DEPENDENCE RELATION IN PREGEOMETRIES

RAMI GROSSBERG AND OLIVIER LESSMANN

Abstract. The aim of this paper is to set a foundation to separate geometric model theory from model theory. Our goal is to explore the possibility to extend results from geometric model theory to non first order logic (e.g. L_{ω_{1},ω}). We introduce a dependence relation between subsets of a pregeometry and show that it satisfies all the formal properties that forking satisfies in simple first order theories. This happens when one is trying to lift forking to nonelementary classes, in contexts where there exists pregeometries but not necessarily a well-behaved dependence relation (see for example [HySh]). We use these to reproduce S. Buechler’s characterization of local modularity in general. These results are used by Lessmann to prove an abstract group configuration theorem in [Le2].

1. Introduction

The notion of forking is at the center of stability theory. Forking is a dependence relation discovered by S. Shelah. It satisfies the following properties in the first order stable case, see [Sh b]:

(1) (Finite character) The type p does not fork over B if and only if every finite subtype q ⊆ p does not fork over B.

(2) (Extension) Let p be a type which does not fork over B. Let C be given containing the domain of p. Then there exists q ∈ S(C) extending p such that q does not fork over B;

(3) (Invariance) Let f ∈ Aut(ℭ) and p be a type which does not fork over B. Then f(p) does not fork over f(B).

(4) (Existence) The type p does not fork over its domain;

(5) (Existence of κ(T)) For every type p, there exists a set B ⊆ dom(p) such that p does not fork B;

(6) (Symmetry) Let p = tp(ā/Bc̄). Suppose that p does not fork over B. Then tp(c̄/Bā) does not fork over B;

(7) (Transitivity) Let B ⊆ C ⊆ A. Let p ∈ S(A). Then p does not fork over B if and only if p does not fork over C and p ↾ C does not fork over B.

Already in the introduction of Chapter III of [Sh b], S. Shelah states what is important about the forking relation is that it satisfies properties (1)–(7). S. Shelah stated another property named by S. Buechler [Bu] the Pairs Lemma (see Proposition 17 for the statement) as one of the basic properties of forking, which was proved in [Sh b] using the Finite Equivalence Relation Theorem. Later Baldwin in his book [Bl] presented an axiomatic treatment of forking in stable theories. This allowed Baldwin to derive abstractly Shelah’s Pairs Lemma from the other properties of forking. Following these ideas, it has now become common to characterize various stability conditions in terms of the axiomatic properties that forking satisfies.

Date: April 25, 2000.

This is part of the second author’s PhD thesis, under the direction of the first author. The second author wishes to thank the first author wholeheartedly for his kind guidance and support.

The second author is supported by the Swiss National Science Foundation.

One of the most difficult directions of pure model theory is the area called by Shelah classification theory for nonelementary classes. A major problem is to find a dependence relation which is as well-behaved as forking for first order theories. See for example [Gr 1], [Gr 2], [GrHa], [GrLe1], [GrLe2], [GrSh 1], [GrSh 2], [HaSh], [HySh], [Ki], [KlSh], [Le1], [MaSh], [Sh3], [Sh47], [Sh 87a], [Sh 87b], [Sh 88], [Sh tape], [Sh 299], [Sh 300], [Sh 362], [Sh 394], [Sh 472], [Sh 576] and [Sh h]. The situation in nonelementary classes is very different from the first order case. In the first order case, the Extension property for forking comes for free; it holds for any theory and is a consequence of the Compactness Theorem. This is in striking contrast with the nonelementary cases; the Extension property is usually among the most problematic and does not hold over sets in general for any of the dependence relations introduced thus far.

A general dependence relation satisfying all the formal properties of forking has thus not been found yet for nonelementary classes. There are, however, several cases where pregeometries appear; i.e. sets with a closure operation satisfying the properties of linear dependence in a vector space. In the first order case, the pregeometries are the sets of realizations of a regular type, and the dependence is the one induced by forking and thus satisfies automatically many additional properties. In nonelementary classes the situation is different.

Here are several nonelementary examples: The first three examples have in common that there exists a rank, giving rise to a reasonable dependence relation. However the Extension property and the Symmetry property fail in general (they hold over sufficiently “rich” sets). The rank introduced for these classes are generalizations of what S. Shelah calls R[·, L, 2]. Intuitively, a formula has rank α + 1 if it can be partitioned in two pieces of rank α with some additional properties that are tailored to each context. It is noteworthy that extensions of Morley rank are inadequate, as partitioning a formula in countably many pieces makes sense only when the compactness theorem holds. In the last example, no rank is known, but pregeometries exist.

Categorical sentences in L_{ω_{1}ω}(Q): S. Shelah started working on this context [Sh47] to answer a question of J.T.Baldwin: Can a sentence in L(Q) have exactly one uncountable model? Shelah answers this question negatively using V=L (and later using different methods within ZFC) while developing powerful concepts. A main tool is the introduction of a rank. This rank is bounded under the parallel to ℵ_{0}- stability. It gives rise to a dependence relation and pregeometries. Later, H. Kierstead [Ki] uses these pregeometries to obtain some results on the countable models of these sentences.

