Introduction to Forking

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An introduction to forking

Daniel Lascar and Bruno Poizat

The Journal of Symbolic Logic / Volume 44 / Issue 03 / September 1979, pp 330 - 350DOI: 10.2307/2273127, Published online: 12 March 2014

Link to this article: http://journals.cambridge.org/abstract_S002248120004826X

How to cite this article:Daniel Lascar and Bruno Poizat (1979). An introduction to forking . The Journal of Symbolic Logic,44, pp 330-350 doi:10.2307/2273127

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T H E J O U R N A L OF SYMBOLIC L O G I C

Vo lume 44, Number 3, Sept. 1979

AN INTRODUCTION TO FORKING

DANIEL LASCAR a n d BRUNO POIZAT

The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [SI]. The principal aim of this paper is to show that it is an easy and natural notion.

Consider some well-known examples of x0-stable theories: vector spaces over Q, algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if s/ £ 88 are subsets of some model Ji and c a point of this model, the fact that the type of c over 88 does not fork over si means that there are no more relations of dependence between c and 86 than there already existed between c and si. In the case of the vector spaces, this means that c is in the space generated by 88 only if it is already in the space generated by si. In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in s/ and 88 have the same order. Of course, these notions of dependence are essential for the study of the'above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case.

What we have to do is the following. Assuming T stable and given si £ 88 and p a type on si, we want to distinguish among the extensions of p to 88 some of them that we shall call the nonforking extensions of p.

In the case where si is a model of T, this is particularly simple, even without assumption of stability. We introduce the following preorder which we call fundamental, on types on models of T: p > q if p can be extended to a type which is isomorphic to an ultrapower of q. In §2 we study this order and we prove that it is complete (Proposition 2.5). In §3 we use it to define two kinds of privileged extensions of p: the heir, which is a generalization of the corresponding concept in [LI], and the dual notion of coheir. In §4, definable types are proved to be exactly those which have only one heir. A type is stable if all its extensions are definable. If p is a stable type on Jl, its heir is the unique extension of/? which is equivalent for the fundamental order (Theorem 4.6). This is an unexpected property because the definition of heir makes use of the parameters in „//, by contrast with the definition of the fundamental order. Another important property is that a stable type has only one coheir, which is its heir (Theorem 4.9). In this particular case, the heir will turn out to be the only nonforking extension of p.

Let us turn now to the case where si is not necessarily a model of T. Given si £ Jl and p € S(s/), we single out among the extensions of p to ..// those that

R e c e i v e d Oc tobe r 10, 1977.

330

© 1979 Association for Symbolic Logic O022-4812/79/44O3-0006/SO6.25

AN INTRODUCTION TO FORKING 331

are maximal for the fundamental order. If p is stable, all these extensions are equivalent (Theorem 5.1); we shall call the bound of p the class of these maximal extensions. A nonforking extension of p is one which has the same bound. Everything else follows from this.

For expository purposes, we carry out in §5 the program outlined above in the particular case where the theory T is stable. For a stable type within an unstable theory, we must prove first a technical proposition (Proposition 7.1) which insures that only the stable parameters are relevant. This proposition, together with the development in §6 enables us to carry out the program outlined in full generality, which we do in §7. Finally, §8 is devoted to a brief study of superstable types. We prove that in this case, the fundamental order is wellfounded and that its foundation rank is the rank U defined in [LI].

None of the results on forking itself are new (they were all proved first by Shelah), but our introduction of forking is entirely different; it seems to us that it is more understandable and conceptually more satisfactory. We introduce a number of new notions that we believe are interesting in their own right, and which allow us to have a good insight into the behaviour of stable types.

The reader of this paper is supposed to have only a general background information in model theory, as can be found in the textbook of Chang and Keisler [CK]. No special knowledge of stability is assumed.

We want to thank M. Dickmann for a close reading of the manuscript and for helping us with our stumbling English.

§1. Notation. In this paper, T will be a complete theory in a language L. To avoid useless complication, we shall suppose L countable; the theorems which involve cardinal computation are stated under this assumption, but they can be easily adapted to the case where L is nondenumerable.

Characters such as..//, J/\ ... will always denote models of T, and the universes of these models will be denoted by M, M\ .... A subset of M will be a pair (A, M~) where A <=, M. Subsets of models of T will be denoted by characters like s/, 33, V, ... and A, B, C, ... will be the first elements of these pairs. If si = (A,..//), we set T(sJ) = {<p e L(A); Jl |= <p).

The notation s/ S 0} means that A ^ B and T(s/) £ T(@) ; J{ is identified with (A/, J/). We say that / is a monomorphism from sd into @ i f / is a map from A into B such that for all <p e L(A), <p e T(s-f) if and only iff(<p) e T{@) (f(<p) is the formula obtained from cp by replacing each ae A occuring in <p by/(a)). So sJ £ Si means that A £ B and that the inclusion map is a monomorphism.

For us, a 1-type over sJ is a complete theory in L{A ij {x}) including T(s<J), where x is a new individual constant. In the same way we define n-types and even bigger types: for any set /, we define the /-types as being the complete theories in L(A IJ {*,; /€ /}) containing T(jtf). In each of these cases, we define a topology on the set of types in the usual way: a basis for the open sets will be {; where = {pe S,{sJ);p f- cp}- If P e SiCsO we introduce sometimes a point cp which realizes p over s/. ltp is a monomorphism from sf to :%, and p e S-^SS), then f(p) = {(p;cpe L(A U {x0}); f(cp)ep}. It should be clear that/0>) e S ^ ) .

332 DANIEL LASCAR AND BRUNO POIZAT

In order to avoid complicated notations, we state and prove the theorems for 1-types. But, in fact exactly the same proof will give us the theorems for /-types (minor and obvious changes are needed when cardinality computation are involved). We shall shamelessly use these generalizations when necessary, even if they are not stated explicitly.

If A is a set, we shall denote by A the set of all finite sequences from A ; jj/4|j is the least cardinal bigger than that of A and x0. The symbol v„ will denote the sequence (v0, vx, ..., vn-{); L„ is the set of formulas whose only free variables are V0, Vj, . . . , V*_i.

§2. The fundamental order. In this section, we will be concerned with types over models of T.

DEFINITION 2.1. Let p e S\{M) and qe Si(Jr). We say that p > q if for all <p(x, v„) e Ln{x) and a e M" such that p (— cp(x. a), there is a b e TV" such that q \— (p{x, b).

Let us say that cp(x, v„) is represented in p if there is some a e M" such that p \— <p(x, a). Thus, p > q if all formulas represented inp are also represented in q.

Obviously, the relation > is a preorder on the class of 1-types over models of T. If we set p ~ q when p > q and q > p, we obtain an equivalence relation whose classes will be called classes of types. Of course, > induces an order on the classes of types. Since a class of types is determined by the set of formulas represented in the types belonging to it, there are no more than 2K° classes.

For types over models containing some fixed set si, we can define the preorder ># in the same way, but by looking at the formulas <p(x, v„) of L{A \J {x}). Thus > ^ and ~ # are finer than > and ~ .

It is quite obvious that if p and q are comparable for > , then p \ 0 = q \ 0 . In the same way, if they are comparable for > sl, then p f s? — q t sZ. Also. If Jt < JV, p e Si(J/), q e S^JV) and p c q, then p > q (and even p >Mq).

As an exercise, let us look at the minimal classes. It is clear that p e Si(.//) is realized in Jl if and only if the formula x — v0 is represented in p. So, the minimal types are realized.

Now, if p e Si(J/) and q e Sx(JV~) are realized by a e M and be N respectively, and if p f 0 = q \ 0 , then p ~ q. Indeed, the same formulas are represented in p and q: if for some c e M", p (— <p{x, c), then Jl t= 3\„ cp(a, v„) and A" (= 3v„<p(b, v„), and for some c' e N", q \— <p(x, c').

This means that the minimal classes are in one-to-one correspondence with the types over the empty set.

In short, if p e SX(J/) and p \— 3vp(x, v), such a sequence v may already exist in the model J/, or not; the existence of such a sequence insures a lower place for p in the fundamental order.

EXAMPLE. Let T be the theory of dense linear orderings without endpoints. Let Jl be a model of J and p e S-^Jl). Then p is entirely determined by the following sets.

Xp = {a; a e M and p \- x = a),

Yp = {a;aeM and p \— x < a}.


