Geometric Simplicity Theory Ivan Tomašić

PhD Thesis University of Edinburgh

Contents

Abstract v

Declaration of authenticity vii

Acknowledgements ix

Introduction 1

Chapter 1. Preliminaries 5 1.1. Conventions and notation 5 1.2. Simplicity 5 1.3. Combinatorial geometries 12 1.4. Polygroups 14

Chapter 2. Germs and group configuration 17 2.1. Partial generic multiactions 17 2.2. Completion and reduction 19 2.3. Composing germs 20 2.4. Group configuration 22

Chapter 3. Almost hyperimaginaries 25 3.1. Ultraimaginaries and almost-hyperimaginaries 26 3.2. Dividing 27 3.3. Almost hyperdefinable polygroups 28 3.4. The core relation 33 3.5. Group chunk theorem 34 3.6. Stabilizers 38

Chapter 4. Blowup 41 4.1. Hyperdefinable blowup 42 4.2. The ‘classical’ blowup construction 44 4.3. Universal blowup 48

Chapter 5. Applications and geometric simplicity 51 5.1. Getting a group from a polygroup 51 5.2. Reconstructing the action 52 5.3. Group configuration in ω-categorical theories 54 5.4. One-based case 56 5.5. Pseudolinearity implies linearity 57

Bibliography 61

Abstract

We prove the group configuration theorem in simple theories, a very abstract result reconstructing a group (action) from a certain independence-theoretic configuration of points, and argue that such a result gives rise to ‘geometric simplicity theory’ (i.e. analogues of methods and results of geometric stability theory).

The proof involves studying the behaviour of multivalued algebraic structures like polygroups and polyspaces, a development of the theory of independence for almost hyperimaginaries, and a sophisticated blowup procedure.

Some of the corollaries of the group configuration theorem we obtain include finding the group associated to a polygroup in a simple theory, interpreting a vector space over a finite field inside a one-based ω-categorical theory of SU-rank 1, and showing how pseudolinearity implies one-basedness under the assumption of ω-categoricity.

Declaration of authenticity

This thesis has been composed by myself.

Important parts of the thesis come from my collaboration with Itay Ben-Yaacov and Frank O. Wagner on [BTW] and some from my collaboration with just Frank O. Wagner ([TW]). My contribution has been clearly indicated throughout the chapters. Moreover, since [BTW] will also form part of Ben-Yaacov’s forthcoming PhD thesis at Université Paris 7, I made an effort to emphasize his substantial contribution.

Ivan Tomašić

Acknowledgements

I am grateful to my supervisor, Professor Angus J. Macintyre for sharing his deep mathematical insight in our discussions, as well as making it possible for me to meet mathematicians from all over the world.

I would like to thank Professor Rami Grossberg for teaching me the basic concepts of model theory and Professor Anand Pillay for frequent discussions concerning simple theories. Special thanks to Itay Ben-Yaacov, whose numerous contributions made our collaboration most fruitful.

My deepest words of gratitude go to Professor Frank O. Wagner who has supported and encouraged me from my first steps in the direction of simple theories, sharing patiently his vast knowledge of the subject. Without him, this thesis would not have been possible.

Finally, I would like to acknowledge the financial support from the ORS Awards Scheme and the Faculty of Science and Engineering of the University of Edinburgh.

Introduction

In the beginning, there was Morley’s theorem from the 1960’s, stating that if a complete countable first order theory T is categorical in an uncountable cardinality, it must in fact be categorical in all uncountable cardinalities.

After that came Shelah’s monumental programme ([S]), concerned with classifying theories in terms of structure/non-structure dichotomies, where a structure theorem states that a theory has relatively few models and thus it makes sense to try and classify its models (Morley’s theorem being the first example of such a theorem), while a non-structure theorem asserts that a theory has a maximal number of models in a given cardinality, their classification thus being hopeless.

On the other hand, Zil’ber developed in the 1970’s a variety of methods and results which became known as geometric stability theory, oriented more towards the qualitative classification of theories. After showing many results concerning ω_{1}-categorical theories, he conjectured that if M is a model of such a theory, then M must essentially be either a vector space, or an algebraically closed field, or a degenerate structure (see [Z1]–[Z6]). We will refer to this conjecture as Zil’ber’s trichotomy. The methods consisted of studying the combinatorial pregeometries arising on strongly minimal sets inside a given structure, and the relationship between the properties (e.g. (non) local modularity, triviality, etc.) of those pregeometries and the structure as a whole. Interpretability of nice algebraic structures (e.g. groups or fields) inside a structure had significant consequences.

In 1980’s, Hrushovski brought geometric stability theory to its peak ([H0]–[H8]). He showed in particular that Zil’ber’s conjecture does not hold in full generality ([H5]), and then later with Zil’ber that it does hold in Zariski geometries [HZ]. Also, his work on the Mordell-Lang and Manin-Mumford conjectures ([H6], [H8]) clearly demonstrated that ‘excursions’ outside the classical model theory of stable structures can answer fundamental questions about the classical structures themselves. Some of the theories he considered we now know to be simple.

Simple theories were defined by Shelah in [S93], but only became a subject of intense investigation after Kim managed to prove symmetry and transitivity of forking in [Ki] and especially after Kim and Pillay’s proof of the Independence Theorem. The program was apparent: to generalise well-understood methods of stability in this new and mysterious context. With more or less difficulty, the usual techniques of forking calculus, canonical bases, orthogonality, regularity, coordinatisation, type-definable groups and Hrushovski’s amalgamation construction were translated into simplicity. There were extremely difficult problems intrinsic to simple theories which were also resolved, e.g. elimination of hyperimaginaries for supersimple theories ([BPW]).

However, for a long time, there was no results reminiscent of geometric stability theory. In particular, there was no group existence theorems. And yet, it was apparent there should be, since at least in all the known examples, speaking loosely, we either have a stable ‘reduct’ controlling forking (e.g. random graph, vector space over a finite field with a bilinear form, algebraically closed fields with an automorphism), or the structure can be ‘embedded’ into a stable one (pseudofinite fields), so the ‘geometry’ should come from these stable structures.

The first attempts on the group configuration theorem (for stable theories first recognised by Zil’ber and then in full generality by Hrushovski) in simple theories by Ben-Yaacov [BY] and myself [To] yielded some partial results and indicated the way to proceed. The collaboration of Ben-Yaacov, Wagner and myself gave a satisfactory solution to the problem in [BTW]. Furthermore, there were several different lines of attack on the binding group theorem by Hart, Shami and Wagner. It seemed for a while that it was just a matter of guessing the right definition of the group of automorphisms, but then Wagner reduced the problem to the group configuration ([W2]), showing that it is indeed the only group existence theorem around. Since then there were other developments in the direction of geometric simplicity, see [Va] and [dPK]. My purpose in this thesis is to present the proof of the group configuration theorem for simple theories, as well as its application in developing geometric simplicity theory, with a view to Zil’ber-type trichotomy for simple theories.