Excellent Scott sentences: In [Sh 87a] and [Sh 87b] Shelah introduces a simplification of the rank of [Sh47]. Shelah identifies the concept of excellent Scott sentences and proves (among many other things) the parallel to Morley’s Theorem for them. Again, this rank induces a dependence relation on the subsets of the models. Later, R. Grossberg and B. Hart [GrHa] proved the existence of pregeometries (regular types) for this dependence relation and used it to prove the main gap for excellent Scott sentences.

Totally transcendental diagrams: In [Le1] Lessmann introduced a rank for ℵ_{0}-stable diagrams. Finite diagrams were introduced by S. Shelah [Sh3] in 1970 (see also [GrLe1] for an exposition). They are classes of models omitting a prescribed set of types, with an additional condition. We call a finite diagram totally transcendental when the rank is bounded. The rank gives rise to a dependence relation on the subsets of the models and pregeometries exist often. This is used to give a proof of categoricity generalizing the Baldwin-Lachlan Theorem. In a work in preparation, [GrLe2], we prove the main gap for totally transcendental diagrams.

Superstable diagrams: In [HySh], Hyttinen and Shelah study stable finite diagrams ([Sh3] or [GrLe1]) under the additional assumption that κ(D) = ℵ_{0}. Such diagrams are called superstable. They introduce a relation between sets A, B and an element a, written a ↓_{B} A. The main result is that the parallel of regular types exist. More precisely, for every pair of “sufficiently saturated” models M ⊆ N, M ≠ N, there exists a type p realized in N − M such that the relation a ↓_{M} C (standing for a ∉ cl(C)) induces a pregeometry among the realizations of p in N.

Thus, pregeometries seem to appear naturally in nonelementary classes, while general well-behaved dependence relations are hard to find. The main goal of this paper is to recover from any pregeometry a dependence relation over the subsets of the pregeometry that satisfies all the formal properties of forking. While it is known that forking generalizes the first order notion of algebraic closure inside a strongly minimal set, recovering a “forking-like” dependence relation from an arbitrary pregeometry has not been done. This is, of course, particularly useful when the pregeometry itself was not induced by forking. In superstable diagrams for example, using our formalism, a good dependence relation can be recovered inside the pregeometry, while the original dependence relation inducing the pregeometry is not as well behaved (see [HySh]).

A similar endeavor was attempted by John Baldwin in the early eighties. In [Bl1], J.Baldwin examined some pregeometries and several dependence relations in the first order case. From a pregeometry, he defines the relation a ⌣ B C, by a ∈ cl(B ∪ C) − cl(B). He

did not however introduce A ⌣ B C, where A is a tuple or a set as opposed to an element,

which we do (see Definition 8). This is a crucial step; it is built-in in the model theory of first order, since forking is naturally defined for types of any arity. To make this more precise, fix T a first order stable theory. Let us write

ā ∗ ; ⌣ ; B C for tp(ā/B ∪ C) does not fork over B.

Inside a regular type p(x) ∈ S(B), the relation a ∈ cl(C) given by a ∗ ; / ⌣ ; B C gives rise to a

pregeometry. But, the relation ā ∗ ; ⌣ ; B C is defined in general whether or not ā and C consist

of elements realizing p. Inside the pregeometry, the relation ā ∗ ; ⌣ ; B C holds (defined with

forking) if and only if the relation ā ⌣ B C holds (defined formally from our definition using the closure operator of the pregeometry). This is a consequence of the Pairs Lemma, which holds for first order simple theories. When we start from an abstract pregeometry (or an abstract dependence relation), we do not have the formalism of types or the Pairs Lemma. Therefore the relation ā ⌣ B C has to be introduced for tuples, using the relation a ⌣ B C for

elements. As a consequence, suppose we are given the corresponding notion of a regular type p ∈ S(B) in a nonelementary context. Suppose there is some ambient dependence

relation, written A ∗ ; ⌣ ; B C such that over realizations of p the relation a ∈ cl(C), given by

a ∗ ; / ⌣ ; B C, induces a pregeometry. Then, the truth value of the relation ā ∗ ; ⌣ ; B C (given from

the ambient dependence relation) and ā ⌣ B C (defined from the closure operation in the

pregeometry) may not coincide. They will coincide only if the Pairs Lemma holds for the dependence relation (and this fact is not known in general for nonelementary cases). Therefore, this abstract formalism allows us to introduce for nonelementary classes a (possibly) better dependence relation, inside the pregeometry.

As an illustration of the value of this general relation, we present S. Buechler’s characterization of local modularity with parallel lines (see [Bu]) in this general context. This also has esthetic value as it allows one carry out this work in the general context of combinatorial geometry, without logic.

Finally, we add a short section with some easy set-theoretic results.

In a follow-up paper [Le2], Lessmann presents an abstract framework where, using the “anchor relation” defined is this paper, he derives a generalization of Zilber-Hrushovski group configuration theorem. We expect this result to have a potential for the classification theory of nonelementary classes.

We would like to thank John Baldwin and Saharon Shelah for valuable comments on a draft of this paper.

2. Preliminaries

We recall a few standard and well-known facts about pregeometries. The notation is standard. We write Ab for A ∪ {b}.

