ℵ_{0}-categorical structures with a predimension

David M. Evans

1 Predimensions

Our aim in this paper is to give an axiomatic framework for the non-modular simple ℵ_{0}-categorical structures discovered by Ehud Hrushovski ([4] and [7]). This allows us to verify some of their properties (weak elimination of imaginaries, simplicity, the nature of forking, CM-triviality and stable forking) in a uniform way, and to show that these properties are preserved by ‘iterations’ of the construction (Section 3). In particular, all the simple ℵ_{0}-categorical structures produced have the following property, which we refer to as CM-triviality (in the stable context, it is easily seen to be equivalent to the definition in ([6], Proposition 10), cf. Remarks 2.22):

Suppose E_{1}, E_{2} are algebraically closed and independent over E_{1}∩E_{2}. Let D ⊆ E_{1}∪E_{2} be such that acl(D) ∩ (E_{1} ∪ E_{2}) = D and let D_{i} = D ∩ E_{i}. Then D_{1} and D_{2} are independent over D_{1} ∩ D_{2}.

Here, algebraic closure includes imaginaries, and independence is in the sense of non-forking. The axiomatization we give postulates properties of independence with respect to a predimension: a real-vauled submodular function on the type of a finite subset. One of the axioms essentially states that CM-triviality holds with respect to this notion of independence. In the actual constructions, this axiom holds for the following reason. The structure is built up as a sequence of ‘free amalgamations’ of finite structures. The predimension detects when something is a free amalgam, and the amalgamation procedure is such that a subset of a free amalgam is again a free amalgam.

It would of course be of considerable interest to know whether there are (super)simple ℵ_{0}- categorical structures which are not CM-trivial, and presumably this is related to the question of whether there are non-CM-trivial strongly minimal sets which do not interpret an infinite field (see [2], [10]).

Axiomatizations of these constructions in various contexts and levels of generality have been given before (cf. [1], [12] for example), but the approach we give here is different in several respects.

There is also overlap between this work and the Doctoral Thesis of Massoud Pourmahdian [11], and of course much of what we do here is at least implicit, and much is explicit, in Hrushovski’s work. In contrast to [7] and [12] the axiomatization makes no explicit mention of generic structures or free amalgamation. These notions are replaced here by axioms guaranteeing d-free extensions of types, and a version of the independence theorem over finite algebraically closed sets. The axioms (P1-P5) can all be checked on finite subsets of the countable model.

In the remainder of this introduction we summarise the axioms and main results. The second section contains the proof of the main theorems, and in the third section we give a slightly more general version of Hrushovski’s construction (showing how it can be ‘iterated’). We then show that the stable forking conjecture holds for these structures. The final section contains a few further observations about the structures produced.

The Author would like to thank MSRI for its hospitality in June 1998 when a preliminary version of this paper was written. The material on stable forking was written in March 2000.

1.1 Summary of the axioms and results

For convenience, we collect together the axioms.

Throughout, M is a countably infinite ℵ_{0}-categorical structure and N a large, highly saturated, highly homogeneous elementary extension of M. The letters A, B will usually be used to denote finite subsets of M or small (that is of cardinality less than the degree of saturation of N) subsets of N; c will be a finite tuple. We denote by acl(A) the algebraic closure of A in N. If A is a finite subset of M, then by the Ryll-Nardzewski Theorem this is the union of the finite Aut(M/A)- orbits on M, and is finite. We assume that there is an Aut(M)-invariant ℝ^{≥0}-valued function d_{0} defined on the finite subsets of M. By ℵ_{0}-categoricity we may assume that d_{0} is also defined on finite subsets of N.

Axioms (P1), (P2) say what we mean by a predimension. The function d which is determined by d_{0} is called the corresponding dimension function. Axiom (P4) postulates existence of d-non-forking extensions over finite sets, and (P5) is the d-independence theorem over finite algebraically closed sets. Axiom (P3) looks contrived, but is what gives CM-triviality. It is used only in the proof of Lemma 2.11, where it can be replaced by (P3′) and (P7) (see Remark 2.12). The point of using (P3) in preference to these, is that it is checkable on finite subsets of the countable model. Axioms (P6) and (P7) follow from (P1-P5). Note also that (P7) holds automatically if the subgroup of ℝ generated by {d(A) : A ⊆ M finite } is discrete. (P1) For finite subsets A and B of M

d_{0}(A ∪ B) ≤ d_{0}(A) + d_{0}(B) − d_{0}(A ∩ B).

In other words, d_{0} is submodular. If equality holds here, we say that A, B is a modular pair. We shall always assume that (P1) holds.

If A, B are finite subsets of M we write A ≤ B to mean that d_{0}(A) < d_{0}(B′) whenever A ⊂ B′ ⊆ B. We write A ≤ M if A ≤ B for all finite subsets B of M which contain A. (P2^{−}) For all finite A ⊆ M there exists a finite B ≤ M containing A.

Note that as M is ℵ_{0}-categorical this is a first-order property (the size of B can be bounded in terms of the size of A). It follows from Lemma 2.1 that for any finite A there is a least B ≤ M containing it. Denote this by cl(A). Let d(A) = d_{0}(cl(A)) and note that this is equal to min(d_{0}(B) : A ⊆ B). It is clear that cl(A) ⊆ acl(A). The next axiom is a strengthening of (P2^{−}). (P2) For all finite A we have cl(A) = acl(A).

If c is a (finite) tuple of elements of M let d(c/A) = d(cA) − d(A), and similarly define d_{0}(B/A) = d_{0}(B ∪ A) − d_{0}(A). Note that A, B is a modular pair if and only if d_{0}(A/B) = d_{0}(A/A ∩ B). (P3) For any finite A_{2} ≤ M and c there exists ϵ > 0 such that if B is finite with A_{1}, A_{2} ≤ B ≤ M, and acl(cA_{i}) ∩ B = A_{i}, and

d_{0}(acl(cA_{1})) + d_{0}(B) − d_{0}(A_{1}) − d_{0}(acl(cA_{1}) ∪ B) < ϵ

then acl(cA_{1}) ∩ acl(cA_{2}), A_{2} is a modular pair. (P3′) Suppose X, Y ≤ M is a modular pair of finite sets. If A ≤ X ∪ Y then A ∩ X, A ∩ Y is a modular pair. (P4) If A ≤ B ≤ M are finite and c is a finite tuple of elements of M with acl(cA) ∩ B = A, then there exists c_{1} with

(i) tp(c_{1}/A) = tp(c/A);

(ii) d(c_{1}/B) = d(c_{1}/A);

(iii) if C ≤ B is such that acl(cC) ∩ A = C ∩ A, then acl(c_{1}C) ∩ A = A ∩ C.

(P5) Let A ≤ M and B_{1}, B_{2} ≤ M be finite and such that B_{1} ∩B_{2} = A and d(B_{1}/B_{2}) = d(B_{1}/A). Suppose c_{1}, c_{2} are tuples such that tp(c_{1}/A) = tp(c_{2}/A) and d(c_{i}/B_{i}) = d(c_{i}/A). Then there exists c with tp(c/B_{i}) = tp(c_{i}/B_{i}) and d(c/A) = d(c/B_{1} ∪ B_{2}).

For (possibly infinite) S ⊆ N we let

d(c/S) = inf(d(c/A) : A is a finite subset of S).

(P6) Let A ⊆ B be small algebraically closed subsets of N and let c be a finite tuple. Then there exists c_{1} such that tp(c/A) = tp(c_{1}/A), d(c_{1}/A) = d(c_{1}/B), and acl(c_{1}A) ∩ B = A. (P7) If a, b are finite tuples and A is a small subset of N such that d(a/Ab) = d(a/A) and acl(aA) ∩ acl(bA) = A, then for every finite A_{0} ⊆ A there exists a finite subset A_{0} ⊆ A_{1} ⊆ A with d(a/A_{1}b) = d(a/A_{1}).

The main results can be stated as follows. if (P1, P2, P5, P6, P7) hold then T h(M) is simple, and d describes forking (Corollary 2.20). If (P1-P5) hold then (P6) and (P7) hold (Corollary 2.21), and moreover T h(M) is CM-trivial (Corollary 2.15). Weaker versions of (P3) and (P4) suffice to prove weak elimination of imaginaries (Lemma 2.9). The discrete case, leading to supersimple structures, is more straightforward: see Section 2.8.

2 Consequences of the axioms

2.1 Consequences of the axioms for the predimension

Lemma 2.1 Assume (P1) and (P2^{−}) hold.

(i) If A ≤ B and X ⊆ B then A ∩ X ≤ X. (ii) If A ≤ B and B ≤ C then A ≤ C. (iii) If A, B ≤ M then d(A ∪ B) ≤ d(A) + d(B) − d(A ∩ B). (iv) For any finite A, B, C, d_{0}(A ∪ B/C) ≤ d_{0}(A/C) + d_{0}(B/C) − d_{0}(A ∩ B/C).

(v) If A ≤ B ≤ M then d(c/A) ≥ d(c/B).

Proof. These are well-known and, apart from (iii) and (v), are all consequences of (P1).

(i) Let A ∩ X ⊂ Z ⊆ X. So A ⊂ A ∪ Z and A ∩ Z = A ∩ X. Then by (P1) we have d_{0}(A) + d_{0}(Z) − d_{0}(A ∩ Z) ≥ d_{0}(A ∪ Z). As A ≤ B we therefore get d_{0}(Z) > d_{0}(A ∩ X), as required.

(ii) Let A ⊂ X ⊆ C. If X ⊆ B then we already have d_{0}(X) > d_{0}(A). On the other hand if X ⊈ B, then

d_{0}(X) + d_{0}(B) − d_{0}(X ∩ B) ≥ d_{0}(X ∪ B) > d_{0}(B)

so d_{0}(X) > d_{0}(X ∩ B) ≥ d_{0}(A), as required.

(iii) By (i) and (ii), A ∩ B ≤ M. The inequality now follows as d(A ∪ B) ≤ d_{0}(A ∪ B).

(iv) d_{0}(A ∪ B/C) = d_{0}(A ∪ B ∪ C) − d_{0}(C). By (P1) this is at most d_{0}(A ∪ C) + d_{0}(B ∪ C) − d_{0}((A ∩ B) ∪ C) − d_{0}(C), and this is d_{0}(A/C) + d_{0}(B/C) − d_{0}(A ∩ B/C).

(v) By (iii), and the fact that cl(cA) ∩ B ≥ A we have:

d(cA) + d(B) ≥ d(cB) + d(cl(cA) ∩ B) ≥ d(cB) + d(A),

as required. □

For A ⊆ B, write A ≤^{∗} B if d_{0}(A′) ≥ d_{0}(A) for all A ⊆ A′ ⊆ B. Replacing the strict inequalities in the above proofs by weak inequalities gives:

Lemma 2.2 Assume (P1) holds.

(i) If A ≤^{∗} B and X ⊆ B then A ∩ X ≤^{∗} X. (ii) If A ≤^{∗} B and B ≤^{∗} C then A ≤^{∗} C. □

We say that X, Y ≤ M form a modular pair if d_{0}(X ∪ Y) = d_{0}(X) + d_{0}(Y) − d_{0}(X ∩ Y).

Lemma 2.3 Assume (P1) and (P2^{−}) hold. Let c be a tuple of elements of M and A ≤ B ≤ M be finite sets. Then d(c/A) = d(c/B) if and only if the following three conditions hold:

(I1) cl(cA) ∩ B = A;

(I2) cl(cA), B is a modular pair in M;

(I3) d(cB) = d_{0}(cl(cA) ∪ B). In particular, in this case we have cl(cA) ∪ B ≤^{∗} cl(cB).

Proof. d(c/A) = d(c/B) if and only if d_{0}(cl(cA)) − d_{0}(A) = d_{0}(cl(cB)) − d_{0}(B). But

d_{0}(cl(cB)) − d_{0}(B) ≤ d_{0}(cl(cA) ∪ B) − d_{0}(B)
≤ d_{0}(cl(cA)) − d_{0}(cl(cA) ∩ B),

by submodularity. Moreover as A ≤ cl(cA) ∩ B we have d_{0}(cl(cA) ∩ B) ≥ d_{0}(A), with equality if and only if A = cl(cA) ∩ B. Thus d(c/B) = d(c/A) if and only if there is equality at all three places in the above, hence the result. □

2.2 Weak canonical bases

In this section we consider a predimension satisfying (P1), (P2^{−}) and the extra axiom: (P3^{−}) Suppose A_{1}, A_{2} ≤ B ≤ M and c is a tuple in M such that cl(cA_{i}) ∩ B = A_{i} (for i = 1, 2). If cl(cA_{1}), B is a modular pair, then so is cl(cA_{1}) ∩ cl(cA_{2}), A_{2}.

