Subsets of superstable structures are weakly benign

Bektur Baizhanov Institute for Informatics and Control Almaty, Kazakhstan^{∗} John T. Baldwin Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago^{†}

Saharon Shelah^{‡} Department of Mathematics Hebrew University of Jerusalem Rutgers University

February 19, 2003

Baizhanov and Baldwin [1] introduce the notions of benign and weakly benign sets to investigate the preservation of stability by naming arbitrary subsets of a stable structure. They connect the notion with work of Baldwin, Benedikt, Bouscaren, Casanovas, Poizat, and Ziegler. Stimulated by [1], we investigate here the existence of benign or weakly benign sets.

Definition 0.1 1. The set A is benign in M if for every α, β ∈ M if p = tp(α/A) = tp(β/A) then tp_{∗}(α/A) = tp_{∗}(β/A) where the ∗-type is the type in the language L^{∗} with a new predicate P denoting A.

2. The set A is weakly benign in M if for every α, β ∈ M if p = stp(α/A) = stp(β/A) then tp_{∗}(α/A) = tp_{∗}(β/A) where the ∗-type is the type in language with a new predicate P denoting A.

∗Partially supported by CDRF grant KM2-2246.

†Partially supported by NSF grant DMS-0100594 and CDRF grant KM2-2246.

‡This is Publication 815 in Shelah’s bibliography. This research was partially supported by The Israel Science Foundation.

Conjecture 0.2 (too optimistic) If M is a model of stable theory T and A ⊆ M then A is benign.

Shelah observed, after learning of the Baizhanov–Baldwin reductions of the problem to equivalence relations, the following counterexample.

Lemma 0.3 There is an ω-stable rank 2 theory T with ndop which has a model M and set A such that A is not benign in M.

Proof: The universe of M is partitioned into two sets denoted by Q and R. Let Q denote ω ×ω and R denote {0, 1}. Define E(x, y, 0) to hold if the first coordinates of x and y are the same and E(x, y, 1) to hold if the second coordinates of x and y are the same. Let A consist of one element from each E(x, y, 0)-class and one element of all but one E(x, y, 1)-class such that no two members of A are equivalent for either equivalence relation. It is easy to check that letting α and β denote the two elements of R, we have a counterexample. In this case, the type p is algebraic. Algebraicity is a completely artificial restriction. Replace each α and β by an infinite set of points which behave exactly as α, β respectively. We still have a counterexample. In either case, α and β have different strong types. This leads to the following weakening of the conjecture.

Conjecture 0.4 (Revised) If M is a model of stable theory T and A is an arbitrary subset of M then A is weakly benign.

We give here a proof of Conjecture 0.4 in the superstable case. There are two steps. In the first we show that if (M, A) is not (weakly) benign then there is a certain configuration within M. (This uses only T stable.) The second shows that this configuration is contradicted for superstable T. Note that if (M, A) is not weakly benign, neither is any L^{∗}-elementary extension of (M, A) so we may assume any counterexample is sufficiently saturated.

1 Refining a counterexample

In this section we choose a specific way in which sufficiently saturated pair (M, A) where Th(M) is stable, fails to be weakly benign. Fix (M, A), a κ^{+}-saturated of a stable theory T where κ = κ^{|T |} is regular.

We introduce some notation. Recall that A is relatively κ-saturated in M if every type over (a subset of A) whose domain has cardinality less than κ and which is realized in M, is also realized in A. First note that for any c ∈ M − A, there is a pair (M_{1}, A_{1}) such that A_{1} is relatively κ-saturated in A; A_{1} ∪ c ⊆ M_{1} and M_{1} is independent from A over A_{1}; A_{1} and M_{1} have cardinality κ and M_{1} is κ-saturated. For this, choose A_{0} ⊂ A with c independent from A over A_{0} and |A_{0}| < κ (which follows since κ ≥ |T | ≥ κ(T)). Then extend A_{0} to a subset A_{1} of A with cardinality at most κ which is relatively κ-saturated in A. Finally, let M_{1} ≺ M be κ-prime over A_{0} ∪ c. We have shown the following class K_{c} is not empty.

Notation 1.1 1. For any c ∈ M, let K_{c} be the class of pairs (M_{1}, A_{1}) with c ∈ M_{1} ≺ M such that A_{1} is relatively κ-saturated in A; A_{1} ∪ c ⊆ M_{1} and M_{1} is independent from A over A_{1}; A_{1} and M_{1} have cardinality κ and M_{1} is κ-saturated with |M_{1}| ≤ κ.

