The number of types in simple theories

ANNALS OF PURE AND APPLIED LOGIC Annals of Pure and Applied

The number

Logic 9X (1999)

69-86

of types in simple theories’ Enrique Casanovas *

Received

27 April 1998; wxived

in revised form

Communicated

I September

1998

by A. Wilkic

Abstract We continue work of Shelah on the cardinality of families of pairwise incompatible types in simple theories obtaining characterizations of simple and supersimple theories. WC develop a local analysis of the number of types in simple theories and we find a new example of a simple unstable theory. @ 1999 Elsevier Science B.V. All rights reserved. ,-lAfS

cbss~jicntion:

03C45

Ke~~~o~ds: Model theory; Simple theories;

Low theories

1. Introduction In [9] Shelah introduced simple theories as a generalization of stable theories where forking has still some nice properties. In Kim’s thesis [2] this class of theories was studied and it was shown that in simple theories independence is symmetric and transitive and forking = dividing. As a consequence a substantial part of the forking machinery developed by Shelah for stable theories is valid in the wider context of simple theories. More surprisingly, Kim and Pillay (see [2, 31) were able to show that simple theories are in fact the class of theories where a notion of independence satisfying some basic propcrtics can bc defined. Morcovcr there arc several cxamplcs of simple unstable theories that are of interest in their own right, mainly the random graph, pseudofinite fields, smoothly approximable structures and algebraically closed fields with a generic automorphism. So it seems now that stable theories are going to be replaced by simple theories in many foundational investigations in pure model theory. The interested reader can find a good overview of the situation in [6]. The present paper is devoted to extending some early work of Shelah about the number of types in simple theories. ’ Partially supported by grant PB94-0854 * E-mail: [email protected].

of DGICYT

016%0072/99/$-see front matter @ 1999 Elsevier PII: SOl68-0072(98)00050-5

and grant HA1996-0131

Science B.V. All rights reserved

E. Casanovusl

70

obtaining

Annals of Pure and Applied Logic 98 (1999)

this way several new characterizations

well as a new example The number

of a simple unstable

of simple and supersimple

q= cp(x,y)

by the condition

theories as

theory.

of types played a major role in Shelah and subsequent

theory, both from a local or a global point of view. Locally of a formula

6946

work in stability

one can define stability

that for any infinite

set A there are at

most (A] many complete q-types over A, i.e, IS,(A)I d IA]. Globally, one can define A-stability of a theory T by the condition that for any set A of cardinality 62 there are at most II many complete n-types over A, i.e, I&(.4)1 63,. Moreover, the classes of theories in the stability hierarchy are usually defined in terms of the number of types. A small number of types corresponds to definability of types, to the nonexistence of certain combinatorial structures like orderings or trees and to the nonexistence of long forking chains. Simple unstable theories have the independence property and therefore for every cardinal /1 there exists a set A with IAl = ,? and IS( = 2’. But there are nonsimple theories with the independence property, so we cannot expect to find a dividing line between simple and nonsimple theories by looking at the cardinality of S(A). However there is a different way of counting types, taking in consideration both the cardinality 2 of the set of parameters and the cardinal@ K of the (partial) types. Shelah discovered in [9] how to use this notion to find a different behavior between simple and nonsimple theories. In [9], Theorem 0.2, we can read: Theorem 1.1. The following conditions on T are equivalent: (1) T is nonsimple;

(2) for every 2, K such that 35” = 2, there are /1”pairwise contradictory m-types of power ICover a set A of cardinality 3,; (3) there is a set A, and a set S of m-types over A ofpower x such that ISI > IAll’l+z+ 21T1+%and no C E 6 realizes > x types from S. A proof can be found in [8], Theorem III 7.7, or, with some simplifications, in [2], Proposition 2.20. The main ingredient in the proof is an application of the Erdiis-Rado theorem. proof.

Many results

in the present

paper are obtained

by developing

ideas of this

In this paper we study the function NT(rc, 2) which assigns to every pair K, J. the supremum of the cardinalities IP] of families P which consists of pairwise incompatible partial types of size
E. CasanovasI Annals of Pure und Applied Logic, 98 ( 1999) 6946 Definition

2.5).

This allows

us to find in Theorem

71

2.7 a local characterization

of

simplicity. The well-known plies NT,(x,

examples

of simple unstable

theories

A) < i. + 2” for every 40. The questions

NT: (K, A)
are supersimple

and this im-

arise of what is the meaning i) > ,! + 2” is possible

is given in Proposition

of in a

3.4. In Section 5

of a simple theory with NT,(K, i) > i + 2” for some cp.ti. A. This

example has a property which seems to be stronger:

the formula

cp divides c( times for

every c(
of a certain kind of trees and to the non existence

of

2. Counting types In the following T is a complete theory with infinite models and L is the language of T. By consistent we always mean consistent with T. We work inside a very big, very saturated model of T, the monster model of T. All the parameters needed belong to the monster model and all the models considered are elementary submodels of the monster model. We use x, y,z, . . , for finite tuples of variables and a, b, c.. . . for finite tuples of parameters. If a is an ordinal number and (b,: i
of the same length,

a -_I! b means

that they have the

same type over A, that is, tp(a/A) = tp(biA). Definition 2.1. Let k>2 be an integer. A set of formulas is called k-inconsistent if all its subsets of k elements are inconsistent. We say that the formula cp= cp(x, y) E L has the tree property with respect to k if there is a sequence (a,\: s E <“‘CO) such that for all s E <“u, {(~(x,a.~-;): i ,2 if there are different a,, (i < CO), such that a; z.1 a for all i and {cp(x, ai): i
72

E. Casanovas/

indiscernible respect

to some k. It follows sequence

A partial

and q(x)

Logic 98 (1999)

6946

over A. It is said that cp(x,a) divides over A if it divides

A-indiscernible sistent.

