On Countable Simple Unidimensional Theories

On Countable Simple Unidimensional TheoriesAuthor(s): Anand PillaySource: The Journal of Symbolic Logic, Vol. 68, No. 4 (Dec., 2003), pp. 1377-1384Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/4147768 .

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1378 ANAND PILLAY

In the remainder of this section we will give precise definitions, as well as recall some facts which will be used later in the proof. For general facts about simple theories, SU-rank, canonical bases, ..., see Wagner's book [9]. T will denote a complete theory in a language L. We work in a big saturated model MR of T unless we say otherwise. We work freely in Teq. a, b,

.... denote tuples of small length from

Meq. Likewise for sets A, B, .... A partial type I(x, b) is said to divide over a set A if there is an A-indiscernible sequence ( bi : i < w ) of realizations of tp(b/A) such that U{ Z(x, bi) : i < co } is inconsistent. T is (by definition) simple (supersimple), if for every B and finite tuple a, there is a subset A of B of cardinality at most ITI (cardinality < co) such that tp(a/B) does not divide over A. For T simple we will use "dividing" and "forking" interchangeably.

Let us now assume T to be a complete simple theory in a language L. T is said to be unidimensional if any two nonalgebraic types are nonorthogonal. That is, whenever p(x), q(y) are complete nonalgebraic types over sets A, B respectively then there is C D A U B and nonforking extensions, p'(x) of p and q'(y) of q respectively, over C, such that for some realizations a of p' and b of q', tp(a/C, b) forks over C.

Although we will not really make explicit use of the Do ranks, we will give the definition. Let 0(x, y) be an L-formula, and y/(x, a) a formula with parameters a. Then Do(y/(x, a)) > 0 if y/(x, a) is consistent, Do(Vy(x, a))

_ a + 1 if for

some b, D•(/(x,

a) A 4(x, b)) _

a and q5(x, b) divides over a, and for limit 6, D(/((x, a)) 6 if it is > a for all a < 6. We say that the Do rank is finite and definable if (i) Do(x = x) < o and (ii) for any n and L-formula y/(x, z) there is an L-formula X(z) such that for all a, I X(a) if and only if Do(Vy(x, a)) = n.

The L-formula 4(x, y) is said to be low if there is k < co such whenever (bi : i < co ) is indiscernible and { ((x, bi) : i < oI } is inconsistent, then { 1(x, bi) : i < k } is inconsistent. In [1] it is proved that 0(x, y) is low just if DO(x = x) < W. T is said to be low if every L-formula is low.

Now we define the relations Qo,,. Let q(x, y) and y/(x, z) be L-formulas. Then

Qo,, holds of b if whenever I 0(a, b) then y/(a, z) does not fork over b. We say that Q0,, is definable if there is an L-formula X(y) such that for all b, QO, ,(b) if and only if X(b).

Now we recall the lovely pairs. Let N < M be an elementary pair of models of T, and K a cardinal > I TI+. We call this pair (M, N) n-lovely if (i) for any A c M of cardinality < n. and p(x) E S(A) (where x is a finite tuple of variables), some nonforking extension of p(x) over A U N is realized in M, (ii) if p as in (i) does not fork over N then p(x) is realized in N.

By a lovely pair (of models of T) we mean a I TI+-lovely pair. Let Lp be L together with a new unary predicate P. Any pair (M, N) of models

of T will be considered as an Lp-structure by taking N as the interpretation of P. It is a relatively straightforward fact that any two lovely pairs of models of T are elementarily equivalent, as Lp-structures. So TP, the common Lp-theory of lovely pairs, is complete.

L) denotes the language obtained by adjoining to Lp•

a new relation symbol R4(x) for each L-formula q(x, y). T4 will be the LP-theory obtained by adding to Tp the sentences Vx(R4(x) - 3y(P(y) A #(x, y)). (So any model of Tp has a unique expansion to a model of T+.)



ON COUNTABLE SIMPLE UNIDIMENSIONAL THEORIES 1381

q(x, y) E L (y varying), there is oa(y) E 10(y) such that V, (x) E p(x). So (as T is countable), S;(Tp) is a Ga-subset of S,(Tp).