2. For any a, b in M which realize the same type over A, let K_{a,b}^{1} be the set of tuples ⟨A_{1}, M_{a}, M_{b}, N_{a}, g⟩ such that (M_{a}, A_{1}) and (M_{b}, A_{1}) are in K_{a}, K_{b} respectively, g is an isomorphism between M_{a} and M_{b} (subsets of M) over A_{1} (taking a to b), N_{a} contains M_{a} and is saturated with cardinality κ, and N_{a} is independent from A over A_{1}.

3. Let K_{a,b}^{2} be the set of tuples ⟨A_{1}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b}^{1} such that g is an isomorphism between M_{a}^{eq} and M_{b}^{eq} over A_{1}^{eq}.

4. We will write K^{i} to denote either K^{1} or K^{2}. Note the only difference between them is that K^{2} has a more restrictive requirement on the isomorphism g.

Note that the last clause of item 2 implies that N_{a} is independent from A over N_{a} ∩ A and that N_{a} ∩ A = A_{1} = M_{a} ∩ A. Moreover, if ⟨A_{1}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b} and B ⊆ A with |B| ≤ κ then there is an ⟨A′_{1}, M′_{a}, M′_{b}, N′_{a}, g′⟩ ∈ K_{a,b} with A_{1}∪B ⊆ A′_{1}. (Just include B when making the construction from the first paragraph of this section to show K_{a,b} is nonempty). We need a couple of other properties of K_{a,b}. Note that K_{a,b} is naturally partially ordered by coordinate by coordinate inclusion.

Lemma 1.2 Every increasing chain from K_{a,b}^{i} of length δ a limit ordinal less than κ^{+} has an upper bound in K_{a,b}^{i}.

Proof. If the cofinality of the chain is at least κ_{r}(T), just take the union (in each coordinate). We check that N_{a}^{δ}, A are independent over A^{δ}: By induction, for every α < β < δ, tp(N_{a}^{α}/A) does not fork over A_{1}^{β} (by monotonicity of nonforking). Hence if δ is a limit ordinal, tp(N_{a}^{δ}/A) does not fork over A_{1}^{δ}.

But if the cofinality is smaller the union may not preserve κ-saturation. In this case, let ⟨A′_{1}, M′_{a}, M′_{b}, N′_{a}, g′⟩ denote the union of the respective chains; each has cardinality κ. Choose A_{1} ⊆ A with |A_{1}| = κ and such that A_{1} is relatively κ-saturated in A and A_{1} contains A′_{1}. Then let the bound be ⟨A_{1}, M_{a}, M_{b}, N_{a}, g⟩ where M_{a} is κ-prime over M′_{a} ∪ A_{1}, M_{b} is κprime over M′_{b} ∪ A_{1}, g is the induced isomorphism extending g′ and N_{a} is any κ-saturated elementary extension of M_{a} ∪ N′_{a} in M with N_{a} independent from A over A_{1}. □_{1.2}

Lemma 1.3 If t = ⟨A_{1}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b}^{i} and p ∈ S(M_{a}) is non-algebraic, orthogonal to A and p ⊥̸ tp(N_{a}/M_{a}), then there is t′ = ⟨A′_{1}, M′_{a}, M′_{b}, N′_{a}, g′⟩ ∈ K_{a,b}^{i} with t′ extending t and tp(N_{a}/M′_{a}) forking over M_{a}.

Proof. Since M is κ^{+}-saturated, we can find d ∈ M realizing p such that tp(d/N_{a}) forks over M_{a} and d′ ∈ M realizing g(p). Now, construct t′ by letting A′_{1} = A_{1}, M′_{a} be κ-prime over M_{a} ∪ {d}, M′_{b} be κ-prime over M_{b} ∪ {d′}, g′ be an extension of g taking d to d′, and N′_{a} ≺ M any κ-saturated extension of M′_{a} ∪ N_{a}. We need to show that M′_{a} and M′_{b} are independent from A over A′_{1}. For this, note that since p ∈ S(M_{a}) is orthogonal to A (a fortiori to A_{1}) and A is independent from M_{a} over A_{1}, d is independent from A over M_{a}. Since M′_{a} is κ-prime over M_{a} ∪ {d}, it follows that M′_{a} is independent from A over A′_{1}. An analogous argument shows M′_{b} is independent from A over A′_{1}. Since d ∈ M′_{a}, we have fulfilled the lemma. □_{1.3}

For any ordinal μ and any sequence ⟨a_{i} : i < μ⟩ and any finite w ⊆ μ, a_{w} denotes ⟨a_{i} : i ∈ w⟩. We require one further technical notion.