This may seem slightly contrived. Note however that it is implied by the following (take X = cl(cA_{1}), Y = B and A = cl(cA_{2}) ∩ (X ∪ Y) = (cl(cA_{1}) ∩ cl(cA_{2})) ∪ A_{2} in (P3′) to get (P3^{−})).

(P3′) Suppose X, Y ≤ M is a modular pair, and A ≤ X ∪ Y. Then A ∩ X, A ∩ Y is a modular pair.

If we read ‘are independent over their intersection’ in place of ‘is a modular pair,’ then we see that this corresponds to CM-triviality of the predimension. Moreover note that in the above situation B ∩cl(cA_{1})∩cl(cA_{2}) = A_{1} ∩A_{2} = A_{0}. So cl(cA_{0}), A_{2} is a modular pair, by the following lemma.

Lemma 2.4 Suppose (P1) holds and A, B ≤ M is a modular pair. If A ∩ B ≤ A′ ⊆ A, then A′, B is a modular pair.

Proof. We have

d_{0}(A) + d_{0}(B) − d_{0}(A ∩ B) = d_{0}(A ∪ B) = d_{0}((A′ ∪ B) ∪ A) ≤
d_{0}(A′ ∪ B) + d_{0}(A) − d_{0}(A′).

Thus d_{0}(A′∪B) ≥ d_{0}(A′)+d_{0}(B)−d_{0}(A∩B) = d_{0}(A′)+d_{0}(B)−d_{0}(A′∩B). So by submodularity, we have equality here. □

Lemma 2.5 Assume (P1, P2^{−}, P3^{−}). Let A_{1}, A_{2} ≤ B ≤ M, and suppose c is such that d(c/A_{1}) = d(c/B) and cl(cA_{2}) ∩ B = A_{2}. Let A_{0} = A_{1} ∩ A_{2}. Then d(c/A_{0}) = d(c/A_{2}).

Proof. We show that (I1-3) of Lemma 2.3 hold for (c, A_{0}, A_{2}).

For (I1) note that

A_{1} ∩ A_{2} ⊆ cl(c(A_{1} ∩ A_{2})) ∩ A_{2} ⊆ cl(cA_{1}) ∩ A_{2} ⊆ A_{1} ∩ A_{2},

using (I1) for (c, A_{1}, B).

For (I2) we can use (P3^{−}) and the remark following it (after observing that cl(cA_{1}) ∩ B = A_{1}, by Lemma 2.3).

For (I3), first note that as (I3) holds in (c, A_{1}, B), we have cl(cA_{1})∪B ≤^{∗} cl(cB). Intersecting this with cl(cA_{2}) and applying Lemma 2.2 gives:

(cl(cA_{1}) ∩ cl(cA_{2})) ∪ A_{2} ≤^{∗} cl(cA_{2}).

Thus d_{0}(A_{2} ∪(cl(cA_{1})∩cl(cA_{2})) = d(cA_{2}). So it remains to show that cl(cA_{1})∩cl(cA_{2}) = cl(cA_{0}).

By (P3^{−}), as above we have that A_{2} and cl(cA_{0}) form a modular pair, and their intersection is A_{0}. Thus we have

d_{0}(A_{2}) + d_{0}(cl(cA_{0})) − d_{0}(A_{0}) = d_{0}(A_{2} ∪ cl(cA_{0})) ≥ d(A_{2}c)
= d_{0}(A_{2} ∪ (cl(A_{1}c) ∩ cl(A_{2}c))).

By (P3^{−}) again, this equals

d_{0}(A_{2}) + d_{0}(cl(cA_{1}) ∩ cl(cA_{2})) − d_{0}(A_{0}).

So we get d_{0}(cl(cA_{0})) ≥ d_{0}(cl(cA_{1}) ∩ cl(cA_{2})). But clearly cl(cA_{0}) ≤ cl(cA_{1}) ∩ cl(cA_{2}) so in fact we have equality (by definition of ≤). □

Definition 2.6 Let B ≤ M and c be a finite tuple from M. Suppose there exists A_{0} ≤ B such that for A ≤ B, d(c/A) = d(c/B) if and only if A_{0} ⊆ A. We refer to this A_{0} as the weak canonical base of c over B, and denote it by cb(c/B). If cb(c/B) exists for all such c, B, we say that (M, d_{0}) has weak canonical bases.

Corollary 2.7 Suppose that (P1, P2^{−}, P3^{−}) hold. Then (M, d_{0}) has weak canonical bases. □

2.3 CM-triviality over finite sets

Corollary 2.8 Suppose M satisfies axioms (P1, P2^{−}, P3^{−}). Let A ≤ B ≤ M and suppose c is a tuple of elements of M such that cl(cA) ∩ B = A. Then cb(c/A) ≤ cb(c/B).

Proof. Let B_{0} = cb(c/B). Then d(c/A ∩ B_{0}) = d(c/A) by Lemma 2.5. So B_{0} ≥ cb(c/A), as required. □

2.4 Weak elimination of imaginaries

We impose the following extra axioms to obtain weak elimination of imaginaries. Strengthening (P4^{−}) will give that M is simple and d describes non-forking. (P4^{−}) Let A ≤ B ≤ M and c a tuple of elements of M. Then there exists a tuple c_{1} from M such that tp(c/A) = tp(c_{1}/A) and d(c/A) = d(c_{1}/B). (P5) Let A ≤ M and B_{1}, B_{2} ≤ M be such that B_{1} ∩ B_{2} = A and d(B_{1}/A ∪ B_{2}) = d(B_{1}/A). Suppose c_{1}, c_{2} are tuples such that tp(c_{1}/A) = tp(c_{2}/A) and d(c_{i}/B_{i}) = d(c_{i}/A). Then there exists c with tp(c/B_{i}) = tp(c_{i}/B_{i}) and d(c/A) = d(c/B_{1} ∪ B_{2}).

Obviously (P4^{−}) corresponds to existence of ‘d-non-forking extensions’ and (P5) to the the Independence Theorem over finite algebraically closed sets. We use them to prove weak elimination of imaginaries. It is also easy to see that (P4) and (P2^{−}) imply (P2).

Lemma 2.9 Suppose (M, d_{0}) has weak canonical bases and satisfies axioms (P1, P2, P4^{−}, P5). Let e ∈ M^{eq}. Then e ∈ dcl^{eq}(acl^{eq}(e) ∩ M). In particular, this holds if M satisfies (P1, P2, P3^{−}, p4^{−}, P5).

Proof. The proof follows Hrushovski’s proof of weak elimination of imaginaries for PAC structures (over a submodel) ([5], Proposition 3.1).

There exists a ∅-definable equivalence relation E on a set of tuples from M such that e = b/E, for some tuple b. Let C = acl^{eq}(e) ∩ M, and let Q be the Aut(M/C)-orbit containing b. It is enough to show that E(b, b′) holds for all b′ ∈ Q. Step 1. There exist b_{1}, b_{2} ∈ Q such that E(b_{1}, b_{2}) and d(b_{1}/b_{2}C) = d(b_{1}/C).

First, it follows from Neumann’s Separation Lemma that there exist Aut(M/Ce)-translates b_{1} and b_{2} of b such that

acl(b_{1}C) ∩ acl(b_{2}C) = C.

Choose these so that d(b_{2}/b_{1}C) is as large as possible (this can be done, as there are only finitely many Aut(M/Ce)-orbits on Q^{2}, by ℵ_{0}-categoricity). Note that E(b_{1}, b_{2}). Now let b_{3} have the same type as b_{2} over Cb_{1}, and be such that d(b_{3}/b_{1}b_{2}C) = d(b_{3}/b_{1}C) (by (P4^{−})). Then E(b_{3}, b_{1}) and therefore E(b_{3}, b_{2}). Moreover, by Lemma 2.3 and (P2), we have

acl(b_{3}b_{1}C) ∩ acl(b_{1}b_{2}C) = acl(b_{1}C).

Thus

acl(Cb_{3}) ∩ acl(Cb_{2}) ⊆ acl(Cb_{3}) ∩ acl(Cb_{1}) = C.

It follows (by choice of b_{1}, b_{2}) that d(b_{3}/b_{2}C) ≤ d(b_{2}/b_{1}C) = d(b_{3}/b_{1}b_{2}C). So by Lemma 2.1 (v), we have d(b_{3}/b_{1}b_{2}C) = d(b_{3}/b_{2}C). Also, d(b_{3}/b_{1}b_{2}C) = d(b_{3}/b_{1}C), by construction. So by existence of weak canonical bases,

cb(b_{3}/ acl(b_{1}b_{2}C)) ⊆ acl(b_{1}C) ∩ acl(b_{2}C) = C.

So d(b_{3}/b_{1}C) = d(b_{3}/C). This completes Step 1. Step 2. E(b′, b′′) holds for all b′, b′′ ∈ Q with d(b′/b′′C) = d(b′/C).

By Step 1, there exist b_{1}, b_{2} ∈ Q such that E(b_{1}, b′), E(b_{2}, b′′) and d(b_{1}/b′C) = d(b_{1}/C), d(b_{2}/b′′C) = d(b_{2}/C). Thus by (P5) there exists b_{3} ∈ Q of the same type as b_{1} over Cb′ and the same type as b_{2} over Cb′′. In particular, E(b_{3}, b′) and E(b_{3}, b′′). So E(b′, b′′). Step 3. E(b_{1}, b_{2}) holds for all b_{1}, b_{2} ∈ Q.

By (P4^{−}), there exists b_{3} ∈ Q such that d(b_{3}/b_{1}C) = d(b_{3}/b_{2}C) = d(b_{3}/C). So by step 2, E(b_{3}, b_{1}) and E(b_{3}, b_{2}). Thus E(b_{1}, b_{2}). □

Remarks 2.10 The proof works under a significantly weaker assumption than (P5). Let C be a finite algebraically closed subset of M and Q the set of realisations of a complete n-type over C (so an Aut(M/C)-orbit on M^{n}). We say that a non-empty R ⊆ Q×Q is a d-independent relation over C, if it is Aut(M/C)-invariant, and has the property that for all (b, b′) ∈ R, d(b/Cb′) = d(b/C). It is connected if the (undirected) graph on Q with the pairs in R as edges is connected. If this is the case for all C, Q, R, then the above proof works (assuming existence of weak canonical bases, and (P1, P2, P4^{−})).

2.5 Dimension over infinite sets

We work in a large model N of T h(M). Note that as d_{0}(A) depends only on tp(A/∅) (for a finite subset A of M), we can consider d_{0} and d as defined on finite subsets of N. For (possibly infinite) S ⊆ N we let

d(c/S) = inf(d(c/A) : A ⊆ S finite ).

We shall be assuming (P1, P2) throughout. We would like to have analogous results to Corollaries 2.7 and 2.8 without assuming B there to be finite. This appears to require a strengthening of Axiom (P3^{−}). (P3) For any finite A_{2} ≤ M and c there exists ϵ > 0 such that if B is finite with A_{1}, A_{2} ≤ B ≤ M, and acl(cA_{i}) ∩ B = A_{i}, and

d_{0}(acl(cA_{1})) + d_{0}(B) − d_{0}(A_{1}) − d_{0}(acl(cA_{1}) ∪ B) < ϵ

then acl(cA_{1}) ∩ acl(cA_{2}), A_{2} is a modular pair.

Note that this implies (P3^{−}), and that it implies the same property for finite subsets of N. Analogously with Lemma 2.5 we have:

Lemma 2.11 Assume axioms (P1, P2, P3) hold. Let A_{2} ≤ N be finite, let c be a finite tuple in N, and suppose X, A_{2} ≤ B ≤ N. If d(c/X) = d(c/B), acl(cX) ∩ B = X, and acl(cA_{2}) ∩ B = A_{2}, then d(c/A_{2}) = d(c/A_{2} ∩ X).

Proof. We show that there exists a finite set A_{1} with A_{2}∩X ⊆ A_{1} ≤ X such that acl(cA_{1})∩B = A_{1}, and

d(cA_{2}) = d_{0}(acl(cA_{1}) ∩ acl(cA_{2})) + d_{0}(A_{2}) − d_{0}(A_{1} ∩ A_{2}).

Then, as A_{1} ∩ A_{2} = A_{2} ∩ X and acl(cA_{1}) ∩ acl(cA_{2}) ≥ acl(c(A_{1} ∩ A_{2})) we have

d(c/A_{2}) = d(acl(cA_{1}) ∩ acl(cA_{2})/A_{1} ∩ A_{2}) ≥ d(c/A_{2} ∩ X).

But, (by Lemma 2.1) d(c/A_{2}) ≤ d(c/A_{2} ∩ X), so we have equality, as required.