(ii) First, if p(x) E S* (Tp), then pIL is nonalgebraic. (For otherwise any realization of p(x) would be in Mo which is included in P). Now suppose that q(x) E Sx,na(T). Then by lovelieness of (M, P), some L-nonforking extension of

q(x) over P is realized in M by a' say. Note that a' V P so q* = tpLp (a') E S* (Tp). By Fact 1.3, if a" E M realizes any nonforking extension of q over P, then for any L-formula q(x, y), (M, P) = 3y E P (04(a', y)) +- 3y E P(0(a", y)). Hence, as

Tp- has quantifier-elimination (by Fact 1.1), tpL,(a") = tpL (a') = q*. So q* is the unique element of

Sx (Tp) which maps to q. -d

Let us endow the set Sx,na (T) with the topology T coming from the bijection

7rSx (T): S (Tp) --+ Sx,na(T) given by 2.2, where S (Tp) is given the subspace topology as a subset of the Stone space Sx(Tp). For a pair (0(x, y), y/(x, z)) of L-formulas, let U(4,,) be the set of p E Sx,na (T) such that for some (any) a realizing p, 0b(a, y) does not L-fork over 0 and Vy(a, z) does L-fork over 0.

LEMMA 2.3. The sets U(O,,) are a basis for the topology T on Sx,na (T).

PROOF Let 0 (x, y), yV(x, z) be L-formulas, and let O6,,.(x) be the Lp-formula 3y E P (4(x, y)) A -dz E P (y(x, z)). Note that any finite conjunction 60,,1, (x) A ... A

60,,,, (x) is equivalent (in Tp) to some 6,•1,,.

So, as Tjj has quantifier- elimination in Lp, any Lp-formula 6(x) is equivalent to a finite disjunction of formulas of the form 63, (x).

Now let p* E Sx(Tp) and let p(x) = p*IL. By Fact 1.3, p(x) E U0,, if and only if 360,(x) E p* (x). So the lemma follows from the definition of the T-topology. -4

We now bring in the rest of our assumptions. For now we work in T. The proof of the next lemma is no different from the stable case, but we give details for completeness.

LEMMA 2.4. Let p(x) E Sx,na(T). Then there are 0-definable functions f (x) and g(u, w1, .., Wn) such that

for some b and realization a of p such that a is independent from b (*)f,g over 0, f (a) V acl(0) and there are realizations cl, ... Cn of po such

that g(b, cl,..., cn) = f(a).

PROOF. By unidimensionality, p(x) is nonorthogonal to po(w). So there are B and a realizing a nonforking extension of p over B such that a forks with c over B for some realization c of po (necessarily independent from B over 0). Then the canonical base of Lstp(cB/a) is in the definable closure of a Morley sequence of

stp(cB/a) and also in acl(a). Under our assumptions, this canonical base is a set of elements of ieq. Let d V acl(0) be in Cb(stp(cB/a), and let d' be its set of a-conjugates. Then d' e dcl(a) \ acl(0) and also d' is in the definable closure of a set (clB1i..., cnBn) where (ciBi)i is an a-independent set of realizations of

tp(cB/a). In particular a is independent from b = B1 U ... U Bn over 0 and all the ci realize po. We may assume that b is finite. So d' = f(a) = g(b, cl,..., c,), for some 0-definable functions f and g. -

For each pair f, g of 0-definable functions, let Wf,g be the set of p(x) E Sx,na (T) such that (*)f,g holds for p.

LEMMA 2.5. Each Wf,g is a closed subset of Sx,na (T) for the z-topology.



1382 ANAND PILLAY

PROOF. Let p(x) E Sx,na(T) be in the z-closure of Wf,g. So for each formula /(x) E p(x) and formulas VI(x, u),..., ,Vr(x, u) such that each Wi(a, u) forks over 0 whenever a realizes p(x), there is q(x) E Wf,g such that 0(x) E q and each Vi(a', u) forks over 0 whenever a' realizes q(x). In particular, there are b and a' such that 0 q(a') A

-,i(a', b) A ..A -yr,(a', b), and such that f(a') V acl(0)

and f (a') = g(b, Cl,..., c,) for some realizations ci of po. By compactness p(x) E

Wf,g. COROLLARY 2.6. For some f, g, Wf,g has nonempty interior in the z-topology.