Definition 1.4 We say M_{a} is A-full in M if for any N κ-prime over M_{a}A and for any C_{0} ⊆ M_{a}, |C_{0}| ≤ |T |, C_{1} ⊆ A with |C_{1}| ≤ |T |, and C_{2} with C_{0} ⊆ C_{2}, C_{1} ⊆ C_{2} ⊆ N, and |C_{2}| ≤ |T |, there is an elementary map f taking C_{1}C_{2} into M_{a} over C_{0} with f(C_{1}) ⊆ A and if C_{2} is independent from A over C_{1} then f(C_{2}) is independent from A over f(C_{1}).

We prove a characterization of a weakly benign pair; a similar result for benign (using K^{1} instead of K^{2} also holds. In view of the counterexample in given in the introduction, weakly benign is the interesting case.

Lemma 1.5 Use the notation of 1.1. Suppose (M, A) is κ^{+}-saturated where κ = κ^{|T |} is regular and T = Th(M) is stable. The following are equivalent.

1. (M, A) is not weakly benign.

2. There exist A_{∗}, M_{a}, N_{a}, M_{b}, g contained in M with a ∈ M_{a}, b ∈ M_{b} such that:

(a) ⟨A_{∗}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b}^{2} and M_{a} ≠ N_{a}.

(b) M_{a} is A-full in M.

(c) N_{a} is independent from A over A_{∗}.

(d) tp(N_{a}/M_{a}) is orthogonal to every nonalgebraic type in S(M_{a}) which is orthogonal to A.

(e) If d ∈ N_{a} − M_{a}, there is no d′ ∈ M which realizes g(tp(d/M_{a})) and such that d′ is independent from A over M_{b}.

(f) M_{a} and M_{b} are isomorphic over A_{∗} by a map g taking a to b and preserving strong types over A, i.e. g ↾ (A^{∗})^{eq} is the identity.

Note that, by general properties of orthogonality, we could rephrase item c) as: tp(N_{a}/M_{a}) is orthogonal to every type in S(M_{a}) which is orthogonal to A_{∗}.

Proof of Lemma 1.5: First we show that condition 2) implies condition 1). By condition 2a), there is an a′ in N_{a}−M_{a}. Note that since A^{∗} is relatively κ^{+}-saturated in A and M_{a} (M_{b}) is independent from A over A^{∗}, M_{a} ∩ A = M_{b} ∩ A = A^{∗}. It follows that (g ∪ id) ↾ acl(A^{eq}) is an elementary map in L^{eq}. Let a = ⟨a_{i} : i < κ⟩ enumerate M_{a} − A with a_{0} = a; denote g(a_{i}) by b_{i} so b = ⟨b_{i} : i < κ⟩ enumerates M_{b}. For any finite set of L-formulas Δ and finite subset w of κ, let ϕ_{Δ,w}(x; a′, a_{w}, b_{w}) be the L^{∗}-formula which assert that xb_{w} and a′a_{w} realize the same Δ-type over A. For any finite w, a_{w} and b_{w} realize the same L-type over A.

Now, let q = {ϕ_{Δ,w}(x; a′, a_{w}, b_{w}) : 0 ∈ w ⊂_{ω} κ, Δ ⊂_{ω} L}. Putting 0 ∈ w guarantees a, b are in any relevant a_{w}, b_{w}. So q is a set of κ L^{∗}-formulas with free variable x and parameters from M_{a} ∪ M_{b} ∪ {a′}. If q is finitely satisfied in (M, A), then q is realized in M by some b′, since M is κ-saturated as an L^{∗}-structure. But since a′ is independent from A over M_{a}, b′ realizes the unique nonforking extension of g(tp(a′/M_{a})) to M_{b} ∪ A contradicting condition d). If q is not finitely satisfiable, there is a formula ϕ_{Δ,w} which demonstrates the L^{∗} type of a_{w} and b_{w} over A are different.

To show the converse, we suppose that a and b realize the same (strong)-type over A but that there is an a′ such that there is no b′ ∈ M with aa′ ≡_{A,L} bb′.

We will use the following basic fact:

Fact 1.6 1. If A_{1} is relatively κ-saturated in A and C is independent from A over A_{1}, then CA_{1} is relatively κ-saturated in CA.