Definition 1. We say that (W, cl) is a pregeometry if W is a set and cl: P(W) → P(W) is a function satisfying the following four properties

(1) (Monotonicity) For every set X ∈ P(W) we have X ⊆ cl(X);

(2) (Finite Character) If a ∈ cl(X) then there is a finite set Y ⊆ X, such that a ∈ cl(Y);

(3) (Transitivity) Let X, Y ∈ P(W). If a ∈ cl(X) and X ⊆ cl(Y) then a ∈ cl(Y);

(4) (Exchange Property) For X ∈ P(W) and a, b ∈ W, if a ∈ cl(Xb) but a ∉ cl(X), then b ∈ cl(Xa).

We always assume cl(∅) ≠ W.

The next two basic properties are standard and easy.

Fact 2. If (W, cl) is a pregeometry and B ⊆ C ⊆ W, then cl(B) ⊆ cl(C).

Fact 3. If (W, cl) is a pregeometry and B ⊆ W, then cl(cl(B)) = cl(B).

Definition 4. Let (W, cl) be a pregeometry.

(1) For X ⊆ W, we say that X is closed if X = cl(X);

(2) I ⊆ W is independent if for every a ∈ I, we have a ∉ cl(I − {a});

(3) We say that I ⊆ A generates A, if cl(I) = cl(A);

(4) A basis for a set A ⊂ W is an independent set I generating cl(A);

(5) For X ⊆ W, the dimension of X, written dim(X), is the cardinality of a basis for cl(X).

Fact 5. Using the axioms of pregeometry, one can show that for every set, bases exist and that the dimension is well-defined see for example Appendix in [Gr]

Definition 6. Let G = (W, cl) be a pregeometry.

(1) A bijection f: W → W is an automorphism of G if for every a ∈ W and A ⊆ W we have

a ∈ cl(A) if and only if f(a) ∈ cl(f[A]).

We denote Aut_{A}(G) the set of automorphisms of G fixing A pointwise.

(2) We say that G is homogeneous if for every a, b ∈ W and A ⊆ W, such that a ∉ cl(A) and b ∉ cl(A) there is an automorphism of G, fixing A pointwise and taking a to b.

Remark 7. While most of geometric model theory requires that the geometries are homogeneous, in this paper homogeneity is not assumed.

3. Dependence in Pregeometries

In this section, we introduce the main concept of the paper.

Definition 8. Let (W, cl) be a pregeometry. Let A, B and C be subsets of W. We say that A depends on C over B, if there exist a ∈ A and a finite A′ ⊆ A (possibly empty) such that

a ∈ cl(B ∪ C ∪ A′) − cl(B ∪ A′).

If A depends on C over B, we write A / ⌣ ; B C;

If A does not depend on C over B, we write A ⌣ B C.

Remark 9. An alternative definition with A′ = ∅ does not permit a smooth extension to sets A ⌣ B C when A is not a singleton.

Remark 10. A ⌣ B C if and only if A ∪ B ⌣ B C ∪ B. Hence, we will often assume that

B ⊆ A ∩ C.

We now prove that the properties of forking in simple theories hold with this formalism, directly from the axioms of a pregeometry.

Proposition 11 (Finite Character). Let (W, cl) be a pregeometry. Let A, B and C be subsets of W. Then

A ⌣ B C if and only if A′ ⌣ B C′,

for every finite A′ ⊆ A and finite C′ ⊆ C.

Proof. If A / ⌣ ; B C, then there exist a ∈ A, and a finite A′ ⊆ A such that

a ∈ cl(B ∪ C ∪ A′) − cl(B ∪ A′).

By Finite Character, there exist a finite C′ ⊆ C such that a ∈ cl(B ∪ C′ ∪ A′). Hence A′ / ⌣ ; B C′, by definition.

For the converse, if there exist a finite A′ ⊆ A and a finite C′ ⊆ C such that A′ / ⌣ ; B C′, then we can find a ∈ A′ and A′′ ⊆ A′ such that

a ∈ cl(B ∪ C′ ∪ A′′) − cl(B ∪ A′′).

Since C′ ⊆ C, we have a ∈ cl(B ∪ C ∪ A′′), by Fact 2. Hence, A / ⌣ ; B C, by definition.

Proposition 12 (Continuity). Let (W, cl) be a pregeometry. Let ⟨C_{i} | i < α⟩ be a continuous increasing sequence of sets in W, and A, B ⊆ W.

(1) If A ⌣ B C_{i} for every i < α, then A ⌣ B ⋃_{i<α} C_{i}.

(2) If C_{i} ⌣ B A for every i < α, then ⋃_{i<α} C_{i} ⌣ B A.

Proof. By the finite character.

Proposition 13 (Invariance). Let G = (W, cl) be a pregeometry. Let A, B and C be subsets of W and let f ∈ Aut(G). Then

A ⌣ B C if and only if f(A) {⌣ \atop f[B]} f(C).

Proof. Immediate from the definitions.

Proposition 14 (Monotonicity). Let (W, cl) be a pregeometry. Let A, B and C be subsets of W. Suppose A ⌣ B C.

