A Note on Finitely Generated ModelsAuthor(s): Anand PillaySource: The Journal of Symbolic Logic, Vol. 48, No. 1 (Mar., 1983), pp. 163-166Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2273329 .

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THE JOURNAL OF SYMBOLIC LoGic Volume 48, Number 1, March 1983

A NOTE ON FINITELY GENERATED MODELS

ANAND PILLAY

Introduction. A model M (of a countable first order language) is said to be finitely generated if it is prime over a finite set, namely if there is a finite tuple a in M such that (M, a) is a prime model of its own theory. Similarly, if A c M, then M is said to be finitely generated over A if there is finite a in M such that M is prime over A U a. (Note that if Th(M) has Skolem functions, then M being prime over A is equivalent to M being generated by A in the usual sense, that is, M is the closure of A under functions of the language.) We show here that if N is a model of an c)- stable theory, M -< N, M is finitely generated, and N is finitely generated over M, then N is finitely generated. A corollary is that any countable model of an )-stable theory is the union of an elementary chain of finitely generated models. Note again that all this is trivial if the theory has Skolem functions.

The result here strengthens the results in [3], where we show the same thing but assuming in addition that the theory is either nonmultidimensional or with finite aT. However the proof in [3] for the case aT finite actually shows the following (which does not assume &-stability): Let A be atomic over a finite set, tp(d/A) have finite Cantor-Bendixson rank, and B be atomic over A U a. Then B is atomic over a finite set. This result is not improved upon in the present note.

In fact the results in this note follow quite easily from some facts on regular types and a result in [2], which we assemble in the next section.

Notation is standard. We tend to work in a big saturated model. For basic facts on stability see [4]. Any other facts we introduce as we go along. Recall also that a countable model M is prime over a subset A of M if and only if every finite tuple in M realises an isolated type over A.

Preliminary results. We assume from now on that T is an )-stable theory. M, N, etc. will denote models of T, and A, B, etc. subsets of models.

DEFINITION I (SHELAH [4]). (i) Let p E S1 (M), and Dp(x) be a formula in p. (qp may contain parameters from M.) Then (p, sp) is strongly regular if p is not realised in M, and there is N >- M in which p is realised such that if b E N - M, and N l= (b), then b realises p.

(ii) Let p E S1(A), (p(x) E p, and p be stationary. Then (p, (p) is strongly regular if there is M :DA, such that (p', sp) is strongly regular, where p' is the nonforking ex- tension of p over M.

(iii) p E S1(A) is strongly regular if there is sq e p such that (p, sq) is strongly reg- ular.

Received January 7, 1981. @ 1983, Association for Symbolic Logic

0022-48 12/83/4801-0015/$01.40

163

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164 ANAND PILLAY

REMARK 2. (i) In (i) above we can specify that N be a model which is prime over M U {a}, where a realises p. Such an N is written as M(p) or M(a).

(ii) (i) and (ii) of Definition 1 are consistent with each other, due to the following fact: Let p E S1(M), p(x) E p, M -< M' and p' be the nonforking extension of p over M'. Then (p, sp) is strongly regular if and only if (p', sp) is strongly regular. (See [1J.)

LEMMA 3 (LASCAR [1] AND SHELAH [4]). Suppose that p E S1(M) and (p, p) is strong- ly regular. Let a realise p and tp(b/M) = Pl, where Pi : p and (p E Pl Then a and b are independent over M.

COROLLARY 4. Let p E S1(A), (p, p) strongly regular. Let a realise p and tp(b/A) -Pl, where p EiPi andp # Pl. Then a and b are independent over A.

PROOF. Let a and b be as in the hypothesis. Let M :DA be a model such that {a, b} and M are independent over A. Then, in particular, tp(alM) does not fork over A. So by Definition 1 and Remark 2(ii), (tp(a/M), (p) is strongly regular. Clearly tp(b/M) # tp(a/M) and k-q(b). Thus by Lemma 3, a and b are independent over M. So a and b are independent over A.

