E R R E D / A N N D'&TENTION ~E

DE CE GRADE

NAME OF SUPERVISOR/NOM DW DIRECTEUR DE T H ~ S E

A'

Lachlan

1

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-

National Library of Canada

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Direction du catalogage. Division des theses canadiennes

Ottawa, Canada KlAON4 L

L

-

.

~ 7


nEw

'

Some c o n j e c t u r e s regarding t h e complexity of t h e o r i g s s a t i s f y i n g .

r e s t r i c t i o n s on language and number of countable models a r e formulated

and discussed.

A theory- T

0

i s constructed which has n i n e countable

models and a nonprincipal 1-type containing i n f i n i t e l y many 2-types. A theory

T

1

i s c o n s t r u c t e d which has f o u r countable models and an

i n e s s e n t i a l extension

T

2

having i n f i n i t e-l y many countable models. 1

(iii)

ACKNOWLEDGMENTS

The author wishes f i r s t and foremost t o express h i s thanks f o r

* the guidance and encouragement given by h i s s e n i o r s u p e r v i s o r Professor A.H. have been made.

Lachlan, without which very l i t t l e p r o g r e s s would The National Research Council of Canada provided

f i n a n c i a l support f o r most o f t h e time t h e author was engaged i n t h i s , work. .Lp

Thanks a r e a l s o do t o t h e a u t h o r ' s o f f i c e mates over t h e y e a r s , k

Robert Lebeuf, J i m Dukarm and Ron Morrow. discussion of t h e d i f f i c u l t i e s of t h e day.

0

They endured many hours of P a r t i c u l a r thanks areLdue

t o Ron who s u f f e r e d through t h e w r i t i n g up and provided a w i l l i n g e a r

&

.t

-and f r i e n d l y support throughout t h a t time. S p e c i a l thanks g o t o Dolly Rosen who performed t h e n e a r ' m i r a c l e of transforming my handwritten manuscript i n t o something l e g i b l e .

*

TABLE OF CONTENTS

.

/ '

Page

Abstract Acknowledgments

C h a p t e r 1.

N o t a t i o n and P r e l i m i n a r i e s jr

C h a p t e r 2.

Ehrenfeucht-like

theories

'$1 D e f i n i t i o n s

C h a p t e r 3.

$2

The (Theorem

$3

P r o o f o f Lemma 2.2

$4

P r o o f .of Lemma 2.3

$5

summary

2-generic

structures

D

C h a p t e r 4.

Q u a n t i f i e r eliminable graphs .

rT

$1 Examples $2

,

Definable equivalence r e l a t i o n s

'$3 Tournaments $4

Undirected graphs

$5

~ i n k ultrahomogeneous e graphs

C h a p t e r 5.

The Theory

c h a p t e r 6.

C 1 , C4 and t h e Theory

To

$1 C1 a n d C4 $2 Conclusion

The Theory

T

1

T

1

>

This work i s concerned

with

those countable complete t h e o r i e s -

which have a f i n i t e number of countable models.

An e l e g a n t c h a r a c t e r -

i z a t i o n of those t h e o r i e s which have one countable model i s b t h e theorem of Engeler [ 3 ] , Ry11-Nardzewski

191 and Svenonius [ll]:

a theory i ~ o , - c a t e g o r i c a l j u s t i n case f o r each 0

a f i n i t e number of n2types,

n

there are

The work of Vaught [12, p. 3201 shows t h a t

no countab3e complete theory -have

e x a c t l y two comt&e

models.

*>

Work of Baldwin and Lachlan [ 2 ] and Lachlan [61 shows t h a t a count*

e

complet'e theory with a f i n i t e number of countable models b u t more than

n

e

annot be s u p e r s t a b l e .

I n a l e t t e r t o Professor Lachlan, Shelah f

u n j e c t u r e d t h a t no such theory could be s t a b l e . conjecture remains open.

To our knowledge t h e

F u r t h e r progress on t h e general problem

appears t o be very d i f f i c u l t .

We s h a l l r e s t r i c t o u r s e l v e s t o t h e o r i e s

which a r e simple i n complexity of language.