(1) If A′ ⊆ A and C′ ⊆ C, then A′ ⌣ B C′;

(2) If B′ ⊆ C, then A {⌣ \atop B ∪ B′} C.

Proof. (1) Suppose that A′ / ⌣ ; B C′. Let a ∈ A′ and A^{∗} ⊆ A′ finite such that

a ∈ cl(B ∪ C′ ∪ A^{∗}) − cl(B ∪ A^{∗}).

Then, by Fact 2, we have a ∈ cl(B ∪ C ∪ A^{∗}) − cl(B ∪ A^{∗}). But a ∈ A and A^{∗} ⊆ A, so

A / ⌣ ; B C.

(2) Suppose A / ⌣ ; B ∪ B′ C. Let a ∈ A and A′ ⊆ A finite such that
a ∈ cl(B ∪ B′ ∪ C ∪ A′) − cl(B ∪ B′ ∪ A′).

Since B′ ⊆ C, we have cl(B ∪ B′ ∪ C ∪ A′) = cl(B ∪ C ∪ A′). Also, cl(B ∪ A′) ⊆ cl(B ∪ B′ ∪ A′). Hence a ∈ cl(B ∪ C ∪ A′) − cl(B ∪ A′). Therefore A / ⌣ ; B C.

The role of A′ becomes clear in the next proof.

Proposition 15 (Symmetry). Let (W, cl) be a pregeometry. Let A, B and C be subsets of W. Then

A ⌣ B C if and only if C ⌣ B A.

Proof. Suppose that A / ⌣ ; B C. Choose a ∈ A and a finite A′ ⊆ A such that

a ∈ cl(B ∪ C ∪ A′) − cl(B ∪ A′). (*)

By Finite Character and (*), there exist c ∈ C and a finite (and possibly empty) C′ ⊆ C such that

a ∈ cl(B ∪ C′ ∪ c ∪ A′) and a ∉ cl(B ∪ C′ ∪ A′). (**)

Therefore, by the Exchange Property, we have

c ∈ cl(B ∪ C′ ∪ A′ ∪ a).

But c ∉ cl(B ∪ C′ ∪ A′), (**). Hence,

c ∈ cl(B ∪ C′ ∪ A′ ∪ a) − cl(B ∪ C′ ∪ A′).

Therefore, C / ⌣ ; B A′, for some finite subset A′ of A. Hence, C / ⌣ ; B A, by Finite Character.

Proposition 16 (Transitivity). Let (W, cl) be a pregeometry. Let A, B, C and D be subsets of W such that B ⊆ C ⊆ D. Then,

A ⌣ C D and A ⌣ B C if and only if A ⌣ B D.

Proof. Suppose first that A / ⌣ ; B D. Choose a ∈ A and a finite A′ ⊆ A such that

a ∈ cl(D ∪ A′) − cl(B ∪ A′).

Either a ∈ cl(C ∪ A′), and so

a ∈ cl(C ∪ A′) − cl(B ∪ A′),

which implies that A / ⌣ ; B C. Or a ∉ cl(C ∪ A′), and therefore

a ∈ cl(D ∪ A′) − cl(C ∪ A′),

which implies that A / ⌣ ; C D.

The converse follows by Monotonicity since B ⊆ C ⊆ D.

The following is proved in [Sh b] directly using the finite equivalence relation theorem. The proof that it follows from the other axioms of forking is due to J. Baldwin. We present it here for completeness.

Proposition 17 (Pairs Lemma). Let G = (W, cl) be a pregeometry. Let A, B, C and D be subsets of W such that C ⊆ B ∩ D. Then

A ∪ B ⌣ C D if and only if A {⌣ \atop C ∪ B} D ∪ B and B ⌣ C D.

Proof. Notice first, that by definition

(*) A {⌣ \atop C ∪ B} D ∪ B if and only if A {⌣ \atop C ∪ B} D.

Therefore, by Symmetry and (*), it is equivalent to show that

D ⌣ C A ∪ B if and only if D {⌣ \atop C ∪ B} A and D ⌣ C B,

which is true by Transitivity.

Remark 18. Let (W, cl) is a pregeometry. Let A, B, C and D be subsets of W. Then

AD ⌣ B C if and only if A ⌣ B CD.

Proof. Suppose A ⌣ B CD. Then, by Monotonicity we have A ⌣ B D. Therefore, by Sym-

metry, we have D ⌣ B A. By Transitivity, we have A {⌣ \atop BD} CD. Hence, AD ⌣ B C by Con-

catenation.

For the converse, suppose that A ⌣ B CD. Then by Symmetry we must have

CD ⌣ B A. Hence, by the first paragraph, we know that C ⌣ B AD, so by Symmetry, also

AD ⌣ B C.

This finishes the list of usual properties of forking. We now prove a few propositions relating closure and ⌣.

Proposition 19 (Closed Set Theorem). Let (W, cl) be a pregeometry. Let A, B and C be subsets of W. Then

A ⌣ B C if and only if A′ {⌣ \atop B′} C′,

provided that cl(A ∪ B) = cl(A′ ∪ B′), cl(B) = cl(B′) and cl(C ∪ B) = cl(C′ ∪ B′).