Annals of’ Pure and Applied

over A if and only if there is an

(a,: i
type p(x)

divides

that qo(.x,a) divides

over A with

divides over A if for some formula

over A. Finally,

a partial

type p(x) forks

q(x),

p(x)

I- q(x)

over A if there are for-

q,(x) such that p(x) b (q,(x) V . V q,,(x)) and each cp;(x) dimulas pi(x),..., vides over A. As shown in [2, 31, forking and dividing are the same thing in simple theories. Definition fOtnl

2.3. Let A = {qi(x, yi):

CJ?i(X,U)or

i EL}.

A A-formula

over A is a formula

1 cp;(x, a) with a E A and i E I. A A-type

over A is a consistent

of the set of

A-formulas over A. A A-type over A is positive if contains only positive A-formulas over A, i.e., formulas of the form Cpi(x,a) with a E A and i E I. In the case A = {cp(x, y)}, we talk of a (positive) q-formula over A. Let K,A be infinite cardinal numbers. For finite A we define NTd(lc,i) as the supremum of the cardinalities lPI of families P which consist of pairwise incompatible A-types of size
/z contains

at most i many

formulas.

Hence

we may always

assume

K
By [8], Lemma 11.2.1 we know that for any finite A we can find a formula $A such that each A-type of size K over A (with /Al 32) is equivalent to a positive &-type of size K over A and every positive *A-type of size K over A is equivalent to a A-type of size K over A. Hence we can transfer many results obtained for NT: to NTd. Finally, which

we define NT(K,A)

consist

cardinality Moreover, KG/t+

of

pairwise

as the supremum incompatible

of the cardinalities

types

of

size


lPl of families

P

over

of

a

set

/2. Now NT(x, 2) d (ITI + i)” and if T is stable NT(K, 2) < 3,1ri. if T is stable in 2, then NT(K, 1) < 2. We can always assume that

ITI.

Our first lemma

shows how to go from chains

and from trees to such chains of formulas.

of formulas

which divide to trees

It will be used many times throughout

the

paper. Lemma 2.1. Let c( be an ordinal number. For (rpi(x, y;): i
and Jar

E Cusclnoc~sIAnnuls

Moreover

in point (cI,,P, I: i-cx). Proof.

We prove

inductively.

(2)

we

Lun

udd

that

,fiv

first (1) + (2) with the moreover

The choice of 0~1 is arbitrary

is limit. Let us consider

7 .J

qf Pure nrd Applied Logic 98 i 1999) 69-86

all ye.\‘t “w, (u,,:,_ ,: i
clause.

We construct

the tree

as is the choice of a,, for 17E ‘COif i < x

the case i + 1 and assume

we have obtained

(a\: s E < ‘co).

Suppose 11t ‘CO.We want to define cl,, , for any I E CO.We can take as an inductive assumption the fact that (~,,~irl: j
,j
Since cp,(x,b,) divides over {b,: j
with respect to IT,, ~,(.x,c,,) di-

vides over { u,/I, _ I : j c i} with respect to M,. Therefore there are (ali /: I t 01) such jI 1, , ,: ,j < i}) and { cp,(.~,cl,! , ): 1<(II} is n,-inconthat tp(a,, ,:~{u,~I,~l: j
Let i.>2~‘~~‘~L~.By compactness

there arc parameters

that for all 17E “i,, { +QN, a,, I(+, ): i < a} is consistent {(Pl(.U. 0, ,):j < ;_} is n,-inconsistent.

By induction

(a,: SE s ‘i)

and for all i
such

and all s t ‘i,

on /)‘< x we can easily

see that

since 1.>2”‘+ IL’, for every such fi
2.4. Let x be an ordinal

number

and let ,4 = { cpl(x. y;):

i < x} be a set

of formulas in the language L. If (h,: i
WC say and for

all i < #a, cpi(x,h,) divides over b,,. If moreover cp;(s, h,) divides over h I always with respect to a fixed k
(6, : i < c() such that (cp(s, 6, ): i < a) is a dividing

respect to k). This property is not symmetric specific separation of variables.

Remark 2.2. Let 43= cp(.~,y) EL. 1. The following are equivalent: (a) cp has the tree property with respect to k. (b) For every n
(r) times with respect to k.

(d) For every x, cp divides x times with respect to k. 2. The following are equivalent: (a) cp has the tree property. (b) cp divides

LC)~times.

(c) For every 1, cp divides

c( times.

chain (with

in X, y, so we should take care on the

E. Casanovas/ Annals of Pure and Applied Logic 98 (1999) 6946

74

Proof.

By Lemma 2.1 and compactness.

The next lemma

0

shows how to get many types from a dividing

The idea of the proof appears in [8], Theorem

chain of formulas.

III 7.7.