In Chapter 1, we deal with prerequisites needed to develop further chapters, to make the exposition as self-contained as possible. Thus, on one hand, we discuss simple theories, independence relations and combinatorial pregeometries arising within, and on the other, we define and study basic properties of polygroups, because even though they appear quite naturally, we cannot really consider them to be standard mathematical objects.

In Chapter 2, we develop the theory of germs of type-definable generic partial multiactions needed for the proof of the group configuration theorem. In the stable case, germs can be developed quickly and efficiently, as shown in the introduction to the chapter. In the simple case, however, the situation is very intricate so the whole chapter is dedicated to it, based on [BY]. It is shown how a polygroup chunk arises from the usual algebraic quadrangle (group configuration), thus completing the first step towards constructing a group. In the stable case, at this stage a group chunk would be derived, and a direct application of the Hrushovski-Weil style group chunk theorem would yield a group. A polygroup chunk proves to be much more troublesome.

To overcome the difficulties with the polygroup chunk, in Chapter 4, we present a variety of blowup procedures which give a group chunk, where the group chunk theorem will apply and provide us with the sought-after group. The construction is of algebraic-geometric nature (hence the name) and has a surprising similarity to the reconstruction of the division ring from a projective geometry (maybe not so surprising if we remind ourselves that the group configuration is a highly abstract analogue of the very same classical construction). Three variants of the construction due to myself are given (it should be noted here that the idea by Ben-Yaacov of using the core relation in a ‘blowup-like’ context resulted in the first successful blowup construction): the first is as hyperdefinable as possible, the second is written to look like von Neumann’s proof from [vN], and the third to be as universal as possible–in fact, to have the usual universal property of blowing-up from algebraic geometry.

Unfortunately, the construction of Chapter 4 works only up to a certain invariant relation, thus giving what we call a gradedly almost hyperdefinable, and not necessarily hyperdefinable group chunk, so in Chapter 3 we develop the machinery (our presentation modifies slightly Ben-Yaacov’s notes for the purpose of [BTW]) needed to handle almost hyperimaginary elements, gradedly almost hyperdefinable polygroups and polygroup chunks. In particular, gradedly almost hyperdefinable polygroups are defined, as well as their generic elements, whose existence and basic properties are shown (for hyperdefinable polygroups in supersimple theories this was originally done in [To1]). The group (space) chunk theorem is proved in this category as well, based on ideas of [To], Ben-Yaacov noting it preserves almost hyperdefinability.

In Chapter 5 we give a few applications of the group configuration framework. Firstly, we show how each gradedly almost hyperdefinable polygroup has a group closely related to it, solving a fundamental problem of classical hypergroup theory. There are two approaches to the problem, the first by Wagner and myself, the second by Ben-Yaacov. We then demonstrate that no problems with almost definability arise in an ω-categorical theory where the group configuration gives an interpretable group (action). This result is due to myself. Then, we find a vector space interpreted in an one-based SU-rank 1 simple theory and argue that this partially solves the well-known stable forking hypothesis, and also leads towards a Zil’ber trichotomy-type result for simple theories. And, finally, we reconstruct the original group action and prove that pseudolinearity implies one-basedness under the assumption of ω-categoricity. These last three applications are due to Wagner and myself.

CHAPTER 1 Preliminaries

This chapter is devoted to fixing notation and stating the facts necessary for the rest of the thesis. Even though it is impossible to give a brief outline of the theory of simplicity without repeating e.g. the entire thesis of Kim ([Ki]) and quite a few research papers, or the first several chapters of Wagner’s book ([W]), in Section 1.2 we attempt to at least set out the facts we will need, assuming the reader is somewhat familiar with basic first order model theory and stability. In Section 1.3 we study combinatorial (pre)geometries which are important for developing geometric stability (and later simplicity) theory. In Section 1.4 we expound the definitions and basic notions of the classical theory of polygroups, since most mathematicians are not likely to have encountered it.

1.1. Conventions and notation

Let T be a first order theory with infinite models in a language L, ℭ the corresponding monster model (highly saturated and strongly homogeneous model of the theory) and ℭ^{eq} the associated enrichment by imaginaries.

Arbitrary small (we do not wish to be precise about the meaning of ‘small’ which is after all a common practice in stability theory, mostly it will mean ‘of cardinality less than |ℭ|’) tuples of elements of ℭ^{eq} will usually be denoted by a, b…, and ‘small’ subsets of ℭ^{eq} will be denoted by A, B… The concatenation of tuples a and b is usually written as ab (or occasionally aˆ b), and the union of sets A and B is sometimes written simply as AB. The group of automorphisms of ℭ fixing a set A is denoted by Aut_{A}(ℭ). We write a ≡_{A} b to express that tp(a/A) = tp(b/A). Our assumptions on ℭ allow us to use principles of the kind: a ≡_{A} b if and only if there is an automorphism of ℭ^{eq} fixing A and taking a to b. By dcl(A) we denote the set of all elements of ℭ^{eq} which are fixed by any automorphism fixing A, and acl(A) is used for the set of elements of ℭ^{eq} which have only finitely many conjugates by automorphisms fixing A.

Furthermore, we write A ≈ B for A ∩ B ≠ ∅, which becomes extremely useful when dealing with multivalued algebraic objects. If R is an equivalence relation, we write A ≈_{R} B if A/R ∩ B/R ≠ ∅, and A =_{R} B if A/R = B/R as sets. The partitive set of A is denoted by ℙ(A), and the nonempty subsets of A by ℙ^{∗}(A).

1.2. Simplicity

I leave to several futures (not to all) my garden of forking paths. JORGE LUIS BORGES, The Garden of Forking Paths

The concept of forking was introduced by Shelah in early 1970s, and a smooth theory was developed in a stable setting in [S], showing that it gives a good notion of independence in the sense that if tp(a/B) does not fork over A (⊆ B), then a satisfies no more dependency relations with elements of B than with elements of A. In 1977, Lascar and Poizat gave an alternative approach to forking [LPo] which replaced Shelah’s ‘combinatorial’ definition, and resulted in popularization of forking, followed by books by Pillay [P1], Lascar [L4], and papers by Harnik and Harrington [HH] and Makkai [Mk].