We can distinguish several families of types. (1) Xp # 0 : p is realized in Jl. (2) Xp = 0 and there exists c0 e M such that y , = {a; a e M and a0 < a}. Let us call this type a$. {!') Xp = 0 and there exuts a$eM such that 5^ = {a; ae M and a0 < a}. Let us call this type OQ. Q) Xp= 0,YP* 0,YP* M, and we are not in family (2) or (2'). (4) Yp = 0 ; s e t /> = +co. (5) y# = M ; s e t p = - o o . The types in family (2) or (2') will be called rational types, those in family (3),

irrational types. It is easy to see that + oo and — oo are maximal and incomparable; below

them are the rationals and the irrationals in the same class; then the types which are realized are in the same minimal class.

REMARK. It follows from the fact that the language is countable that every set wellordered by < or by > is countable.

DEFINITION 2.2. Let ^ be an ultrafilter on a set I and for each ;' e 7, p{ e Sx{Jli). Then Up,-/® >s the type over WJl^ defined by: for all <j>{x,vn) e L{x) and a e (ft.//;/®)", ViPi\°U \-<p(x, a) if and only if

The fact that this is a type is obvious. It could be defined as the type realized by (c,)lC/ over \\J/il<%, where each c,- realizes/>, over .,//,-. If the „//,• are all equal to Jl, and the pt all equal top , we will write M* instead of Jll\Hl and/?* instead of pJj%.

THEOREM 2.3. Let p e Si(Jl) and q € Sx{Jr). Thenp > q if and only if there exist an ultrafilter <>U, J'"' < J/~%, and an isomorphism from Jl onto J/" which maps q* r Jl" onto p.

PROOF. First, from the definition of q* it is clear that if a formula is represented in #*, it is already represented in q. So q* ~ q.

Suppose the second condition true. Then q* \ Jr' > q, and of course p ~ q* t Jr'. Sop > q.

Suppose now p > q. Let / be the set of finite subsets of p, and tfl an ultrafilter which contains the sets {s; s e I and "* such that p f— cp(x, a) implies qw \— <p{x, cs). Then, we will be done, because the complete diagram of Jl is included in p, so Jl" = {cb; be M} is an elementary submodel of Jr<iu, and the map b •*» cb is an elementary map from Jl onto Jr' which carries qm \ Jr' onto p.

For each se I, the sentence /f^s, e L„(x) and a = (a0, •••, a»-i), where a, are distinct elements of M. Since q < p, there is a' = (a'0, ..., a'„_]) in N* such that q \- 6{x, a'). Define cb(s) as being aj if b is equal to a,, and whatever you like if b is different from all the a,. Clearly ifp f— <p(x, a), then

{s; q \- p(x, ca{s)} 2 {s; ^q if and only if there exist an ultrafilter <%, an elementary submodel Jl" of J/"* including j>/, and an jtf-isomorphism from Jl onto Jl" which maps p* r Jf'onto p.


PROOF. Just apply the above theorem to the theory T(jrf). • The following proposition states that the orders < and <^ are complete. PROPOSITION 2.5. (1) If F is a closed subset of S^Jl) (and s4 £ j / ) ? then there

is an element of Fmaximal, among elements of F, for thepreorder > (>,»»). (2) For the orders > and > rf, every linearly ordered set of classes of types has a

unique least upper bound and a unique greatest lower bound. PROOF. (1) We will apply Zorn's lemma. Let {p,-; i 6 /} be a subset of F which is

linearly ordered by > . It suffices to find p e F bigger than all the p,. The sets {_/'; p , > p j for i e I generate a proper filter on / ; let % be an ultrafilter

extending it, p' — Up;l$l and p = p' \ Jl. Let us prove that p' > p,-, for all i e /. If cp(x, v„) is represented in p', it has to be represented in some p; with j e {}';p? > Pi}- But then, it is represented in p,. Thus/?' > p,-; but of course p > p', so p > Pi for all i e /.

It only remains to be proved that p e F. But, for all cp e L(M \J {x0}), p \— <p if and only if/?' \- cp, if and only if {/; Pi I— <p} e <& sop is in the topological closure of (p,•; / e /} and we are done since F is a closed set.

(2) The proof is almost the same. Let us prove for example the existence of the least upper bound for the ordering > . Let / be a linearly ordered set of classes of types; for each ;' e /, let p{ e i, />, e Si(J/^, and consider again an ultrafilter °U on / containing the sets {j; pj > /?,•} for i e /. Then it is easily seen that the class of ViPi\°U is the least upper bound we were looking for. •

§3. Heir and coheir. DEFINITION 3.1. Let/? e S\{,,U) and $# 2 Jl. An extension q ofp to s-/ is an heir

of p if for all <p(x, v„) e L(M J {x}) and a e A" such that q (— <p(x, a), there exist m e M" such that p \— cp(x, rfi).

Thus if Ji < Jir and p e Si(J/), the heirs of/? on A' are exactly those types q over Jf such that p ~ M q. We see also that q is an heir of p if and only if there exists an elementary embedding of Jf into an ultrapower Jlm of Jl which extend the canonical embedding, and which makes jf an extension of q.

It would seem that /?* is a privileged extension of p to Jl*. However, we must be cautious because Jlv can be ..//-isomorphic to Jl*', but /?*' not equal to /?*. For example assume G.C.H. and let Jl be a dense ordering, p an irrational type and Jl' a saturated extension of Jl of cardinality 2im; Jl' is .//-isomorphic to „//* where Ql is any ||M||+-good ultrafilter over M. But there are several extensions (in fact at least 2liMt:) /?' of/? over _//' which can be viewed as /?*, that is such that there exists an .//-isomorphism from Jl' onto Jl% which carries p* ontop'.

PROPOSITION 3.2. Le/ y// s j ^ , p e Si(.y//), ^ e S^.r/). r/ze/7 ^ « an heir of p if and only if for every JV including stf, q has an extension to .. V which is an heir of p.

PROOF. The second condition implies obviously the first one. So, suppose that q is an heir of p.

Consider the set

0 = {—\ip(x, a); cp(x, vn)L(M\J {x}), a e M", and , which, in turn, follows from the consistency of


<b{x, a) A x(b) A <j>0, where <f>0 e <£, 6{x, v„) e L„(x), a e .4" and 9 1— c6 {x, a), ^ ( v j e Lm, i? e A/™ and J," t= ^(6). We may moreover suppose that m > n and that 6 extends a, so Z> = a H t>i.

Since 9 is an heir of /?, we can find a' e M" such that/) (— p(jc, a') A 3vm_„ ^(a' f] i-m_„). and since „// < ^ " we can find V e Mm extending a' such that Jl f= £(&')• Whatever interpretation we give to the new constants occurring in <p0, we get a subset of/? which is consistent. •

We should remark that in the above proposition, we can change "for every Ar

including s/"1 by "for some J," including „</". COROLLARY 3.3. Let p e Sx{Jl) and SB 2 sJ 2 Jl. Then p has an heir on si, and

every heir over s/ can be extended to an heir on SB. It is also clear that if Jl < J<~ c j . p e S\(Jl) and q e S^j / ) , then: If 9 is an heir of/?, then 9 f J>r is an heir of/?. If q \ Jl is an heir of/) and g is an heir of q \ Jl", then q is an heir of p. But it can happen that q is an heir of/? but not of q \ Jr. It is also easy to see that if <% is an ultrafilter on /, and for each / e /, „//,- < Ari,

p{ e S(Jl{), q{ e SL/T^) and q,- is an heir of/?,, then Wq^ is an heir of UPi/^l- Moreover />* is always an heir of/?.

EXAMPLE. Let us consider again the theory of dense linear orderings without endpoints; and let „-// -< JT be two models of this theory. Then it can be easily seen that:

For a0 6 M, the type a^ has a unique heir on J', which is again aj . If/? is an irrational type over .//, then every extension of/? to Jl" which is not

realized in Jr is an heir of p. This extension can be irrational or rational. An irrational type can be equivalent (for ~ ) to a rational type but cannot be the

heir of it. This provides us with an example of a type p e S{Jl) an extension q of p to Jir where p ~ q, but it is not the case that p ~ M q.

Also, we can find Jl < Jlx < Jl2, p e S(Jl) irrational, px e S(„//i) a rational extension of/?, and p2

e S(Jl2) an irrational extension of/?i. Then/?2 is an heir of/?, but not of/?!.