Let δ > 0. There exists a finite B_{1} ≤ B containing A_{2} with d(c/B_{1}) − d(c/B) < δ. As d(c/X) = d(c/B) there exists a finite X_{1} ≤ X with d(c/X_{1}) ≤ d(c/B_{1}). Without loss, we may assume that A_{2}∩X ⊆ X_{1} and (as acl(cX)∩B = X) that acl(cX_{1})∩B = X_{1}. Let B_{2} = acl(X_{1}B_{1}). We may assume that acl(cB_{2}) ∩ B = B_{2}.

Note that d(c/X_{1}) − d(c/B_{2}) < δ, that is

d_{0}(acl(cX_{1})) − d_{0}(X_{1}) − d_{0}(acl(cB_{2})) + d_{0}(B_{2}) < δ.

But

d_{0}(acl(cB_{2})) ≤ d_{0}(acl(cX_{1}) ∪ B_{2}) ≤ d_{0}(acl(cX_{1})) + d_{0}(B_{2}) − d_{0}(X_{1}).

By the above, this is less than d_{0}(acl(cB_{2})) + δ. So acl(cX_{1}) and B_{2} are within δ of being a modular pair, and d_{0}(acl(cX_{1}) ∪ B_{2}) − d_{0}(acl(cB_{2})) < δ. Let Z_{1} = acl(cX_{1}) ∪ B_{2}. Note that acl(Z_{1}) = acl(cB_{2}). Then d_{0}(Z_{1} ∪ acl(cA_{2})) ≤ d_{0}(Z_{1}) + d_{0}(acl(cA_{2})) − d_{0}(Z_{1} ∩ acl(cA_{2})), and as Z_{1} ⊆ Z_{1} ∪ acl(cA_{2}) ⊆ acl(Z_{1}), we have d_{0}(Z_{1} ∪ acl(cA_{2})) ≥ d_{0}(acl(cB_{2})) > d_{0}(Z_{1}) − δ. Thus d_{0}(Z_{1} ∩ acl(cA_{2})) − d_{0}(acl(cA_{2})) < δ. Now, Z_{1} ∩ acl(cA_{2}) = (acl(cX_{1}) ∩ acl(cX_{2})) ∪ A_{2}. Thus

0 ≤ d_{0}((acl(cX_{1}) ∩ acl(cA_{2})) ∪ A_{2}) − d_{0}(acl(cA_{2})) < δ.

But this inequality relates to the fixed finite set acl(cA_{2}). So if we choose δ small enough here, we must have d_{0}((acl(cX_{1}) ∩ acl(cA_{2})) ∪ A_{2}) = d_{0}(acl(cA_{2})). Recall that acl(cX_{1}) and B_{2} are within δ of being a modular pair. Moreover, acl(cX_{1}) ∩ B_{2} = X_{1} and acl(cA_{2}) ∩ B_{2} = A_{2}. Let ϵ be as given by (P3) for A_{2} and c, and choose δ < ϵ. Then acl(cX_{1}) ∩ acl(cA_{2}), A_{2} is a modular pair. This gives what we need (let A_{1} = X_{1}). □

Remark 2.12 As stated, axiom (P3) looks rather technical. The only point where it is used (as opposed to (P3^{−}), which itself follows from the more readily intelligible (P3′)) is in the proof of the above lemma. Note however that the lemma holds if we assume (P3^{−}) and (P7). Indeed, with the notation as in the statement of the lemma, by (P7) there exists a finite A_{1} ⊇ X ∩ A_{2} contained in X with d(c/A_{1}A_{2}) = d(c/A_{1}). Let B_{1} = acl(A_{1}A_{2}). Then, as A_{1} ∩ A_{2} = A_{2} ∩ X, the required result follows from Lemma 2.5.

Definition 2.13 Let B be an algebraically closed subset of N. Consider the subset

⋃{cb(c/A) : A ≤ B finite, acl(cA) ∩ B = A}.

Note that if B is finite then Corollary 2.8 states that this is equal to cb(c/B). Thus, we denote this set by cb(c/B) for arbitrary B, and refer to it as the weak canonical base of c over B.

Corollary 2.14 Suppose (P1, P2, P3) hold. Suppose B_{1} ≤ B ≤ N satisfies acl(cB_{1}) ∩ B = B_{1}. Then d(c/B_{1}) = d(c/B) if and only if cb(c/B) ⊆ B_{1}.

Proof. First suppose cb(c/B) ⊆ B_{1}. Given a finite C ≤ B, there exists a finite A with C ≤ A ≤ B and acl(cA) ∩ B = A. Then d(c/C) ≥ d(c/A) = d(c/A_{0}) where A_{0} = cb(c/A) ⊆ B_{1}. Thus d(c/C) ≥ d(c/A_{0}) ≥ d(c/B_{1}). So d(c/B_{1}) ≥ d(c/B) = inf(d(c/C) : C ⊆ B finite) ≥ d(c/B_{1}), and we therefore have equality.

Conversely, suppose d(c/B_{1}) = d(c/B), A ≤ B is finite and acl(cA) ∩ B = A. By Lemma 2.11 d(c/A) = d(c/B_{1} ∩ A), so cb(c/A) ⊆ B_{1} ∩ A ⊆ B_{1}, by Corollary 2.7. □

Corollary 2.15 Suppose (P1-P3) hold. Suppose A ≤ B ≤ N and acl(cA) ∩ B = A. Then cb(c/A) ⊆ cb(c/B).

Proof. By definition, it is enough to prove this when A is finite. But this is then Lemma 2.11. □

2.6 Existence of d-non-forking extensions

We now wish to impose conditions guaranteeing that T h(M) is simple and that the dimension function d detects forking. In particular we want to pass from the local condition (P4^{−}) (over finite sets) to a global one (over arbitrary sets) via some sort of compactness argument. Making this work in general appears to require a strengthening of (P4^{−}), and curiously CM-triviality is involved in the proof.

(P4) If A ≤ B ≤ M are finite and c is a finite tuple of elements of M with acl(cA) ∩ B = A, then there exists c_{1} with

(i) tp(c_{1}/A) = tp(c/A);

(ii) d(c_{1}/B) = d(c_{1}/A);

(iii) if C ≤ B is such that acl(cC) ∩ A = C ∩ A, then acl(c_{1}C) ∩ A = A ∩ C.

Once again, we work in a large highly saturated model N of T h(M). We shall show that the following holds.

(P6) Let A ⊆ B be small algebraically closed subsets of N and let c be a finite tuple. Then there exists c_{1} such that tp(c/A) = tp(c_{1}/A), d(c_{1}/A) = d(c_{1}/B), and acl(c_{1}A) ∩ B = A.

Lemma 2.16 Suppose (P1, P2, P3^{−}, P4) hold. Then (P6) holds.

Proof. Let A ≤ B ≤ N be small and c a finite tuple from N. By Neumann’s lemma, we may assume that acl(cA) ∩ B = A. We must show that there is some c_{1} with tp(c/A) = tp(c_{1}/A) and d(c_{1}/B) = d(c_{1}/A). We first do this in the case where B = A ∪ b for some finite tuple b.

Note first that for every finite X ⊆ A there exists a finite A_{1} with X ⊆ A_{1} ⊆ A such that

acl(cA_{1}) ∩ A = A_{1} and acl(bA_{1}) ∩ A = A_{1}.

Indeed, let A_{1} = acl(bcX)∩A. Then acl(cA_{1})∩A ⊆ acl(bcX)∩A = A_{1}, and likewise acl(bA_{1})∩A = A_{1}.

Now let A = {A_{i} : i ∈ I} be the directed set of finite A_{1} with the above properties (X varying). Consider the set of formulas (in the language for N with constant symbols added for elements of A) expressing that

tp(c_{1}/A) = tp(c/A)

and, for i ∈ I,

d(c_{1}/A_{i}) = d(c_{1}/A_{i} ∪ b).

We show that this is consistent. Let A_{1}, …, A_{r} ∈ A. There exists A′ ∈ A with A_{i} ≤ A′ for i = 1, …, r.

By (P4), there exists c_{1} with with the same type over A′ as c, and such that d(c_{1}/ acl(bA′)) = d(c_{1}/A′), and moreover acl(c_{1}C)∩A′ = C ∩A′ whenever C ≤ acl(bA′) is such that acl(cC)∩A′ = C ∩ A′. Consider (for i = 1, …, r) C = acl(bA_{i}). Then (by definition of A) C ∩ A′ = A_{i} and acl(cC) ∩ A′ = acl(cbA_{i}) ∩ A′ = A_{i} = C ∩ A′. So acl(c_{1}C) ∩ A′ = C ∩ A′ = A_{i}. We want to show cb(c_{1}/C) ⊆ A_{i}. Let Y = acl(c_{1}C) ∩ acl(bA′). Then acl(c_{1}Y) ∩ acl(bA′) = Y, so by Corollary 2.8 cb(c_{1}/Y) ⊆ cb(c_{1}/A′b) ⊆ A′. Thus cb(c_{1}/Y) ⊆ acl(c_{1}C) ∩ acl(bA′) ∩ A′ ⊆ A_{i}. So d(c_{1}/Y) = d(c_{1}/A_{i}) and as A_{i} ⊆ C ⊆ Y we have d(c_{1}/C) = d(c_{1}/A_{i}), as required.

This establishes the consistency of the above set of formulas. Let c_{1} realise them. Then of course tp(c_{1}/A) = tp(c/A) and

d(c_{1}/A ∪ b) = inf(d(c_{1}/A_{i} ∪ b) : i ∈ I)
= inf(d(c_{1}/A_{i}) : i ∈ I) = d(c_{1}/A).

Moreover, by Lemma 2.3 acl(c_{1}A_{i}) ∩ acl(c_{1}A_{i}b) = A_{i}. Thus acl(c_{1}A) ∩ acl(c_{1}Ab) = A.

Finally, the statement for arbitrary B ⊇ A follows by the above and compactness, by allowing b to range over all finite tuples from B. □

2.7 Simplicity

The terminology in the following lemma is that of Kim and Pillay ([8], Definition 4.1). It is explained in the proof of the lemma.

Lemma 2.17 Let Γ′ be the collection of triples (c, B, A) where c is a (finite) tuple from N, and A ⊆ B are small subsets of N such that d(c/B) = d(c/A). Denote by Γ the subset of this where additionally acl(cA) ∩ acl(B) = acl(A).

(a) If d_{0} satisfies axioms (P1) and (P2), then Γ, Γ′ have the properties of invariance under automorphisms, local character, finite character, symmetry and transitivity.

(b) If additionally d_{0} satisfies (P6), then Γ, Γ′ are notions of independence.

Proof. (a) We check the axioms of Kim and Pillay for Γ′. The modifications required for Γ are easy and omitted.

(i) Invariance under automorphisms: this is clear.

(ii) Local character: this follows from separability of ℝ^{≥0}.

(iii) Finite character: Suppose (c, B, A) ∈ Γ′. Then d(c/A′) = d(c/A) for all A ⊆ A′ ⊆ B. On the other hand, if (c, B, A) ∉ Γ′, then d(c/B) < d(c/A), so inf(d(c/S) : S ⊆ B finite ) < d(c/A). Thus for some finite S ⊆ B, we have d(c/S) < d(c/A). So d(c/S ∪ A) < d(c/A), that is, (c, S ∪ A, A) ∉ Γ′.

(v) Symmetry: Suppose (c, A ∪ B, A) ∈ Γ′. We must show that (b, A ∪ c, A) ∈ Γ′ for all finite tuples b from B. It is enough to show that for any finite A_{0} ⊆ A and ϵ > 0 there exists a finite A_{2} ⊇ A_{0} contained in A such that d(b/A_{2}) − d(b/A_{2}c) < ϵ. But there exists A_{2} with A_{0} ⊆ A_{2} ⊆ A and d(c/A_{2}) − d(c/A) < ϵ. So as d(c/Ab) = d(c/A) ≤ d(c/A_{2}b) ≤ d(c/A_{2}) we get d(c/A_{2}) − d(c/A_{2}b) < ϵ. Thus

d(cA_{2}) − d(A_{2}) − d(cbA_{2}) + d(A_{2}b) < ϵ

whence d(b/A_{2}) − d(b/A_{2}c) < ϵ, as required.

(vi) Transitivity: Suppose A ⊆ B ⊆ C. Then d(c/C) ≤ d(c/B) ≤ d(c/A), so (c, C, A) ∈ Γ′ if and only if (c, C, B) ∈ Γ′ and (c, B, A) ∈ Γ′.

(b) All that is missing is the existence of Γ and Γ′ -non-forking extensions. This is exactly what (P6) gives for Γ. For Γ′, use Neumann’s separation lemma first to get the extra hypothesis for (P6). □

It is convenient to isolate the following property.