Then, in [S93], Shelah introduced a new class of theories generalizing stable theories called simple, and noticed that the theory of forking might behave well in this context, but he didn’t prove all the good properties of forking and was forced to introduce the concept of weak dividing which allowed him to solve a combinatorial problem he considered in that paper. However, work of Hrushovski in certain algebraic examples of simple theories [H3], Hrushovski and Pillay [HP1], [HP2] and Hrushovski and Chatzidakis [ChH], showed that there is indeed a good notion of independence in all cases, and an important result called the Independence Property was isolated.

The greatest contribution to the theory of forking in simple context came with work of Kim and Pillay, when first Kim proved the symmetry of dividing in presence of simplicity in [Ki], and then Kim and Pillay were able to prove all the relevant good properties, including the above mentioned Independence Theorem, see [KP], [KP1], [Ki1]. Some of the important problems that have been investigated are the equivalence of the notions of Lascar strong type and strong type, connected to the problem of eliminating hyperimaginaries ([PP], [BPW], [LP], [P5]), and the existence of canonical bases ([HKP]) in ℭ^{eq}.

A natural question, motivated by the lack of examples, posed by Hart and others, was whether forking in simple theories (under certain reasonable assumptions like elimination of hyperimaginaries) is in some way represented or determined by stable formulae; this became known as the stable forking hypothesis. Needless to say, all the known simple theories satisfy this in one way or another. Kim an Pillay offer a solution to an interesting approximation to the problem, as well as a better understanding of canonical bases in the supersimple case, see [KP2]. Also, we shall prove in Chapter 5 that certain structures have stable reducts ‘preserving’ forking, thus showing a very strong form of stable forking.

We start with Shelah’s original definition of forking in terms of dividing in an arbitrary theory.

Definition 1.2.1. · A formula φ(x, b) divides over A, if there exist {b_{i}|i < ω} and k < ω such that for every i < ω, tp(b_{i}/A) = tp(b/A) and the set {φ(x, b_{i})|i < ω} is k-contradictory. · A (partial) type p divides over A if there is a formula φ(x, b) with p ⊢ φ(x, b) and φ(x, b) divides over A. · A formula φ(x, b) forks over A if there are n < ω and {φ_{i}(x, b^{i})|i < n} such that φ(x, b) ⊢ ⋁_{i<n} φ_{i}(x, b^{i}) and for every i < n, φ_{i}(x, b^{i}) divides over A. · A (partial) type p forks over A if there exists a formula φ(x, b) such that p ⊢ φ(x, b) and φ(x, b) forks over A.

Definition 1.2.2. A theory T has the tree property, if there is a formula φ(x, y) with the tree property, i. e., there are k < ω and {a_{η}|η ∈ ^{ω>}ω} such that

· for every η ∈ ^{ω}ω, the set {φ(x, a_{η↾l})|l < ω} is consistent;

· for every η ∈ ^{ω>}ω, the set {φ(x, a_{ηˆ n})|n < ω} is k-contradictory.

Shelah originally defined a theory to be simple, if it did not have the tree property. Let us write A | ⌣_{C}^{f} B (to be read ‘A is (forking-)independent from B over C’) if

for every finite a ∈ A, tp(a/BC) does not fork over C. Kim has shown that in theories without the tree property, forking coincides with dividing and it is possible to prove properties listed in the definition below for this notion of independence. There are many beautiful accounts of this, see [Ki], [KP], [KP1], [P4], or [W]. We shall, however, adopt an axiomatic approach to simple theories.

Definition 1.2.3. A theory is simple if it has an independence relation, i.e. a ternary relation | ⌣ on subsets of the monster model satisfying:

[Invariance] | ⌣ is invariant under automorphisms of ℭ;
[Finite Character]  A | ⌣_{E} B if and only if for all finite a ∈ A, b ∈ B, a | ⌣_{E} b;
[Transitivity]  A | ⌣_{E} BC if and only if A | ⌣_{E} B and A | ⌣_{EB} C;
[Symmetry]  A | ⌣_{E} B if and only if B | ⌣_{E} A;
[Extension]  for all A, B, E, there is A′ with A′ ≡_{E} A and A′ | ⌣_{E} B;

[Local Character] for every finite a and all B, there is E ⊆ B with |E| ≤ |T |

such that a | ⌣_{E} B.

[Independence Theorem] if M is a model, a_{0} ≡_{M} a_{1}, a_{i} | ⌣_{M} b_{i} for i < 2 and b_{0} | ⌣_{M} b_{1},

there is a | ⌣_{M} b_{1}b_{2} with a ≡_{Mb_{0}} a_{0} and a ≡_{Mb_{1}} a_{1}.

A theory is supersimple, if in ‘Local Character’ above, E can in fact be found finite. The property replacing the Independence Theorem in the characterization of stable theories is Stationarity: for a model M and a set B, if a_{0} ≡_{M} a_{1} and a_{0} | ⌣_{M} B, a_{1} | ⌣_{M} B, then a_{0} ≡_{MB} a_{1}.

A justification for such a definition can be found in [KP] (or [W], 2.6.1):

Theorem 1.2.4. Let T be simple in the sense of 1.2.3. Then T does not have the tree property and | ⌣ = | ⌣^{f}.

One of the particularly useful properties of independence which can be derived from the others, is invariance under algebraic closure, i.e. A | ⌣_{C} B if and only if acl(AC) | ⌣_{acl(C)} acl(BC).

Definition 1.2.5. An infinite sequence ⟨a_{i} : i < α⟩ is indiscernible over a set A (sometimes called A-indiscernible), if for every n < ω, for every i_{0} < ⋯ < i_{n−1} < α and j_{0} < ⋯ < j_{n−1} < α, tp(a_{i_{0}}, …, a_{i_{n−1}}/A) = tp(a_{j_{0}}, …, a_{j_{n−1}}/A).

Indiscernible sequences of arbitrary length can be found (and the existing ones can be made longer) inside our model by the well-known combinatorial argument using Ramsey’s theorem, giving this subject a combinatorial/set theoretic flavour. The following lemma, however, a complete proof of which can be found in [GIL], can be used in producing indiscernible sequences with special properties:

Lemma 1.2.6. For every set A and sequence ⟨a_{i} : i < ℶ_{(2^{(|T |+|A|+λ)})^{+}}⟩ of tuples of length λ, there exists an A-indiscernible sequence ⟨b_{n} : n < ω⟩ such that for every n < ω there are i_{0} < ⋯ < i_{n−1} with tp(b_{0}, …, b_{n−1}/A) = tp(a_{i_{0}}, …, a_{i_{n−1}}/A).

Definition 1.2.7. We say that a set X is independent over A, if for every x ∈ X, x | ⌣_{A} X − {x}.

Using the properties of independence, it is easy to see that a sequence ⟨a_{i} : i < α⟩ is A-independent if and only if for each i < α, a_{i} | ⌣_{A}⟨a_{j} : j < i⟩.