PROPOSITION 3.4. Le? V/ < ./T, and pe SX(J*). There exists Jl', Jl < Jl' < A~ such that \\M'l = \\M\\andpisanheirofp \ Jl'.

PROOF. It is clear that we can find a model Jlx, Jl < Jlx < JV such that every formula with parameters in ..// and represented in /? is represented in p \ Jlh and moreover such that | | M J = ||M||. Construct a model Jl2, Jl\ < Jh < -H* s u c h that every formula with parameters in Jlx and represented in /? is represented in p t Jl2, and so on. Then we take Jl' = (J„ ...//„. Q

DEFINITION 3.5. Let Jl £ SB, p e Sx{Jl) and let /?' be an extension of/? to S3, realized by some point a in some J," including &}. We say that p' is a coheir of /? if t{es, Jl U {a}) is an heir of /(^,. , / /) .

REMARK. We could have said equivalently for every b e B, t{b, Jl |j {a}) is an heir of t(b, Jl).

PROPOSITION 3.6. Let Jl s S3; the set of \-types over SB which are coheirs of their restrictions to Jl is exactly c\{t(c, S3); ceM} (the topological closure of {t(c,S$);ceM}).

PROOF. Let q e SX(SS), p = q \ Jl, and let a realize q. It is clear that for any


(p{x, v„) £ L(M (J {x}) and b e B", q\- <p(x, b) if and only if t(b, Jl U {a}) h <p(a, x„).

Thus, if q is a coheir of p and q e (<p(x, b)}, there exists a' e M such that t(b, Jl) f- ^)(a', xK), and this means that t{a', 33) e (<p(x, h)}.

Conversely, if q e cl{t(c, 38), c s M}, and if t(b, Jl U_ {"}) h- p(a, JC„), then q e {(p(x, b)} hence, for some a' e M, t(a', 33) e < >(x, £)>, which means that f(i, Jl) \- <p(a', xn). D

COROLLARY 3.7. Lef /> e S(Jl) and Jl £ j2/. Then the set of coheirs of p on s/ has cardinality at most 22,11 !.

PROOF. The space cl{/(c, sf), c e J/} is a HausdorfT space with a dense set of cardinality \\M\\ so there are no more than 2llMil clopen sets, and no more than 22llMi points. •

COROLLARY 3.8. Let Jl s si s J1, p e ^(y/) a«J g a co/ie/r o / p on &l. Then there exists an extension r ofq to SS which is a coheir of p.

PROOF. Let X= {t(c, sa);ce Jl) and Y = {r(c, 28); ceJl}. Then cl Y is a compact subspace of Si{33) so its image in Sx{s/) under the operation induced by restriction is a closed set and includes X, so it includes X, and this is exactly what we had to prove. •

It is also clear from the definition that if Jl s s4 £ S8, p e Si(-B) and p is a coheir of p t Jl, then/? r s? is a coheir of p [ Jl.

EXAMPLE. Let us look again at the dense linear orders without endpoints. Let ji < jr.

Consider first the type + co over Jl. Then it has only one coheir on Jf, which is the type larger than every element of Jl (and of course than every element of Jf smaller than some member of Jl) and smaller than every element of Jf larger than every element of Jl.

Let us consider now p e S(Jl), p irrationnal. Suppose that the set P of elements of Jr realizing/) is nonempty. Then a coheir of p may be larger than every element in P, or smaller than every element in P, and so we get two coheirs.

REMARK. The bound 22 given in Corollary 3.7 is the best one (see example in[P]).

§4. Stable types over models. DEFINITION 4.1. Let p e Si(Jl). We say that p is definable if for any <p(x, v„) e L(x)

there exists 6(v„) e L(M) such that for any a e M", p (— <p(x, a) if and only if Jl f= (fid).

THEOREM 4.2. Let pe S(Jl). Then p is definable if and only if for any Jf > Jl, p has only one heir on Jf.

PROOF. Suppose first p definable. For any <p(x, v„) e L(x), let ^(v„) e L(M) be the formula given by the Definition 4.1. Then the formulas —\{(p(x, v„) <->• <pv(yn)) are not represented in p, thus they are not represented in q whenever q is an heir of p on Jr. So, for any a e N", q f— (p(x, a) if and only if Jf |= (b<p{d), and this leaves only one choice for q.

Conversely, suppose that for any Jf > Jl, p has only one heir on Jf. Let L' be the new language obtained by adding, for each <p(x, v„) € L(x) a new «-ary predicate


symbol A9(vn) and let Jl' be the expansion of Jl where the interpretation of Af is exactly {a; a e M" and p \~ <p(x, a)}. Let T' be the theory of Jl' in the language L'(M). "We claim that A9 is defined implicitly by 7".

Indeed, if Jf' is a model of 7", then Jf = Jf' t L is an elementary extension of „//, and it is easy to see that the set

{<p(x, a); <p(x, v j 6 L(x), deN" and ^ ' [= A9(a)}

is a type over Jf, and moreover an heir of p on ^T. Thus, for any model Jflt of T(Jl), since /? has only one heir on JfY, it follows

that Jf1 can be extended in at most one way to a model of T'. By Beth's definability theorem (see [CK, p. 86] for example), each A9 is explicitly definable, that is, there exist <fi9(y„) e L(M) such that

which means that, for any a e Mn, p (— (fix, a) if and only if Jl |= ^p(a). D PROPOSITION 4.3. Suppose that p e S(„//) is not definable. Then, for any cardinal

X > I! M H, fAere ;'s >" > .// SMC/I f/iaf )| N \\ = A awd/? has at least X+ heirs on Jf. PROOF. We first show that every heir of p has two extensions which are again

heirs of p. So let px be an heir of p over Jl\ > Jl- Since p is not definable, there are Jl' > Jl, and p' and p" two heirs of /? on Jl'. By Corollary 2.4, there is some ultrafilter <fy, Jl[ < Jl0" and an ^//-isomorphism from Jli onto Jl[ mapping />* r Jl'i, onto pi. We may as well suppose that Jlx — Jl[ and/?! = /?* \ Jl\. Now „//* < ^/ '* , and clearly/?'* and/?"* are heirs of/?*, and of/>, and they extend pv

(Caution: p"* and /?"* are not necessarily heirs of pi\) Let fi = inf{a; 2« > A}, and 5 = (Ja ordinai{0, 1}". We remark that ft < X, and

for all a < fj.,2" < X. We construct by induction on a < p. a model ^//a > Jl, and for each 5 e {0, 1}" an heir ps of p on .y/a such that

(1) a > a' implies Jla > .//„/. (2) If 5 extends .s', then />s extends /v-(3) If J, 5' e {0, 1}». ^ * s', then A # A . (4) HAf.il < X. We begin with Jl0 = .///, and p0 = /?. At limit stages, we just take Jla = l j ^ < a Jl$

and for s e {0, 1}", ps ~ [J{ps>; s extends 5'}. Clearly ps is again an heir of p. At stage a + 1, for every s e {0, 1 }a, there exist V/s > Jla and two extensions

p's and />j of/? on Jls which are heirs of/?. Since we only need a finite number of parameters to distinguish between ps and p[ we may suppose that ]|Ms|| = ||Ma|| < X. We can include all the Jls for s e {0, \}a in one model Jla+\, and since ||{0, l}"!! < X, we can require that || Ma+i \\ < X. It suffices now to take ps-0 and pri heirs of p's and p"s on Jla+\.

Thus \\Mp\\ < X, and p has 2^ > X heirs on *///,,. D THEOREM 4.4. Let p 6 S^Jl). The following conditions are equivalent. (1) Every extension of p over a model of T is definable. (2) For any Jf > Jl, p has at most HAHI"0 extensions to Jf. (3) There is a cardinal X > \\M\\ such that for every Jr > Jl such that \N\ = X,

p has at most X extensions to Jf.


In this case, we will say that/7 is stable. A theory is stable if every type over every model is stable.

PROOF. (1) -* (2) A definable type over JT is given by a map from L(x0) into L(N); there are no more than ||N||*° such maps.

(2) -+ (3) Just take for X a cardinal of the form p*>. (3) -+ (1) In view of Proposition 4.3, it is enough to show that if p has a non-

definable extension, it has a nondefinable extension over a model Jl' of the same cardinality as Jl.