(P7) If a, b are finite tuples and A is a small algebraically closed subset of N such that d(a/Ab) = d(a/A) and acl(aA) ∩ acl(bA) = A, then for every finite A_{0} ⊆ A there exists a finite subset A_{0} ⊆ A_{1} ⊆ A with d(a/A_{1}b) = d(a/A_{1}).

Lemma 2.18 If (P1, P2, P3) hold, then (P7) holds.

Proof. Let B = acl(Ab) and B_{1} = acl(abA_{0})∩B. Then acl(aB_{1})∩B = B_{1}, so by CM-triviality (Corollary 2.15) and hypothesis we have cb(a/B_{1}) ⊆ cb(a/B). By Corollary 2.14 this is contained in A. Thus if we let A_{1} = B_{1} ∩ A we have cb(a/B_{1}) ⊆ A_{1} and in particular d(a/A_{1}b) = d(a/A_{1}). □

We are now in a position to prove simplicity. It is tempting to try to do this by showing that the Independence Theorem holds over a model (for the notion of independence Γ), and then invoke Theorem 4.2 of [8]. But there appear to be difficulties passing from (P5) to IT over a model via compactness. Instead, we proceed directly.

Lemma 2.19 Let A ⊆ B be small subsets of N and c a tuple of elements from N.

(a) Assume (P1, P2, P6, P7) hold. If d(c/B) < d(c/A) then tp(c/B) divides over A. If acl(cA) ∩ acl(B) ≠ acl(A) then tp(c/B) divides over A.

(b) Assume (P1, P2, P5, P6, P7) hold. If d(c/B) = d(c/A) and acl(cA) ∩ acl(B) = acl(A), then tp(c/B) does not divide over A.

(c) Assume that (P1, P2, P5, P6, P7) hold. If tp(c/B) does not divide over A, it does not fork over A.

Proof. (a) The first part follows exactly as in the proof of Claim I of Theorem 4.2 of [8] using Γ′. For the second part, suppose A is algebraically closed and the inequality is witnessed by ϕ(c, b, a), b ∈ acl(B) \ A. By Neumann’s separation lemma, there are b_{i} (i < ω) with tp(b_{i}/A) = tp(b/A) and acl(Ab_{i}) ∩ acl(Ab_{j}) = acl(A) if i ≠ j. Thus {ϕ(x, b_{i}, a) : i < ω} is 2-inconsistent.

(b) We modify the proof of Lemma 4.8 of [7]. Without loss, assume acl(A) = A. Suppose, for a contradiction, that there is ϕ(x, b) ∈ tp(c/B) and an infinite A-indiscernible sequence (b_{i} : i ∈ I) with b_{0} = b and {ϕ(x, b_{i}) : i ∈ I} being k-inconsistent (for some natural number k). Then d(c/Ab) = d(c/A) and acl(cA) ∩ acl(Ab) = A, by assumption, so by (P7) we may assume that b is an extension of a tuple a of elements of A, such that d(c/b) = d(c/a).

We may assume that the indexing set I is ℤ. Let B_{1} = acl(a, b_{i} : i ≤ 0). Then (b_{i} : i > 0) is indiscernible over B_{1}, and as there is an automorphism of N fixing a, b_{i+1} and taking …, b_{0}, …, b_{i} to …, b_{0}, we have

d(b_{i+1}/B_{1}b_{1} … b_{i}) = d(b_{i+1}/B_{1})

for i ≥ 0.

By extending the B_{1}-indiscernible sequence (b_{i} : i ≥ 1) and applying the Erdös-Rado Theorem, we may assume that X = acl(B_{1}b_{i_{k}}) ∩ acl(B_{1}b_{i_{1}} … b_{i_{k−1}}) is constant for all 1 ≤ i_{1} < … < i_{k}, and (b_{i} : i ≥ 1) is X-indiscernible. Then d(b_{k}/Xb_{1} … b_{k−1}) = d(b_{k}/X) and acl(Xb_{k}) ∩ acl(Xb_{1} … b_{k−1}) = X. So by (P7) again, there is a finite B_{0} ⊆ X which contains a and is such that d(b_{k}/B_{0}b_{1} … b_{k−1}) = d(b_{k}/B_{0}). By indiscernibility (over X), we therefore have d(b_{i+1}/B_{0}b_{1} … b_{i}) = d(b_{i+1}/B_{0}) for 1 ≤ i < k.

Let c′ be such that tp(c′b_{1}/A) = tp(cb/A). Thus ϕ(x, b_{1}) ∈ tp(c′/b_{1}) and d(c′/b_{1}) = d(c′/a). Let c′′ be such that tp(c′/b_{1}) = tp(c′′/b_{1}) and d(c′′/Xb_{1}) = d(c′′/b_{1}) (by (P6)). So d(c′′/B_{0}b_{1}) = d(c′′/b_{1}) = d(c′′/a) = d(c′′/B_{0}) and ϕ(x, b_{1}) ∈ tp(c′′/b_{1}). Thus there exist c_{1}, …, c_{k} such that d(c_{i}/B_{0}b_{i}) = d(c_{i}/B_{0}), tp(c_{i}/B_{0}) = tp(c′′/B_{0}) and ϕ(x, b_{i}) ∈ tp(c_{i}/b_{i}) (simply take tp(c_{i}b_{i}/B_{0}) = tp(c′′b_{1}/B_{0})). So by (P5) (applied k − 1 times) there exists c_{0} with tp(c_{0}/B_{0}b_{i}) = tp(c_{i}/B_{0}b_{i}) for i = 1, …, k. But this contradicts the inconsistency of {ϕ(x, b_{i}) : i = 1, …, k}.

(c) Suppose ϕ ∈ tp(c/B) and ϕ → ϕ_{1} ∨ ⋯ ∨ ϕ_{r}, where each ϕ_{i} divides over A. Let B_{1} ⊇ B be algebraically closed and include the extra parameters needed for these ϕ_{i}. By (P6) we may assume that d(c/B_{1}) = d(c/B) and acl(cB)∩B_{1} = B. By (a), d(c/B) = d(c/A) and acl(cA)∩B = acl(A). Thus d(c/B_{1}) = d(c/A) and acl(cA) ∩ B_{1} = acl(A). As ϕ ∈ tp(c/B_{1}) we must have ϕ_{i} ∈ tp(c/B_{1}) for some i. So tp(c/B_{1}) divides over A, which contradicts (b). □

Corollary 2.20 Suppose (P1, P2, P5, P6, P7) hold. Then T h(M) is simple and, for small algebraically closed A, B and any tuple c,

tp(c/B) does not fork over A ⇔ d(c/A) = d(c/B) and acl(cA) ∩ B = A.

Proof. The last statement follows at once from Lemma 2.19. Simplicity then follows from separability of ℝ. □

Corollary 2.21 Suppose (P1-P5) hold. Then T h(M) is simple and, for small algebraically closed A, B and any tuple c,

tp(c/B) does not fork over A ⇔ d(c/A) = d(c/B) and acl(cA) ∩ B = A.

Proof. This follows from the above and Lemmas 2.16 and 2.18. □

Remarks 2.22 If (P1-P5) hold, then T h(M) is CM-trivial, as defined in the introduction. Indeed, suppose E_{1}, E_{2} are algebraically closed and independent over E_{1} ∩ E_{2}. Let D ⊆ E_{1} ∪ E_{2} be such that acl(D) ∩ (E_{1} ∪ E_{2}) = D and let D_{i} = D ∩ E_{i}. We must show that D_{1} and D_{2} are independent over D_{1} ∩ D_{2}. So let c be a tuple of elements in D_{1}. We know (by Corollary 2.21) that d(c/E_{2}) = d(c/E_{1} ∩ E_{2}) and acl(c(E_{1} ∩ E_{2})) ∩ E_{2} = E_{1} ∩ E_{2}. Thus by Corollary 2.14, cb(c/E_{2}) ⊆ E_{1} ∩ E_{2}. Now, acl(cD_{2}) ∩ E_{2} = D_{2}, so by Corollary 2.15, cb(c/D_{2}) ⊆ cb(c/E_{2}) and it follows easily (using Corollary 2.21 again) that c and D_{2} are independent over D_{1} ∩ D_{2}.

The statement of Corollary 2.15 is closer to the usual definition of CM-triviality in a simple theory (with elimination of hyperimaginaries), expressed as a property of canonical bases: if A ⊆ B are algebraically closed and acl(cA) ∩ B = A, then Cb(c/A) ⊆ acl(Cb(c/B)).

2.8 Supersimplicity

In this subsection, we suppose that the set of values of d(c/A) (where c is a finite tuple from M and A is a finite subset of M) is well-ordered. In this case we say that d is discrete. For example, this happens if the subgroup of ℝ generated by {d_{0}(A) : A ⊆ M finite } is discrete. It is clear that if d is discrete then for all c and small B ≤ N there is a finite A ≤ B with d(c/B) = d(c/A). Moreover, Lemma 2.3 shows that acl(cA) ∩ B = A. So by Corollary 2.21, if (P1-P5) also hold then T h(M) is supersimple and of finite SU-rank. However, more is true.

Lemma 2.23 Suppose (M, d_{0}) satisfies (P1), (P2) and d is discrete.

(i) If (P4^{−}) holds, then (P6) holds.

(ii) (P7) holds.

Proof. (i) A compactness argument similar to (but easier than) Lemma 2.16.

(ii) Trivial. □

Corollary 2.24 Suppose (M, d_{0}) satisfies (P1, P2, P4^{−}, P5) and d is discrete. Then T h(M) is supersimple of finite SU-rank. Moreover, for small algebraically closed A, B and a tuple c,

tp(c/B) does not fork over A ⇔ d(c/A) = d(c/B) and acl(cA) ∩ B = A.□

Remarks 2.25 There is clearly a converse to this. Suppose M is supersimple, ℵ_{0}-categorical and of finite SU-rank. For finite A ⊆ M define d_{0}(A) = SU(tp(A/∅)). This satisfies (P1, P2, P4^{−}), and if M has weak elimination of imaginaries, then (P5) also holds (strong type and Lascar strong type over finite sets coincide in ℵ_{0}-categorical theories, so the independence theorem holds over finite algebraically closed sets, by ([8], Theorem 5.8)).

Solutions to the following might shed some light on the problem of finding supersimple ℵ_{0}- categorical structures which are not CM-trivial.

Question 2.26 Find ℵ_{0}-categorical structures (with weak elimination of imaginaries) without the strict order property which have a predimension satisfying (P1, P2, P4^{−}), but not (P3). If this can be done, impose the extra condition of having weak canonical bases for the predimension.

3 Examples

In this section we give a slight modification of Hrushovski’s construction from [4] and [7]. Throughout, M_{0} is a countable ℵ_{0}-categorical L-structure having quantifier elimination in a relational language, and y_{0} is a predimension on M_{0} (with corresponding dimension y) satisfying (P1-P5). We assume that y(X) → ∞ as |X| → ∞ (for X ⊆ M_{0}), and, for convenience, we assume that y(acl(∅)) = 0. For example, M_{0} could be a pure set, and y_{0}(X) the cardinality of X; or M_{0} could be a vector space over a finite field (Morleyized, to give quantifier elimination in a relational language), and y_{0}(X) the dimension of the subspace spanned by X. It could also be a structure produced by a previous application of this construction (with its language enriched to give the quantifier elimination).

Now let L^{+} be a first-order language extending L by relation symbols, with finitely many of each arity. Let C′_{0} be the collection of finite L-structures A such that: the reduct A | L to L is isomorphic to an algebraically closed substructure of M_{0}; none of the new relations involve elements of acl_{M_{0}}(∅); and for all n-ary relation symbols R in L^{+} \ L,

A ⊧ R(x_{1}, …, x_{n}) → ⋀_{i≠j} x_{i} ≠ x_{j}.

(We could also insist that some of the relations be invariant under permutation of coordinates.) Let C′′_{0} be the set of substructures of elements of C′_{0}.

Let α > 0 and α_{R} ≥ 0 (for each such R) be real numbers. For A ∈ C′_{0} and B ⊆ A let

d_{0}(B) = αy(B) − ∑_{R} α_{R} | R[B] |

(the sum ranges over the new relation symbols, and | R[B] | is the number of realisations of R in B). Note that this depends only on the L^{+}-isomorphism type of B (and not on A) by quantifier elimination in the restricted language. For B ⊆ C ⊆ A ∈ C′_{0} write

B ≤_{d} C ⇔ d_{0}(B) < d_{0}(D) for all B ⊂ D ⊆ C.