Lemma 2.3. Let A = {qi(x, y;): i < tc} be a set of formulas in the language L and assume there exists a dividing chain (cpj(x,b,): i 2”, there exists a family P of 1,”many pairwise incompatible positive A-types of size < 2” over a set of cardinality A. If (cpi(x,bi): i < K) is a dividing chain with respect to 2, the assumption 3,“>2” is not necessary and we obtain types of size < ic. Proof. Assume A<” = 3, and /2” > 2”. By Lemma 2.1 and compactness there are parameters (a,Y:s C’i) and natural numbers (k;: i< K) such that for every i < K and every s E’& {cpi(x, a,Y-i): j < ;1} is &inconsistent and for every q E”& pV = {(P~(x,a,li+, ): i < K} is consistent. If I 2” 3, and IJ,,, pV is consistent, then I is a finitely branching tree, so it can be seen as a subset of ‘w and therefore 111<2”. Thus, if we extend p,, to a maximal consistent union of p,,‘s, we obtain a set of cardinality <2”. Let P be the set of such extensions. Since 1””>2”, 1PI= A”. In case k; =2 we do not need to make the extension Definition

and we put P = {pa: n ~“1).

0

2.5. Let q = cp(x, y) EL. We define the nth conjunction of cp as the formula A q(x, Y,,). The separation of variables made in this

l”\n~(X;Y,,...,Y,,)=cp(x,Yl)A~~’

definition will be understood in what follows. More generally, if A = {(pi(x, yi): i E I} is a set of formulas in the language L, by A A we denote the set of all conjunctions of the form $(x; yi, , . . . , y,, ) = vi, (x, yi, ) A . . . A cpi,,(x, yi,, ) for Jo, . , j, E 1. Thus A{cp(s Y)} = {A,Z cp: n 9 11. The next step is types. This is done present paper. The [8]. Here is where

to show how to get a dividing chain starting from a big family of in the next lemma. It will be applied in different contexts in the proof is an adaptation of parts of the proof of Theorem III 7.7 of the Erdos-Rado theorem appears.

Lemma 2.4. Let A = {(p,(x, yl): i < K} a family of formulas in the language L and assume there is a family P of pairwise incompatible positive A-types of size IAl IAl IAl”, we may assume that for every CI,/?, i, q+(x, yf) = q$(x, ya) = (pi(x, y,). Since K > ITI and J. > 2K, we may assume a” E at’ for all CI,p < /2. We inductively define a sequence (hi: i < ,u) of

E. Casanovas1 Annals

finite subsets hi of K such that for some c(
1E

75

of Pure and Applied Logic 98 11999) 6946 /j,t,,,

hi} with respect to 2. Suppose we have obtained

cp,(.x,a;)

divides

hi for every ,j
Since (Al
Since pr U p/j is inconsistent,

there exists a finite set

h( { CC/I}) C K such that

is inconsistent.

Now A3(2”)+,

to obtain an infinite

so we may use the Erdiis-Rado

set I C 3. (in fact (11a~+)

theorem (2“)~. -

( K~.)t

such that for all pairs {K p}, {~‘,fi’}

taken from I, h({a,~})=h({a’,~‘}). We define h, as the common value h({cc,P}) of all pairs (2. a} in 1. If c( E I, then A, E {I,cp,(x.a;) divides over U,__;{a;: 1 E hi} with respect to 2. This construction is independent of the choice of 1. since for any other b
is from Shelah and it is stated explicitly

in [2], Remark 2.21.

Proposition .formula

2.5. Zf CJJ = cp(x, y) EL has the tree property, then ,fbr some n < 1’). the A,, q~(x; yl.. . , y,,) has the tree property ivith respect to 2.

Proof. Suppose cp has the tree property. Choose K 3 WI regular. By Remark 2.2 cp(s, y) divides ti times. Choose i, such that F” = i. and i” > 1” + 2” (for instance take 1. = J,Jti)). By Lemma 2.3 there exists a family P of pairwise incompatible positive q-types of size 62” over a set A of cardinality 1” such that lPI = i“. Thus (PI > IAl”’ + 2”‘ and we may apply Lemma 2.4 to get a dividing chain with respect to 2 (cpl(x, bi): i < (01) where every qi(.r. y, ) is of the form A,,, y for some n,. Obviously for some n there exists an infinite subset 12 WI such that n = n, for every i E I. Then A,, cp divides respect to 2.

w times with respect U

The next theorem

to 2 and therefore

it has the tree property

shows that what really works to characterize

with

locally the property

of having a small number of types is the negation of a weak form of the tree property: no conjunction of the formula has the tree property. Theorem 2.6. For cp= cp(x, y) EL the following are equitlalent. (1) For all n
ivith respect

(4) For some regular K 3 LOI,for all i,, NT,(ti, j,) d i.“’ + 2“. (5) There are ti and 2 such that i, = 2 SC’iand NT$( K, j.) < I”. Proof. The equivalence between points (1) and (2) follows from Proposition 2.5. By Lemma 2.4 for p = 01 and Remark 2.2, (2) implies (3). Obviously (4) follows from

76

E. CasanovasI Annals qf Pure and Applied Logic 98 (19991 69-86

(3) and (5) follows from (4) (take K>W~ regular and 2=x,(~)). (5) and prove (2). Assume A, CP divides pairwise

some A,, cp has the tree property

o times with respect

incompatible

positive

to 2. By Lemma

A, q-types

a set of cardinality

with respect to 2. Then

2.3 we obtain

each instance

incompatible

positive

Our next result is a local characterization

/1 such

of A, CJ in instances

q-types

i, such that IPI = /2”, which is a contradiction.