The main emphasis w i l l be

on t h e o r i e s with more than one countable model, b u t we a l s o p r e s e n t some r e s u l t s which a r e s t r i c t l y concerned with

o -categorical theories. 0

We s h a l l f i r s t give a b r i e f account of t h e h i s t o r y of t h e o r i e s with more than one model. For f o u r t e e n y e a r s t h e only widely known examples of t h e o r i e s with a f i n i t e number of countable models b u t more than one were those due t o Ehrenfeucht [12, •˜61 .

The archetype of t h e example i s the rational

where numbers under t h e usual o r d e r , and t h e of

Q

6 o s e union i s

Q

.

( ~ ~+ 13 )countable m d e l s ,

D

m

a r e d i s j o i n t dense s u b s e t s

The theory o f t h i s s t r u c t u r e w i l l have I n a l i k e manner c e r t a i n o t h e r countable

r

a model of dense

o r d e r t y p e s can be d i s t i n g u i s h e d by c o n s t a n t s i n

--

l i n e a r o r d e r t o give a s t r u c t u r e whose theory w i l l have f i n i t e l y many -

countable mode 1s. C e r t a i n tkchniques can then be applied t o known examples t o yield others.

For example t h e c o n s t a n t s i n Ehrenfeucht's example may rn

'be replaced by unary p r e d i c a t e s which determine i n i t i a l segments of A t t h e expense of e l j m i n a t i o n of q u a n t i f i e r s

the order.

takes an equivalence r e l a t i o n respect t o a binary r e l a t i o n .are densely ordered by c l a s s with

(n+l)

, and

R

members.

,

which i s a congruence r e l a t i o n wit

E R

.

.

T h e equivalence c l a s s e s under

f o r each

For

n< m

( n + l )-members .precedes t h a t with R

an example

For t h i s example one

with a f i n i t e language can be given [12, $61.

n

E

t h e r e is e x a c t l y one

t h e equivalence c l a s s with

(m+l)

i n t h e o r d e r i n g induced by

Another way of forming a new theory i s t o c o n s t r u c t a d i s j o i n t

union.

Given two t h e o r i e s

models, r e s p e c t i v e l y

uo '

u1

which have

p r e d i c a t e symbols and and

of

W

{p0

, P1 ,

c

Let

P

0

a new c o n s t a n t symbol.

i s t h e u n i o n o f t h e languages of

c}.

2)

The nonlogical axioms of

RV

0'

. .v n

-t

A

j%

P .v

for

l j

p r e d i c a t e symbol of

.

U

i

,i

U

0

W

R

and

m W

, Pl

having .

= 0,l

that

Then t h e language U

1

with

a r e t h e following:

an

nem

be new .unary

We may ass*

have no nonlogical symbols in common.

U1

,

we can form t h e d i s j o i n t union

countable models i n the following manner.

Uo

n

(n+l)-ary

li

\-

L

\

-

A

3)

\

--

--

-

-1~ v . - i f ~ ~ , . . . ~=v c 1 j n

f u n c t i o n symbol o f

-

for and

Ui

'-.

p p

an

f

--

--

(n+l)-ary

j 5 n\.;i

= 0,.1

C

6

4)

The r e l a t i v i z a t i o n t o axiom o f

U ,

P

of each n o n l o g i c a l

i

-

for

i

h

With " l i n k i n g " of t h e .copi,es i n t h e d i s j o i n t union some f u r t h e r c o n t r o l A l i n k between an

on t h e number of c o u n t a b l e models may be e x e r t e d . m- t y p e

p

and an n- t y p e

and for every formula

q

i s an

r

(m+n)- t y p e

>p

r

such t h a t

cp

Linking t h e n r e f e r s t@t h e a d d i t i o n of s u i t a b l e l i n k s .

,

Yet a n o t h e r

4

way t o proceed i s t o form t h e p r o d u c t

of two t h e o r i e s

Uo'ul

Up ,U1

-

\

which have a f i n i t e number o f c o u n t a b l e y o d e l s , s a y E s s e n t i a l l y i t s models are o b t a i n e d from'pode'ls

Formally we assume t h a t A new con'stant

,

c

of

respectively. and

U

0 by a copy o f =N

P

r e s p e c t i v e l y by r e p l a c i n g each member of

M,N

m,n

M

U

1

.

have no n o n l o g i c a l symbols i n common.