2. If A_{1} is relatively κ-saturated in A and D is κ-atomic over A_{1}, D is independent from A over A_{1}.

The following lemma essentially shows 1) implies 2) of Lemma 1.5.

Lemma 1.7 There is a t = ⟨A_{∗}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b}^{2} such that

A N_{a} ≠ M_{a},

B tp(N_{a}/M_{a}) is orthogonal to every nonalgebraic type in S(M_{a}), which is orthogonal to A.

C If d ∈ N_{a} − M_{a}, there is no d′ ∈ M which realizes g(tp(d/M_{a})) and such that d′ is independent from A over M_{b}.

D M_{a} is A-full.

Proof. Try to construct by induction a sequence ⟨t_{α} : α < κ^{+}⟩ where t_{α} = ⟨A_{∗}^{α}, M_{a}^{α}, M_{b}^{α}, N_{a}^{α}, g^{α}⟩ of elements of K_{a,b}^{2} which are increasing in the natural partial order, continuous at limit ordinals of cofinality greater than κ_{r}(T) and with a′ ∈ N_{a}^{0}.

1. If α is an even ordinal there are several cases.

(a) Suppose condition B fails, i.e. for some d ∈ N_{a}, p = tp(d/M_{a}) is nonorthogonal to some stationary type q ∈ S(M_{a}) which is orthogonal to A. Then by Lemma 1.3, there is t′ = ⟨A′_{∗}, M′_{a}, M′_{b}, N′_{a}, g′⟩ ∈ K_{a,b}^{2} with t′ extending t and tp(N_{a}/M′_{a}) forks over M_{a}.

(b) Suppose condition B holds.

i. If α is a limit ordinal of cofinality κ, stop.

ii. If α is a limit ordinal of cofinality < κ or α is a successor ordinal, let t_{α+1} = t_{α}.

2. α is an odd successor ordinal. Choose an auxiliary M̂_{a}^{α} κ-prime over M_{a}^{α}A. Choose A_{∗}^{α+1}, M_{a}^{α+1}, M_{b}^{α+1} such that A_{∗}^{α} ⊆ A_{∗}^{α+1} ⊆ A, |A_{∗}^{α+1}| = κ and so that

(M_{a}^{α+1}, A_{∗}^{α+1}) ≺_{L_{(}|T |^{+},|T |^{+}} (M̂_{a}^{α}, A)

and M_{a}^{α+1} is κ-prime over M_{a}^{α}A_{∗}^{α+1}. This is possible since κ = κ^{|T |}. In particular, M_{a}^{α+1} is independent from A over A_{∗}^{α+1}. The κ-primeness allows us to easily construct M_{b}^{α+1} and g_{α+1}. Now choose N_{a}^{α+1} to be a κ-saturated extension of M_{a}^{α+1} that is independent from A over A_{∗}^{α+1}

3. If α is a limit ordinal choose t_{α} by Lemma 1.2.

We cannot carry out this construction for κ^{+} steps. If we did, by clause 1) of the construction at each limit α with cf(α) = κ, clause B) fails. Thus, M_{a}^{α+1} depends on N_{a}^{α} over M_{a}^{α} for all such α, which contradicts stability. (If we were dealing with finite sequences, the bound would be κ(T); since we deal with sets of cardinality κ, the bound is κ^{+}.)

Fix α where the construction stops. We have constructed t_{α} = ⟨A_{∗}^{α}, M_{a}^{α}, M_{b}^{α}, N_{a}^{α}, g^{α}⟩ but for any choice of t_{α+1} ∈ K_{a,b}^{2}, M_{a}^{α+1} is independent from N_{a}^{α} over M_{a}^{α}. Note that each member of t_{α} = ⟨A_{∗}^{α}, M_{a}^{α}, M_{b}^{α}, N_{a}^{α}, g^{α}⟩ is the union of the respective member of t_{β} over β < α. We claim this t_{α} is a t satisfying the conditions of the lemma.

For clause A note N_{a}^{α} ≠ M_{a}^{α} since a′ ∈ N_{a}^{α} and a′ cannot be in the domain of g^{α} by the original choice of a′. Since the construction stopped clause B, holds.