Definition 1.2.8. A sequence ⟨a_{i} : i < α⟩ is a Morley sequence over A if it is both A-indiscernible and A-independent.

The existence of Morley sequences is shown by refining a long independent sequence (which exists by Extension) using 1.2.6. These sequences play a crucial role in Kim’s arguments.

Definition 1.2.9. Let A be a set. The group of Lascar strong automorphisms of ℭ over A is the subgroup of Aut_{A}(ℭ) generated by all automorphisms fixing some model M ⊇ A, usually denoted LAut_{A}(ℭ). Two tuples a and b have the same Lascar strong type over A, denoted a ≡_{A}^{L} b, if they are conjugate by a Lascar strong automorphism over A. The equality of Lascar strong types over A is clearly an equivalence relation, the equivalence class of a denoted by lstp(a/A).

It can be shown that LAut_{A}(ℭ) is a normal subgroup of Aut_{A}(ℭ), and that equality of Lascar strong types over A is the finest bounded A-invariant equivalence relation, see [W]. The notion of Lascar strong type was studied in [L3]. Lascar points out that in stable theories, lstp(a/A) = lstp(b/A) if and only if stp(a/A) = stp(b/B). In any case, it is clear that the role of strong types from stability is taken over by Lascar strong types in simplicity since it is possible to prove the Independence Theorem for Lascar strong types ([KP], see also [Sha] and [P4] for more elegant proofs):

Theorem 1.2.10. Let a_{0} ≡_{A}^{L} a_{1}, a_{i} | ⌣_{A} b_{i} for i < 2 and b_{0} | ⌣_{A} b_{1}, there is a | ⌣_{A} b_{1}b_{2} with a ≡_{Ab_{0}} a_{0} and a ≡_{Ab_{1}} a_{1}.

While canonical bases in stable theories exist in ℭ^{eq}, in simple theories it became apparent ([HKP]) that we need to expand somewhat our universe.

Definition 1.2.11. Let x and y be a tuples of (possibly infinite, but small) order type α, and let E(x, y) be a type-definable equivalence relation. We call E countable if the partial type defining E is, and finitary if α is finite. A hyperimaginary element of type E is just an equivalence class a_{E} for some a of length α. A hyperimaginary is countable or finitary if the corresponding relation is.

Lemma 1.2.12. In any complete theory, every type-definable equivalence relation is the intersection of countable equivalence relations.

It should be noted that it is no longer possible to treat hyperimaginaries as elements of the structure, as it was for imaginaries, see [W], 3.1.6. In spite of that fact, we denote by ℭ^{heq} the collection of all countable hyperimaginaries, and it is still possible to make sense of (Lascar strong) types of hyperimaginaries and develop the theory of independence satisfying the same axioms as the ones mentioned above; we refer the reader to [W].

The closure operators from ℭ^{eq} extend naturally, since every automorphism of ℭ extends uniquely to ℭ^{heq}.

Definition 1.2.13. (1) We say that a ∈ dcl(A) for hyperimaginary a and A, if a is fixed by all the automorphisms fixing A. As an object, we define dcl(A) to be the set of all countable hyperimaginaries which are fixed under all A-automorphisms (the reason for allowing only countable hyperimaginaries being our desire to keep dcl(A) small, which is not a serious restriction by 1.2.12).

(2) We say that a ∈ bdd(A) if a has only boundedly many conjugates under Aautomorphisms. We let bdd(A) be the set of all countable hyperimaginaries which are bounded over A.

The bounded closure takes on the role of algebraic closure in the hyperimaginary universe; in particular, a | ⌣_{c} b if and only if bdd(ac) | ⌣_{bdd(c)} bdd(bc). Furthermore, in

a simple theory, a ≡_{A}^{L} b if and only if a ≡_{bdd(A)} b.

Definition 1.2.14. The SU-rank, sometimes called the Lascar rank, is the least function from the collection of all types (over parameters in the monster model) to On ∪{∞}, such that for every α, SU(p) ≥ α + 1 if there is a forking extension q of p with SU(q) ≥ α.

The properties of SU-rank can be summarised in ([W], Section 5.1):

Proposition 1.2.15. SU-rank in a simple theory T has the following properties (where ⊕ denotes the Cantor commutative sum of ordinals):

(1) SU is automorphism invariant;

(2) q ⊢ p implies SU(q) ≤ SU(p);

(3) SU(p) = 0 if and only if p is bounded;

(4) if SU(p) < ∞ and α ≤ SU(p), then p has an extension q with SU(q) = α;

(5) if q ⊢ p and SU(q) < ∞, then q is a nonforking extension of p if and only if SU(q) = SU(p);

(6) T is supersimple if and only if SU(p) < ∞ for every (real) type p;

(7) SU(a/bA) + SU(b/A) ≤ SU(ab/A) ≤ SU(a/ba) ⊕ SU(b/A);

(8) if a | ⌣_{A} b, then SU(ab/A) = SU(a/A) ⊕ SU(b/A).

Let us consider now the definability of various objects discussed above. Regarding definability of independence (see [W], 2.3.15, 3.2.9), we will use the following without explicit mention. Sometimes it is possible to show definability even if p is not complete ([W], proof of 4.7.1).

Lemma 1.2.16. The condition ∃x[x ⊧ p ∧ x | ⌣_{A} b ∧ Φ(x, b)], for a complete (hyperimaginary) type p over A and any partial type Φ(x, y), is type-definable.

For the definability of equality of Lascar strong types for hyperimaginaries, see e.g. [W], 3.2.12:

Lemma 1.2.17. The relation LS(y, x; y′, x′), true if x = x′ and y ≡_{x}^{L} y′ is a hyperdefinable equivalence relation.

We say that a type p(x, a) (where a is a hyperimaginary element) is an amalgamation base if (the hyperimaginary version of) the Independence Theorem holds over p, i.e. any two nonforking extensions of p over a-independent sets can be amalgamated. In particular, all Lascar strong types are amalgamation bases.

Definition 1.2.18. Let p be an amalgamation base.

· The amalgamation class of p is P_{p} = {r|r is an amalgamation base and there are amalgamation bases p_{0}, …, p_{n} such that p = p_{0}, r = p_{n}, and for every i < n, p_{i} and p_{i+1} have a common nonforking extension}. · We call a set B of (hyper)imaginaries a canonical base for p, if every automorphism fixes P_{p} setwise if and only if it fixes B pointwise.

It is clear that any two canonical bases for a fixed amalgamation base p are interdefinable, so we sometimes abuse the language and speak about ‘the’ canonical base, denoted by Cb(p). The following very important theorem is from [HKP].