We now work towards the following: PROPOSITION 5. Let p E S1(M), Dp E p, (p, p) strongly regular. Let A be a subset of

M such that (i) p is definable over A (so p does not fork over A, andp I A is stationary), (ii) the parameters from p are in A, and (iii)p r A is not realised in M.

Then p I A F- p (i.e. any element which realises p I A realises p). One part of this is: LEMMA 6. Let p, p, and A satisfy the hypotheses of Proposition 5. Then p IA F

p r (A U TM). PROOF. Let a realise p r A. Let bj-jb2^ **bn be a tuple from q M. We first

show (*) tp(a/A U b) does not fork over A. Let b' be b1jb2 * * * -bn-1. Then by induction tp(a/A U b') does not fork over A.

But tp(a/A) is stationary, so tp(a/A U b') = p I (A U b'). Thus (tp(a/A U b'), (p) is strongly regular. But bn satisfies sp, and moreover tp(bn/A U b') 0 tp(a/A U b') (for otherwise bn would realise p I A, which would contradict hypothesis (iii) in Proposition 5). Thus, by Corollary 4, a and bn are independent over A U b'. It follows that a and b are independent over A, namely (*) is established.

Thus, if a realises p I A, then tp(a/A U qAm) does not fork over A. But tp(a/A) is stationary. Thus tp(a/A U qAm) is precisely p I (A U rpM). So Lemma 6 is proved.

The following is proved in a more general situation in [2]. We give here another (although similar) proof.

LEMMA 7. Let p e S1(M), and <p(x) e p. Then p does not fork over foM, and moreover p I f M is stationary.

PROOF. We will find a realisation a of p such that tp(a/M) does not fork over M. Let N and M1 be models such that both N and M are elementary substructures

of M1, N contains pM, and N and M are independent over pM. Let us expand M1 to M' by adding predicates P and Q, where P is interpreted as the universe of M

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FINITELY GENERATED MODELS 165

and Q as the universe of N. Now we can find an elementary extension M2 of M which contains an element a such that the (L-) type of a over M is p, and more- over M2 1= P(a). (This is by compactness, as every formula in p is satisfied in M.) In particular, a satisfies SD(x). However, by construction we have M1' 1 (Vx)((p(x) A P(x) -- Q(x)). Thus M2 1= (Vx)(QD(x) A P(x) -- Q(x)). Thus

(* *) M2' 1= Q(a). Let M2 be the L-reduct of M2. Let N2 be the interpretation of Q in M2. Then it is easy to check that in M2, tp(M/N2) is the heir of tp(M/N). But tp(M/N) does not fork over pM. Thus M and N2 are independent over qM. By (**), a is in N2. Thus a and M are independent over qM. As a realises p we have shown that p does not fork over qM.

Let us denote p r fpM by q. If q were not stationary, there would be Pi e S1(M), a nonforking extension of q, such that p # Pi. Let a realise p, and b realise Pi Then by the finite equivalence relation theorem, there would be E(x, y), a finite equiv- alence relation over pM, such that # -'E(a, b). But a satisfies ~o(x). So it is easy to find an element c in M which satisfies oD(x) such that l= E(a, c). But then E(x, c) e q, so kE(b, c) and so kE(a, b), which is a contradiction. So q is stationary.

COROLLARY 8. Let p e S1(M) and p(x) e p. Let A c M be such that the parame- tersfrom p are in A, and also q M C A. Then p I A F- p.

PROOF. Let a realise p I A. Let q = tp(a/M). Also oD(x) e q. So by Lemma 7, q does not fork over pM, and q r pM is stationary. Thus q does not fork over A, and also q r A = p I A is stationary. Similarly p does not fork over A. Thus

p = q.

Proposition 5 is now an immediate consequence of Lemma 6 and Corollary 8.