For clause C, we must show that if d ∈ N_{a} − M_{a}, there is no d′ ∈ M which realizes g(tp(d/M_{a})) and such that d′ is independent from A over M_{b}. Fix d ∈ N_{a} − M_{a}; if such a d′ exists, choose M_{a}^{α+1}, M_{b}^{α+1} contained in M prime over M_{a}^{α}d and M_{b}^{α}d′ respectively. We easily extend g^{α} to g^{α+1} mapping M_{a}^{α+1} to M_{b}^{α+1}. By the construction, A_{∗}^{α} is relatively κ-saturated in A. So, M_{a}^{α} ∪ {d} and A are independent over A_{∗}^{α} by monotonicity, as N_{a}^{α} is independent from A over A_{∗}^{α}. Now by Fact 1.6 1), M_{a}^{α} ∪{d} is relatively κ -saturated inside M_{a}^{α} ∪{d}∪A. Whence, by Fact 1.6 2) M_{a}^{α+1} and A are independent over M_{a}^{α} ∪ {d}. By transitivity of nonforking, M_{a}^{α+1} and A are independent over A_{∗}^{α}. Similarly, since d′ is independent from A over M_{b}, M_{b}^{α+1} is independent from A over A_{∗}^{α}. But now, d ∈ (M_{a}^{α+1} ∩ N_{a}^{α}) − M_{a}^{α} soN_{a}^{α} depends on M_{a}^{α+1} over M_{a}^{α} and we have violated the choice of α.

Finally we verify clause D: M_{a} is A-full. Choose N, which is κ-prime over AM_{a}. Then N can be embedded over AM_{a} into M̂_{a}^{α} = ∪_{i<α}M̂_{a}^{i}. By the Tarski union of chains theorem (using clause 2) of the construction), (M_{a}^{α}, A ∩ M_{a}^{α}) ≺_{L_{|T |^{+},|T |^{+}}} (M̂_{a}^{α}, A). Let C_{0}, C_{1}, C_{2} ⊆ N satisfy the hypotheses of the definition of A-full. The elementary submodel condition easily allows us to define the required function f. □_{1.7}

2 The Superstable Case

The aim of this section is to prove that if M is a model of a superstable theory and A ⊂ M, then (M, A) is weakly benign. This is a generalization of a result of Bouscaren [2], who showed, in our terminology that every submodel of a superstable structure is benign.

Theorem 2.1 If M is a model of a superstable theory and A ⊂ M, then (M, A) is weakly benign.

Proof. We work in M^{eq}. Without loss of generality, assume (M, A) is κ^{+}-saturated for a regular κ satisfying κ^{|T |} = κ. By Lemma 1.5 if (M, A) is not weakly benign, there exist A_{∗}, M_{a}, N_{a}, M_{b}, g contained in M satisfying the conditions of Lemma 1.5 and with ⟨A_{∗}, M_{a}, M_{b}, N_{a}, g⟩ ∈ K_{a,b}^{2}.

Since M_{a} is properly contained in N_{a}, we can choose c ∈ M_{a} and ϕ(x, c) to have minimal D-rank among all formulas with ϕ(N_{a}, c) ≠ ϕ(M_{a}, c). Then for any d^{∗} ∈ ϕ(N_{a}, c) \ ϕ(M_{a}, c), p^{∗} = tp(d^{∗}/M_{a}) is regular. Without loss of generality again, we can fix d^{∗}, which does not fork over c and so that p^{∗} has the same D-rank as ϕ(x, c) and tp(d^{∗}/c) is stationary. By clause c) of Lemma 1.5, p^{∗} is not orthogonal to A_{∗}. So, there is a q′ ∈ S(M_{a}) which does not fork over A_{∗} and is nonorthogonal and so non-weakly orthogonal to p^{∗}. Fix C ⊆ A_{∗} with |C| ≤ |T | and c is independent from A_{∗} over C. Without loss of generality tp(d^{∗}/A_{∗}c) ⊥̸^{w} q ↾ (A_{∗}c) and tp(d^{∗}/Cc) ⊥̸^{w} q ↾ (Cc). Let P = {p : p is regular, stationary, and nonorthogonal to p^{∗}}. P is based on B = acl^{eq}(C), i.e. every automorphism of M fixing B maps P to itself.