Theorem 1.2.19. In a simple theory, every (hyperimaginary) amalgamation base p over A has a hyperimaginary canonical base A_{0}, i.e., A_{0} has the following properties:

(1) p does not fork over A_{0}.

(2) p ↾_{A_{0}} is an amalgamation base.

(3) If q is an amalgamation base over B such that p and q have a common nonforking extension, then A_{0} ⊆ dcl(B).

(4) If q is over B and p and q have a common nonforking extension, then A_{0} ⊆ bdd(B).

The proof goes roughly by defining a generically transitive relation on the conjugates of parameters of A, identifying A and A′ if p(x, A) and p(x, A′) have a common nonforking extension. Then, the canonical base will be the class of A and it will be a hyperimaginary by the following lemma ([W], 3.3.1).

Lemma 1.2.20. Suppose R_{0} is type-definable reflexive symmetric relation on a partial type π, generically transitive in the sense that whenever a, a′, a′′ ⊧ π with a′ | ⌣_{a} a′′ and R_{0}(a, a′) and R_{0}(a, a′′) hold, then R_{0}(a′, a′′) holds. Then, the transitive closure R of R_{0} equals the 2-step iteration of R_{0} and is thus a type-definable equivalence relation. Furthermore, R(a, b) holds for some a, b ⊧ π if and only if there is some c ⊧ π with c | ⌣_{a} b and c | ⌣_{b} a such that R_{0}(a, c) and R_{0}(c, b) hold.

Definition 1.2.21. A theory T is one-based, if a | ⌣_{bdd(a)∩bdd(b)} b for any a, b.

A characterisation of one-based theories is given by ([W], 3.5.18):

Proposition 1.2.22. A theory is one-based if and only if every real type tp(a/A) is based on bdd(a), i.e. Cb(a/A) ∈ bdd(a).

We might ask ourselves whether hyperimaginaries are really neccessary in simple theories, motivating the following definition. We say that a theory admits elimination of hyperimaginaries if every hyperimaginary is interdefinable with a sequence of imaginary element. Now we can state all the known results connecting these concepts. In [BPW], using a quite involved analysability argument, the authors managed to prove:

Theorem 1.2.23. Any supersimple theory admits elimination of hyperimaginaries.

Elimination of hyperimaginaries, being equivalent to the statement that on the set of realizations of a fixed complete type, every type-definable equivalence relation is an intersection of definable equivalence relations, was proved for stable theories already in [PP]. Also, the equivalence of Lascar strong types and strong types was proved by Buechler in [Bu] for a certain class of simple theories he called low.

Now, elimination of hyperimaginaries clearly implies the existence of canonical bases in ℭ^{eq}, and the converse is also true for simple T, as shown in [LP]. Also, elimination of hyperimaginaries easily implies the equivalence of Lascar strong types and strong types (since, of course, acl = bdd becomes true).

The definition of canonical bases as given above is not completely satisfactory, especially when compared to the stable case, where canonical base of a stationary type p(x) ∈ S(A) in ℭ^{eq} is given by Cb(p) = dcl(⋃{⌜d_{p}xφ(x, y)⌝|φ(x, y) ∈ L(T)}), where by d_{p}xφ(x, y) we denote the φ-definition of p and by ⌜X⌝ we denote the canonical parameter or name of a definable set X.

In their efforts to understand the stable forking hypothesis, one form of which could say that, in a simple theory, canonical bases of amalgamation bases can be obtained as unions of names of definitions corresponding to stable formulae (similarly as above in stable theories), Kim and Pillay proved a very close approximation to it in [KP2].

Definition 1.2.24. (1) A complete type p over A is foreign to an A-invariant family of partial types Σ, if for all a ⊧ p, B | ⌣_{A} a, and realisations c̄ of possibly forking extensions of types in Σ over B, we always have a | ⌣_{AB} c̄.

(2) A type is regular if it is unbounded and foreign to all its forking extensions.

Clearly, any type p with SU(p) = 1 is regular, since all its forking extensions are bounded types.

Having fixed the terminology above, let us proceed by giving examples of simple theories. Firstly, all the stable theories are simple, including e.g. the theory of the trivial structure (with just equality), theory of vector spaces over division rings, algebraically closed fields, separably closed fields, etc. Below are some examples of simple unstable theories.

Example 1.2.25. The theory of the random graph, i. e., the theory of an irreflexive symmetric binary operation such that for all n, for all distinct x_{1}, …, x_{n}, y_{1}, …, y_{n} there is z such that R(z, x_{i}) for i ∈ {1, …, n} and ¬R(z, y_{i}) for all i ∈ {1, …, n}. This theory is ω-categorical, with quantifier elimination, and it can be shown that any complete 1-type does not divide over a finite set (in fact over the empty set if p is not algebraic). Thus, by 1.2.3, this theory is simple (in fact supersimple of SU-rank 1). It is not stable because the formula R(x, y) has the order property (it is even more obvious that it has the independence property).

Example 1.2.26. Let V be an infinite vector space over a finite field, and ⟨ , ⟩ a nondegenerate bilinear form on V. Then, (V, ⟨ , ⟩) is unstable, because using a variant of the Gramm-Schmidt procedure we can find an orthogonal basis and then easily a formula having the independence property. It is supersimple of SU-rank 1 and ω-categorical.

Example 1.2.27. A field F is pseudofinite if F is perfect, for every n > 1, F has a unique algebraic extension of degree n and F is pseudo-algebraically closed (PAC), i. e., every (absolutely irreducible) variety over F has an F -rational point. It can be shown that these properties are first-order axiomatizable; call the resulting theory Psf. Psf was shown to be decidable by Ax, Duret [Du] proved it unstable, the structure of definable sets was then studied in detail by Chatzidakis, Macintyre and van den Dries in [ChMv]. In pseudofinite fields, dimension corresponds to SU-rank and is actually the algebraic-geometrical dimension of the Zariski closure, and so forking corresponds to algebraic independence. This was essentially known to Hrushovski in [H3], where he proved the Independence Theorem for types over relatively algebraically closed subfields (after adding a fixed suitable set of constants). From this, simplicity is clear and it also follows that lstp = stp, canonical bases exist in the real world, which also implies elimination of imaginaries.

Example 1.2.28. ACFA is the model companion of the theory of fields with a distinguished automorphism σ (i.e. the theory of existentially closed fields with an automorphism). The existence of the model companion is due to van den Dries, Macintyre and Wood ([Ma]), and a deep model-theoretic analysis of ACFA can be found in [ChH].