UO,U1

two unary p r e d i c a t e s

P

of-P 1

and two unary funcs I

t i o n symbols

p ,p 0

1

a r e added.

The n o n l o g i c a l &ions of

1) The r e l a t i v i z a t i o n t o

axiomsof

U

for

i

P

0

1

are:

of t h e n o n l o g i c a l

i

i = 0 , 1 z

2)

U *U

,>

,

Axioms a s s u r i n g t h a t o b t s i d e

P

the nonlogical

'

i

OQ

. .

symbols o f a)

RvO,.

U

i

.: ,Vri

symbol of

have t r i v i a l i n t e r p r e t a t i o n . -+

Ui

A

j5n

P .u ~j

and

for

i = 0,l-

R

an n-ary r e l a t i o n

u,

=-


., X n- 11 .

d

we w r i t e

I t i s assumed

t h a t no c l a s h o f q u a n t i f i e r s r e s u l t s , and u n l e s s t h e c o n t e x t i m p l i e s IS

otherwise t h a t t h e f r e e variables of

~ ( x )occur among

x0

,...,xn-1

.

-

~f

A

,

x

&----

A

y , a r e two f i n i t e s e q u e n c e s t h e n

L

is

q(x,y)

2 n d

n

i s t h e operation of concatenation of f i n i t e sequences.

x

for

L

.

cx>

If

---

WFwrite

x'.

i s a f i r s t o r d e r l a n g u a g e and

L

new

then

s e t o f formulae whose f r e e v a r i a b l e s o c c u r among

is t h e

L

n

v ~ r . g . f n-1 v

*

is

.

b

L

0

is t h u s the set of sentences f o r

.

Z

L (A) n

denotes

and

1

5

p,

0,

or

among mappings.

n

t o range @,

a formula

@

1

.

@

denotes

(L(A))

X I @, $J, 0 ,

We s h a l l u s e lower c a s e Greek l e t t e r s among formulae and

where

-

i s the l e n g t h of

lh(G)

y)

cp (x

A

Let

I

b e a complete t h e o r y w i t h language

T

sequence o f

new c o n s t a n t symbols.

n

r

A set

. ,

i s a n n-type i n

L

j u s t i n case

consistent extension o f , T

. r

be a

of formulae i n -.

T

x

ahd l e t

~ [ r =] T u {cp(x) : qCr)

L

n

is a

i s a complete n - t y p e j u s t i n c a s e

i_

T[T] ,

i s complete.

1-type i n

T

.

We r e s e r v e t h e l e t t e r

A , T, @

Upper c a s e Greek

p

t o d e n o t e a complete

w i l l n o r p a l l y d e n o t e complete

4

n-types.

that

Unless o t h e r w i s e s p e c i f i e d n-type w i l l mean complete n-type.

is a s t r u c t u r e f o r

If

M

.*

is valid i n

M

.

If

L

r

and

c L

q€L(M) then

M

then

I=

l?

M

I=

means

cp M

means

I=

"

cp

. M T h a s the u s u a l meaning. I f -a i s a n n - t u p l e q ( a ) } , tp(a is a n n-type i n T ~ ( M ) . i n M t h e n tp(a) = { c p C ~:~ M , - -17tp(a,b) = t p ( a b). We say t h a t a realizes q i f qctp(a) . I

7

for all

cpCT

I(T,K) -

,. .

-::..

*'

models o f

T

cardinality.

i s t h e c a r d i n a l i t y o f t h e s e t o f isomorphism t y p e s o f of

power

K

.

If

S

is a s e t ,

IS[

Countable means i n f i n i t e and c o u n t a b l e .

denotes its

P

,

is an inessential extension o f t h e complete theory

T'

i s complete and an extension of

T'

from

by f i n i t e l y

L

c o n s t a n t symbols.

A

A c A

j u s t i n case

and

-L

9 E A

t h e r e i s some

such t h a t whenever

,

\

A

realizes

9 i s s a i d t o generate

In t h i s case

-

.a

,

E M

then

m,n

z-

.

s*

.

t@(a) and

is

let

@

generate

Then

w i l l g e n e r a t e t h e type of

??