Proof. It is clearly enough to prove one direction. Furthermore, by Symmetry, it is enough to show that A ⌣ B C implies A {⌣ \atop B′} C′. Suppose that A / ⌣ ; B′ C′. Let a ∈ A and A^{∗} ⊆ A be

such that

a ∈ cl(B′ ∪ C′ ∪ A^{∗}) − cl(B′ ∪ A^{∗}).

But, it follows from the assumption that cl(B′ ∪ C′ ∪ A^{∗}) = cl(B ∪ C ∪ A^{∗}) and cl(B′ ∪ A^{∗}) = cl(B ∪ A^{∗}). Therefore

a ∈ cl(B ∪ C ∪ A^{∗}) − cl(B ∪ A^{∗}),

which implies that A / ⌣ ; B C.

Remark 20. In view of the previous result, when A ⌣ B C, we can first choose a basis B′

of B, and choose A′ ⊆ A and C′ ⊆ C, independent over B (or equivalently B′), such that cl(A∪B) = cl(A′ ∪B) and cl(C ∪B) = cl(C′ ∪B), and thus A′ ⌣ B C′ and also A′ {⌣ \atop B′} C′.

Proposition 21. Let (W, cl) be a pregeometry. Let A, B and C be subsets of W.

A ⌣ B C implies cl(A ∪ B) ∩ cl(C ∪ B) = cl(B).

Proof. Certainly cl(B) ⊆ cl(A ∪ B) ∩ cl(C ∪ B). Suppose that the reverse inclusion does not hold, and let a ∈ cl(A ∪ B) ∩ cl(C ∪ B) such that a ∉ cl(B). Then a ∈ cl(C ∪ B) − cl(B), so cl(A ∪ B) / ⌣ ; B C. But the previous proposition implies that A / ⌣ ; B C,

which is a contradiction.

Remark 22. In view of the definition and symmetry, when we look at A ⌣ B C, we will

generally assume that B ⊆ A and B ⊆ C. Further, because of the closed set theorem, we may assume that A, B and C are closed, and finally, that B = A ∩ C.

4. Buechler’s Theorem

This section is devoted to reproducing a theorem of S. Buechler in general pregeometries. The point is to illustrate the following idea: Any geometric property valid in the first order case which uses only basic properties of forking is in fact valid in every pregeometry, in particular in nonelementary examples.

We list a few more definitions.

Definition 23. Let (W, cl) be a pregeometry.

(1) (W, cl) is called modular if for every closed subsets S_{1} and S_{2} of W we have

dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) = dim(S_{1}) + dim(S_{2});

(2) (W, cl) is called locally modular if for every closed subsets S_{1} and S_{2} of W we have

dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) = dim(S_{1}) + dim(S_{2}),

provided that S_{1} ∩ S_{2} ≠ ∅.

Definition 24. Let (W, cl) be a pregeometry.

(1) A closed set L ⊆ W is a line if dim(L) = 2;

(2) Two disjoint lines L_{1} and L_{2} are parallel if dim(L_{1} ∪ L_{2}) = 3.

Definition 25. Let G = (W, cl) be a pregeometry and A ⊆ W. Define the localization of G at A, written G_{A} = (W_{A}, cl_{A}), by

W_{A} = W − A and cl_{A}(X) = cl(X ∪ A) − A, for X ⊆ W_{A}.

Remark 26. It is easy to see that if G is a pregeometry, then G_{A} is a pregeometry. In G_{A}, we denote the dimension of X by dim(X/A).

Remark 27. If G = (W, cl) is locally modular, then G_{A} is modular for any finite subset A of W − cl(∅).

Proposition 28. Let (W, cl) be a pregeometry. Let S_{1}, S_{2} be finite dimensional closed sets satisfying S_{0} = S_{1} ∩ S_{2}. Then,

S_{1} {⌣ \atop S_{0}} S_{2} if and only if dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) = dim(S_{1}) + dim(S_{2}).

Proof. Suppose first that S_{1} {⌣ \atop S_{0}} S_{2}. Let I be a basis for S_{0}, and let I_{i} ⊇ I be a basis for

S_{i} for i = 1, 2. Clearly, cl(S_{1} ∪ S_{2}) = cl(I_{1} ∪ I_{2}). We claim, in addition, that I_{1} ∪ I_{2} is independent. Otherwise there is a ∈ cl(I_{1} ∪ I_{2} − {a }). Without loss of generality, we may assume that a ∈ I_{1}. Now, since I_{1} is independent, a ∉ cl(I_{1} − {a }), thus

a ∈ cl(I_{1} ∪ I_{2} − {a }) − cl(I_{i} − {a }), for i = 1, 2.

We may also assume that a ∉ I. To see this, assume that a ∈ I. Choose I′_{i} ⊆ I_{i} − I, minimal with respect to inclusion, such that a ∈ cl(I′_{1} ∪I′_{2} ∪I −{a }), I′_{i} ≠ ∅, for i = 1, 2. By the Exchange Property, there is b ∉ I, such that

b ∈ cl(I′_{1} ∪ I′_{2} ∪ I ∪ {b }) ⊆ cl(I_{1} ∪ I_{2} − {b }).