If c′′ ∈ M realizes tp(c/A^{eq}) and d′′c′′ realizes r = tp(d^{∗}c/B), then tp(d′′/c′′) is regular and nonorthogonal to p^{∗}. We can find ⟨c_{i} : i < ω⟩ in M_{a} with c_{0} = c which are indiscernible over B and which are based on B. The r(x, c_{i}) are regular, pairwise nonorthogonal, and all nonorthogonal to P and each r(x, c_{i}) is not weakly orthogonal to q′ ↾ (Bc_{i}). Note r(x, c_{i}) ⊂ p^{∗}. Let r_{i} ∈ S(M) denote the nonforking extension of r(x, c_{i}) to S(M). By Section V.4 of [3], there is a q ∈ S(B), which is P-simple and k < ω such that w_{P}(q) > 0 and q(M) ⊆ acl(B ∪ ⋃_{i<k} c_{i} ∪ ⋃_{i<k} r(M, c_{i}). (This q is actually q′/E for an appropriate definable (over B) equivalence relation; compare V.4.17(8) of [3].)

Let q^{+} denote the unique nonforking extension of q to S(M), p_{a}^{+} denote the unique nonforking extension of p^{∗} to S(M), and p_{b}^{+} denote the unique nonforking extension of g(p^{∗}) to S(M). Clearly, p_{a}^{+} ↾ (M_{a} ∪ A) is a nonforking extension of the stationary type p^{∗} and realized by d^{∗}; so it is equivalent to p_{a}^{+} ↾ acl(M_{a} ∪ A).

Remark 2.2 Note (g ∪ id_{A})(p_{a}^{+} ↾ (M_{a} ∪ A) = p_{b}^{+} ↾ (M_{b} ∪ A) ∼ p_{b}^{+} ↾ acl(M_{b} ∪ A) is omitted in M.

We use the next lemma several times.

Lemma 2.3 If A^{eq} ⊆ N_{1} ⊆ N_{2} ⊆ M and N_{1}, N_{2} are |T |^{+}-saturated then

w_{P}(q(N_{2}), N_{1}) = w_{P}(q(N_{2}), q(N_{1})A^{eq}).

Proof. Fix b ∈ N_{1} and choose D ⊆ q(N_{1})A^{eq} with |D| ≤ |T | such that tp(b/q(N_{1})A^{eq}) does not fork over D. If tp(b/q(N_{2})A^{eq}) forks over D, there are finite d_{1} ⊆ q(N_{2}) and d_{2} ⊆ A^{eq} such that tp(b/BDd_{1}d_{2}) forks over D. But there is a d′ ∈ q(N_{1}) realizing stp(d_{1}/Dbd_{2}), which contradicts tp(b/q(N_{1})A^{eq}) does not fork over D.

So tp(b/q(N_{2})A^{eq}) does not fork over q(N_{1})A^{eq}. Since b was arbitrary in N_{1}, tp(N_{1}/q(N_{2})A^{eq}) does not over q(N_{1})A^{eq}. By symmetry of forking, tp(q(N_{2})/N_{1}A^{eq}) does not fork over q(N_{1})A^{eq}. Since A^{eq} ⊆ N_{1} we finish. □_{2.3}

The proof now proceeds by a series of claims. The key idea is that w_{P}(q(M), A^{eq}) can be calculated as either w_{P}(q(M), q(M_{b}) ∪ A^{eq}) + w_{P}(q(M_{b}), A^{eq}) or as w_{P}(q(M), q(M_{a}) ∪ A^{eq}) + w_{P}(q(M_{a}), A^{eq}). We will calculate both ways to obtain a contradiction. We begin with the M_{a} side.

Claim 2.4 If dim(r_{0} ↾ A_{∗}c_{0}, M_{a}) is finite, then w_{P}(q(M_{a}), A_{∗} ∪ ⋃_{i<k} c_{i}) is finite.

Proof. If u is a finite subset of ω, since the r_{i} are regular, it is easy to show that for each i, dim(r_{i} ↾ (A_{∗}c_{i}), M_{a}) is finite iff dim(r_{i} ↾ (A_{∗} ∪ c_{i} ∪_{j∈u} c_{j}), M_{a}) is finite. Since the r_{i} ↾ (A_{∗}c_{i}c_{j}) are regular and pairwise not weakly orthogonal

dim(r_{i} ↾ A_{∗}c_{i}c_{j}, M_{a}) = dim(r_{j} ↾ A_{∗}c_{i}c_{j}, M_{a}).

The previous two sentences imply: dim(r_{i} ↾ A_{∗}c_{i}, M_{a}) is finite iff dim(r_{j} ↾ A_{∗}c_{j}, M_{a}) is finite. So if dim(r_{0} ↾ A_{∗}c_{0}, M_{a}) is finite then w_{P}(⋃_{i<k} r_{i}(M_{a}, c_{i}), A_{∗} ∪ ⋃_{i<k} c_{i}) is finite; whence w_{P}(q(M_{a}), A_{∗} ∪ ⋃_{i<k} c_{i}) is finite. □_{2.4}

Now we drop the ⋃_{i<k} c_{i} in the conclusion.