If

r

then

i s a 2-type,

i s s a i d t o be i n

A formula 8 -categorical 0

r

of m-types

$

-

b

I1

p,q

-c

tp(g)

over

1-types with

pxq

p

C

r

and

q

C

Tlt1

(

r

1

.

with one f r e e v a r i a b l e

-just i n ease k r each which- c o n t a i n

.

v

0

i s s a i d t o be,

there a r e a f i n i t e number

1C, ( v1. )

for

i < m

.

A graph is a s t r u c t u r e f o r t h e language with one b i n a r y r e l a t i o n

symbol

R

.

If

G

is a graph

i s t h e vertek-s e t of

IGI

members a r e c a l l e d v e r t i c e s while

R

G

G

,

and

i s t h e edge s e t and members a r e t

c a l l e d edges.

Chapter 2 Ehrenfeucht-like t h e o r i e s *

cs

-I

.

-

-

L

~hfhitions

1.

-4-

In t h i s c h a p t e r we g i v e some r e s u l t s which i n d i c a t e a d i s t i n c t i o n I

L

between two t y p e s of t h e o r i e s i n a small language

-

those with only

e

r e l a t i o n symbols and c o n s t a n t symbols and t h o s e which a l l o w f u n c t i o n

The main theorem u s e s s e v e r e r e s t p i c t i o n s on t h e rider o f c o u n i m e models and t h e language.

Before i n t r o d u c i n g t h e s e however we s h a l l

p r e s e n t some more g e n e r a l r e s u l t s , and t h e n o t i o n of b e i n g " l i k e "

an Ehrenfeucht s t r u c t u r e . Vaught's argument of 112; p.3201 t h a t no c o u n t a b l e complete t h e o r y h a s e x a c t l y two isomorphism t y p e s o f c o u n t a b l e models may b e -modified u s i n g h i s Theorem 3.5

[12; p. 3111 on t h e e x i s t e n c e o f prime models t o

give t h e following observatign, I(T,w) = 3

If

then

has countablemodels

T

M

M

0'

1'

andM

C

where n-type

M

0

A

i s prime, -

M1

i s s a t u r a t e d , and f o r each n o n p r i n c i p a l

-a

t h e r e i s a sequence

-

M

i s prime over a.

a b l e models. a b l e model realizes

prime.

A

Thus i f

of

T

.

and

1; C N

B u t then

= A

N

0 T

and

must b e isomorphic t o

N

T

and

and 'any

have prime and s a t u r a t e d count-

such t h a t

N

A

t h e r e i s a couqt-

i d prime

over

and

cannot b e s a t u r a t g d and i%cannot be

is a prime m d e l o f

M

s a t u r a t e d model of M

T

Thus given a n o n p r i n c i p a l n-type

N

tp(a

such t h a t

€ M

The m o d i f i c a t i o n i s t o n o t e t h a t

complete i n e s s e n t i a l e x t e n s i o n of

b

/

These d i f f e k e n c e s a r e r e l a t e d t o c o n j e c t u r e s C 1 , C 2 , &d C3,

symbols.

T

, M1*

is a countable

M

i s the t h i r d c o u n t a b l e model of

T

.

We c a l l

T

M

the middle m o d e l of

.

- -

[m

h o t h e r r e s u l t W c X WE a i s c o v e r e d in7Ependently by - ~ e n d a ,

'-

-

and t h e author- i s : Let

U

be a c o u n t a b l e complete t h e o r y w i t h

Then i f

A

i s any n o n p r i n k i p a l m.-type t h e r e i s a 2m-type

Lemma 2 . 1 I(U,w)

= 3.

while

T

mtm

is not.

)

Proof.

Let

A

i n f i n i t e l y many m-gypes s i n c e

-

a

many m-types o v e r

-

a

b e prime o v e r

M

,

i.e.

realizing

.

There a r e

i s n o n p r i n c i p a l and hence i n f i n i t e l y

-

in

A

Thus t h e r e i s a 2n-type

Th ( M , a ) .

.

' 2 A which is n o t p r i n c i p a l o k e r A -, n -

- 1

~hbose a

realizing

il

~

such t h a t , a

and - I

a

fl

g

since

.