P of

a family

of size
that JP(= 3.“. If in each such type we decompose of rp, we obtain a family P of pairwise

Now we assume

of size
of simplicity.

Theorem 2.7. The follo~+zg are equivalent. (1) T is simple. (2) No formula in T has the tree property with respect to 2. (3) For all jnite A, .for all K,A, NT~(K, j.)
Proof.

It is a consequence

of Theorem

A we can code A-types by positive The

equivalence

[9], Theorem

between

2.6 since, by [8] Lemma 11.2.1, for every finite

tiA--types for some $A.

(1)

and

(4)

in

the

0

next

theorem

appears

in

0.2.

Theorem 2.8. The ,following are equivalent. (1) T is simple.

(2) For all %,A, NT(K, i.)bAlrl + 2”. For some regular K>(TI+,for all 1, NT(~,A)d/ll~i +2”. (4) There are K, A such that A:= iv’” and NT(ti, iL) < A”. (3)

Proof.

(2) follows

(,” +2”)lri

from (1) by Theorem

= iiri +2”.

Obviously

rc>jT(+ regular and iti= since NT,(K,I)
I,(K)).

2.7 since NT(rc,i)<

&,,[,

NTf,l(ti,

j_)<

(3) follows from (2) and (4) follows from (3) (take The implication

(4)+

(1) follows from Theorem

2.7

U

As observed in [2], Remark 2.18, if T is not simple, then some 40= cp(x, v) where x is a single variable has the tree property. For the same reason we can restrict ourselves in Theorem 2.7 and Theorem 2.8 to types in a single variable.

3. Characterizing

supersimplicity

Definition 3.1. The theory T is supersimple if for any p E S(A) there exists a finite subset A0 CA such that p does not fork over Ao.

Supersimple supersimple

theories

are introduced

theories are just superstable

T is supersimple

theories. It is implicit

if and only if there is no dividing

This will be used in what follows. dividing

in [2, 51. They are simple.

Moreover

stable

in Chapter V of [2] that

chain of formulas

of length

CO.

The next lemma states that it is enough to look at

chains with respect to 2.

Lemma 3.1. !f there exists a dividing chain (cp,(x. a, ): i < co), then there esists trlso LI dividing chuin ($,(.Y, 6,): i < CO)M+threspect to 2. Moreover. if A = { cp,(.u,J’,): ; <(I)} HY ccm jnd the ,formulas $,(x,z,) in /j A. Proof.

Suppose the formulas cp,(x. a;) arc given and let i. > 21’ I-?” be a cardinal number

such that i < j.“‘. By Lemma 2.3 we get a family P of i.” many pairwise incompatible positive /l-types of size 62”’ over a set A of power i. The rest follows from Lemma 2.4. since we have IPl = 2.“‘> iA( ‘ “’ + 2“ for K = 2”. 2 It is well known that superstable types: a theory T is superstable we give an analogous

theories can be defined in terms of the number

if for every A, IS(A)] 621”

characterization

of supersimple

of

+ Iill. In the next theorem

theories.

Theorem 3.2. The ,follon.irugare equir&~tlt. ( 1) T is .supersimple. (2) For all ti. i., NT(k-.J.)<2~r1’ ” + i. (3) For sotnc ti, for all A, NT( K, A) < 21T’+” + i.. (4) For smw K, i,. NT( K, j_) < 2”. Proof. ( 1) =A (2). Assume we have a family P which consists of pairwise incompatible types of size 21’~+‘~ + j.. We can apply Lemma 2.4 to obtain a dividing chain of length o. But this contradicts the supersimplicity

of T.

(3) follows directly from (2) and (4) follows from (3). We prove (4) 3 ( I ). Suppose 7’ is not supersimple. This implies the cxistcnce of a dividing sequence (cp,(.u. h, ): i < (II). By Lemma 3.1 we may assume that in fact cp,(~, 6, ) divides over b,,

with respect to 2.

By Lemma 2.3 we get a family P of i.’ ’ many pairwise incompatible over a set of cardinality 3.. This contradicts (4). ~~1

types of size < (‘1

By the same arguments given in the end of the last section, Theorem 3.2 the function NT( K, j.) to types in a single variable. Example

3.3. The theory

a unstable supersimple disjoint sets A. B AR(x.i)A ii I

of the random

theory.

A lR(x,i) ICB

we could restrict

graph is the most well-known

cxamplc

in

of

It is the theory of a graph R such that for any finite

.

E. CasanovaslAnnals

78

The number directly.