But, if a ∉ I, then cl(I_{1} − {a }) = cl(I ∪ I_{1} − {a }) so

a ∈ cl(I_{2} ∪ (I_{2} − {a})) − cl(I ∪ (I_{2} − {a })),

which means that S_{1} / ⌣ ; S_{0} S_{2}, a contradiction. Hence I_{1} ∪ I_{2} is independent. Therefore

dim(S_{1} ∪ S_{2}) = |I_{1} ∪ I_{2}|. But |I_{1} ∪ I_{2}| + |I| = |I_{1}| + |I_{2}|, so

dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) = dim(S_{1}) + dim(S_{2}).
For the converse, suppose S_{1} / ⌣ ; S_{0} S_{2}. Let a ∈ S_{1} and A_{1} ⊆ S_{1} such that

a ∈ cl(S_{2} ∪ A_{1}) − cl(S_{0} ∪ A_{1}). (*)

Choose a such that A_{1} has minimal cardinality. This implies that A_{1} ∪ {a} is independent over S_{0}, and A_{1} is independent over S_{2}. Thus, we can pick a basis I_{0} for S_{0}, and extend I_{0} ∪ A_{1} ∪ {a} to a basis I_{1} of S_{1}. Now choose I′_{2} disjoint from I_{0}, such that I_{0} ∪ I′_{2} is a basis of S_{2}. But, I_{0} ∪ A_{1} ∪ {a} ∪ I′_{2} is not independent by (*). Hence

dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) < dim(S_{1}) + dim(S_{2}),

which finishes the proof.

In the previous section, we showed that in any pregeometry, there is a relation that satisfies all the properties that forking satisfies in the context of simple theories. This allows us to show a theorem of Buechler [Bu], originally proved for stable theories, when the pregeometry comes from forking.

Theorem 29 (Buechler). Let G = (W, cl) be a pregeometry. Then G is locally modular if and only if G_{A} has no parallel lines for every finite A ⊆ W, such that A ⊈ cl(∅).

Proof. Suppose first that there is a finite A ⊆ W, such that A ⊈ cl(∅) and G_{A} contain parallel lines. Thus, let L_{1} and L_{2} be disjoint lines in G_{A} such that dim(L_{1} ∪ L_{2}/A) = 3. Let L′_{i} = cl(L_{i} ∪ A) for i = 1, 2. Then A ⊆ L′_{1} ∩ L′_{2}, so L′_{1} ∩ L′_{2} ⊈ cl(∅), L′_{i} is closed for i = 1, 2, and

dim(L′_{1} ∪ L′_{2}) + dim(L′_{1} ∩ L′_{2}) ≠ dim(L′_{1}) + dim(L′_{2}).

This shows that G is not locally modular.

For the converse, suppose that G is not locally modular. Then there are closed S_{1} and S_{2} subsets of W such that S_{1} ∩ S_{2} ⊈ cl(∅) and

dim(S_{1} ∪ S_{2}) + dim(S_{1} ∩ S_{2}) ≠ dim(S_{1}) + dim(S_{2}).

We may assume that S_{1} and S_{2} are finite dimensional. Let S_{0} = S_{1} ∩ S_{2}. By Proposition 28, this implies that S_{1} / ⌣ ; S_{0} S_{2}.

Let D be the set of pairs of integers ⟨d_{1}, d_{2}⟩ such that there are closed sets S_{1} and S_{2} such that

• S_{0} = S_{1} ∩ S_{2} and S_{0} ⊈ cl(∅);

• d_{1} = dim(S_{1}/S_{0}) and d_{2} = dim(S_{2}/S_{0});

• S_{1} / ⌣ ; S_{0} S_{2}.

By assumption D ≠ ∅. Choose ⟨d_{1}, d_{2}⟩ minimal with respect to the lexicographic order. We claim that ⟨d_{1}, d_{2}⟩ = ⟨2, 2⟩. Note that this is enough to prove the theorem since cl_{S_{0}}(S_{1} − S_{0}) and cl_{S_{0}}(S_{2} − S_{0}) are parallel lines in G_{S_{0}}.

Certainly, d_{1} > 1. Otherwise, dim(S_{1}/S_{0}) = 1 and since S_{1} / ⌣ ; S_{0} S_{1} there must

exist a ∈ S_{1} − S_{0} such that a ∈ cl(S_{2}) − cl(S_{0}). Since S_{2} and S_{0} are closed, we have a ∈ S_{1} ∩ S_{2} − S_{0}, a contradiction, since S_{1} ∩ S_{2} = S_{0}.

We now show that d_{1} < 3. Suppose d_{1} = dim(S_{1}/S_{0}) ≥ 3. We will show that this contradicts the minimality of d_{1}. We first show that

(*) S_{1} ∩ cl(S_{2}a) = cl(S_{0}a), for any a ∈ S_{1} − S_{0}.

First, notice that S_{0}a ⊆ S_{1} and S_{0}a ⊆ cl(S_{2}a), so

S_{1} ∩ cl(S_{2}a) ⊇ cl(S_{0}a), for any a ∈ S_{1} − S_{0}.