' M

b

for otherwise

1e s t a b l i s h e s

I"

- n

I-

Consider t h e t y p e

-

i s prime o v e r

r1

realizes

a

= tp(a

7m,m

but

and -I

2

%&$

ep

is p r i n c i p a l over

A

cannot b e p r i n c i p a l over

A

,

3

would be p r i n c i p a l o v e r

t h e lemma.

r

a ).

(r) -

i s prime o v e r

M

A

by Lemma 1.1

.

This

@

=3

T b e a c o u n t a b l e complete t h e o r y which h a s a b i n a r y r e l a t i o n

Let *

=

symbol

R

.

%

For

cp C L

define

1

lcpq = ((Rxy A Ry XI

We say t h a t property

i

d

( 7

E

Rxy A 1 Ryx)) A cp(x)

holds o f

cp

l9 (ii)

1

A

. i

-

i s an e q u i v a l e n c e r e l a t i o n on

kT

cp(y)

i f the f o l l o w i n g s e v e S

conditions a r e s a t i s f i e d :

i .e .

A

cp

.

lqyyl A ~ x y A 1~ y + x RX y A l 11

i s a congruence r e l a t i o n w i t h r e s p e c t t o

~ xy 11

pi

i.e .

X~

Rv ir A 1 Rv v 0 1 1 0

(Rzy A 7 Ryz)) i.e.

(R;

where

z

i s dense on

v A 1 Rv v ) 0 1 1 0

rp/xip

.

*

(ftxz h 7 Rzx) )

where

z

i s a new

(cp(z) /\ (Rzx A 7 Rxz) )

where

z

i s a new

32 ( c p f z )

32

-

i s a new v a r i a b l e

A

variable. i.e.

0

Rv v A 1 Rv v 1 0 0 1

i s "without e n d p o i n t s " on

cp/kip

.

and :

( v i i - ) Either f o r each p r i n c i p a l 1-type a r e a t most f i n i t e l y many .l-types

$

A

t h a t t h e r e i s a 2-ty&

((Rv V A T R v v ) V 1 0 0 1

i n pxq X'Py

o r f o r each p r i n c . i p q l 1-type

containing

p q

there

cp

containing

such

cp

containing

%

v ) 0 1 p

there

a r e a t most f i n i t e l y many 1-types

q

conta

ng

cp

such

1 A

t h a t t h e r e is a 2-type

in

pxq

containing

1 4

Roughly speaking p r o p e r t y of

T

E

h o l d s of

t h e s t r u c t u r e o b t a i n e d by r e s t r i c t i o n t o

Ehrenfeucht example:

R

N

i f i n each model

cp

N

resembles t h e

i s a dense o r d e r i n g o f e q u i v a l e n c e c l a s s e s ,

and p r i n c i p a l t v p e s are almost a r r a n s e d i n a seauence.

~~~~~

A model

of t h e complete t h e o r y

M

-

-

whose language i n c l u g e s

T

t J'

t h e b i n a r y r e l a t i o n symbol

f i n i t e number o f formulae

E

(i) property

is s a i d t o be

R

q0;.

...$I t n

holds of

I n t h i s case w e a l s o say t h a t

q

T

i

L

for

is

1

E-like

i f there are a

such t h a t :

i 5 n

E-like.

-

The Theorem

2.

For t h e remainder of t h i s c h a p t e r we assume t h h t t h e o r y i n t h e lahguage w i t h one b i n a r y r e l a t i o n symbol symbols

.

{a : i C } i

We a l s o assume t h a t

T

i s a complete

T R

and c o n s t a n t

admits e l i m i n a t i o n

*

of q u a n t i f i e r s And t h a t it h a s t h r e e countable models.

Our main r e s u l t i s t h e f o l l o w i n g : There i s

Theorem 2 . 1 such t h a t

1V

iTn

.

h o l d s of . q . i

.

cp

i

nCw

and t h e r e a r e formulae

i s w - c a t e g o r i c a l and f o r each 0

i5n

CPOt-.

.

property

Ll

E

The proof of t h i s theorem r e s t s on t h e f o l l o w i n g two lemmas whose p r o o f s a r e d e f e r r e d t o $ 3 and $ 4 . Lemma 2 . 2

Let

'

Then t h e r e i s a formula

holds of

3

E L

1

qCp

b e c o n t a i n e d i n a nonprdncipal l - t y p e such that

kT cp

-+ Q'

land p r o p e r t y

p

.

E

.

Lemma 2 . 3

There are o n l y f i n i t e l y many n o n p r i n c i p a l l - t y p e s i n .2-

The f o l l o w i n g i s immediate:

T

.

- -

Lemma 2.4'1'1f

tp(b)

i s an n-tuple

i n a model

N

A

7-

. (A)

i s t h e unique n-type

such t h a t Y

v

i

= v

j

~ A i f f b= b i j w

i

v

then

T

,

tp(b.1 I

2

~ b1 . 3b : f o r

By Lemma 2.4 and

Proof of t h e theorem from t h e lemmas, Ryll-Nandzewski's

n-j,~

F A if; N

j

of

theorem t h e r e must b e a n o n p r i n c i p a l 1-type i n

T

.


For

n

where

i s an isomorp&sm

K n

n E o

and f o r

n

n Co

C o n s t r u c t by r e c u r s i o n l

P (0) = 0 n

is

A

.

r e a l l y needed i n Lemma 3 . 4 Example 3 . 1

C

.

where

n = (m : m=n)

isomorphic

L

P (m) = m-1 n

and

The C-generic model i s t h e n

to,Pw> =

The f o l l o w i n g lemma p r o v i d e s a p a r t i a l c o n v e r s e t o Lemma 3.4 and s t r e n g t h e n s t h e c o n n e c t i o n between classes o f - s t r u c t u r e s w i t h AP and t h e o r i e s which admit e l i m i n a t i o n o f q u a n t i f i e r s .

\ Lemma 3.5

be a denumerable s t r u c t u r e s u c h t h a t Th (M)

M

Let

a d m i t s e l i m i n a t i o n o f q u a n t i f i e r s , and s u c h t h a t e v e r y f i n i t e s u b s e t of

i s contained i n a f i n i t e substructure o f

M

c l a s s o f f i n i t e s t r u c t u r e s which c a n b e embedded i n h a s t h e amalgamation p r o p e r t y , and

for

Proof.

Let

i = 0,l

.

A, Bo' in

A

0'

B

I

E C

Since

0

Let M

.

C

be t h e

Then

C

i s 2-generic.

andlet

f

,i

: A - t B

i

b e embeddings

There i s n o l o s s o f " . g e n e r a l i t y i n assuming t h a t

are substructures of

B1 B

A, B

M

,

.

M

A

and

B 1

M

and t h a t

f

0 I are f i n i t e and L

is t h e inclusion o f

i s f i n i t e t h e r e are

open formulas which f i x t h e i r isomorphism t y p e s a s s u b s t r u c t u r e s of But

A,

f

(A)

1

s a t i s f y the same open formula and s o

•’1

e l e m e n t a r y s i n c e Th (M) admits e l i m i n a t i o n o f q u a n t i f i e r s f

-1 1

can b e e x t e n d e d t o a n isomorphismeof

B

1

M.

is i n f a c t

.

But t h e n

and an e x t e n s i o n

B'

1

of

'

A

.

Now

shdws t h a t

(B

0

M

U B ') 1

C

generates a m e m b e r o f

.

his argument a l s o

s a t i s f i e s t h e condtions of Lemma 3 . 1 and i s 4