So let q e S(Jf) be a nondefinable extension of p, and let J/"x > J" such that q has two heirs qx and q2 on JT\. By Proposition 3.4, there is a model . / / ' , . / / -< Jl' < Jf, such that q is an heir of p' = q \ Jl', and ||Af'|| = \\M\\. Thus qx and q2 are also heirs of p', and we are done. •

REMARK. From this, it is easy to see that for any tf and S3, t{sJ, SS) is stable if and only if for any as A, t(a, S8) is stable.

PROPOSITION 4.5. Let p e S(Jl) be a stable type and q < p, q e S(Jl"). Then q is stable.

PROOF. Suppose that q has an extension r with two heirs rl and r2; for some ultrafilter ty, q°" is isomorphic to an extension of p and has an extension r* with two heirs rf and r%. D

This proves that stability is a property of classes of types (in contrast with definability).

PROPOSITION 4.6. Let Jl -< Jf, p e S{Jl), p stable and q an extension of p to JV such that p ~ q. Then q is the heir of p.

PROOF. We shall prove that if p e S(Jl) has two extensions px and p2 to J/" such that/7 ~ pi ~ pi, then for any cardinal X > \\M\\, p has at least X+ ~-equivalent extensions to some Jl" of cardinality X, and thus p is not stable. We do that by analogy to Proposition 4.3, replacing "p, is an heir of />" by "ps ~ p" and using Theorem 2.3 instead of Corollary 2.4. •

From this, we see that if Jl1 < Jl2 < Jl, pe S(Jl) is stable and it is the heir of p \ Jlx, then it is the heir of p f Jl2 (in fact, this already follows from the definability of p r Jl{).

PROPOSITION 4.7. Let p e S(Jl), q e S{Jf) be stable types such that p ~ q. Then p and q have isomorphic ultrapowers.

PROOF. As in Proposition 4.2, let L' be the language obtained from L by adding for each <p(x, v„) e L(x) a new «-any predicate symbole A9.

For p e S(Jl), let Jl(p) be the expansion of Jl to this language, where Jl(p) \= A9(a) if and only ifp t= <p{x, a).

We first claim that if/? is stable and px is the heir of/? on Jlh then Jli{p{) = Jl(p).

This is because for each <p(x, v„) there exist (p9(vn) e L(Jl) such that the formula VvB(^lp(v„) <-• ^(v„)) is satisfied in both Jl(p) and Jliipi).

Second it is clear that J/*(p*) = (Jl(p)T-By Theorem 2.3, there exist an ultrafilter <H,Jl" < Jl0", and an isomorphism from

Jl" onto Jf' mapping p* [ Jl" onto q. So we have

q ~ p* \Jr'~ p ~ p'


and />* is the heir ofp* f Jf'. Thus

Jt(p) = Jt*W) s ^'(j?' r Jf') = Jf(q).

By the Keisler-Shelah theorem (see [S2]), there is an ultrafilter % such that Jt(pf is isomorphic to Jf(qf, so that />* and <7* are isomorphic. •

PROPOSITION 4.8. Let p be a stable type over Jt, Jf a X-saturated extension of Jt (X > ||M||+), and q an extension ofp to Jf which is not the heir ofp. Then, there exist at least X extensions ofp to Jr which are ~ ^-equivalent to q.

PROOF. Let Jtx be such that Jt < Jtx < Jf, \\MX\\ = \\M\\ and q is the heir of q \ Jtx = p\. Of course, px is not the heir ofp.

Claim. For any Jt' such that Jtx < Jt' < Jf, \\M'\\ < X, there exists an extension q' ofp to Jf such that q' ~ Mq and q' f Jt' is the heir ofp.

Let/?' be the heir of p on Jt'. Then/?' > px, so by Theorem 2.4, p' can be embedded in some ultrapower of px. This means that there is Jt" > Jt' and p" an extension of p' to Jt" which is the heir to an ^//-isomorphic copy of px. Clearly we can require that ||Af"|| < X, and also that Jt" < Jf. The heir of p" on JT will satisfy the conditions.

Now, we perform the construction of types qa e S{JT) and of an increasing chain of Jta, for a < A, such that:

(i)qa ~M q-(2) qa \ Jttt is the heir of p. (3) qa T Jta+X is not the heir of p. (4) IIMJ < X. We set q0 — q, Jt0 = Jt, and we already have Jtx. To get qx, or more generally

qa if we are given Jta, we just use the claim. If a is limit, we take Jta — []$<# -dtp- If a -= /3 + 1, we take Jta as being such

that Jtp < Jta < JV, qp is the heir of q? f Ma and \\M$\\ = \\Ma\\. Q THEOREM 4.9. Let p e 5^^/Z) be a stable type. Then for every Jf>Jt,p has only

one coheir on J/" which is its heir. PROOF. We may as well suppose that Jf is A+-saturated, with X = 22'M". We will

argue as follows. Supposing that q e S(J/~) is a coheir ofp, but not the heir, we will construct X^ distinct coheirs ofp on Jf, contradicting Corollary 3.7.

Let Jtx be such that Jt < Jtx <Jf, \M\ = \\MX\\ and q is the heir of q t Jtx = />i- We claim that if qx is the heir of px on any Jfx > .//i then qx is a coheir of/?. This is clear if Jfx -< Jf, because then qx = q \ Jfx. So, it will be true also if Jfx can be embedded in Jf by an ./^-isomorphism, that is if \NX || <, X. In the general case, every restriction of qx to model of cardinality at most X is a coheir of p, and this suffices to make qx a coheir ofp.

By Proposition 4.8, there are X+ extensions of p which are ~ ^-equivalent. Let q' be one of them. By Proposition 4.7 there exists an ultrafilter °lt such that q"* and q°" are .//-isomorphic. But we have just proved that q°" is a coheir ofp; hence so are q"* and g'. D

§5. Stable theories. Let us call a type p e S(JJ / ) 5/aWe if all its extensions over models are stable. We shall be concerned with types over sets which are not necessarily models of T, and we begin our study assuming T stable.

340 PANIEL LASCAR AND BRUNO POIZAT

Let/? £ S{si), and look at the set of classes of types which are extensions of/? over models. A bound of p will be by definition a maximal element of this set for the order > .

We begin by a remark. Let Jl\ and Jl2 be two models containing si and included in a third model. If px e S(Jl{) is an extension of p, then there exists p2 e S(Jl2), also an extension of p, such that p2 > p1: take the restriction to Jl2 of an heir of Pi on J{x |J y//2. This proves that for any Jl 3 jaf, and any £, bound of/?, there is an extension of/? to y/ whose class is precisely f. By Proposition 2.5, /? has at least one bound.

THEOREM 5.1. There is a unique bound for each p. We remark that this is not true without the assumption of stability: look at the

dense linear order. PROOF. Let Jl and Jl\ be two models including si, and suppose that p is realized

by c € Mv Consider r = t(Jl,si) and let r' be an extension of r to Jlh whose class is a bound of r. We may suppose that Jl and Jl\ are included in some big model J,r, and that t(Jl, Jl{) — r'. We claim that, under these conditions, t(c, Jl) is maximum for > among the extensions of p.