6946

of types in this theory is given by NT(lc, 1.) = 2” + 1. We can check it

On the one hand, we have 2 many pairwise

a set of cardinality incompatible

incompatible

algebraic

types over

2. On the other hand, it is easy to find a family of 2” many pairwise

nonalgebraic

we consider

types of power K over a set A of power K: for any B C:A

the type

ps(x)={R(x,a): Finally,

of Pure and Applied Logic 98 (1999)

aEB}U{~R(x,a):

a~A\B}U{xfa:

UEA}.

we show that if P is a family of pairwise incompatible

nonalgebraic

types over

a set of power A and each p E P is of power
{j
aj EAT} +

2 be the mapping

defined

by: f(j)=0

if and only if

R(x,aj) E pI. Each f; belongs to the collection Fn(/2,2, K+) of all partial mappings from a subset of j_ of power 2”. This means that, using Theorem 3.2, we have a modeltheoretic proof that Fn(& 2, K+) has the (2”)+-chain condition. If we try to eliminate the model theory in the proof and we look only to the combinatorics, we discover a quick proof of this fact from the Erdiis-Rado theorem. This proof is as follows. Assume IS a family of pairwise incompatible functions in Fn(3,, 2, K’). Write {fi: i<(2”)+} fi={($,f,(c$)): j<~}. If j,l<~ and d={j,Z}, we put ,Ad=fi r{c$,c(;}. Given different i, i’ < (2’)+, compatible

with j$”

there are always two ordinals j,,,,, I,,;!
theorem

(2”)+

-+

(K+);, there is an infinite Z C (2”)+ (in fact 1112 K+) and a two-element set A such that Aj, if = A for any different i, i’ E I. But it is clear that this is not possible if 111~4. This ends the proof. Observe finally that the fact that Fn(& 2, p) has the (2<“)+-chain condition follows easily from the case p = K+. Unlike superstability, supersimplicity has some consequences in the number types. The general bound for simple theories given in Theorem 2.6 is NT,+(K,~)@’

of local

+ 2h.

One can ask whether the exponentiation and prove that in fact

1_“’ is really necessary

or one can do better

E. Casanoonsl Annals of‘ Pure wd Applied Logic 98 (1999)

After Theorem

79

6946

3.2 we know that this is really the case for supersimple

theories. On the

other hand, if we start from a stable theory then we have (as shown in the next section) that NTi( ti, A) 6 j.. Therefore superstability In the next proposition global whether

characterization

of simplicity

the condition

given in Theorem

or not the case NT;(K, i) > /i + 2” is possible

power i ” occurring

in the upper bound

level. In Section 5 we give an example NT:(o), iL) = jJ” for any /1. Proposition

3.4.

(1) For all ti,k

of a simple

are equirahnt

NT;(K,A)<~.

+ 2”.

(3) For some ti,k,

on NT: ( IL i 1.

NT,: (K, i)
2.8 cannot in a simple

could hide the presence

The ,following

(2) For some I;, ,for all i,, NT:(~,i.)bi,

(4)

does not add new information

we try to characterize

theory

be used to decide theory since the of jb”’ at the local

and a formula

cp with

.fkw cp= cp(.x,y) EL.

+ 2“.

NTz(ti,A)
There are no dividing chains of the ,form (A,,, rp(x, b,): i < w).

(5) There are no diaiding chains b$Ytizrespect

to 2 sf the ,form (A,, cp(.r.b,): i<(j)).

Proof. (I)=+ (2) and (2) + (3) are clear. (1) =+ (4). Suppose (4) is false and take K = 2”’ and Ab>2“ such that I”’ > i.. types of size By Lemma 2.3 we get a family of 3.“’ many pairwise incompatible ~2”’ over a set of power 3.. Each formula cp-formulas. NTG(ti,i)=

in these types is a conjunction

of positive

If we consider now the types consisting of these cp-formulas we see that /_“‘>i+2”. (4) =+ (5) is clear and (3) + (5) is also clear by Lemma 2.3.

We prove now (5) + (1). Let P be a family of pairwise incompatible positive (P-types of size IAl + 2“. Lemma 2.4 gives us a dividing chain with respect to 2 of length COmade of instances ‘This contradicts (5). 0

of conjunctions

of up.

4. The stable case The functions NT,,(ti, 1”) and NT(ti, i) can also be used to characterize the more basic stability classes. But NT,(ti, i) does not give information about the stability of cp. It is easy to find an unstable cp with NT,( ti, A) < i, for all K, i.. The relation R(.r, J’) in the random graph is such an example. Recall that a formula cp= C&X,y) is stable if for every infinite set A there arc at most (Al (p-types over A, i.e., I.S,,,(A)l< 1~1. This is equivalent to the nonexistence of a sequence of parameters (a,: s E <“‘2) such that for every q ~“‘2, { cp(x, a,,~,)““‘: i t: PI} is consistent, where $” = $ and $’ = T$. Therefore, by compactness, if cp is not stable then for all 1, such that 1. = 2 < ’ there exists a set A of cardinality i. such that I%(A )I = 2’.

SO

E. CusanovaslAnnals

Proposition

of Pure and Applied Logic 98 (1999)

69.46

4.1. The following are equivalent for 40= cp(x,y) EL.

(1) cp is stable. (2) For all IC,/~,NT,(lc,/2)61.. (3) For some L such that /1=2<“,for Proof. By the previous

remarks.

all K, NT,(rc,1.)<2”.

0

4.2. ( 1) T is stable in 3, if and only if for all K, NT(K, 1”)= 1. The ,following are equivalent: (i) T is stable. (ii) For all ti,i, NT(~,1)<‘3.1”1. (iii) For some 3, such that 2
Proposition (2)

T is stable in J if and only if for any set A of cardinality 3,, IS(A)/ 6iL. This condition is clearly equivalent to NT(>_ + ITI, n) di. The equivalence (i) H (ii) in (2) follows from the fact that T is stable if and only if for any set A of

Proof.