It is easy to see that d_{0} satisfies (P 1) and if B ≤_{d} C then B | L ≤_{y} C | L. So (by Lemma 2.1) for any B ⊆ A ∈ C′_{0} one can define cl_{d}^{A} (respectively cl_{y}^{A}) to be the smallest subset C of A containing B such that C ≤_{d} A (respectively C ≤_{y} A). Clearly cl_{y}^{A}(B) ⊆ cl_{d}^{A}(B). Let C_{0} = {A ∈ C′_{0} : cl_{y}^{A′}(∅) ≤_{d} A′ for all A′ ⊆ A}.

Definition 3.1 For x ≥ 0 let κ(x) = min(1, d_{0}(D/A) : D ∈ C_{0}, A ≤_{d} D, A ≠ D, y(D) ≤ x).

For example, if y, α and α_{R} are integer valued, then κ(x) = 1.

Definition 3.2 Let f : ℝ^{≥0} → ℝ^{≥0} be a continuous function with f(x) → ∞ as x → ∞ whose right derivative f′ exists at all points, and is ≤ 1. Suppose further that 0 < f′(x) ≤ κ(x)/x and f′(x) is monotone decreasing for all x. Let

C_{f} = {X ∈ C_{0} : d_{0}(B) ≥ f(y(B)) for all B ⊆ X}.

Lemma 3.3 (Amalgamation Lemma) Let A, B_{1}, B_{2} ∈ C_{f} and β_{i} : A → B_{i} be embeddings with β_{i}A ≤_{d} B_{i}. Then there exists C ∈ C_{f} and embeddings γ_{i} : B_{i} → C with:

(i) γ_{1}β_{1} = γ_{2}β_{2}; (ii) γ_{i}B_{i} ≤_{d} C; (iii) γ_{1}B_{1} ∩ γ_{2}B_{2} = γ_{1}β_{1}A and d_{0}(C) = d_{0}(B_{1}) + d_{0}(B_{2}) − d_{0}(A).

Proof: To ease notation, assume that the β_{i} are inclusions and B_{1} ∩ B_{2} = A. By (P4) for y_{0}, we can assume that there exists and L-structure C^{−} isomorphic to an algebraically closed subset of M_{0}, such that B_{i} | L ≤_{y} C^{−} and

y(B_{1} ∪ B_{2}) = y(C^{−}) = y(B_{1}) + y(B_{2}) − y(A).

Now let C be the L^{+} structure obtained by expanding C^{−} by the relations on B_{1} and B_{2} (and no others). We call this a free amalgam of B_{1} and B_{2} over A. It is clear that B_{i} ≤_{y} C and part (iii) of the Lemma holds.

To show that B_{1} ≤_{d} C, let D = cl_{d}^{C}(B_{1}). By y-modularity (and Lemma 2.4 for y), y(D) ≥ y(D ∩ (B_{1} ∪ B_{2})) = y(B_{1}) + y(D ∩ B_{2}) − y(A). As any instance of a new relation in D lies in D ∩ (B_{1} ∪ B_{2}) this gives d_{0}(D) ≥ d_{0}(B_{1}) + d_{0}(D ∩ B_{2}) − d_{0}(A). If A ⊂ D ∩ B_{2} then d_{0}(A) < d_{0}(D ∩ B_{2}), and so d_{0}(D) > d_{0}(B_{1}), contradicting the definition on d-closure. Thus D ∩ B_{2} = A, so y(D) = y(B_{1}), whence D = B_{1}.

Next, we show that if D ⊆ C then d_{0}(D) ≥ f(y(D)). First, suppose that D ≤_{y} B_{1} ∪ B_{2}. We may assume D ⊈ B_{i}. Then by CM-triviality in M_{0} we have y(D) = y(D ∩ B_{1}) + y(D ∩ B_{2}) − y(D ∩ A). Thus d_{0}(D) = d_{0}(D ∩ B_{1}) + d_{0}(D ∩ B_{2}) − d_{0}(D ∩ A). A familiar argument, which we repeat for convenience, then shows that d_{0}(D) ≥ f(y(D)). For X ∈ C′′_{0} let ⟨X⟩ = (y(X), d_{0}(X)). Let D_{0} = D ∩ A and D_{i} = D ∩ B_{i}. Then

⟨D⟩ = ⟨D_{0}⟩ + (⟨D_{1}⟩ − ⟨D_{0}⟩) + (⟨D_{1}⟩ − ⟨D_{0}⟩).

So the line segment in the (y, d_{0})-plane joining ⟨D_{0}⟩ and ⟨D⟩ lies above the line segment joining ⟨D_{0}⟩ and ⟨D_{1}⟩, say. As D_{0} ≤_{d} D_{1} the latter has slope at least κ(y(D_{1}))/y(D_{1}) ≥ f′(y(D_{1})). As f′ is non-increasing, the half line joining ⟨D_{0}⟩ and ⟨D_{1}⟩ lies above the graph of f, so the same is true of ⟨D_{0}⟩ and ⟨D⟩. In particular d_{0}(D) ≥ f(y(D)).

Now let D be any subset of C. We may assume that D ≤_{y} C. Let D_{0} = D ∩ (B_{1} ∪ B_{2}). Then d_{0}(D) = y(D) − y(D_{0}) + d_{0}(D_{0}). By the above d_{0}(D_{0}) ≥ f(y(D_{0})) and so it remains to show that y(D) − y(D_{0}) ≥ f(y(D)) − f(y(D_{0})). But this follows as f′(x) ≤ 1. □

Lemma 3.4 Let E ∈ C′_{0} and X, Y ⊆ E be a y-modular pair. Suppose that any instance of a relation of L^{+} \ L on X ∪ Y is in X or in Y, and that X ∩ Y ≤_{d}^{∗} X. Then Y ≤_{d}^{∗} X ∪ Y.

Proof. The hypotheses imply that if X∩Y ⊆ T ⊆ X then d_{0}(Y ∪T) = d_{0}(Y)+d_{0}(T)−d_{0}(X∩Y) ≥ d_{0}(Y). □

Theorem 3.5 There exists a countable L^{+}-structure M_{f} with a predimension d_{0} satisfying (P1) and (P2) and having the following properties:

(i) The set of isomorphism types of finite ≤_{d}-closed substructures of M_{f} is C_{f} (and d_{0} agrees with the d_{0} defined above). (ii) There exist finite X_{1} ≤_{d} X_{2} ≤_{d} ⋯ with M_{f} = ⋃_{i<ω} X_{i}. (iii) If β : X → Y is as embedding of structures in C_{f} with βX ≤_{d} Y, then for all embeddings δ : X → M_{f} with δX ≤_{d} M_{f} there exists an embedding γ : Y → M_{f} with γβ = δ and γY ≤_{d} M_{f}.

Moreover, M_{f} is determined by the above properties. It is ℵ_{0}-categorical and any isomorphism between finite algebraically closed substructures of M_{f} extends to an automorphism of M_{f}.

Proof. Existence and uniqueness of a countable structure M_{f} with properties (i), (ii), (iii) is standard, using Lemma 3.3, and we omit this. Moreover, using these properties we obtain ≤_{d}- homogeneity: any isomorphism between finite structures A, B ≤_{d} M_{f} extends to an automorphism of M_{f}.

For ℵ_{0}-categoricity, note that for finite subsetes X ⊆ Y of M_{f}, if d_{0}(Y) ≤ d_{0}(X), then y(Y) ≤ f^{−1}(d_{0}(X)) ≤ f^{−1}(αy(X)). Thus, there exists Z ≤_{d} M_{f} containing X with y(Z) ≤ f^{−1}(αy(X)). So, because y(X) → ∞ as |X| → ∞, we have that | cl_{d}^{M_{f}} (X)| is uniformly bounded by a function of |X|. This, together with ≤_{d}-homogeneity, gives ℵ_{0}-categoricity.

For (P2) we need to know that if A ≤_{d} M_{f} and b ∉ A then b ∉ acl(A). Let B = cl_{d}(Ab). Then A ≤_{d} B ≤_{d} M_{f} and A, B ∈ C_{f}. By Lemma 3.3 there exist B_{i} ≤_{d} M_{f} (for i ∈ ω) with pairwise intersection A and tp(B_{i}/A) = tp(B/A) for all i. Thus tp(b/A) is non-algebraic. □

It should be noted that the reduct of M_{f} to L is isomorphic to M_{0} (this is an easy backand-forth argument using quantifier elimination). Also, for X ⊆ M_{f} we have d(X) → ∞ as |X| → ∞.

Theorem 3.6 (i) The predimension d_{0} on M_{f} satisfies (P1-P4).

(ii) If additionally f(3t) ≤ f(t) + κ(t) for all t ≥ 0, then d_{0} has property (P5), and M_{f} is simple.

Remarks 3.7 Suppose there exists c_{0} > 0 such that κ(x) ≥ c_{0} for sufficiently large x. Then the condition on f in (ii) simplifies to f(3t) ≤ f(t) + c_{0} for sufficiently large t. Thus, if M_{0} is supersimple and we take y to be given by SU-rank, there is no problem in finding such a function f. So the construction can be iterated to produce supersimple structures (all of which will be CM-trivial). It is possible, however, that there is no such c_{0}, and then the existence of a suitable function f is not guaranteed. In fact, an obvious additional condition is that the series ∑_{k<ω} κ(t_{0}3^{k}) should diverge (for sufficiently large t_{0}).

Proof of 3.6. (i) Note that A, B is a d_{0}-modular pair if and only if it is a y-modular pair, and the only instances of relations R of L^{+} \ L with α_{R} > 0 lie entirely in A or entirely in B. Now we show (P3) holds. For fixed A_{2} ≤_{d} M_{f} and c, we must show that there exists ϵ > 0 such that if A_{1}, A_{2} ≤_{d} B ≤_{d} M_{f}, acl(cA_{i}) ∩ B = A_{i} and A_{1} and B are within ϵ of being a d_{0}-modular pair, then C_{0} = acl(cA_{1}) ∩ acl(cA_{2}) and A_{2} is a d_{0}-modular pair. Note that this fails to be modular only if it is not y-modular, or there is a relation R(a)(with α_{R} > 0) holding on C_{0} ∪ A_{2} with a not contained entirely within one of the parts. But the latter possibility can be excluded by taking ϵ < α_{R} for all R ∈ L^{+} \ L of arity at most | acl(cA_{2}) |. To exclude the possibility that the pair is not y-modular, note that acl(cB) is L-isomorphic to an algebraically closed subset C_{1} of M_{0}. So it will suffice to take ϵ no greater that the ϵ guaranteed by (P3) in M_{0} for (the finitely many possibilities of) the pair C_{0}, A_{2}.

For (P4) first note that by (P4) in M_{0} and the properties (i) and (iii) of M_{f} in Theorem 3.5 we may assume that y(c/B) = y(c/A). Let D = acl(cB). Then using Lemma 3.3 and (ii) of Theorem 3.5 there exists D_{1} ≤_{d} M_{f} and an L-isomorphism ϕ : D → D_{1} such that ϕ | acl(cA) is an L^{+}-isomorphism, ϕ | B is the identity, and there are no instances of relations of L^{+} \ L on D_{1} except those lying in ϕ(acl(cA)) or B. Let c_{1} = ϕ(c). Then (i), (ii) of (P4) hold, and if C ⊂ B then acl(c_{1}C) = ϕ(acl(cC)), which gives (iii) of (P4).

(ii) The proof is an elaboration of the proof of Lemma 4.14 in [7].

Note that by Theorem 3.5, the type in M_{f} of a tuple c over a finite set X is determined by the quantifier-free type of acl(cX) over X. We use the same notation as in the statement of (P5). Let ψ : acl(c_{1}A) → acl(c_{2}A) be an elementary map which is the identity on A and ψ(c_{1}) = c_{2}. By (P4) and (P5) in M_{0}, there exists E ∈ C′_{0} and L^{+}-embeddings

ϕ_{12} : acl(B_{1}B_{2}) → E
ϕ_{j3} : acl(c_{j}B_{j}) → E for j = 1, 2

such that:

• ϕ_{12} | B_{j} = ϕ_{j3} | B_{j} (for j = 1, 2);

• ϕ_{13}(x) = ϕ_{23}(ψ(x)) for all x ∈ acl(c_{1}A) = B_{3};

• E_{ij} := im(ϕ_{ij}) ≤_{y} E;

• E_{ij} ∩ E_{kj} = E_{j} = ϕ_{ij}(B_{j}) if j = 1, 2, and = ϕ_{13}(B_{3}) if j = 3;

• E_{1}, E_{2}, E_{3} are y-independent over their intersection E_{0} and if C = ⋃ cl_{y}(E_{i}E_{j}), then the E_{ij} are y-independent over C; • any instance of an L^{+} \ L-relation on E is entirely in some E_{ij}.