Let (F, σ) ⊧ ACFA be saturated. For A ⊆ F, let ⟨A⟩ be the smallest subfield of F closed under σ. Chatzidakis and Hrushovski introduce the following relation of independence: B | ⌣_{A} C if ⟨AB⟩ is algebraically independent from ⟨AC⟩ over ⟨A⟩, and prove the Independence Theorem for types p(x) ∈ S(A) where A equals the field theoretic algebraic closure of ⟨A⟩. It follows that ACFA is simple and this notion of independence coincides with nonforking. It can also be seen that canonical bases exist in the real world, which implies elimination of imaginaries. In fact, ACFA is supersimple.

Example 1.2.29. Theories with ‘generic’ predicates and/or automorphisms, as studied in [ChP]. We begin with a complete theory T with quantifier elimination in a language L. Let P (resp. σ) be a new predicate (resp. function) symbol. If T has a model companion in L(P) (resp. L(σ)), we call it T_{P} (resp. T_{σ}). It can be shown that if T is simple, T_{P} is simple and if T is stable, T_{σ} is simple.

Thus, the ACFA example above is a special case of this construction.

Example 1.2.30. A smoothly approximable structure, as introduced by Lachlan, is a countable relational ω-categorical structure M which is the union of an increasing chain of finite homogeneous substructures of M (such a substructure is a subset A of M such that for every finite tuples a, b ∈ A, tp_{M}(a) = tp_{M}(b) if and only if there is an automorphism of M fixing A setwise and taking a to b). By [CHL], ω-categorical, ω-stable structures are smoothly approximable. A monumental treatise of smoothly approximable structures is currently in preparation by Hrushovski and Cherlin [CH].

A rank notion giving rise to a notion of independence can be defined for which the Independence Theorem over algebraically closed sets can be proved and thus we get simplicity.

It is a remarkable fact that in spite of all the work being done in simple theories, the above list of examples remains exhaustive; there are (virtually) no more examples of simple unstable theories, apart from Hrushovski’s amalgamation construction modified for simple theories, see [H7], [E], [Pou] (where stable forking for structures obtained from the amalgamation construction is shown), for general framework compare with [W1]; there’s also an extensive work of Baldwin on the topic, for bibliography see http://www.math.uic.edu/~jbaldwin/. In particular, we haven’t got any ‘bad’ examples where lstp = stp, elimination of hyperimaginaries or stable forking property would fail.

1.3. Combinatorial geometries

In this section we recall some definitions regarding combinatorial geometries needed for understanding the concepts of geometric stability (and simplicity) theory. For more details, we refer the reader to [P3].

Definition 1.3.1. A (combinatorial) pregeometry (or a matroid) is a pair (S, cl) consisting of a set S and a closure operation cl : ℙ(S) → ℙ(S) such that:

(1) X ⊆ cl(X);

(2) cl(cl(X)) = cl(X);

(3) if a ∈ cl(Xb) − cl(X) then b ∈ cl(Xa) (Steinitz exchange);

(4) if a ∈ cl(X) then a ∈ cl(Y) for some finite Y ⊆ X. The pregeometry (S, cl) is a geometry, if cl(∅) = ∅ and cl({a}) = {a} for every a ∈ S. It is homogeneous if for every closed subset X of S and a, b ∈ S − X, there is an automorphism of S (a cl-preserving permutation of S) fixing X and taking a to b.

Remark 1.3.2.

(1) To any pregeometry (S, cl) we can associate a canonical geometry (S′, cl′), where S′ = {cl({a}) : a ∈ S − cl(∅)}, and for X ⊆ S, cl′({cl({a}) : a ∈ X}) = {cl({b}) : b ∈ cl(X)}.

(2) If (S, cl) is a pregeometry and A ⊆ S, we can localize at A to obtain (S, cl_{A}), where cl_{A}(X) := cl(AX) for X ⊆ S.

(3) We invoke the reader’s knowledge of linear algebra in order to remark that Steinitz exchange axiom gives rise to notions of independence and dimension. Thus, if we are in a pregeometry (S, cl), we say that a set A is independent over a set B, if for every a ∈ A, a ∉ cl((A − {a}) ∪ B). We say that A_{0} ⊆ A is a basis for A over B, if A_{0} is independent over B and A ⊆ cl(A_{0}B). All bases for A over B will have the same cardinality denoted dim(A/B). We say that A is independent from B over C if dim(A′/CB) = dim(A′/C) for all finite A′ ⊆ A. This is easily shown to be a symmetric relation.

Definition 1.3.3. Let (S, cl) be a pregeometry. It is said to be:

· trivial or degenerate, if for every X ⊆ S, cl(X) = ⋃{cl({a}) : a ∈ X}; · modular, if for any closed sets X, Y ⊆ S, X is independent from Y over X ∩ Y, or, equivalently, if for any finite-dimensional closed sets X and Y, dim(X) + dim(Y) = dim(XY) + dim(X ∩ Y); · locally modular, if some localization at a point is modular; · projective, if it is non-trivial and modular;

· locally finite, if cl(A) is finite for any finite A.

Example 1.3.4. Let S be a nonempty set and let cl(A) = A for any A ⊆ S. Then (S, cl) is a trivial homogeneous (pre)geometry.

Example 1.3.5. Let F be a division ring and V be a κ-dimensional vector space over F. For A ⊆ V, let cl(A) := span_{F} (A). Then (V, cl) is a homogeneous modular pregeometry and its associated geometry is called (κ − 1)-dimensional projective geometry over F. If we let affcl(A) be the smallest F -affine subspace of V containing A, we get a geometry (V, affcl) which is locally modular (localisation at 0 takes us into the situation above) but not modular.

Example 1.3.6. If K is an algebraically closed field of infinite transcendence degree over its prime subfield, and if we let cl(A) be the field-theoretic algebraic closure of A in K, (K, cl) is a homogeneous geometry which is not locally modular.

The above are the classical examples of pregeometries which are stable. We can add some random structure to make them simple unstable and less well-behaved, in view of the examples given in the previous section; see also [Va] and [dPK]. Moreover, we have the following.

Lemma 1.3.7. Let p be a (complete) regular type over A in a simple theory and let D be the set of realisations of p. For B ⊆ D, we let cl(B) := {b ∈ D : b |̸ ⌣_{A} B}. Then, (D, cl) is a pregeometry.

Proof. The only difficult part is showing that cl(cl(B)) ⊆ cl(B). If b ∈ cl(cl(B)), then b |̸ ⌣_{A} cl(B), so there is c̄ ∈ cl(B) such that b |̸ ⌣_{A} Bc̄ and for every i, c_{i} |̸ ⌣_{A} B. If b | ⌣_{A} B, it follows by regularity that b | ⌣_{AB} c̄ and b | ⌣_{A} Bc̄, which is a contradiction.