(1) By definition,

cardinality /2, IS(A)/, < 1 lrl. The equivalence (i) H (iii) follows from Proposition 4.1 and the fact that T is stable if and only if every cp is stable in T. (3) follows from the fact that T is superstable if and only if for any set A of cardinality i,, IS( 621’1 $2”. Finally

(4) follows

for every 2.

from (1) and the fact that a countable

cd-stable theory is %-stable

0

In [2], Remark 2.21, it is shown what amounts to (with our notation) the following characterization of stability: T is stable if and only if there are K, A such that jU= ;1<’ and NT(&/I)
4.3. For countable T, T is stable tf and only if NT(K, 2) is independent

from K, i.e., .for all injinite ICI,~1, I_, NT(KI, A) =NT(tiz,

2).

Proof. Assume first T is stable. By Proposition 4.2. for all K, 2, NT(K, 3,)
K,/z, NT(K,,?)=i"'.

If T is superstable,

then by Proposition

4.2. for all ~,i,

NT(K, 1) < 2’” + L If moreover T is not o-stable then there exists a countable set A such that IS( = 2”’ and therefore for all K,& NT(K, %)=2”’ + 3.. Finally, if T is o-stable, then by Proposition 4.2. for all 1c.1, NT(K,/2) = 2. Assume now that NT(rc,;l) is independent from K. Then for all K, A, NT(lc, A) = NT(o, 3.) d 2”‘. By Proposition 4.2. T is stable. 0

5. An example In this last section we present an example of a simple T with a formula cp that divides w times. Therefore NT~(K, iL) > /2’“. In fact the formula CPdivides x times for

81

E. Cusanoous/ Annals qf Pnrr and Applied Logic 98 i 1999) 69-86

every

ordinal

Remark

CI
2.2, a theory

As mentioned

This is maximal T is simple

in the introduction

The language

L of T consists

for a formula

in a simple

if and only if no formula T is a nonlow

theory,

in T divides

since

by

(III times.

theory in the sense of [I].

of a binary predicate

R and unary predicates

I, P and

Z,, for n E cl),, n 3 1. The axioms of T are as follows: (1) The universe

is the disjoint

union of I and P.

(2 ) I,, C I and I,, n I,,, = 0 for n # m. (3) RCZxP. (4) Mp E P 3Yi E I,, R(i, p). (5) Each Z,, is infinite. (6) If A, B are finite disjoint

~PEP

AR(i,p)A

! /E.I

A

-R(i,p)

.

IEB

(7) If A, B are finite disjoint

A R(i,p)A pt.4

subsets of I and for all n 3 1, IA n I,, 1
subsets of P, then:

A -R(i,p) /EK

Notice that the axioms imply that P is infinite. Observe also that axioms (6) can be written in first-order language in the following way: if nl < “’
It is preserved

under union of

chains. We leave it to the reader to check that all the axioms hold in any model of TO which is existentially closed for To. From the completeness is the theory of all existentially closed models of To. Proposition

5.1. The given list of axioms

is complete

of T it will follow that T

and the mlgehrrtic closure in T

is given his

Moreover,

if A is algehraicaliy

(dent in T(A) Proof.

to a quanti$er-free

We give a back-and-forth

cl(/+AuU{i~Z,,:

3arA

closed ecery formula formula

over A.

argument

for proving

in one vauiuble ovet’ A is eyuiv-

the completeness.

Let

R(i,u)}.

In a model of T we call the elements of I indices and the elements of P points. Let us call a set A closed if cl(A) = A. We work between wI -saturated models and with

82

E. Casanovas1 Annals of Pure and Applied Logic 98 (1999)

countable

partial

isomorphisms

(for the language

range. Let f be such a partial isomorphism. point or index. Let A = dom f. countable

R(i,p)},

these sets are countable.

Let us consider

Q’= {p~AflP:

-R(i,p)}

G(x) = {I(x) A ~Zn(x>: n 2 1) U {R(x,f(p)):

The consistency

~~Q’Mxff(j):

I,. All

realizing Q}

~EJ>.

of Q(x) follows by compactness

B, C finite disjoint

P E

since I,, is infinite, A is now the case i EI\ U, Z,.

and J=(AnI)\U,

What we need is to find f(i)

U{lR(xtf(p)):

and

A is closed. To add a new index i E I,

which is always possible

and the models are wi-saturated.

Let Q= {~EA~P:

with closed domain

We show how to extend f adding a new

By hypothesis

it is enough to take ,f(i) E In\,f(A),

R, P,I,l,,)

6946

since our axioms prove that for any

subsets of P and any n,

3”i E Z\(Il U. . .UI,,)

A

R(i,p)A

A

lR(Lp)

pEC

( PEB

. 1

By cul-saturation we can realize Q(x). Finally, we consider the case of adding a new point p E P. Let J be the (countable) set of all indices i E U,, I, such that R(i, p). By what we have shown we can add all these indices to f. K’={iE(AnI)\U,Z,7:

So we may assume J CrA. Let K = {i E (A n I)\ U, I,: 1 R(i, p)} and Q = A n P. We need to realize

@(u)={P(u)}U{R(f(i),u): U {U#f(q): The consistency

3”p~P i

iEK’}

qEQ,.

of Q(u) follows by compactness

B, C finite disjoint

Again

iEJUK}U{lR(f(i),u):

R(i, p)},

since the theory proves that for any

subsets of I such that for every n, IA nI,, 1
AR(i,p)A

A -R(i,p)

;EB

IEC

by oi-saturation

i

we can take as f(p)

a realization

of Q(u).