By Theorem 3.5 it is enough to prove:

(i) E_{ij} ≤_{d} E;

(ii) E ∈ C_{f};

(iii) d(E_{3}/E_{1}E_{2}) = d(E_{3}/E_{0}).

Let F = ⋃ E_{ij}. Clearly F ≤_{d}^{∗} E. Let i, j, k denote the indices 1, 2, 3 in some order (and of course, E_{ij} = E_{ji}, by definition). Claim 1. y(F) = y(E_{ik}) + y(E_{ij} ∪ E_{kj}) − y(E_{i} ∪ E_{k}).

To see this, note that y(F) = y(F/C) + y(C) and use the above y-independence properties to get y(F) = ∑_{s<t} y(E_{st}) − ∑_{s=1}^{3} y(E_{s}) + y(E_{0}). But also E_{ij}, E_{kj} is a y-modular pair, as is E_{i}, E_{j}. The claim follows. Claim 2. E_{ij} ∪ E_{kj} ≤_{d}^{∗} F.

Note that the d-independence conditions on B_{1}, B_{2}, acl(Ac_{l}) imply E_{i} ∪E_{k} ≤_{d}^{∗} E_{ik} (by Lemma 2.3). So the claim follows from Claim 1 and Lemma 3.4. Claim 3. E_{ij} ≤_{d} E_{ij} ∪ E_{kj}.

As E_{j} ≤_{d} E_{kj} this follows from the amalgamation lemma. Claim 4. E_{ij} ≤_{d} F.

Let E_{ij} ⊂ X ≤ F. Without loss of generality, X ∩ (E_{kj} \ E_{ij}) ≠ ∅. Then by Claim 3, d_{0}(E_{ij}) < d_{0}(X ∩ (E_{ij} ∪ E_{kj})), and by Claim 2 and Lemma 2.2 d_{0}(X ∩ (E_{ij} ∪ E_{kj})) ≤ d_{0}(X).

We can now prove (i). Let E_{ij} ⊂ X ⊆ E. As F ≤_{d}^{∗} E we have d_{0}(X ∩ F) ≤ d_{0}(X). First suppose E_{ij} ⊂ X ∩ F. Then by Claim 4, d_{0}(E_{ij}) < d_{0}(X ∩ F). On the other hand, suppose X ∩ F = E_{ij}. As all instances in E of relations of L^{+} \ L are in F, we have d_{0}(X) − d_{0}(E_{ij}) = y(X) − y(E_{ij}). But E_{ij} ≤_{y} E so we again get d_{0}(X) > d_{0}(E_{ij}).

(ii) Let g = f^{−1}. Making the substitution t = g(x) the extra condition on f gives x+κ(g(x)) ≥ f(3g(x)), thus g(x + κ(g(x))) ≥ 3g(x).

Now let D ⊆ E. We must show that d_{0}(D) ≥ f(y(D)), that is, y(D) ≤ g(d_{0}(D)). As in the proof of the amalgamation lemma, we may assume that D ≤_{d} F.

Let D_{ij} = D ∩ E_{ij} and d_{ij} = d_{0}(D_{ij}). Suppose, without loss of generality, that d_{12} is the largest of these. By the amalgamation lemma E_{ij} ∪ E_{kj} is a substructure of an element of C_{f} so we may assume that D ≠ D_{12} ∪ D_{23}. Thus D ∩ E_{1} ⊂ D_{13}. Let D^{1} = D_{12} ∪ D_{13}. By Claim 2 and Lemma 2.2 we have D^{1} ≤_{d}^{∗} D. Moreover, D^{1} is a free amalgam over D ∩ E_{1} therefore:

d_{0}(D^{1}) = d_{12} + d_{0}(D_{13}/D ∩ E_{1}) ≥ d_{12} + κ(y(D_{13})).

Also,

y(D_{13}) ≤ g(d_{13}) ≤ g(d_{12}),

so

d_{12} ≤ d_{0}(D^{1}) − κ(g(d_{12})).

The assumption on g implies g(d_{12} + κ(g(d_{12}))) ≥ 3g(d_{12}). Thus we have:

y(D) ≤ ∑ y(D_{ij}) ≤ 3g(d_{12}) ≤ g(d_{0}(D^{1})) ≤ g(d_{0}(D)),

the last inequality following as D^{1} ≤_{d}^{∗} D. This completes the proof of (ii).

(iii) We leave the reader to show this. □

Remark 3.8 It is useful to have a version of this construction where the conditions on the function f are only assumed to hold from some point onwards. It is then necessary to check by hand ‘small’ cases of the amalgamations in Lemma 3.3 and Theorem 3.6. For example, suppose in Lemma 3.3 that we only have f′(x) ≤ κ(x)/x for x ≥ n_{0}. Then for the Lemma, we also have to check that if A ≤_{d} B_{1}, B_{2} ∈ C_{f} and y(B_{i}) ≤ n_{0} then d_{0}(B_{1}) + d_{0}(B_{2}) − d_{0}(A) ≥ f(y(B_{1}) + y(B_{2}) − y(A)). Similarly, if the additional condition in Theorem 3.6 (ii) is only known to hold for t ≥ n_{1} then the amalgamations in the proof have to be checked to lie in C_{f} when d_{0}(D) ≤ n_{1}.

Lemma 3.9 Suppose M_{0} is stable and for all A ⊆ B ∈ C_{f}, if y(A) < y(B) then d_{0}(A) ≠ d_{0}(B). Then M_{f} is stable.

Proof: Note that (P1-P4) hold even without the extra condition in Theorem 3.6. We show uniqueness of d-non-forking extensions of types over algebraically closed sets. Thus (P5) holds in a very strong sense, and Corollary 2.21 shows that d-non-forking is actually non-forking. So we have stability. Moreover, note that by property (P7) (Lemma 2.18) it is enough to check the uniqueness over finite algebraically closed subsets of M_{f}.

So let A ≤_{d} B ≤_{d} M_{f} be finite and c_{1}, c_{2} have the same type over A, and be such that d(c_{i}/B) = d(c_{i}/A). Let C_{i} = acl(c_{i}A). Then (by Lemma 2.3) d_{0}(acl(c_{i}B)) = d_{0}(C_{i} ∪ B) and C_{i}, B is a d_{0}-modular pair. So by assumption y(acl(c_{i}B)) = y(C_{i} ∪ B), and this equals y(C_{i}) + y(B) − y(A). Furthermore, there are no L^{+} \ L-relations on acl(c_{i}B) outside C_{i} and B (note that the assumption implies all α_{R} are non-zero). It follows that d_{0}(cl_{y}(c_{i}B)) = d_{0}(C_{i}) + d_{0}(B) − d_{0}(A) = d(C_{i}B). Thus, by our assumption again, cl_{y}(C_{i}B) = acl(c_{i}B).

As M_{0} is stable with weak elimination of imaginaries (Lemma 2.9) and quantifier elimination, it follows that the L-types of C_{1} and C_{2} over B are the same (they are non-forking extensions of tp^{M_{0}}(C_{i}/A) and A is algebraically closed in M_{0}). So the L-types of cl_{y}(C_{1}B) and cl_{y}(C_{2}B) are the same. By the construction of M_{f} and the above, it then follows that c_{1} and c_{2} have the same L^{+}-type over B. □

Remarks 3.10 It is clear that without the extra condition in the above lemma, we do not have uniqueness of non-forking extensions of types over algebraically closed sets. As the structures have weak elimination of imaginaries, we do not have stability.

As is well-known, the extra conditions in Lemma 3.9 can be guaranteed as follows. Without loss, assume α = 1 and let K be the subfield of ℝ generated by {y(A) : A ⊆ M_{0} finite}. Let R be the set of new relations in L^{+} \ L and suppose that {1} ∪ {α_{R} : R ∈ R} is K-linearly independent. It follows that if Y ⊆ X ∈ C′_{0} and d_{0}(X) = d_{0}(Y), then R[X] = R[Y] for all R ∈ R and y(X) = y(Y).

4 Stable forking

Recall that a simple L-theory T has stable forking if forking is witnessed by stable formulas. More precisely, whenever N ⊧ T, c̄ is a tuple from N, and A ⊆ B ⊆ N are such that tp(c̄/B) forks over A, there exists a stable L(A)-formula ϕ(x̄, ȳ) such that some ϕ(x̄, b̄) ∈ tp(c̄/B) forks over A. We show that the simple L^{+}-theories T h(M_{f}) constructed in Section 3 have stable forking. We first give the formulas which will witness forking; then we prove stability of these formulas (relative to T h(M_{f}), of course). The formulas we use are just the ones which express d-independence in Lemma 2.3: we shall refer to these as dimension formulas for d. Using the notation of Section 3, the point is that if the dimension formulas for y are stable, then the dimension formulas for d are stable. So iterations of the construction in Section 3 starting off with a stable structure (or even a modular simple structure) result in structures with stable forking. Thus it seems that all ℵ_{0}-categorical simple structures which are currently known have stable forking.

It should be noted that Pourmahdian [11] has an alternative and far more elegant approach to proving stable forking. At present, this approach works (in a wider context) when M_{0} is a pure set (i.e. y is simply cardinality). It would be very interesting to generalise these methods.

Henceforth, the notation will be as in Section 3, but we rehearse it slightly before proceeding. We have an L-structure M_{0} with a predimension y (which satisfies P1-P5). Introducing new relation symbols we obtain L^{+}, and using a suitable auxiliary function f we define a predimension d_{0}, and a ≤_{d}-amalgamation class with Fraïssé limit M_{f}. Conditions on f ensure that d_{0} on M_{f} satisfies (P1-P5). In particular M_{f} is simple and d detects forking (Theorem 3.7 and Corollary 2.21). We let M = M_{f} and work in a large model N of T h(M), as before. Note that M_{f}|L ≅ M_{0}, so N|L ⊧ T h(M_{0}). Also, if a pair of finite subsets A, B is d_{0}-modular, then it is y-modular.

By ℵ_{0}-categoricity, algebraic closure is uniformly definable in N. Explicitly, for each n < ω there is an L^{+}-formula β(x_{1}, …, x_{n}, y) such that for all a_{1}, …, a_{n}, b ∈ N,

N ⊧ β(a_{1}, …, a_{n}, b) ⇔ b ∈ acl(a_{1}, …, a_{n}).

Thus, for a finite subset X of N, and all m, n < ω there exist L^{+}(X)-formulas ϕ_{1}(x̄, ȳ), ϕ_{2}(x̄, ȳ), ϕ_{3}(x̄, ȳ) (with x̄ and ȳ being tuples of variables of lengths m and n respectively) such that for all m-tuples c̄ and n-tuples b̄ from N:

N ⊧ ϕ_{1}(c̄, b̄) ⇔ acl(c̄X) ∩ acl(b̄X) = acl(X)
N ⊧ ϕ_{2}(c̄, b̄) ⇔ acl(c̄X), acl(b̄X) is a modular pair in N with intersection acl(X)
N ⊧ ϕ_{3}(c̄, b̄) ⇔ d_{0}(acl(c̄b̄X)) = d_{0}(acl(c̄X)) + d_{0}(acl(b̄X)) − d_{0}(acl(X)).

We refer to these formulas as dimension formulas for d (of course, we have such formulas for each parameter set X). We also use L^{+}(X) formulas α(x̄, ȳ, z̄) such that

α(x̄, ȳ, z̄) ⇔ acl(Xx̄ȳ) ∩ acl(Xȳz̄) = acl(Xȳ).

Henceforth, we do not put bars over tuples.

The following definition is rather technical. The point of it is to isolate a property which appears in an argument in [6] (cf. Proposition 16 there).

Definition 4.1 Say that (M, d_{0}) is quasi-flat if the following holds. Suppose (D_{ij} : j < i < ω) are finite and algebraically closed and

• D_{ij} ∩ D_{ij′} = C̃_{i} is constant for distinct j, j′ < i

• D_{ij} ∩ D_{i′j} = B̃_{j} is constant for distinct i, i′ > j

• D_{ij} ∩ D_{i′j′} = X̃ is constant if i ≠ i′ and j ≠ j′.

Moreover, assume that d_{0}(C̃_{i}) = c̃, d_{0}(B̃_{j}) = b̃, d_{0}(D_{ij}) = d̃ for i > j are independent of i, j, and the same holds for the y-values of these sets. Let ã = d_{0}(X̃). Then d̃≥ c̃ + b̃ − ã.

Remark 4.2 A pure set and a vector space are quasi-flat.

Lemma 4.3 If (M_{0}, y) is quasi-flat, then (M, d_{0}) is quasi-flat.