Claim 2.5 dim(r_{0} ↾ A_{∗}c_{0}, M_{a}) is finite implies w_{P}(q(M_{a}), A_{∗}) is finite.

Proof. Find d ⊆ q(M) such that ⋃_{i<k} c_{i} is independent from A_{∗} ∪ q(M) over A_{∗} ∪ d. Now, as tp(d/A_{∗}) is P-simple, w_{P}(q(M_{a}), A_{∗}) = w_{P}(q(M_{a}), A_{∗}d) + w_{P}(d, A_{∗}). The second term is finite and w_{P}(q(M_{a}), A_{∗}d) = w_{P}(q(M_{a}), A_{∗}d ∪ ⋃_{i<k} c_{i}) by the independence. But, w_{P}(q(M_{a}), A_{∗}d ∪ ⋃_{i<k} c_{i}) = w_{P}(q(M_{a}), A_{∗} ∪ ⋃_{i<k} c_{i}) − w_{P}(d, A_{∗} ∪ ⋃_{i<k} c_{i}). Now the first of the last two terms is finite by Claim 2.4 (since dim(r_{0} ↾ A_{∗}c_{0}, M_{a}) is finite) and the second by the finiteness of d so w_{P}(q(M_{a}), A_{∗}) is finite . □_{2.5} Claim 2.6 dim(r_{0}, M_{a}) is finite.

Note that p_{a}^{+} ↾ (Bc_{0}) = r_{0} ↾ (Bc_{0}). Choose by induction a_{α} ∈ M_{a} so that a_{α} realizes p_{a}^{+} ↾ A_{∗}^{eq} ∪ g(c_{0}) ∪ {a_{β} : β < α} for as long as possible to construct: I = ⟨a_{α} : α < α^{∗}⟩. Clearly α^{∗} < |M_{a}|^{+}, but in fact α^{∗} is finite. As, since M_{a} is independent from A over A_{∗}, I is a set of indiscernibles over A. Since M is κ^{+}-saturated, if I is infinite ⟨g(a_{α}) : α < α^{∗}⟩ can be extended to a set J of indiscernibles over A contained in M_{b} with cardinality κ^{+}. Then all but at most κ members of J realize p_{b}^{+} ↾ (M_{b} ∪ A) contradicting Remark 2.2 that p_{b}^{+} ↾ (M_{b} ∪ A) is omitted in M. □_{2.6}

Now, easily we have

Claim 2.7 w_{P}(q(M_{a}), A_{∗}) = w_{P}(q(M_{a}), A^{eq}) is finite.

The equality holds by the independence of M_{a} and A over A_{∗}. The finiteness follows from Claim 2.6 and Claim 2.5. □_{2.7}

The next claim involves both M_{a} and M_{b}.

Claim 2.8 Suppose w_{P}(q(M_{a}), A_{∗}) is finite and N ≺ M is κ-prime over M_{b}A. Then w_{P}(q(N), q(M_{b})A) = 0.

Proof. Since w_{P}(q(M_{a}), A_{∗}) is finite, and A, M_{a} are independent over A_{∗},we can choose finite D ⊆ q(M_{a}) with w_{P}(q(M_{a}), A_{∗}) = w_{P}(q(M_{a}), A) = w_{P}(D, A_{∗}) = w_{P}(D, A).

Now assume for contradiction that w_{P}(q(N), q(M_{b})A) > 0. Let N′ ≺ M be κ-prime over M_{a} ∪ A, so there is g^{+} ⊇ g ∪ id_{A} which is an isomorphism from N′ onto N. Then there is a finite D_{2} ⊆ q(N′) with w_{P}(D_{2}, M_{a}A) > 0. Choose C_{0} ⊆ M_{a}, |C_{0}| ≤ |T | with DB ⊆ C_{0} and C_{1} ⊆ A with |C_{1}| ≤ |T | so that D_{2} is independent from M_{a}A over C_{0}C_{1} and is the unique nonforking extension of tp(D_{2}/C_{0}C_{1}) to S(M_{a}A) which is realized in M. Recall that M_{a} is A-full and apply the Definition 1.4 of A-full with C_{0}C_{1}D_{2} playing the role of C_{2} to obtain an embedding f. Then, f(D_{2}) ⊆ q(M_{a}) and f(D_{2}) is independent from C_{0}A over C_{0}f(C_{1}). Thus,

w_{P}(f(D_{2}), AD) = w_{P}(D_{2}, AD) ≥ w_{P}(D_{2}, q(M_{a})A) > 0.