Observe

that

f U {(p, f (p))} is closed. This ends the proof of the completeness. It is clear that cl(A) 5 acl(A). Assume A = cl(A). To show that we have quantifier elimination in T(A) for formulas in one variable it is enough to prove that if two objects a, b have the same atomic type over A they have the same type over A. Without loss of generality, A is countable. Looking at the proof of completeness we can see that the mapping which is the identity on A and takes a to b is a partial isomorphism that can be extended to a countable partial isomorphism f’ with closed domain and range. This is easy if a,b ~1: f’= f U {(qb)}. In case a, b E P we need first to extend J to a closed partial isomorphism f * by adding all i E U,, Z, such that R(i, a) (and all j E l_l, I,? such that R(j, b)) and then we put f’ = f * U {(a, b)}. Since f’ belongs to the family of partial isomorphisms considered above, it follows that f’ is an elementary

E Cusanovasl

mapping noticing

sf Pure and Applied Loyic 98 (1999) 69-86

Annals

and thus tp(a/A)=

tp(b/A).

many objects with the same atomic

0

To get full quantifier

elimination

we need to add one more kind of definable relations: ,,i

I ?...1

1

Now we can finish the proof of cl(A)=acl(A

that for any a $ cl(A) there are infinitely

type over cl(A) as a.

S nw~‘.n,h~p

83

p,n,q,

,...,

/pw,

4d)+@~%I,,

i

Hi p,)A

A

/=I

-R(i,q,)

/= I

1

But we do not need to use this fact in what follows. Proposition

5.2. T has elimination

of quantifiers

in the hnguuge

R, P, I, I,,. S’ir.nr’.‘i-i.

Proof. Argue as in the proof of Proposition 5.1 working in 01 -saturated models now with finite partial isomorphisms with respect to the language R, P.Z, I,,,Sf7’,nr’.f’.i, and with not necessarily

closed domain.

17

Our next goal is to show that T is simple. By [5], Theorem 4.2, to do that, it suffices to show that there exists an property over a model (also we will prove that T has the set. By [4], Corollary 9, this Observe Proposition

independence relation in T with the jLee amalgamation called the independence theorem over u model). In fact, free amalgamation property over any algebraically closed implies that in T Lascar strong type equals strong type.

that after the characterization 5.2, we know that

of the algebraic

closure

in

T given

in

acl(A) = U acl(a). 1IE.1 Proposition

5.3. T is simple.

Proof. We define: A J, B if and only if acl(A) (‘ This condition

n acl(B) C_acl(C).

is here equivalent

erties are easy to check: 1. Invariance under automorphisms: .L,.,,., 2. Local CCB 3. Finite

to acl(AC)nacl(BC)

C acl(C).

if J’ is an automorphism

The following

prop-

and A -1, B, then j(A)

f(B). character: for every finite tuple a and every set B there is a countable subset such that aL,B. character: If a is a finite mple and for all finite tuples b in B. a _L,, b, then

aLcB. 4. Monotonicity: if A C B C C and D J,,4 C, then D _1, C. 5. Transitivity: if A C B C C, D _L, B and D _1, C, then D -1, C. 6. Symmetry: if A -1, B, then B _L, A.

E. Casanovasl Annals of’ Pure and Applied Logic 98 11999) 69-86

It remains

only to prove that L has the two following

1. Extension: a=Ba’

properties:

for all sets B 5 C, for every finite tuple a there is a tuple a’ such that

and a/LB C.

2. Free amalgamation

over algebraically

closed sets: for every two finite tuples a, b

and every A, B, C such that C LA n B, A _1, B, C = acl(C), b L,

a --(‘ b, a .J,, A and

B, there exists a tuple c such that c =A a. c GB b and c -1, AB.

The extension

property

follows from the well-known

fact that in any theory if A, B, C

are sets such that A n acl( C) = 0, we can find always A’ such that A fc A’ and A’ n acl (BC) = 0. We

will

prove

in the

next

lemmas

the

free

amalgamation

property

holds too. Lemma 5.4. It is enough to prove the free amalgamation closed sets for elements, i.e., for tuples of length 1.

property

over algebraically

Proof. Assume the free amalgamation property over algebraically closed sets holds for elements. We show it holds for any closed sequence Z= (a;: i< n), i.e., a sequence such that for every i
the case CI+ 1. Let Z= (ai:

i c x) and b= (bi: i
closed sequences and assume a, b are elements such that acl(a) C: {a} U {ai: i sequence

C and an element

ZFC b, 5-1,

A and “L,

c such that Cc --A Za,

B, by the inductive

that c zA a, C-B b and Z-L, AB. Let dl,dI

Cc

_Bbb,

hypothesis

and Cc L, AB.

there is a sequence

Since C such

be such that ?d, -_A Za and Ed2 5~3 bb.

Then dl =crdz, dl J_,L,,A and d2 _L,, AB. Since CC is algebraically closed and clearly AZ J,,, Bc, we can apply the assumption to obtain an element d such that d s.4~ d 1, d--B; dz and d _L,, AB. It follows that cd m aa,cd fB bb, and cd -1,. AB. Lemma 5.5. The free amalgamation

property

over algebraically

0

closed sets holds.