Hence, if (*) does not hold, it is because for some a ∈ S_{1} − S_{0}, there exists

b ∈ (S_{1} ∩ cl(S_{2}a)) − cl(S_{0}a).

By definition, this implies that { a, b } / ⌣ ; S_{0} S_{2}.

Let S′_{1} = cl(S_{0}ab). Then S′_{1} ∩ S_{2} = S_{0} and S_{0} ⊈ cl(∅). Furthermore S′_{1} / ⌣ ; S_{0} S_{2}.

But dim(S_{2}/S_{0}) = d_{2} and dim(S′_{1}/S_{0}) = 2 < 3 ≤ d_{1}, which contradicts the minimality of d_{1}. Therefore, (*) holds.

Now, since S_{1} / ⌣ ; S_{0} S_{2}, there exist a ∈ S_{1} and a finite A ⊆ S_{1} such that

a ∈ cl(S_{2} ∪ A) − cl(S_{0} ∪ A). (**)

But A ⊈ S_{0}. Otherwise, by (**) we have a ∈ cl(S_{2})−cl(S_{0}). This shows that a ∈ S_{2}−S_{1} since S_{2} and S_{0} are closed. But a ∈ S_{1}, so a ∈ (S_{1} ∩ S_{2}) − S_{0} = ∅, which is impossible. Hence, there is b ∈ A − S_{0}. Then, since Ab = A, we have

a ∈ cl(S_{2} ∪ A) − cl(S_{0}b ∪ A).

Hence S_{1} / ⌣ ; S_{0} ∪ b S_{2}.

Now consider S′_{2} := cl(S_{2}b). Then, S_{1} / ⌣ ; S_{0} ∪ b S_{2} implies that S_{1} / ⌣ ; S_{0} ∪ b S′_{2}. By

(*) we have S_{1} ∩ S′_{2} = cl(S_{0}b). Finally, dim(S_{1}/(S_{0}b)) < dim(S_{1}/S_{0}) = d_{1} and d_{2} = dim(S_{2}/S_{0}) = dim(S′_{2}/S_{0}b). This contradicts the minimality of d_{1}. We prove similarly that d_{2} = 2, which finishes the proof.

5. Some “set theory”

In this section, we gather several observations with a set-theoretic flavor. The next theorem is a generalization of a lemma from J. Baumgartner, M. Foreman and O. Spinas [BFS]. Although the proof is easy, it does not follow from the analog theorem involving models as we do not have control over the cardinality of the closures. The value of this theorem is that it makes it possible to attach a club as an invariant of the pregeometry.

Theorem 30. Let G = (W, cl) be a pregeometry. Suppose dim(W) = λ is regular and uncountable. Let I = { a_{i} | i < λ } and J = { b_{i} | i < λ } be bases of W. Then

C = { i < λ : cl({ a_{j} | j < i }) = cl({ b_{j} | j < i }) }

is a closed and unbounded subset of λ.

Proof. We first show that C is closed. Let δ = sup(δ ∩ C). Then, for any i < δ there is i_{1} ∈ C such that i < i_{1} < δ. Hence, by definition of C

cl({ a_{j} | j < i_{1} }) = cl({ b_{j} | j < i_{1} }). (*)

Lemma 4 and (*) implies that a_{i} ∈ cl({ b_{j} | j < δ }). Hence,

{ a_{j} | j < δ } ⊆ cl({ b_{j} | j < δ }),

and therefore

cl({ a_{j} | j < δ }) ⊆ cl({ b_{j} | j < δ }),

by Fact 2 again. The other inclusion is similar and so

cl({ a_{j} | j < δ }) ⊇ cl({ b_{j} | j < δ }).

This shows that δ ∈ C, by definition of C.

We now show that C is unbounded in λ. Let i < λ be given. We construct i_{n} < λ for n ∈ ω increasing with i_{0} = i such that

(1) cl({ a_{j} | j < i_{n} }) ⊆ cl({ b_{j} | j < i_{n+1} }) if n is even;

(2) cl({ b_{j} | j < i_{n} }) ⊆ cl({ a_{j} | j < i_{n+1} }) if n is odd.

This is enough: Let i(∗) = sup{ i_{n} | n ∈ ω }. Then i(∗) < λ since λ is regular uncountable. Further cl({ a_{j} | j < i(∗) }) = cl({ b_{j} | j < i(∗) }), since if i < i(∗), then there is i_{n} with n even such that i < i_{n}, so

a_{i} ∈ cl({ a_{j} | j < i_{n} }) ⊆ cl({ b_{j} | j < i_{n+1} }) ⊆ cl({ b_{j} | j < i(∗) }),

hence

cl({ a_{j} | j < i(∗) }) ⊆ cl({ b_{j} | j < i(∗) }).

The other inclusion is proved similarly. Thus i < i(∗) ∈ C, which shows that C is unbounded.