Let Jl' ^ si, and let/?' be an extension of/? to ^//'. We want to inject Jl' in Jr

in such a way that t(c, Jl') = /?'. This is of course possible. It suffices that Jl' realizes a given type gx over si U {c}- Let q2 be an extension of qx to . / / / j , and 9 be a coheir of q2 over Jlx U -//• Now, if Jl' < Jf realizes q over Jlx \j Jl, then t(c, Jl') = /?', and moreover t{Jl, Jlx (J „//') is an heir of ?(^7, . / / !) . So every formula represented in t(Jl, Jl') is represented in t(Jl, Jlx SJ .//') and also in r(^/, Jli). From the maximality of t(Jl, Jl{), we deduce that t(Jl, Jl') ~ ?(.///, y/j), and since the theory is stable, t(Jl, Jli U Jl') is the heir of t(Jl, Jl') (Proposition 4.6). So t{c, Jl IJ Jl') is the coheir of t(c, Jl') and hence, it is also its heir. We infer that every formula represented in t{c, Jl) is represented in p' = t(c, Jl'). •

We shall denote /3(/?) the bound of p. DEFINITION 5.2. Let si s 38, and p e S{@). We say that p does not fork over si

\f £(p) = %p \ si). The following are obvious. (1) If/? e S(si), p does not fork over si. (2) If si S 38 £ <£ and /? e S(&) does not fork over si, then /? does not fork

over 38 and /? f ^ does not fork over si. (3) Let sal <=,38 <=,<£ and /> e S(<<?). If/? does not fork over 38 and /? C ^ does not

fork over si, then p does not fork over si. (4) If j ^ c & c # and /? e S(^?) does not fork over si, then it has an extension

to <<? which does not fork over si. (5) If Jl G si and p e SCs/), then p does not fork over Jl if and only if/? is the

heir of/? f Jl. PROPOSITION 5.3. Let si £ ji, bx andb2 be elements ofJl. Then t(bx, si U {h})

forks over si if and only ift(b2, si U { 1}) forks over si. PROOF. The proof resembles that of Theorem 5.1. Our construction takes place in

some big model Jf. Let /?,• = t{b{, si), i = 1,2. Suppose that t{b2, si U {b\}) does not fork over si. This means that there exists


M\ < JV containing si and bh and such that the class of t{b2, Jl\) equals (3(p2). There exists also J/2, si <= Jit <= Jf such that the class of t(b1,J/2) equals (3(pi), and moreover, we may suppose that t(Jl2, Jl\ U {b2}) is the coheir of t(Jl2, Jl\), that is t(b2, Jlx U J/2) is the heir of t(b2, Jlx).

We know that t(b2, {fej} 'J Jl2) is the heir of t(b2, Jl\) and thus t(b2, J/2) > t{b2, J/i). From the maximality of t(b2, Jl\), we get that t{b2, Jl\) ~ t(b2, Jl2) and that t{b2, {bi} U Jh) is the heir of t(b2,J/2); this implies that t(bh J/2 (J {62}) is the heir of f(6b y/2) . We may find J/' => .///2 U {b2} such that t{bx,Jl') is the heir of t(bx, J/2), and thus t(bt, si U {&2}) d ° e s not fork over si. •

The following proposition proves in particular that our notion of forking is the same as Shelah's (see [L2] e.g.).

PROPOSITION 5.4. Let sil £ @,pe S(si), p' an extension ofp to @. The following conditions are equivalent.

(1) p' does not fork over si. (2) For any J/, J/' such that si £ Jl < J/' and SS £ J/', there exists an exten

sion px ofp to J/ whose heir on Jl', say p[, is an extension ofp'. (3) There exist J/,Jl' such that si £ Jl < Jl', 35 £ Jl', and t{@, Jl) does not

fork over si, and an extension px ofp to Jl, whose heir over Jl', say p[, is an extension ofp'.

PROOF. (l)-»(2) L e t ^ be an extension of p' whose class is fi(p), a.ndpl = p[ \ Jl. Then px ~ p[, and p[ is the heir ofpx.

(2) -+ (3) Clear. (3) -»(1) Suppose that c realizes p[. Then t{3$, Jl 1J {c}) is the heir of t{3$, Jl),

and thus, does not fork over si. It follows that t(38, si U {c}) does not fork over si, and hence neither does t(c, 3$). D

COROLLARY 5.5. Let p e S(si), Jl 3 si. The extensions ofp to Jl which do not fork over si are exactly those which are maximum for >^.

PROOF. Consider the theory T(si) in the expanded language L(A). From the second characterisation of the preceeding proposition, an extension of p to Jl does not fork relatively to T(sf) if and only if it does not fork relatively to T. •

COROLLARY 5.6. Let p e S(si), si £ Jl, px and p2 be two nonforking extensions ofp to Jl. Then there exists an ultrafilter % such that pf and p% are si-isomorphic.

PROOF. From Corollary 5.5, we know that/?] ~s?p2. Apply Theorem 4.7. Q COROLLARY 5.7. Let si £ 38. The set of elements of S{38) which do not fork over

si is closed. PROOF. Consider the third characterisation in Proposition 5.4. The elements of

S(Jl') which are heirs of their restriction to Jl form a closed set (by Proposition 3.6, since they are also coheirs of it). The set of elements of S(@) which do not fork over si is the image of this set by the continuous map induced by the restriction to si. •

COROLLARY 5.8. Let si £ 3$ and suppose thatpe S(3S) forks over si. Then there exists a finite subset &0 of 38 such that p \ si U 38^ forks over si.

PROPOSITION 5.9. Let p e S(si). There exists sit, £ si, si0 countable, such that p does not fork over si0.

PROOF. Suppose the contrary. Then we can define an increasing chain {3$a; a <


Xj) of countable subsets of s/ such that for all a < «i, p \ 83a+\ forks over 83a. Thus fi(p I 83a) is a strictly decreasing chain of length KU which is impossible. (Remember that T is countable.) •

PROPOSITION 5.10. Let s4 £ 83, pe S{jrf). There are no more than 2K° nonforking extensions of p to 83.

PROOF. Clearly, it suffices to prove this when 83 is a model of T. Let sf0 £ J > •s/0 countable, be such that p does not fork over „</0, and .y/0 countable such that

If/?' is a nonforking extension of/? to 83, then /?' does not fork over s/0, and hence it is the heir of p' [ J/0. Thus there are no more than [|SC//o)|| < 2K° such extensions. D

Let cp(x, v„) e L(A \J {x}) and p e S(&/). The set {cp(x, a); ae A", <p(x, a)ep} will be called the 95-type of p. By compactness, the above proposition implies:

COROLLARY 5.11. Let s/ <= 83, pe S^jtf), and <p{x, v„) e L(A U {x}). Then the set ofcp-types of nonforking extensions ofp to 83 is finite.

THEOREM 5.12 (OPEN MAP THEOREM). Let sJ £ 88, and set F = {pe S{83); p does not fork over s/}. Then the map fform F into S(jsf) which carries p into p t sJ is an open continuous onto map.

PROOF. The only thing which has not been proved already is that / i s an open map. For this it is clearly enough to prove the theorem when 89 = Ji is a model of T.

Let O = {p; pe F and p \- <p} for <p e L(M U {x}) be a basic clopen set of F. We want to show that f[0] is an open set in S(&). Let O* = / _ 1 [ / [ 0 ] ] . Clearly, f[0] = f[0*], f[CO*\ = C(f[0]) and since F is a compact space and / continuous, it is sufficient to prove that O* is an open set in F. Up e O*, there exists/?' e O such that p I sJ = p' \ sJ. We infer (Proposition 4.7) that there is some ultrafilter <fy such that /?* is ^-isomorphic to p'm. Let g be the ^-isomorphism mapping />* into /»'*. Then p« (- g(<p).

Let F' = {qesiJ/f); q does not fork over sJ), O' = {qeF'; q (— g( o)} and let / ' be the map from F' into F which carries g into q t Jl. It is clear that / ' is a homeomorphism, so that/'[£>'] is an open set in F. We know that p' ef'\0'\ and it remains to show that/'[£>'] £ O*.

Let /?i e / ' [0 ' ] . Then pi has an extension to .//*, say qu belonging to O', and ai \- g(<p)- Hence gfai) |— 9, and g(q{) \ Jl eO; this implies that px e O*, since g(qx) t s/ = qits/ - pit s/. D

We will say that an equivalence relation is finite if it has a finite number of classes. Let R be an equivalence relation definable with parameters in s/. It is clear that if R is finite, then there is a representative of each of its classes in every model including sf.

THEOREM 5.13 (FINITE EQUIVALENCE RELATION THEOREM). Let s? £ . / / , p e S{s/)

and po and pi be two distinct nonforking extensions of p to Jl'. Then there exists a finite equivalence relation R, definable with parameters in ,c/, such that p0(x0) A />l(*l) \--IR(XQ, XJ) .

PROOF. Let Jl' be a ||M^-saturated extension of Jl. Since p0 # pv there exist some <p(x0, v„) e L(x0) and me M" such that pQ \— cp(x0, m) and px \— —1 (p(x0, m). In Jl' define the relation: R(c, c') if and only if every nonforking extension of


t(m, si) to si U [c, c'} contains the formula <p(c, x„)*->• cp{c', xn). By Proposition 5.3, this is true, if and only if every nonforking extension of t((c, c'), si) to si U {m} contains the formula (p(x0, m) *-* <p(xh m).