PROOF. This is by definition of the z-topology, Lemma 2.2 (i), and the fact that Baire category holds for Ga subsets of compact spaces. -

Let U be a nonempty basic T-open subset of Sx,na (T) which is contained in Wf,g, where g = g(u,

wl,..... wn) say. Let z be a variable corresponding to the range sort

of f. Let f(U) = { tPL(f(a)) : tp(a) E U }. It is clear, by the definition of Wf,g that f (U) c Sz,nm (T). Sz,na (T) is equipped also with a T-topology.

LEMMA 2.7.

(i) f (U) is an open subset of Sz,na (T) in the z-topology. (ii) For any q(z) E f (U), SU(q(z)) < n.

PROOF.

(i) Let U = U(4(x,y),i(x,w)) n Sx,na(T). Note that by lowness of T, there are L-formulas Xi(x) for i < o- such that for any a E A, 0(a, y) does not L-fork over 0 if and only if L ViXi(x). For p(x) E Sx,na(T) let p* E S*(Tp) be the Lp-type of a realization in (M, P) of some L-nonforking of p(x) over P. We know that p* is uniquely determined by p(x) (by 2.2 (ii)). Let U* = { p* : p(x) e U }. Let

f(U*) = { tpL,(f(a)) :tpLp(a) E U* }. Note that f(U*) C S;(Tp).

CLAIM. f ( U*) is a open subset of S; (Tp) in the Stone topology.

PROOF OF CLAIM. We must find an open subset of Sz (T) whose intersection with S* (Tp) is precisely f (U*). Consider the open subset W of Sz (Tp) given by Vi(3x (f(x) = z A Xi(x) A -3w E P (V(x, w))). Suppose first that q*(z) e W n Sz (Tp), realized by d. So by choice of the Xi, there is c E (M, P) such that f(c) = d, q(c, y) does not L-fork over 0 and = -3]w E P (V(c, w)). Lovelyness of (R, P) together with the last clause implies that yr(c, y) L-forks over P and thus L-forks over 0. Let c' be such that tpL(c', d) = tpL(c, d) and c' is L-independent from P over d. So f(c') = d, 4(c', y) does not fork over 0, ,V(c', w) forks over 0 and c' is L-independent from P over 0 (and clearly c' V aclL (0)). So tPL, (c') E U*, and thus tPLp (d) = q* E f (U*).

Conversely, if q* E f(U*), realized by d and f(c) = d where tpLp(d) E U*, then we know that = Vi Xi(c) and = -3w E P (y/(c, w)). Hence q* E W. This proves the claim.

Note that f(U) = { q(z) E Sz,,na(T) : q*(z) E f(U*) }, and so is z-open in Sz,na (T) by the Claim and the definition of the z-topology.

(ii) If q(z) E f(U), then are b, and d realizing q independent of b such that d E dcl(b, cl,..., cn) for some realizations cl,..., c, of po. So SU(q(z)) < n. -



ON COUNTABLE SIMPLE UNIDIMENSIONAL THEORIES 1383

Working in the space Sz,na (T) with its z-topology, f (U) contains some nonempty basic open set of the form

U0(z,y),V(z,v) = { q(z) E Sz,na(T) : (a, y) does not fork

over 0 and Vy(a, v) forks over 0 for all (some) a realizing q }. LEMMA 2.8. There is bo E Mo such that U(z,bo),,(z,v) C Sz,na(T) is nonempty. PROOF. Let q(z) E U/(z,y),v(z,v).

Let b be such that some nonforking extension

q'(z, b) of q(z) contains 0(z, b). CLAIM. For some (any) d realizing q'(z, b), yV(d, v) forks over b. PROOF OF CLAIM. Otherwise, there is d realizing q'(x, b) and c such that

Vi(d, c), and c is independent from d over b. But then d is independent from c over b and so over 0. This contradicts the fact that yi(d, y) forks over 0 for all d realizing q(z), proving the claim.