Proof. Write I = {(i, j) : 1 ≤ j < i ≤ r}. If ∅ ≠ S ⊆ I let D_{S} = ⋂_{(i,j)∈S} D_{ij}. By definition of d_{0}:

d_{0}( ⋃_{(i,j)∈I} D_{ij}/X̃) = y(⋃ D_{ij}/X̃) − ∑_{R} α_{R}(|R[⋃ D_{ij}] \ R[X̃]|)  (1)
≤ y(⋃ D_{ij}/X̃) − ∑_{∅≠S⊆I}(−1)^{|S|+1}y(D_{S}/X̃) + ∑_{∅≠S⊆I}(−1)^{|S|+1}d_{0}(D_{S}/X̃).

Note that if | S |≥ 2 then D_{S} = C̃_{i}, B̃_{j} or X̃. Moreover, the first happens iff all elements of S have first coordinate i; the second happens iff they all have second coordinate j. For a particular i ≥ 3, the contribution of the first case to the sums on the right hand side of the above inequality is

(( \begin{matrix}i − 1 \\ 2 \end{matrix} ) − ( \begin{matrix}i − 1 \\ 3 \end{matrix} ) + ⋯) = (i − 2).

Thus the total contribution is

(1 + ⋯ + (r − 2)) = \frac{(r − 1)(r − 2)}{2}.

Likewise for the B̃_{j}.

Let y(D_{ij}) = δ, y(B̃_{j}) = β, y(C̃_{i}) = γ, y(X̃) = α. Then

∑_{S≠∅}(−1)^{|S|+1}y(D_{S}/X̃) = \frac{r(r − 1)}{2}(δ − α) − \frac{(r − 1)(r − 2)}{2}(β + γ − 2α)

and

∑_{S≠∅}(−1)^{|S|+1}d_{0}(D_{S}/X̃) = \frac{r(r − 1)}{2}(d̃− ã) − \frac{(r − 1)(r − 2)}{2}(b̃ + c̃ − 2ã).

Moreover, by submodularity

y(⋃ D_{ij}/X̃) ≤ r(δ − α).

Thus the inequality (1) gives

d_{0}( ⋃_{(i,j)∈I} D_{ij}/X̃) ≤ r(δ − α) + \frac{r(r − 1)}{2}(d̃− ã − δ + α) − \frac{(r − 1)(r − 2)}{2}(c̃+ b̃ − 2ã − γ − β + 2α).

This is quadratic in r and non-negative, thus the coefficient of r^{2} on the right-hand side here is non-negative. So

d̃− c̃ − b̃ + ã ≥ δ − β − γ + α.

But δ = y(D_{ij}) ≥ β + γ − α, by quasi-flatness of y. So

d̃− c̃ − b̃ + ã ≥ 0.

□

Lemma 4.4 (1) If the dimension formulas for (M_{0}, y) are stable, then the dimension formulas for (M, d) are stable.

(2) If also (M_{0}, y) is quasi-flat, then the formulas α(x, y, a) are stable.

Proof. (1) The proof is in 3 parts, dealing with each of the three types of formula in turn. In each part, we suppress the subscript on the formula. We will assume that X is finite, algebraically closed, and that (c_{i}b_{i} : i ∈ ℤ) is an X-indiscernible sequence such that ⊧ ϕ(c_{i}, b_{j}) if and only if i ≤ j. Write C_{i} = acl(c_{i}X) and B_{i} = acl(b_{i}X). Indiscernibility (and finiteness of B_{i}, C_{i}) implies that we may assume that B_{i} ∩ B_{j} = B′ and C_{i} ∩ C_{j} = C′ are independent of i, j (if i ≠ j), and also that B_{i} ∩C_{j} is constant and is contained in B′ ∩C′. In fact, we may assume that C_{i} ∩B_{j} = X for all i, j. Case 1: Assume ϕ = ϕ_{1}. Thus C_{i} ∩ B_{j} = X iff i ≤ j. This is a contradiction to the above remarks. Case 2: Assume ϕ = ϕ_{2}. As C_{i} ∩ B_{i} = X, we have C′ ∩ B′ = X and so C_{i} ∩ B_{j} = X for all i, j. Thus C_{i}, B_{j} is a modular pair if and only if i ≤ j. Using that C′, B_{j} is a modular pair we get that C_{i}, C′ ∪ B_{j} is d_{0}-modular iff i ≤ j. As the corresponding dimension formula for y is stable, C_{i}, C′ ∪ B_{j} is y-modular for all i, j. By definition of d_{0}, it follows that there is some L^{+} \ L-relation R with α_{R} > 0, and for every j < 1 a realisation of R in C_{1} ∪ B_{j} that is not contained in C_{1} or C′ ∪ B_{j}. It follows that for large enough r, C′ ∪ B_{0} ∪ ⋯ ∪ B_{−r} ≰_{d} C_{1} ∪ B_{0} ∪ ⋯ ∪ B_{−r}. So acl(C′ ∪ B_{0} ∪ ⋯ ∪ B_{−r}) ∩ C_{1} properly contains C′. Using indiscernibility (and finiteness of C_{1}) this contradicts the definition of C′. Case 3: We are assuming (for a contradiction) that d_{0}(acl(C_{i} ∪ B_{j})) = d_{0}(C_{i}) + d_{0}(B_{j}) − d_{0}(X) if and only if i ≤ j. In particular if i ≤ j we have that C_{i}, B_{j} is a modular pair with intersection X. Indiscernibility and the previous case then imply that if i > j, then C_{i}, B_{j} is also a modular pair, and d_{0}(acl(C_{i} ∪ B_{j})) < d_{0}(C_{i}) + d_{0}(B_{j}) − d_{0}(A).

Let D_{ij} = acl(C_{i}B_{j}) (for i > j). Indiscernibility implies that we may assume

• D_{ij} ∩ D_{ij′} = C̃_{i} is constant for distinct j, j′ < i

• D_{ij} ∩ D_{i′j} = B̃_{j} is constant for distinct i, i′ > j

• D_{ij} ∩ D_{i′j′} = X̃ is constant if i ≠ i′ and j ≠ j′.

Moreover, we may assume that d_{0}(C̃_{i}) = c̃, d_{0}(B̃_{j}) = b̃, d_{0}(D_{ij}) = d̃ for i > j are independent of i, j. Stability of the dimension formulas (for y) implies that

y(D_{ij}) ≥ y(cl_{y}(C̃_{i}B̃_{j})) = y(C̃_{i}) + B̃_{j} − y(X̃).

So the proof of Lemma 4.3 gives that

d_{0}(acl(C̃_{i}B̃_{j})) = d_{0}(C̃_{i}) + d_{0}(B̃_{j}) − d_{0}(X̃) = d_{0}(C̃_{i} ∪ B̃_{j})

if i > j.

Now, we may assume that (C̃_{i}B̃_{i} : i < ω) is X̃-indiscernible. So by cases (1) and (2) C̃_{i}, B̃_{j} is a modular pair for all i, j. Recall that we are assuming that if i > j then

d(C_{i}B_{j}) < d_{0}(C_{i}) + d_{0}(B_{j}) − d_{0}(X).

Note also that acl(C̃_{i}B̃_{j}) = acl(C_{i}B_{j}). Thus, if i > j,

d_{0}(C̃_{i}B̃_{i}) = d_{0}(C̃_{i}) + d_{0}(B̃_{i}) − d_{0}(X̃) = d_{0}(C̃_{i}) + d_{0}(B̃_{j}) − d_{0}(X̃) = d_{0}(acl(C̃_{i}B̃_{j}))

= d_{0}(acl(C_{i}B_{j})) < d_{0}(C_{i}) + d_{0}(B_{j}) − d_{0}(X) = d_{0}(C_{i}) + d_{0}(B_{i}) − d_{0}(X) = d_{0}(C_{i}B_{i}).

So d_{0}(C̃_{i}B̃_{i}) < d_{0}(C_{i}B_{i}). But d(C_{i}B_{i}) = d(C̃_{i}B̃_{i}) ≤ d_{0}(C̃_{i}B̃_{i}). So we have a contradiction.

(2) Suppose ((c_{i}b_{i}) : i < ω) is Xa-indiscernible and such that

acl(c_{i}b_{j}X) ∩ acl(b_{j}aX) = acl(b_{j}X) ⇔ i ≤ j.

Let E_{ij} = acl(c_{i}b_{j}X) for i ≤ j. We may assume that E_{ij} ∩ E_{i′j} = B_{j} (for distinct i, i′), E_{ij} ∩ E_{ij′} = C_{i} (for distinct j, j′) and E_{ij} ∩ E_{i′j′} = X (for i ≠ i′ and j ≠ j′) as before. By quasi-flatness (and indiscernibility) we get that

d(C_{i}B_{j}) = d_{0}(C_{i}) + d_{0}(B_{j}) − d_{0}(X)  (2)

for all i ≤ j. (Quasi-flatness gives d(E_{ij}) ≥ d_{0}(C_{i})+d_{0}(B_{j})−d_{0}(X) for i ≤ j. As E_{ij} = acl(C_{i}B_{j}) we have equality here.) Thus, by part (1), this holds for all i, j. Claim: We may replace c_{i}, b_{i} by C_{i}, B_{i} (containing X) satisfying equation 2. First, note that, with the above notation, if i ≤ j then acl(C_{i}B_{j}) = acl(c_{i}b_{j}X) = E_{ij} and B_{j} = E_{ij} ∩ acl(B_{j}a). Next, note that if i > j then acl(C_{i}B_{j}) ∩ acl(B_{j}a) ≠ B_{j}. In fact, let x ∈ (acl(c_{i}b_{j}X) ∩ acl(b_{j}aX)) \ acl(b_{j}X). Then x ∈ acl(C_{i}B_{j}) ∩ acl(B_{j}a). Take i′ < j. As x ∉ E_{i′j} we have x ∉ B_{j}, as required. This gives the claim.

So we may assume that c_{i} = C_{i}, and b_{i} = B_{i}, and we have equation 2 holding for c_{i}, b_{j}.

Now let D_{ij} = acl(c_{i}b_{j}) for i > j. As before we may assume that D_{ij} ∩ D_{i′j} = B̃_{j}, D_{ij} ∩ D_{ij′} = C̃_{i} and D_{ij} ∩ D_{i′j′} = X̃. By assumption, B_{j} ⊂ D_{ij} ∩ acl(B_{j}a) and by indiscernibility D_{ij} ∩ acl(B_{j}a) = D_{i′j} ∩ acl(B_{j}a). Thus B_{j} ⊂ B̃_{j}.

By quasi-flatness, we get that

d(C̃_{i}B̃_{j}) = d_{0}(C̃_{i}) + d_{0}(B̃_{j}) − d_{0}(X̃)  (3)

for all i > j. So by part (1) again, this holds for all i, j.

Thus, if k ≥ i > j:

d(C̃_{i}B̃_{j}) = d(C̃_{i}) + d(B̃_{j}) − d(X̃) = d(C̃_{i}) + d(B̃_{k}) − d(X̃) = d(C̃_{i}B̃_{k})
≥ d(c_{i}b_{k}) = d(c_{i}) + d(b_{k}) − d(X) = d(c_{i}) + d(b_{j}) − d(X).

So by submodularity we have equality everywhere here, and in particular acl(C̃_{i}B̃_{k}) = acl(c_{i}b_{k}) for all i, k. So B̃_{j} ⊆ E_{ij} for i ≤ j, whence B̃_{j} ⊆ B_{j}, which is a contradiction. □

Lemma 4.5 Let A, B be small algebraically closed subsets of N and c a tuple from N. Suppose tp(c/B) forks over A and tp^{L}(c/B) does not fork over A. Then there exists a dimension formula ϕ_{j}(x, b) and some α(x, b, a) (where the parameters X, a are from A, and b is from B), such that α(x, b, a) ∧ ¬ϕ_{j}(x, b) ∈ tp(c/B) forks over A.

Proof. We have that cb(c/B) ⊈ A, so there is a finite Y ⊆ B with acl(cY) ∩ B = Y such that cb(c/Y) ⊈ A. Let X = A ∩ Y, and let b enumerate Y. Then tp(c/Xb) forks over X.

It follows that α(x, b, a) ∈ tp(c/B) for all tuples a from A, and ¬ϕ_{j}(x, b) ∈ tp(c/B) for some j = 1, 2, 3.

Note that as acl(cb) ∩ B = Y we have acl(cb) ∩ A = X. Thus (working temporarily in the reduct to L) acl^{y}(cY) ∩ B = Y, whence cb^{y}(c/Y) ⊆ A ∩ Y = X, so tp^{L}(c/Ab) does not fork over A.