Claim 1. The relation R is definable with parameters in si. Let F be the set of 2-types over si \J {m} which do not fork over si, and / the canonical map from F onto S2(si). By definition, for any c, c' e M', we have: not R(c, c') if and only if there is some extension of t((c, c'), si) to si U {™} belonging to Fand containing —<(<p(x0, m) *-* <p(xi, m)); that is, if and only if/((c, c'), si) e / [ 0 ] , where

O = {qeF;q\- ->(<p{x0, m) «-> <p(xh m))}.

By the open map theorem, there is some d>(x0, x}) e L(A \J {x0, xi}) such that not R(c, c') if and only if r((c, c'), si) \— 6(x0, xt), if and only if Jl' N= </>(c, c').

Claim 2. R is an equivalence relation. It is obviously symmetric and reflexive. Let us prove that it is transitive: thus, let c, c' and c" be such that R{c, c') and R{c', c"). Let q be any nonforking extension of t(m, si) to si \J {c', c"}; then we may extend q to qx over si U {c, c', c"} in such a way that qx does not fork over si, and then 9i I si U {c, c'} and <?i f si U {c'> c"} do not fork over „•=/. Then

qx V- <p(c, x„) *-* <p{c'\ x„), 91 1- p(c', x„) <-* y>(c", x j

and s o ? h p(c, x j •*->• y>(c", x„). C/a/m 3. i? is finite. Suppose the contrary; there exists a sequence (c,; z" e co) in

.// of inequivalent elements; hence for i # j , there is a nonforking extension q{j of t(m, si) to Y/ such that qitJ |——'( >(c,-, x„) +->• ^(c, x j ) . Since any two integers are separated by a set of the form {n; <p{c„, x„) e q{J} with / ^ j , there is an infinite number of them. Thus the set of p-type q, •_,- is infinite, which is impossible.

Claim 4. We want p0(x0) A P\(xi)\— -iR(x0, x{). It suffices to prove that Po(x0) A P\(xi) A ^ ^ ( X Q , XJ) is consistent. Indeed, if this is the case, there is a representative in Jl of each equivalence class, and for some c, c' in Jl we have />o(*o) I - -^(ô* c) andp^xj) I— R(xx, c'). We must havc / / |= —<R(c, c') and then poC o) A Pi(xy) f— -iR(x0, Xi).

Let c0 e A/' realize p0 and Cj realize the heir of pi over ^# U {c0}. We see that t(cx, Jl U {c0}) does not fork over si; in particular t(cx, si U {™} U {co}) does not fork over si. So it does not fork over si U {co}- By Proposition 5.3, ?(»j, si U {c0, cj}) does not fork over si U {c0}. Since r(/n, JS/ U {c0}) does not fork over si, t(fh, si |J {c0, C]}) does not fork over si. But we know that

Jl' 1= ~>(p(co, w) • - <p(ci, m))

and we infer that Jl' |= —'/?(c0, Cj). D REMARK. Let /? e 5(j^) and let ^// be a model containing si which is homogene

ous enough. The group of ^-isomorphisms of Jl induces a transitive permutation group G on the set P of nonforking extensions of p to Jl. This group will not depend on the model Jl.

From the transitive action of G and Corollary 5.11, it can be seen thatthe number of nonforking extensions ofp to Jl, and in fact to any model extending si, if finite or 2K°.


The group G is profinite: it is possible to develop a Galois theory to characterize the closed subgroups of G (see [P]).

§6. Immediate and coimmediate extensions. In this section, we do not have any assumption of stability.

DEFINITION 6.1. Let si £ 38, p e 5(.c/) and q an extension ofp to 38. We say that q is an immediate extension of p if there exist Ji 2 38 and qx an extension of q to Jl such that qx is maximal with respect to > ^ among the extensions of p to Ji.

We say that q e S(38) is immediate over si if it is an immediate extension of q t si. We shall make some remarks. (1) Suppose # is an immediate extension ofp. Then for any Ji' 2 38, q has an

extension to Ji' which is maximal with respect to > ^ among the extensions of/?. Indeed, let Ji and qx be as stated in the definition. We may suppose that Ji and Ji'

are included in some model JJ~. Let q[ be an heir of qx on Ji U Ji', and q' = q[ T Ji'. Then every formula represented in q' is represented in q[, and also in qu so that Q' 0i- If w e suppose that there exists q" e S(„//') such that q" >^q', then by the same method, we can find q2 e $(Ji) such that q2 > q", and thus q2 > qx, which is impossible. So q' is the extension that we were looking for.

(2) By Proposition 2.5, if p e S(si), then p is an immediate extension of itself, and it admits an immediate extension to 38.

(3) If sJ is a model of T, then the immediate extensions of p are exactly the heirs of p.

(4) If 38 is a model of 7*, the immediate extensions ofp are exactly the maximal (for >^ ) elements of the set of extensions ofp.

(5) Two immediate extensions ofp can be nonequivalent: look, for example, at the linear dense orderings.

PROPOSITION 6.2. (1) Let si c @ c <<f a«^ suppose that p e S(<<f)wimmediate over si. Then p \ 3d is also immediate over si.

(2) An immediate extension ofqe S(si) over 36 can be extended to an immediate extension of q over <£.

PROOF. The first part is clear from the definition. The second assertion is obvious from the first of the preceding remarks. •

DEFINITION 6.3. Let si E 38, pe S(si) and let q be an extension of pto 39, realized by some point c. We say that q is a coimmediate extension of p if t(38, si U {c}) is an immediate extension of t(3ft, si).

We say that q e S(33) is coimmediate over si if q is a coimmediate extension of qt si.

Thus in the case where si is a model of T, the coimmediate extensions are exactly the coheirs. It is also clear that every type is a coimmediate extension of itself.

PROPOSITION 6.4. Let si £ 3$ £ # , p e 5(<<f) awcf suppose that p is coimmediate over si. Then p \ 38 is coimmediate over si.

PROOF. Translate this proposition in terms of immediate extensions. We see that we need some information about the behaviour of (/ Ij 7)-types; this is given in the following lemma which, for the sake of simplicity, we state for 2-types.

LEMMA 6.5. Let q(x0, xx) e S2{si), q' be an extension of q to Ji, px e Sx{Ji), and


suppose that px >^q' fl L(x0, M). Then there exists qy e St(Jl), including px [j q, and such that qx >^q'-

PROOF OF THE LEMMA. Set p' = q' f) L(x0, M). By Proposition 2.4, there is an ultrahlter <ty, Jl' < „//*, and an .^-isomorphism / from Jl onto Jl' such that / ( / / * r JO = p\. It suffices to take for 91 the type/(g'* r JO- •

It follows from the lemma that if q' is an immediate extension of q, then q' f] L(x0, M) is an immediate extension of q |~j L(x0, A). It is clear that it is also true for types over 38' 2 s#, 38' not necessarily a model of T. This is the content of Proposition 6.4. •

PROPOSITION 6.6. Let sZ s 3$ £ <<?, p1 e S(s-Z), pz be a coimmediate extension of pi to 38. Then there exists an extension p3 ofp% to <& which is coimmediate over s#'.

PROOF. Translating the proposition in terms of immediate extensions, we see that we have to prove the following. Let q(x0, xt) e S2(s/') and/*' e S^^?') be an immediate extension of p = q f] L(x0, A'). Then q has an immediate extension over 38' including p'. (Translation: take s/' = sf, <%' = sf L {^}> q = t($, stf), p' = W, s* U {cfc}).)

Clearly, it suffices to prove this when ^ ' is a model of T. So let q'e S2(&') be maximal for >^> among the types including q [j />'. We claim that q' is an immediate extension of q.

If we suppose that there exists qx e S2(3S') such that ql >^> q', then clearly pi ~ qx fi L(x0, Si') >j>q' fl £(*o> ^ ' ) = P'• Since />' is maximal for >^,, /?' ~ ^ /?l5

and by Lemma 6.5, there exists a" e S2{38') including p' jj q and such that q" >&> qy From the definition of q', we get q" ~&> q' ~&' Q\- d

PROPOSITION 6.7. Le; /> e £(.//), <p e £(M |J {*}) be such that p h- <p, and s4 2 (pM. Then p is coimmediate over s/.