This is possible: Given i < λ, we let i_{0} = i. Assume that i_{n} < λ has been constructed. Suppose n is even. For each j < i_{n}, we have that a_{j} ∈ W = cl({ b_{j} | j < λ }) since J is a basis. By Finite Character, there is a finite S_{j} ⊆ λ such that a_{j} ∈ cl({ b_{k} | k ∈ S_{j} }). Let k_{j} = sup S_{j} < λ, so a_{j} ∈ cl({ b_{l} | l ≤ k_{j} }), and by increasing k_{j} if necessary, we may assume that k_{j} ≥ i_{n}. Set i_{n+1} = sup{ k_{j}+1 | j < i_{n} }. Then i_{n+1} < λ since λ is regular and satisfies our requirement. The case when n is odd is handled similarly.

Proposition 31 (Downward Theorem). Let G = (W, cl) be a pregeometry. Let A, B and C be subsets of W. Suppose A ⌣ B C and A′ is a subset of A, of cardinality at most λ, for

λ an infinite cardinal. Then there is B′ ⊆ B of cardinality at most λ such that A′ {⌣ \atop B′} C.

Proof. Let A′ ⊆ A of cardinality λ be given. Let { ⟨a_{i}, A_{i}⟩ | i < λ } be an enumeration of all the pairs such that a_{i} ∈ A′ and A_{i} ⊆ A′ is finite. Such an enumeration is possible since λ is infinite. Since A ⌣ C B, necessarily

(*) a_{i} ∉ cl(B ∪ C ∪ A_{i}) − cl(B ∪ A_{i}), for every i < λ.

Hence, either a_{i} ∉ cl(B ∪ C ∪ A_{i}), or a_{i} ∈ cl(B ∪ A_{i}). If the latter holds, by Finite Character, we can find a finite B_{i} ⊆ B such that a_{i} ∈ cl(B_{i} ∪ A_{i}). We let B_{i} = ∅, if a_{i} ∉ cl(B ∪ A_{i}). Let B′ = ⋃ B_{i}. Then B′ ⊆ B, and |B′| ≤ λ.

We claim that A′ {⌣ \atop B′} C. Otherwise, there exist a ∈ A′ and a finite A^{∗} ⊆ A′, such

that

a ∈ cl(B′ ∪ C ∪ A^{∗}) − cl(B′ ∪ A^{∗}). (**)

Choose i < λ such that a = a_{i} and A^{∗} = A_{i}. Thus, a_{i} ∈ cl(B′ ∪ C ∪ A_{i}), and so by Fact 2 we have a_{i} ∈ cl(B ∪ C ∪ A_{i}). Therefore, by (*) we have that a_{i} ∈ cl(B ∪ A_{i}). Hence a_{i} ∈ cl(B_{i} ∪ A_{i}) by construction. But B_{i} ⊆ B′, and so a_{i} ∈ cl(B′ ∪ A_{i}) by Fact 2. This contradicts (**) since A^{∗} = A_{i}.

Corollary 32. Let G = (W, cl) be a pregeometry. Let A, B and C be subsets of W. Suppose that A, B and C have cardinality at least λ for some λ infinite. If A ⌣ C B, then

we can find A′ ⊆ A, B′ ⊆ B and C′ ⊆ C of cardinality λ, such that A′ {⌣ \atop C′} B′.

Proof. By the previous theorem using monotonicity.

Proposition 33 (Ultraproducts of Pregeometries). Let I be a set and 𝔇 an ℵ_{1}-complete ultrafilter on I. Suppose that (W_{i}, cl_{i}) is a pregeometry for each i ∈ I. Consider W = Π_{i∈I}W_{i} and for a ∈ W and B ⊆ W, define

a ∈ cl(B) if { i ∈ I | a(i) ∈ cl_{i}(B(i)) } ∈ 𝔇.

Then (W, cl) is a pregeometry.

Proof. We only show Finite Character, since all the other axioms of a pregeometry are routine. Suppose a ∈ cl(B). Then J = { i ∈ I | a(i) ∈ cl_{i}(B(i)) } ∈ 𝔇, and by Finite Character of cl_{i}, for each i ∈ J, there is a finite B′(i) ⊆ B(i), such that a(i) ∈ cl_{i}(B′(i)). Let J_{n} = { i ∈ J | B′(i) has n elements}. Then

{ i ∈ J | a(i) ∈ cl_{i}(B′(i)) } = ⋃_{n<ω} J_{n}.

Hence, by ℵ_{1}-completeness, there exist n < ω such that J_{n} ∈ 𝔇. We now write B′(i) = { b_{1}^{i}, …, b_{n}^{i} } for i ∈ J_{n}. Let A = { f_{1}, …, f_{n} } ⊆ B be given by f_{k}(i) = b_{k}^{i} when i ∈ J_{n} and f_{k}(i) ∈ B(i) arbitrary when i ∉ J_{n}. Then

{ i ∈ I | a(i) ∈ cl_{i}(A(i)) } ⊇ J_{n} ∈ 𝔇,

by construction. Hence { i ∈ I | a(i) ∈ cl_{i}(A(i)) } ∈ 𝔇. Thus, a ∈ cl(A) and A is a finite subset of B, which is what we needed.

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16 RAMI GROSSBERG AND OLIVIER LESSMANN

E-mail address, Rami Grossberg: rami@cmu.edu E-mail address, Olivier Lessmann: lessmann@cmu.edu Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh,

PA 15213