Let (b_{i} : i < ω) be an A-indiscernible Morley sequence of tp(b/A) with b = b_{0}. Claim: For j = 1, 2, 3

⋀_{i<ω} (⋀_{a} α(x, b_{i}, a) ∧ ¬ϕ_{j}(x, b_{i}))

is inconsistent.

Suppose not, and c′ is a realisation. Let C = acl(c′X), Y_{i} = acl(b_{i}). Thus acl(CY_{i})∩acl(AY_{i}) = Y_{i} for all i. We consider the cases j = 1, 2, 3 in turn. Case j = 1 Thus C ∩ Y_{i} ≠ X. By independence of the b_{i} over A, acl(AY_{i}) ∩ acl(AY_{j}) = A if i ≠ j, so Y_{i} ∩ Y_{j} ⊆ A ∩ Y_{i} = X. But as C is finite, we may assume that C ∩ Y_{i} = C ∩ Y_{j} (for all i, j). Thus C ∩ Y_{i} = X, a contradiction. Case j = 2 By the above, C ∩ Y_{i} = X, but we are assuming d_{0}(C/Y_{i}) < d_{0}(C/X). Now, y(C/Y_{0}) = y(C/X). As in the proof of Case (2) of the previous lemma, we get that acl(B_{i_{1}} … B_{i_{r}})∩ C ≠ X if r is large enough, and as C is finite, we may assume this intersection is constant. By independence over A, acl(AB_{1} … B_{r}) ∩ acl(AB_{r+1} … B_{2r}) = A, thus

acl(B_{1} … B_{r}) ∩ C ⊆ C ∩ A = X.

This is a contradiction. Case j = 3 By the above, C, Y_{i} is a modular pair with intersection X. By assumption, there is γ > 0 such that d(C/Y_{i}) = d(C/X) − γ. Fix r ∈ ω. Let Y′ = ⋃_{i<r} Y_{i}, D_{i} = acl(CY_{i}) and D′ = ⋃_{i<r} D_{i}. Note that D_{0} ∩ Y_{i} ⊆ A so D_{0} ∩ Y_{i} = X if i > 0. It follows that d_{0}(D_{0}/Y′C) ≤ d_{0}(D_{0}/Y_{0}C), by submodularity.

Then:

d_{0}(D′/Y′) = d_{0}(D′/Y′C) + d_{0}(C/Y′) ≤ rd_{0}(D_{0}/Y′C) + d_{0}(C/Y′) ≤ rd_{0}(D_{0}/Y_{0}C) + d_{0}(C/Y′).

As Y_{0}, C is modular, we get d_{0}(D′/Y′) ≤ d_{0}(C/Y′) − rγ.

So if r is large enough, acl(Y′) ∩ D_{0} ≠ Y_{0}. We may assume (by finiteness of D_{0}) that acl(b_{0} … b_{r−1}) ∩ D_{0} = acl(b_{0}b_{r} … b_{2r−2}) ∩ D_{0}, and so this is contained in acl(Ab_{0}) ∩ D_{0}. We obtain acl(AY) ∩ acl(Y c) ≠ Y, and this is a contradiction, as acl(Y c) ∩ B = Y.

This proves the claim. So by compactness, there is some j = 1, 2, 3 and some a (containing X) such that α(x, b, a) ∧ ¬ϕ_{j}(x, b) ∈ tp(c/B) and which forks over A. □

Putting these together, we have:

Theorem 4.6 Suppose (M_{0}, y) is quasi-flat, has stable dimension formulas and has stable forking. Then the same is true of (N, d). □

5 Further observations

5.1 A rank 1 set without weak elimination of imaginaries

We use the construction of Section 3 to produce an ℵ_{0}-categorical graph M which has SU-rank 2 (this is essentially the unstable pseudoplane from the end of [4]). The point is that this has a definable subset C which, when given the full induced substructure, has SU-rank 1 and does not have weak elimination of imaginaries. In fact, weak elimination of imaginaries requires the inclusion of elements of C^{eq} of SU-rank 2. This is a phenomenon which does not arise in the ω-stable case (or the smoothly approximated case).

In the notation of Section 3, let M_{0} be a pure set with y(X) the cardinality of X (for X ⊆ M_{0}). The language L^{+} has one additional binary relation symbol R and C′_{0} is the class of finite (loopless, symmetric) graphs. The predimension d_{0} is given by d_{0}(B) = 2y(B)− | R[B] | /2 (i.e. twice the number of vertices, minus the number of edges). Let f(0) = 0, f(1) = 2, f(2) = 3, f(3) = 4, f(4) = 4.5 and for x ≥ 5, let f(x) = log_{3}(x)+5−log_{3}(5). (We do not have f′(x) ≤ 1 here, but this condition is only used in the proof if algebraic closure in M_{0} is non-trivial.) Then using Theorem 3.6 and Remarks 3.8 one can check that M = M_{f} has properties (P1-P5). It then follows that M is supersimple of SU-rank 2. Moreover, because of the choice of initial vaues of f, the smallest cycle in M is a pentagon, and points and adjacent pairs of points are algebraically closed in M. So by homogeneity of M for algebraically closed sets, Aut(M) is vertex and edge-transitive. Claim 1: M is of diameter 3.

If a, b are vertices at distance greater than 2, then {a, b} ≤ M. Also if Y is a path with 4 vertices having endpoints a and b, then {a, b} ≤ Y ∈ C_{f}. So by Theorem 3.5 (iii), there is a path of length 3 from a to b in M. Claim 2: Aut(M) has 2 orbits on pairs of non-adjacent vertices.

Homogeneity gives that Aut(M) is transitive on pairs of vertices at distance 3. Suppose a, b are at distance 2. Then there is a unique vertex c adjacent to both a and b, as M contains no 4-cycle. Easily, {a, b, c} = acl(ab), and the claim then follows from the homogeneity.

Now let a be any vertex and C the set of vertices of M adjacent to a. Note that SU(c/a) = 1 for all c ∈ C. Consider the restriction map μ : Aut(M/a) → Sym(C). Claim 3: μ is one-to-one and has closed image.

Let D and E be the sets of vertices at distance 2 and 3 from a. As in Claim 2, Aut(M/a) is transitive on each of these, and on C. Suppose γ ∈ Aut(M/a) fixes every element of C ∪ D. If e ∈ E, there exist distinct d, d′ ∈ D which are adjacent to e. But then e is the unique vertex adjacent to both d and d′, so γ fixes e.

To show that μ is one-to-one and the image of μ is closed it is therefore enough to show that D ∩ dcl(a, c_{1}, …, c_{n}) ≠ ∅ for some c_{1}, …, c_{n} ∈ C. There is an algebraically closed subgraph A of M with vertex set

{a} ∪ {c_{1}, …, c_{10}} ∪ {d_{1}, …, d_{10}}

such that c_{i} ∈ C and d_{i} ∈ D and the other adjacencies are given by

R(c_{i}, d_{i}), R(d_{i}, d_{i+1 (mod 10)}), R(d_{1}, d_{5}), R(d_{4}, d_{8}).

One computes that d_{0}(ac_{1} … c_{10}) = 12, d_{0}(A) = 10. Thus the algebraic closure of a, c_{1}, …, c_{10} contains a point of d_{1}, …, d_{10}, which clearly lies in the definable closure.

Thus the canonical structure on C induced by Aut(M/a) is biinterpretable with (M, a), and is clearly supersimple of SU-rank 1. On the other hand, C does not have weak elimination of imaginaries: the elements of (M, a) can be regarded as living in C^{eq} and acl(ad_{1}) ∩ C = c_{1}, whereas d ∉ acl(ac_{1}).

5.2 Growth rates of algebraic closure

It is shown in [3] that Hrushovski’s construction yields (multiply transitive) ℵ_{0}-categorical structures with arbitrarily fast growth rates of algeraic closure. Whilst it seems clear (using Theorem 3.6) that that structures in ([3], Theorems 1 and 2) can be taken to be supersimple, there appear to be significant difficulties in using the construction to exhibit stable ℵ_{0}-categorical structures with arbitrarily fast growth rates of algebraic closure (see [9] for some attempts to do this). Although it may transpire that these are only techincal difficulties, the following easy observations show that (as far as these constructions are concerned) stability does impose some restrictions.

Definition 5.1 Let k be a natural number, M a structure and X a subset of M. The k-ary algebraic closure of X (in M), denoted by acl^{k}(X), is the smallest subset of M containing X which is closed under taking algebraic closure of k-element subsets. Clearly this is a well-defined closure operation, and acl^{k}(X) ⊆ acl(X).

The construction in ([3], Section 2.1) shows that k-ary algebraic closure can grow arbitrarily quickly in supersimple ℵ_{0}-categorical structures (this does not appear to follow from the simpler argument of ([3], Section 3). With the notation of Section 3, the following shows that growth of k-ary algebraic closure in the structure M_{f} of Lemma 3.9 is constrained by growth of algebraic closure in M_{0}.

Lemma 5.2 Assume the notation and hypotheses of Lemma 3.9. Suppose the function ϕ is such that the algebraic closure of any n points in M_{0} has size at most ϕ(n). Then there exists a constant c such that for all finite subsets A of M_{f} we have | acl^{k}(A) |≤ ϕ(c | A |).

Proof. Let ϵ_{k} be the minimum value for M_{f} of d_{0}(X′) − d_{0}(acl(X)) where | X |≤ k, X ⊆ X′ ⊆ acl(X) and y(X′) < y(acl(X)). Note that ϵ_{k} > 0.

Write acl^{k}(A) = ⋃_{i=0}^{r} A_{i} where A_{0} = A and A_{i+1} = cl_{y}(A_{i} ∪ acl(X_{i})) for some X_{i} ⊆ A_{i} of size k, and assume that acl(X_{i}) ⊈ A_{i} for i < r. Then by definition of d_{0},

d_{0}(A_{i+1}) ≤ d_{0}(A_{i} ∪ acl(X_{i})) ≤ d_{0}(A_{i}) + d_{0}(acl(X_{i})) − d_{0}(A_{i} ∩ acl(X_{i})).

Moreover A_{i} ∩ acl(X_{i}) and acl(X_{i}) are y-closed, so y(A_{i} ∩ acl(X_{i})) < y(acl(X_{i})). Thus d_{0}(A_{i} ∩ acl(X_{i})) ≥ d_{0}(acl(X_{i})) + ϵ_{k}, whence d_{0}(A_{i+1}) ≤ d_{0}(A_{i}) − ϵ_{k}. In particular, as d_{0}(A_{r}) ≥ 0 we have r ≤ d_{0}(A_{0})/ϵ_{k}.

Note that A_{i+1} = cl_{y}(A_{0} ∪ acl(X_{0}) ∪ ⋯ ∪ acl(X_{i})), so | acl^{k}(A) |≤ ϕ(| A | +rt), where t bounds the size of the algebraic closure of k points in M_{f}. As d_{0}(A) is bounded by a constant multiple of | A | (by submodularity) we get the required result. □

References

[1] John T. Baldwin and Niandong Shi, ‘Stable generic structures’, Annals of Pure and Applied Logic 79 (1996), 1–35.

[2] Andreas Baudisch and Anand Pillay, ‘A free pseudospace’, J. Symbolic Logic 65 (2000), 443–463.

[3] David M. Evans and M. E. Pantano, ‘ℵ_{0}-categorical structures with arbitrarily fast growth of algebraic closure’, Preprint, Norwich, 1996.

[4] Ehud Hrushovski, ‘A stable ℵ_{0}-categorical pseudoplane’, Unpublished notes, 1988.

[5] Ehud Hrushovski, ‘Pseudofinite fields and related structures’, Unpublished notes, 1991.

[6] Ehud Hrushovski, ‘A new strongly minimal set’, Annals of Pure and Applied Logic 62 (1993), 147–166.

[7] Ehud Hrushovski, ‘Simplicity and the Lascar group’, Preprint, December 1997.

[8] Byunghan Kim and Anand Pillay, ‘Simple theories’, Annals of Pure and Applied Logic, 88 (1997), 149–164.

[9] M. E. Pantano, Algebraic Closure in ℵ_{0}-Categorical Structures, Ph. D. Thesis, UEA Norwich, 1995.

[10] Anand Pillay, ‘A note on CM-triviality and the geometry of forking’, J. Symbolic Logic 65 (2000), 474–480.

[11] Massoud Pourmahdian, D. Phil. Thesis, Oxford, 2000.

[12] Frank O. Wagner, ‘Relational structures and dimensions’, in Automorphisms of First-Order Structures, R. Kaye and D. Macpherson (eds.), Oxford University Press, 1994, pp. 153–180.

School of Mathematics, University of East Anglia, Norwich NR4 7TJ, England. d.evans@uea.ac.uk 21 September 2000. Revised 29 May 2001.