C-homogeneous,

Therefore

.Z

h a s AP and

M

i s C-generic.

Chapter. 4 Q u a n t i f i e r e l i m i n a b l e graphs 1.

Examples I n t h i s c h a p t e r we apply t h e r e s u l t s of Chapter 3 t o graphs whose

t h e o r i e s admit e l i m i n a t i o n of q u a n t i f i e r .

T h e o r i e s a r e assumed t o be

complete t h e o r i e s w i t h one b i n a r y r e l a t i o n symbol e l i m i n a t i o n of q u a n t i f i e r s . i r r e f l e x i v e , i.e.

model

vv

R

,

and t o admit

We a l s o s t i p u l a t e t h a t each model b e 0

.

1Rv v

0 0

This e n s u r e s t h a t t h e r e i s

o n l y 1-type. I n t h i s s e c t i o n w e s h a l l p r e s e n t s e v e r a l simple examples and some basic definitions. W e f i r s t i n t r o d u c e a b b r e v i a t i o n s f o r some b a s i c formulas.

abbreviations a r e

I,

r,

AO,

Al,

A2'

U

These

d e f i n e d as f o l l o w s :

A xy = df (Rxy A Ryx) V x = y 0 A xy = df (1 Rxy A 1

1 Ryx) V

x = y

S e v e r a l well-known. examples a r e t h e f o l l o w i n g : DO

-

The t h e o r y o f - < Q , o

t h e r a t i o n a l s under t h e u s u a l

ordering. n Ek

-

The t h e o r y of an e q u i v a l e n c e r e l a t i o n w i t h c l a s s e s o f power

GA

-

The

Z(A)

k

, where'

15 n

embedded f o r

n € A c w

.

(n+3)

equivalence

,k5

generic structure for the class

f i n i t e graphs i n which t h e

n

Z(A)

cycle cannot be

of

-

-

DOn

The t h e o r y of t h e d i r e c t produet of a &el a n d ' t h e complete graph on

Given 2.

G

.

of: T

-

"G

U

?

((9.9)

:

.

g f G))

of

DO

.

p o i n t s where

from t h e denumerable model

i s t h e graph with u n i v e r s e

R- = ( G x G ) \ ( R G G

Clearly

n

?'

we may form i t s d u a l

T

*

--

T

and r e l a t i o n

G

-

is t h e t h e o r y of

a l s d admits e l i m i n a t i o n of q u a n t i f i e r s .

DO

G

.

is self dual,

w h i l e t h e d u a l s of t h e o t h e r t h e o r i e s a r e n o t i n c l u d e d i n t h e l i s t . The f o l l o w i n g p r o p o s i t i o n g i v e s a way o f c o n s t r u c t i n g new

-

examples from known ones. Let

Proposition 4 . 1 b e such t h a t

To'T1 Let

(AivOvl

be

b e denumerable models o f

structure (ao = a

((a ,a

1

)

1

C A

~ c I= IA~x~B(

by

0,1,2

A.v v 1 0 1

V

To

A,B

0

i,j,k

T ,T

1

0

and

.

Then

and

(\vovl) T1

respectively.

Define t h e

RC = { ( ( a , b ) ( a l , b l ) ) 0 o f

( b , b ) F RB) V ( ( a o , a l ) 6 R A ) ] 0 1

C R~ A bo = b l ) ) , ]

i n some o r d e r and l e t

Th(C)

.

Define

and

';he

Th(D)

:

structure

D

admit

e l i m i n a t i o n of q u a n t i f i e r s . . 1

Proof:

We p r e s e n t t h e proof t h a t

fiers.

The proof f o r

w e can r e t r i e v e

A

is s i m i l a r .

D

and

Th(C)

B

admits e l i m i n a t i o n o f q u a n t i -

The main i d e a i s t h a t from

i n o r d e r t o c o n s t r u c t enough automorphisms I t s u f f i c e s t o show

i n t h e Wreath p r o d u c t of t h e automorphism groups. A

that i f

a

c,d

C

a r e two sequences o f n-elements o f

C

such t h a t

-

&

1 r

c

and i

;i_

R

for

, mc n

r

cRcm*

and

p

i < n ( Z ' z ~ + ~6 ) ( c B i thesequence

c

1

.

U {cA})

- Z , and

.

I t i s n o t d i f f i c u l t t o extend

I c ~ \(UC

€

B

0' 1

We prove by i n d u c t i o n that

may be extended t o

B

U UB

= UB

be as above.

B

0' 1

0' 1

construGt t h e r e q u i r e d

A,B

and

z

n+l

('0

€ B

so t h e sequence

,z

€

nL1

.

z

n

F B

\ A

1-i

Now

can b e s h o r t e n e d .

-

'

- -

Thus i n each case. t h e s h d r t e s t sequence h a s Ll e n g t h 2 i f Y

That

can be extended t o a l i n e a r o r d e r i s c l e a r .

a

I UC I

We now show by i n d u c t i o n on "Cf '

IC

such t h a t denote

Ic~

on

A

extending

uC ,

0

t h e r e i s t h e 3-type

r

3

j

{v