Note that q'(z, b) is nonalgebraic. Thus b E X { b': there is d g acl(b') such that 0(d, b') holds and V(d, v) forks over b' }. By Fact 1.4, X is 0-definable. Hence there is bo E X n Mo. The formula 4(x, bo) is now over 0. By definition of X there is q"(x) E Sz,na(T) such that q$(x, bo) E q" and such that V(z, v) is not represented in any nonforking extension of q". Hence U0(z,bo),y(z,v) is nonempty. -

We now return to the lovely pair (lM, P). LEMMA 2.9. Let d EM• \ P be such that 0(z, bo) A -3v E P (V/(z, v)) E tpL, (d).

Then tpL (d) has SU-rank < n. PROOF. Let A C P be a countable set such that tpL (d/P) does not fork over A. By

Fact 1.3, yi(d, v) forks over A and so forks over 0. Thus tpL(d) e U0(z,bo),,(z,v)

C

U(z,y),v(z,v) C f (U). By 2.7 (ii), SU(tpL(d)) < n. -d We will now complete the proof of Theorem 2.1 by finding a nonalgebraic formula

in T which has finite SU-rank. Let us first fix some q(z) E U(x,bo),Y(z,,). Let q'(z)

be an L-nonforking extension of q over P. Working possibly in an elementary extension M' of M, extend q'(z) to a complete L-type r(z) over some P U C, such that SU(r) 1. Realize r by d. We may assume that C is countable, and r does not L-fork over C. Let A c P be countable such that tpL (d, C/P) does not L-fork over A. As (R7!, P) is a lovely pair, we can realize some L-nonforking extension of

tpL(d, C/A) over P in M by some d', C' say. Let r'(z) = tpL(d'/A U C'). So SU(r') = 1.

CLAIM I. If d" E l realizes r'(z) and d" aclL(P U C'), then 05(d", bo) and

E-3v E P (V(d", v)). PROOF OF CLAIM I. d" is L-independent from C' U P over C' U A. But C' is

L-independent from P over A. Hence d"C' is is L-independent from P over A. In particular d" is L-independent from P over A. But tpL(d"/A) = tpL(d/A), an L-nonforking extension of q(z). So tpL(d"/P) is an L-nonforking extension of q(z). So - (d", c) for all c E P (and clearly also 0b(d", bo) holds). This proves Claim I. -

As (M, P) is a saturated model of Tp, we can apply compactness to Claim I, to find a formula O(z, c) E r'(z) such that

(*) whenever d" realizes O(z, c), and d" ( aclL (P U C') then (d", bo) and -v c P (y(d", v)).

Note that 0(z, c) is a nonalgebraic L-formula over c.



1384 ANAND PILLAY

CLAIM II. 0(z, c) has SU-rank < n.

PROOF. Let p(z, c) be a complete nonalgebraic L-type over c containing O(z, c). As (A7!, P) is a lovely pair, we can realize some L-nonforking extension of p(z, c) over P U C' in M, by d" say. But then d" V aclL(P U C') and so by (*), 0(d", bo) and -13v E P (q/(d",v)) (and of course d" V P). By Lemma 2.9, tpL(d") has SU-rank < n. As p(z, c) is an extension of tpL(d"), also p(z, c) has SU-rank <n.

Claim II completes the proof of Theorem 2.1.

REFERENCES

[1] I. BEN-YAACOV, A. PILLAY, and E. VASSILIEV, Lovely pairs of models, to appear in Annals of Pure and Applied Logic.

[2] E. HRUSHOVSKI, Countable stable unidimensional theories are superstable, preprint, 1985. [3] , Unidimensional theories are superstable, Annals ofPure and Applied Logic, vol. 50 (1990),

pp. 117-138.

[4] A. PILLAY, Geometric stability theory, Oxford University Press, 1996. [5] B. POIZAT, Paires de structures stables, this JOURNAL, vol. 48 (1983), pp. 234-249. [6] Z. SHAMI, Coordinatization by binding groups and unidimensionality in simple theories, to be

submitted. [7] - , On Kueker simple theories, submitted to this JOURNAL. [8] S. SHELAH, Classification theory, North-Holland, 1990. [9] E O. WAGNER, Simple theories, Kluwer, 1999.

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On Countable Simple Unidimensional Theories

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