PROOF. Let Jl' be a model containing j& in some extension Jf of Jl such that t(Jl', Jl) is a coimmediate extension of t(Jl', st). Let % be an ultrafilter so that p is realized in Jlm, say by c.

Now, look at what happens in Jf*. The point c is the class of (c,)isI (/ support of W) and since Jr^ (= p(c), we may suppose that all c,- e <p-,f, so that c e Jl'm. But ?(y/, •//'*) is an heir of t{Jl, Jl'), so it is an immediate extension of t(Jl, stf). It follows that t(c, Jl) = p is coimmediate over ,</. •

Exactly the same proof yields: PROPOSITION 6.8. Let stf £ . / / , Jl \\Ap-saturated, andp e S(Jl). Let P = {ae M;

t{a, s/) = p \ stf), and let 38 ~=> P. Thenp is coimmediate over 3$.

§7. More on stable types. The aim of this section is to obtain the same results as in §5, but assuming only the stability of p (instead of the stability of the whole theory T). We just give the sketch of what is to be done, leaving for the reader the pleasure of carrying out the details. The generalization is not obvious because in §5 we often used the stability of the parameters. Everything rests upon the following theorem which shows that we can avoid mixing stable and unstable types; remark that in case of a stable theory, this is an easy consequence of the finite equivalence relation theorem and of Proposition 6.8.

THEOREM 7.1. Let p be a stable type over jtf, Jl an \\A^+-saturatedmodel including

346 DANIEL LASCAR AND BRUNO POZAT

s/, and P the set of elements of M realizing p. Ifp0 and pi are extensions of p to Jl such that p0 I" P = pi \ P, then p0 = pv

PROOF. Let Jl' be a sufficiently saturated extension of Jl. By induction on / e co, construct a sequence (c:; i e co) in Jl' as follows.

If/ is even, then c, is an element realizing the heir of p0 over Jl U {c0,..., c,_i}. If / is odd, then c, is an element realizing the heir of p\ over Jl \J {c0, . •., c,-]}. (1) We first claim that the sequence (c„ / e co) is order-indiscernible over P, that

is, if a(0), a(l) , ..., a(«) and /3(0), /3(.l), .... ,3(w) are two strictly increasing sequences of integers, then t((cal0), co(1), ..., ceW), P) = /((c;(0), ^ a ) , ..., c ? w ) , />)•

We shall adopt the following notation. If a is a sequence of integers of length at least/', we set ca(i) = (cai0), ca(X),..., ca(l)). We argue by induction on sup(a(n), f>(n)).

Case 1. a(n) = /3(«). We already know that /(ca(n_D, P) = ?(^(«-i)» ?)• We know that ?(ca(„_D, Jl U {cB(B)}) is the heir of f(co(B_1), . / / ) , and the same for ?(c;3(„_1), ./// U (/3(n)})- So it suffices to prove the following lemma.

LEMMA 7.2. Let q\ and q2 be two definable types over Jl and suppose that qx\ p = q2 \ P. Let X be a set of elements of Jl' realizing p, and let q[ and q'2 be the heirs of ql and qz over Jl «J X. Then q\ \ P \J X = q2 t P U X.

PROOF OF THE LEMMA Suppose the contrary; let d1 and d2 be the maps from L(x) into L(A) such that:

Pi j - tp(x, a) if and only if Jl f= d,{<p)(a) (i = 1, 2).

There exists a finite sequence b = (b0, bh ..., bk) from Jf y P and a formula p(x, vM) e L(x) such that

So the following set is consistent

/>(>'o) U/<Ji) U -• [) P(yt) U {-'î(f)(Jo,Ji> •••,yk)^d2{<pXyQ,yl, -.-,>*)}.

Since . # is ||Y4 -saturated, it is realized i n ^ , and this contradicts the assumption qx\ P = q2\ P. •

Case 2. a(«) < /3(n). By Case 1, it suffices to prove t{caM, P) = t(ca{„„v~cpW, P). t h a t i s / ( c a ( n ) , i p U { W l } ) = t(cp(n),P U {C«(„_!)}).

Again, this is a consequence of Lemma 7.2. Suppose now that/>0 # />j; hence there exist (f>(x0, v„) e L(x0) and m e M" such

that^0 1— ^)(x, m) and^j f— —'^(A:, m). (2) Let 5 and S' be two disjoint finite subsets of co. Then, there exists m' e M'

such that

Ji 1= /^p(c,-, w') A [f\ -xpici, m'). iss ieS'

Indeed, there is a strictly increasing map a from S [j S' into co such that S is mapped on the even numbers and S' is mapped on the odd numbers. Then

M' N Ik <p(caU), m) A tf\ - i ^(cû), m). i e S j ' eS '

But ?((c,-; / e 5 U 5'), />) = f((cff(l,; ie S U 5'), i>) so that

^ " N 3v„[/^ 9>(c,-, v„) A ^ -• <j>{Ci, v„)]. ieS ieS'


(3) Now let / be any cardinal number and introduce the following new individual constants: for each a e X a sequence ya of length n, and for each S cz }, a constant xs. From (2), it follows that the set

(JO(*s) ; 5 c X} U {<P(XS,ya);aeS}u {~^<P(XS,ya);oci s}

is consistent. This shows that p has 2X extensions over a set of cardinality A, which contradicts the stability of p. •

COROLLARY 7.3. Let p0 and px be two stable types over Jt which satisfy a formula (p{x) e L(M U {x}), and such thatp0 X <pM = p1 \ <pM. Thenp0 = px.

PROOF. Let Jt' be an \M\ j-saturated extension of Jt, and p'0, p[ be the heirs of Po, pi respectively over Jt'. Using the definability of p0 and pu it is clear that Po X <pM' — P\ X <pM' • Thus, if p is the set of elements of Jt' realizing p0 [ 0 = Pi X 0 , we see that p'Q X P = p[ X P and by the preceding theorem, p'0 = p[. D

PROPOSITION 7.4. Letpbe a stable type over si, Jt and \\A\\+-saturated model containing si, andp0 a coimmediate extension of p over Jt. Then for every extension p' ofp to any Jt' 2 si, we havep0 >^p'.

PROOF. AS in Theorem 5.1, in some big model JV we can construct isomorphic copies of Jt and Jt' (that we shall call again Jt and Jt'), a model Jt\ and c e Mx in such a way that t(c, Jt) = po, t(c, Jt') = p', and t(Jt, Jtx \J J/) is an heir of t{Jt, Jti).

Call P the set of elements of Jt realizing p. Since every sequence from P realizes a stable type over sJ, we may apply the same argument as in the proof of Theorem 5.1 and get that (fc, Jt' U P) is the heir of t(c, Jt'). Consider the types p0 and the restriction to „// of the heir of/?' = t{c, Jt') on Jt ij Jt'. These two types have the same restriction to P, so they are equal, and we infer that t(c, Jt') < & t(c, Jt). •

COROLLARY 7.5. A stable type has a unique bound. Thus Definition 5.2 makes sense for stable type too, and we see that a coim

mediate extension of a stable type is an immediate and nonforking extension. We now prove the reverse implication.

PROPOSITION 7.6. Let srf £ SS,p be a stable type over si and p' be an immediate extension ofp to 38. Then p' is a coimmediate extension ofp.

PROOF. It is sufficient to consider the case where @ — Jt is a model of T. Let Pi be a coimmediate extension of p to Jt. Then px ~ ^ p' by Proposition 7.4. Thus, for some ultrafilter °U, pf is ^/-isomorphic to p*. Now, px has an extension to Jl* which is a coimmediate extension of p. Using Proposition 7.4 again, we conclude that this extension must be immediate, and it must be the heir of pu

i.e. it equals pf. Hence pf is a coimmediate extension of p, p' too (since it is s/-isomorphic to pf) and hence p' also has this property. •

The following proposition proves that nonforking extensions, immediate extensions and coimmediate extension are exactly the same. The proof is similar to that of Proposition 5.4.

PROPOSITION 7.7. Let s? G g$, p be a stable type over si, and p' be an extension ofp to 38. Then the following conditions are equivalent.

(1) p' does not fork over si. (2) For any Jt, Jt', with si £ Jt < Jt', 38 £ Jt', there exists an extension

Pi ofp to Jt whose heir on Jt' is an extension ofp'.


(3) There exist models Jl, Jl' and a type px such that s? £ Jl < Jl', 35 £ j / \ t(@, Jl) is an immediate extension of t(0B, s/), and px is an extension of p to Jl whose heir over Jl' is an extension of p'.