The main result. Here we will prove: PROPOSITION 9. Let M be finitely generated, and Nfinitely generated over M. Then

N is finitely generated. The following allows us to make a reduction to the case where N = M(p) and p

is strongly regular. LEMMA 10 (LASCAR [1]). Let N be finitely generated over M. Then there are models

M = Mo -< M1 -< * *- < Mn = N, and elements ai in Mi for i = 1, ..., n such that (i) tp(ai/Mi-1) is strongly regular, for i = 1, . . ., n, and (ii) Mi =Mj_(aj)

fori= 1,...,n. (We do not give the proof here, although we remark that it depends heavily on

w-stability.) Now let us proceed with the proof of Proposition 9. So M is finitely generated and

N is finitely generated over M. We want to show that N is itself finitely generated. By Lemma 10, we can clearly assume that N is generated over M by an element whose type over M is strongly regular. So N = M(a) say, and tp(a/M) is strongly regular. We wish to find a finite A c N, such that for all c in N, tp(c/A) is isolated. Again we can make a further reduction.

(I) In order that N be finitely generated, it is enough that there be finite A c M, such that for all c in M, tp(c/A U a) is isolated (in N of course).

(I) is quite simple and is left to the reader. Now let b be a finite tuple in M such that M is prime over b. So M is atomic over

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166 ANAND PILLAY

b, and moreover M is atomic over any finite b' which extends b. We now have two cases to consider.

Case (i). For all finite c in M, tp(a/b U c) is isolated. As for all c in M, tp(c/b) is isolated, it would follow that for any c in M, tp(a-c/b) is isolated, and thus tp(c/a U b) is isolated. By (I) this would finish the proof.

Case (ii). There is c in M, such that tp(a/b U c) is not isolated. So tp(a!A) is not isolated for any finite A in M which contains b U c. By assumption, tp(a/M) is strongly regular. So let (p(x) E tp(a/M), such that (tp(a/M), sp) is strongly regular. Let p = tp(a/M). Now let A be a finite subset of M such that (i) A contains b U c, (ii) A contains the parameters from the formula sp, and (iii) p is definable over A. (This last is by c-stability.) Note M is atomic over A, but tp(a/A) is not isolated. So clearly p, sp, and A satisfy the hypotheses of Proposition 5 of the last section.

(II) Thus p A F- p. Let c be a finite tuple from M. So tp(e/A) is isolated, by a formula 0b(x) say, which

has parameters in A. I assert that 0(x) isolates tp(c/A U a). For let c' satisfy 0b(x) in some big model. Let a' be such that tp(c'^a/A) = tp(c0a'/A). So in particular tp(a'/A) = p I A. Thus by (II), as c E M, tp(a'/A U c) = p I (A U O). So tp(c0a'/A) = tp(&0a/A), and thus tp(c'/A U a) = tp(e/A U a). So we have shown that tp(c/A U a) is isolated, for all c in M. Thus by (I), Proposition 9 is proved.

The following corollary was pointed out to us by D. Lascar. COROLLARY 11. Let M be a countable model (of an co-stable theory). Then M is

the union of an elementary chain offinitely generated models. PROOF. List M as {an: n < Cl}. Define Mn for n < Co, as follows: MO is an elemen-

tary substructure of M which is prime over ao, and Mn+1 is an elementary sub- structure of M which is prime over Mn U {an+1}. The Mn's exist by c-stability. By induction and Proposition 9, M, is finitely generated for all n < co. Clearly M = Un<wMnn

REFERENCES

[1] D. LASCAR, Ordre de Rudin Keisler et Poids dans les theories stables (to appear). [21 D. LASCAR and B. POIZAT, Introduction to forking, this JOURNAL, vol. 44, no. 3 (1979), pp.

330-350. [3] A. PILLAY, Finitely generated models of )-stable theories, Proceedings of the Logic Meeting

at Brussels and Mons in June 1980 Bulletin de la Societe Mathematique de Belgique, Serie B, vol. 23, fascicule I (1981).

[4] S. SHELAH, Classification theory and the number of nonisomorphic models, North-Holland, Amsterdam, 1978.

UNIVERSITt DE PARIS VII

75221 PARIS, FRANCE

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