□

Remark 1.3.8. In particular, if SU(p) = 1 in the above, the closure operation becomes just cl(B) := D ∩ acl(BA). Furthermore, if we denote by G(D) the set of all SU-rank 1 elements (over A) in D^{eq} := dcl(D ∪ A) ⊆ ℭ^{eq}, G(D) is a pregeometry with full acl(· ∪ A) (in ℭ^{eq}).

Proof. The first part is trivial. To see that G(D) is a pregeometry with cl(X) := acl(XA), the property cl(cl(B)) ⊆ cl(B) for every B follows from the fact that acl is a closure operator, and Steinitz exchange follows from symmetry of forking since for a ∈ G(D), as SU(a/A) = 1, a ∈ cl(B) if and only if a |̸ ⌣_{A} B. □

Definition 1.3.9. Let D be a set of realisations of some p over A with SU(p) = 1. A plane curve in D is a Lascar strong type q = lstp(ab/A) with SU(q) = 1, and {a, b, A} pairwise independent. We say that D is k-linear if there is a plane curve q with SU(Cb(q)) = k and for every plane curve q′, SU(Cb(q′)) ≤ k. It is pseudolinear if it is k-linear for some k. If the set is 1-linear, we just call it linear.

For quite some time it was unclear which is the ‘right’ definition of ‘local modularity’ in simple theories. The following result from [dPK] resolves the ambiguity.

Theorem 1.3.10. The following statements are equivalent for a solution set D of an SU-rank 1 Lascar strong type:

(1) D (D^{eq}) is one-based;

(2) D is linear;

(3) G(D) is linear;

(4) G(D) is modular;

(5) G(D)_{A} is linear, for any (some) small A;

(6) G(D)_{A} is modular, for any (some) small A.

1.4. Polygroups

The study of hypergroups and multivalued mathematical structures was initiated in 1934 by Marty [Mar]. It was noticed around that time that certain structures like double coset spaces maintain some group theoretic behaviour. For more information we refer the reader to [Co].

Definition 1.4.1. A hypergroup is a pair (H, ∗) consisting of a set H and a hyperoperation ∗ : H × H → ℙ^{∗}(H) (with each pair of elements (a, b) we associate a nonempty set a ∗ b), such that

· for all a ∈ H, a ∗ H = H ∗ a = H;

· for all a, b, c ∈ H, a ∗ (b ∗ c) = (a ∗ b) ∗ c (as sets).

A hypergroup (H, ∗) is a polygroup if additionally · there is a scalar identity e ∈ H, i. e. for every a ∈ H, a ∗ e = e ∗ a = {a}; · for each b ∈ H there is a unique b^{−1} ∈ H such that for each a ∈ H, a ∗ b^{−1} = {x ∈ H | a ∈ x ∗ b} and b^{−1} ∗ a = {x ∈ H | a ∈ b ∗ x}.

The way to interpret ‘(a ∗ b) ∗ c’ above, since a ∗ b is a set, is to take ⋃_{d∈a∗b} d ∗ c. We do not wish to consider hypergroups any further, because the requirement that something be a hypergroup carries too little algebraic information: just consider the trivial example of (A, ∗), where A ≠ ∅ and a ∗ b := {a, b} (although there are some interesting hypergroups which are not polygroups).

Definition 1.4.2. A polyspace (P, X, ∗) consists of a polygroup (P, ∗), a set X, and a multivalued map ∗ : P × X → ℙ^{∗}(X), such that:

(1) for every a, b ∈ P, x ∈ X, (a ∗ b) ∗ x = a ∗ (b ∗ x);

(2) for every x ∈ X, e ∗ x = {x};

(3) for every a ∈ P, x ∈ X, y ∈ a ∗ x if and only if x ∈ a^{−1} ∗ y.

The next two examples are the principal examples of polygroups and have played an important role in the development of the blowup construction from Chapter 4.

Example 1.4.3. Let G be a group, and H a (not necessarily normal) subgroup. The double coset space G // H is a polygroup with the multioperation HaH ∗ HbH := {HahbH : h ∈ H}.

Example 1.4.4. A projective geometry is an incidence system (P, L, I) consisting of a set of points P, a set of lines L and an incidence relation I ⊆ P × L satisfying the following axioms:

(1) any line contains at least three points;

(2) two distinct points a, b are contained in a unique line denoted by L(a, b);

(3) if a, b, c, d are distinct points and L(a, b) intersects L(c, d), then L(a, c) must intersect L(b, d) (Pasch axiom), as shown in the

figure. a  b c  d

Let P′ := P ∪ {e}, where e is not in P, and define:

- for a ≠ b ∈ P, a ∘ b := L(a, b) \ {a, b};

- for a ∈ P, if any line contains exactly three points, put a∘a := {e}, otherwise a ∘ a := {a, e};

- for a ∈ P′, e ∘ a = a ∘ e := {a}.

Then it is easily verified that (P′, ∘) is a polygroup; compare to 1.3.5 and 5.4.6.

Lemma 1.4.5. Let (P, ∗) be a polygroup. Then the following properties are equivalent to associativity:

(1) (Second form of associativity). For every a, b, c ∈ P and a′ ∈ a ∗ b, c′ ∈ b ∗ c, a′ ∗ c ≈ a ∗ c′.

(2) (Transposition). a ∗ d ≈ b ∗ c if and only if b^{−1} ∗ a ≈ c ∗ d^{−1} (notice the similarity to the Pasch axiom in 1.4.4).

Proof.

(1) Assume the usual associativity, and let a′ ∈ a ∗ b, c′ ∈ b ∗ c. Then, a ∈ a′ ∗ b^{−1} ⊆ a′ ∗ (c ∗ c′^{−1}) = (a′ ∗ c) ∗ c′^{−1}, so we can find d ∈ a′ ∗ c such that a ∈ d ∗ c′^{−1}.

Assume now the second form of associativity, and let d ∈ (a ∗ b) ∗ c. There will be a′ ∈ a ∗ b with d ∈ a′ ∗ c. We have b ∈ a^{−1} ∗ a′, so there will be d′ ∈ b ∗ c ∩ a^{−1} ∗ d, i.e. d ∈ a ∗ (b ∗ c).

(2) is similar. □

Notice that the second form of associativity (for every a, b ∈ P, x ∈ X and every a′ ∈ a ∗ b and x′ ∈ b ∗ x, a′ ∗ x ≈ a ∗ x′) is equivalent to the associativity of the multiaction in polyspaces as well.

Definition 1.4.6. An equivalence relation R on a hypergroup H is regular on the right if for every x, y ∈ H, xRy implies that for every a ∈ H, x ∗ a =_{R} y ∗ a. It is strongly regular on the right if xRy implies that for all a, x′ ∈ x ∗ a and y′ ∈ y ∗ a, x′Ry′. Analogously we define (strong) regularity on the left. An equivalence is (strongly) regular if it is (strongly) regular both on the left and right.