This implies w_{P}(q(M_{a}), A) ≥ w_{P}(Df(D_{2}), A) = w_{P}(D, A) + w_{P}(f(D_{2}), AD) > w_{P}(D, A), which contradicts our original choice of D. □_{2.8}

Claim 2.9 w_{P}(q(M), q(M_{b})A) = 0

Let N ≺ M be κ-prime over M_{b} ∪ A, so p_{b}^{+} ↾ (M_{b} ∪ A) has a unique extension in S(N). If w_{P}(q(M), N) > 0 then for some b ∈ q(M), w_{P}(b, N) > 0 so tp(b/N) ⊥̸ p_{b}^{+}; recall p_{b}^{+} is parallel to p_{b}^{+} ↾ N. So p_{b}^{+} ↾ N is realized in M_{b} contradicting Remark 2.2. Now 0 = w_{P}(q(M), N) which equals w_{P}(q(M), q(N)A^{eq}) by Lemma 2.3. Since A^{eq} ⊆ N_{b} ⊆ N ⊆ M,

w_{P}(q(M), q(M_{b})A^{eq}) = w_{P}(q(M), q(N)A^{eq}) + w_{P}(q(N), q(M_{b})A^{eq}) = 0 + 0 = 0.

The first 0 was noted in the previous sentence and the second is Claim 2.8. □_{2.9} Now calculating with respect to M_{b}, we have:

Claim 2.10 w_{P}(q(M), A^{eq}) = w_{P}(q(M_{b}), A^{eq}) is finite.

Proof.

w_{P}(q(M), A^{eq}) = w_{P}(q(M), q(M_{b})A^{eq}) + w_{P}(q(M_{b}), A^{eq})
= 0 + w_{P}(q(M_{b}), A^{eq}) < ω.

The first equality holds by additivity [3] and Lemma 2.3, the second by Claim 2.9, and the third by the last observation. □_{2.10}

Now we analyze using M_{a}.

Claim 2.11 w_{P}((q(M), q(M_{a}) ∪ A) ≥ 1.

Proof. w_{P}(d^{∗}, M_{a} ∪ A) ≥ 1 since d^{∗} is independent from A over M_{a}. Let N be κ-prime over M_{a}A^{eq}. As tp(d^{∗}/M_{a}A^{eq}) has all its restrictions to set of size less than κ realized in M_{a}A^{eq}, tp(d^{∗}/N) does not fork over M_{a}A^{eq}. Thus, d^{∗} realizes p_{a}^{+} ↾ N. Since p_{a}^{+} ↾ N is not orthogonal to q^{+} ↾ N, there is b ∈ q^{+}(M) which depends on b over N. So w_{P}(b, N) > 0 whence w_{P}(q(M), N) > 0. By monotonicity, w_{P}((q(M), q(M_{a}) ∪ A_{∗}) ≥ w_{P}(q(M), q(N)A^{eq}). But, by Lemma 2.3, w_{P}(q(M), q(N)A^{eq}) = w_{P}(q(M), N) > 0. □_{2.11}

Now we have

w_{P}(q(M), A^{eq}) = w_{P}(q(M), q(M_{a})A^{eq}) + w_{P}(q(M_{a}), A^{eq}) ≥ 1 + w_{P}(q(M_{a}), A^{eq}) < ω. (1)

Here, the first equality is by [3] and Lemma 2.3 and the second by Claim 2.11. The finiteness comes from Claim 2.7. Since g ∪ id_{A^{eq}} is an elementary map, w_{P}(q(M_{a}), A^{eq}) = w_{P}(q(M_{b}), A_{∗}). We substitute in Equation 1, using Claim 2.10:

w_{P}(q(M_{a}), A^{eq}) = w_{P}(q(M), A^{eq}) = w_{P}(q(M_{a}), A^{eq}) + 1,

or subtracting, 0 = 1 so we finish.

References

[1] B. Baizhanov and J.T. Baldwin. Local homogeneity. submitted.

[2] E. Bouscaren. Dimensional order property and pairs of models. Annals of Pure and Applied Logic, 41:205–231, 1989.

[3] S. Shelah. Classification Theory and the Number of Nonisomorphic Models. North-Holland, 1991. second edition.