Using this result, one proves, exactly as in §5 (Corollary 5.7 and Proposition 5.10) that the set of nonforking extensions of a stable type is a closed set of cardinality not bigger than 2X°.

In order to prove the open map theorem and the finite equivalence relation theorem, we need the following results which is kind of dual to Proposition 7.7.

PROPOSITION 7.8. Let s? s B, and suppose that t($8, sJ) is stable. Let p e S(s?) and p' be an extension ofp to SS. Then the following conditions are equivalent.

(1) p' is an immediate extension ofp. (2) p' is a coimmediate extension ofp. (3) For any Jl 3 s/, there exist an extension px ofp and an heir of it on Jl \J 28

which is an extension ofp'. (4) There exist Jl 3 sf such that t{@, Jl) does not fork over s/, an extension p}

ofp to Jl and an heir ofpl onJl\JSS which is an extension ofp'. Now, it is easy to prove the open map theorem. If t{S8, s/) is stable, then F =

{p e S(&); p is an immediate extension of p I s?) is a closed set, and the canonical map from F onto S(jsl) is open.

The finite relation theorem follows exactly as in §5. Note that it suffices to prove it when Jl is \\A ll^-saturated, and then we can separate the type/>0 and px by some formula <p{x0, m), where <p e L(x, A) and m is a sequence of elements realizing p so that t(m, st) is stable.

§8. Rank, superstable types. Let Z be the set of classes of types; Z is a partially ordered set, and we may assign to each of its elements a foundation rank as follows : define by induction on a the relation V(g) > a for £ e Z.

V(£) > a + 1 if there exists rj < £ such that V(n) > a. For a limit, F(£) > a if for every /3 < a, V(£) > jS. By the second condition, it is clear that if K(£) > a is not true for all a, then

there is a greatest ordinal a such that K(£) > a holds. By definition, this ordinal will be called K(f). If F(f) > a is true for all a, we will set K(£) = oo.

If p is a type over a model, V{p) will be K(f), where £ is the class ofp. We will say that £ (respectively p) is ranked if K(£) < oo (respectively V(p) < oo).

Clearly, V(p) = 0 if and only if p is realized. Among the types which do not represent the formula x = v0, there are some minimal elements: those have rank exactly one. Unfortunately, we cannot go any further.

It is obvious that F(f) > (2K°)+ implies K(£) = oo. In fact K(£) > wi implies F(£) = oo (see [P]).

DEFINITION 8.1. A superstable type is a stable type which is ranked. PROPOSITION 8.2. Let pe S(Jl). The following conditions are equivalent. (1) p is superstable. (2) For any Jl' > Jl, p has no more than || M' \\ + 2K° extensions to Jl'. (3) There is a cardinal I > 2Ko + \\M\\ such that for any Jl' > Jl of cardinality

X, p has strictly less than A*° extensions to Jl'. PROOF. (1) -» (2) For each finite set J c Jl', we construct a countable submodel


J/s of Jl' in such a way that s s s' implies Jls £ .//>. Jl0 will be a countable submodel of Jl' such that p f „//0 ~ /? ; we proceed, then by induction on the length of s.

Now, given an extension p' of p to Jl'', consider the set {V(p' f Jls); s finite and 5 c y / ' } . This is a set of ordinals since p' [ Jls < p. Let V(p' f .//,) be the least element of it. Then p' [ Jlt ~ p', and p' is the heir of p' \ Jl,.

So every extension p' is the heir of a type over one of the Jlt, and hence there are no more than 2X° • \\M'\\ such types.

(2) -* (3) Choose a cardinal / of cofmality co larger than 2K" + \\M\\. (3) -+ (1) Let p b e a nonsuperstable type and X > ||M||. We are going to con

struct a model Jl' of cardinality X such that/? has A*0 extensions to Jl'. \fp e S(Jl) is not stable, or is stable but not ranked, we can find Jl' >• Jl and X

extensions of p to Jl' with the same property as p. For nonstable types, use Proposition 4.3; ifp is stable but not ranked, then there is an extension/?' of p such that// is not ranked andp' < p. Then apply Proposition 4.8.

It is clear that, in addition, we may choose Jl' so that \\M'\\ < X. Now, arguing as in Proposition 4.4, we construct an increasing chain (Jl„ ; n e co) beginning with Jl and for each s e X" an extension ps of p to Jln in such a way that if 5 extends 5', and if J, s' e /*, s # s', then ps # ps>. Then {{J{ps; a extends s}; a e Xa} is a set of A" extensions of p to (J„20) Jl„. •

Let us now look at types over an arbitrary set of parameters. A type is supersta t e if every extension of it over every model is superstable.

DEFINITION 8.3. (The rank U) By induction on a, we define the relation U(p) > a.

If a is limit, U(p) > a if and only if U(p) > /3 for all /3 < a. For p e SOO, £/(/>) > a + 1 if and only if for any I, there exists @ ^ ^ such

that/) has at least X extensions q to ^ such that 1/(9) > a-Then we define the rank U in the same way as we introduced the rank V at the

beginning of this section. It is clear that if/?' is an extension of/?, U(p') < U(p). PROPOSITION 8.4. IfU(p) < a. then p is stable. PROOF. We prove by induction on a that every nonstable type has rank bigger

than a: this is an obvious application of Proposition 4.3. • PROPOSITION 8.5. Let p be a stable type over &?. Then U(p) = V(fi(p)). PROOF. Let sf S Jl and let /?' be a nonforking extension of p to Jl. We first

prove by induction on a that U(p) > a implies V(p') > a. This is obvious for a = 0 and a limit. Suppose a = /S 4- 1 ; there exist @ 3 s4 and (2*°)+ extensions of p to 38 whose {/-ranks are bigger than or equal to /3. By Proposition 5.10, one of them, say/?j, forks over si.

Let J>r ^ Si and p[ be a nonforking extension of px to Jf. By induction hypothesis V(p[) > p. But/?i < /?', so V(p') > /3 + 1.

Let us prove now that V(p') > a implies U(p) ^ a. It suffices to show that U(p') > a. Again we argue by induction on a, and again the only case to be considered is a = /3 + 1.

Since V(p') > (} + 1, there exist Jlx > Jl and an extension px of/?' to Jl\ such that K(/?j) > (3 and /?' > px. We may as well suppose Jix A-saturated, so that by

350 DANIEL LASCAR A N D BRUNO POIZAT

Proposition 4.8, there are at least 1. extensions of p' to J(x equivalent \.op', thus of K-rank at least /3. By induction hypothesis, their {/-rank is at least /3, so U(p') > / 3 + 1 . D

This proves that if p is a superstable type, its extensions of the same rank are exactly the nonforking extensions. The superstable types are exactly those whose i/-rank are less than oo.

The rank U satisfies the axioms for a rank notion stated in [LI]. By Proposition 8.5, it is easy to prove that it assigns to each type the least possible value among all functions satisfying the rank notion axioms.

We do not know how to prove directly (i.e. without the use of forking) that a type of i/-rank a (a > co) always has an extension of the same rank.

REFERENCES

[CK] C. C. CHANG and H. J. KEISLER, Model theory, North-Holland, Amsterdam, 1973. [LI] DANIEL LASCAR, Rank and definability in superstable theories, Israel Journal of Mathematics,

vol. 23(1976), pp. 53-87. [L2] ,Definissabilite dans les theories stables, Logique et analyse, 1971-1972, pp. 489-507;

appearing also in Six Days of Model Theory, Proceedings of a Conference at Louvain-la-Neuve, March 1975 (Paul Henrard, Editor).

[P] BRUNO POIZAT, Deviation des types, Doctoral dissertation, Paris, 1977. [SI] SAHARON SHELAH, Classification theory and the number of non-isomorphic models, Studies

in logic and the foundations of mathematics, vol. 92, North-Holland, Amsterdam, 1978. [S2] , Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal

of Mathematics, vol. 10(1971), pp. 224-233.

UNIVERSITE PARIS VII U.E.R. DE MATHEMATIQUES

TOUR 45-55 , 2 PLACE JUSSIEU, 75005 PARIS, FRANCE

UNIVERSITE PARIS VI 11 PARC D'ARDENAY, 91120 PALAISEAU, FRANCE

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Introduction to Forking