Proof. By the previous lemma we only need to show it for elements. Let C = acl(C), C CA n B, A J,, B, and let a, b elements such that a -c b, a _L, A and b & B. In our theory it is enough to show that there is c such that c =_Aa and c Eg b, since it follows from this that c _L, AB. Without loss of generality, A, B are algebraically closed and a, b @C. Hence a @A and b $!B. Again we call indices the elements of I and points the elements of P. Let us suppose first a E Z,,. Then b E I,, Observe that for every point PEA,

1 R(a, p) and for every point p E B, 1 R(b, p). Then @(x)={I,,(x)}U{~R(x,p):

pEAUB}U{x#i:

iEAUB}

is a consistent type and if i is a realization of Q(x), then i zA a and i =_B b. Assume now a is an index and a E I\ U,, I,,. Then b E I\ U, I,,. Since A J_, B, the sets A\C and

85

B\C

are disjoint.

Since a ZE(‘b, this implies

Q(x) = {Z(s) A U{~R(x.

7

Z,,(x): n 3 I}

p): -R(a,

that

u {R(x. p): R(u. p). p t A}

p), p E A} u {R(x. p): R(b. p), p E B}

L{ 7 IQ, p): 7 R(b, p). p E B} u {Ax# i: i E A u B} is a consistent

type. It is enough

to be considered

to take as c any realization

of Q(x). The last case

is that a is a point. Then b is also a point. All the indices

of LI in

A n (J,,I,, belong to C and all the indices of b in B n IJ,, I,, belong to C. Thus they are the same ones. Moreover, the indices in A\C and the indices in B\C are all different. Therefore. @(tr) = {P(u)} U{R(i,u): U{R(i,u):

is a consistent Proposition cYI < toI.

iEA.R(i,u)}

U {-R(i,u):

iEB,R(i,b)}U{TR(i,u):

type. We take as c a realization

5.6. There

is a fovmulu

cp(x, y)

itB,

in A, lR(i,a)} -R(i.b)}U{u#p:

of Q(u).

ptAlJB)

-J

in T ~~hich dicides

r times jkw rwr:\~

Proof. The formula cp(u,_~) =R(x,u) divides w times: if we choose i,, E I,,, it follows that {R(i,,,u): n> l} is consistent and each R(i,,,u) divides over {i,li: m 1 there is at most one type p, such that P,(X) tin(x) and that there are <2” types p, such that pi(x) kI(x) and p,(x) k T/,,(X) for every n. This gives the result p<;““’ + 2”. Consider first the types pi with pi F P(x). For any i

.f;(n)={~~Anz,,_~:

p;kR(a,x)}.

86

E.

Casanovas1 Annals of Pure and Applied Logic 98 (1999) 6946

Clearly there are d 2”’ many such mappings

C A f? I,,+ I of
chooses for every n a subset f(n) pi such that pi E P(x) {R(a,x):

and fi = f.

aErngf}U

1 elements.

which

There are <2” many

Th e reason is that each such pi is axiomatized

{4?(a,x):

and formulas of the form R(a,x) partial mapping g, with domain

J;. Let f: w -+ [A] <(’ be a mapping

aE (Li;nu,7zH)

\mg1.}

U{x#a:

by

UEA,}

and 1 R(b,x) with a, b E (A; n Z)\ U,l Z,. Consider the (A; n z)\ U, Z,, and such that gi(u)=O if piFR(u,x)

and gi(u) = 1 if pi k -R(u,x). It is easy to check that if pi, pj are incompatible then the partial mappings gi,gj are incompatible too. By the (2”)+-chain condition of such mappings

(see [7, Lemma VII.6. lo]) we conclude

that there are only 62”’ many types

pi with h = f. Fix now n > 1 and assume p; t-Z,,(x). Since pL is not algebraic, for any a E Ai. Thus p; is axiomatized {zn(x)}U{~R(x,u):

uEA;}U{x#u:

UEA;}.

If p,i is another such type then p, U p, is consistent. type in P. Consider

finally

the types pi such that pi FZ(x)

Each such pi is axiomatized {Z(x)}U{~l,,(x):

we have pi k 1 R(x, a)

by

Hence there is at most one such and pi k TZ~(X) for every n 3 1.

by

n>l}U{x#u:

uEAi}

with u,b E PnAi. Again by a and formulas of the form R(x,u) and 7 R(x,b) (2”)+-chain condition argument we can conclude that there are only ~2” many such pairwise inconsistent types.

Acknowledgements Thanks to D. Lascar, A. Pillay and M. Ziegler version

for helpful comments

on a previous

of this paper.

References [l] [Z] [3] [4] [S] [6] [7] [8] [9]

S. B. B. B. B. B. K. S. S.

Buechler, Lascar strong types in some simple theories, J. Symbolic Logic, to appear. Kim, Simple first order theories, Ph.D. Thesis, University of Notre Dame, Indiana, 1996. Kim, Forking in simple unstable theories, J. London Mathematical Society, to appear. Kim, A note on Lascar strong types in simple theories, J. Symbolic Logic, 63 (1998) 926-936 Kim, A. Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997) 149-164. Kim, A. Pillay, From stability to simplicity, The Bull. Symbolic Logic 4 (1998) 17-36. Kunen, Set Theory, an Introduction to Independence Proofs, North-Holland, Amsterdam, 1980. Shelah, Classification Theory, North-Holland, Amsterdam, 1978. Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980) 1777203.