It is an immediate consequence of the definition that if an equivalence relation R is regular, then a_{R} ∗ b_{R} ⊆ (a ∗ b)_{R} for all a, b ∈ H.

Lemma 1.4.7. If (P, ∗) is a polygroup and R is regular, then (P/R, ∘) is a polygroup, where a_{R} ∘ b_{R} := (a ∗ b)/R. If R happens to be strongly regular, the quotient will be a group.

The product is clearly well-defined and all the properties of polygroups descend to quotients.

Definition 1.4.8. If (P, ∗) is a polygroup, we call Q ⊆ P a subpolygroup if Q is itself a polygroup with the same multioperation ∗.

Remark 1.4.9. It should be remarked here that in model-theoretic language, the above definition is a substructure in the ‘functional’ sense, since ∗ on Q is the same as on P, which is a much stronger condition than e.g. requiring that (Q, ∗↾_{Q}) be a polygroup, which would yield a substructure in a ‘relational’ sense.

This strong definition allows us to talk about the index of Q in P, as the number of (left) cosets of Q covering P, since it is clear that different cosets are disjoint.

Definition 1.4.10. Let (P, ∗) be a polygroup. A subpolygroup N < P is called normal, if for every a ∈ P, a ∗ N = N ∗ a.

Lemma 1.4.11. Let us say that a and b are equivalent modulo N, if a ∗ N ≈ N ∗ b. This is a regular equivalence relation.

Proof. It is clearly reflexive and symmetric, by normality of N. For transitivity, let a ∗ N ≈ N ∗ b and b ∗ N ≈ N ∗ c. Then b ∈ N^{−1} ∗ (a ∗ N) ∩ (N ∗ c) ∗ N^{−1}, so by the transposition property, a ∗ N ∗ N ≈ N ∗ N ∗ c, i.e. a ∗ N ≈ N ∗ c. If we denote the coset a ∗ N by a_{N}, it is clear that a_{N} ∗ b_{N} ⊆ (a ∗ b)_{N}, showing regularity. □

Thus, we can quotient by normal subpolygroups.

Definition 1.4.12. We shall say that a map φ : P → P′ between two polygroups is a homomorphism, if φ(a ∗ b) ⊆ φ(a) ∗ φ(b) and φ(a^{−1}) = φ(a)^{−1} for all a, b ∈ P. Denoting a_{φ} := φ^{−1}(φ(a)), we say that a homomorphism φ is of

· type 1, if φ^{−1}(φ(a) ∗ φ(b)) = (a_{φ} ∗ b_{φ})_{φ};

· type 2, if (a∗b)_{φ} = φ^{−1}(φ(a)∗φ(b)) (which in turn implies a_{φ} ∗b_{φ} ⊆ (a∗b)_{φ}); · type 3, if φ^{−1}(φ(a) ∗ φ(b)) = a_{φ} ∗ b_{φ}; · type 4, if it is of type 2 and type 3.

It is an isomorphism, if it is bijective and φ(a ∗ b) = φ(a) ∗ φ(b).

Clearly, type j implies type i for i < j, and type 1 is equivalent to the induced structure (P/φ, ∗_{φ}) being a polygroup, where a_{φ} ∗_{φ} b_{φ} := {c_{φ} : c ∈ a_{φ} ∗ b_{φ}}. Also, for type 2, the corresponding equivalence relation (whose classes are a_{φ}) is regular. We shall not need any further results about polygroups in chapters to follow, but let us just mention a few results from [Ja], showing that the theory is reasonably well-behaved.

Proposition 1.4.13. (1) (First isomorphism theorem). Let φ : P → P′ be a surjective homomorphism of type 2 between polygroups P and P′. Then P/ ker(φ) ≅ P′.

(2) (Second isomorphism theorem). In a polygroup, suppose M is a subpolygroup, and N normal subpolygroup. Then ⟨M, N⟩/N ≅ M/M ∩ N, where ⟨M, N⟩ is the subpolygroup generated by M ∪ N.

(3) (Third isomorphism theorem). Suppose M < N are normal subpolygroups of a polygroup P. Then P/N ≅ (P/M)/(M/N).

(4) (Jordan-Hölder theorem). In a polygroup, let M < N be subpolygroups. Suppose M = K_{0} ⊆ ⋯ ⊆ K_{k} = N and M = L_{0} ⊆ ⋯ ⊆ L_{l} = N, where each K_{i} is maximal proper normal subpolygroup in K_{i+1} for i < k, and similarly for L_{i}. Then there is a bijective correspondence between {K_{i+1}/K_{i} : i < k} and {L_{j+1}/L_{j} : j < l} such that the correspondents are isomorphic.

CHAPTER 2 Germs and group configuration

The main tool for proving the group configuration theorem in the stable case is the machinery of germs of definable functions. They can be developed as follows:

Definition 2.1. Let T be a stable theory.

(1) Let p and q be stationary types over ∅. We say that a strong type r(x, y) over some f defines a function from p to q if:

(a) r(x, y) → p(x) ∧ q(y);

(b) if r(a, b), then a | ⌣ f, b | ⌣ f, b ∈ dcl(fa) and in such a case we may write b = f(a); if also a ∈ dcl(fb), we say the function is invertible.

(2) Two functions r_{1}, r_{2} : p → q (defined over f_{1}, f_{2}) are equivalent if there is a | ⌣ f_{1}f_{2} such that f_{1}(a) = f_{2}(a), i.e. if r_{1} and r_{2} have a common nonforking extension, which is a definable equivalence relation, and clearly the class of r, called the germ of r is in fact σ := Cb(r). Furthermore, σ again defines a function p → q since definability descends: if b ∈ dcl(af) and σ = Cb(ab/f), it is still true that b ∈ dcl(aσ).

(3) To see that germs can be composed, let σ : p → q and τ : q → s be germs. If we define τ · σ := Cb(aτ(σ(a))/στ) for some a | ⌣ τσ, this is a germ p → s and the fact that tp(aτ(σ(a))/στ) is stationary implies τ · σ ∈ dcl(τσ).

What are the obstacles in the simple case? Firstly, definability does not descend (the most we can conclude from b ∈ dcl(af), if σ = Cb(ab/f), is that b ∈ acl(aσ)):

Example 2.2. (Pillay) Suppose we have three sorts, called P, Q and R and let Q be a 2-cover of P, i.e. there is a surjection π : Q → P with each fibre of size 2. Write π^{−1}(a) as {a_{1}, a_{2}}. Let S be a random bipartite graph S between P and R and let f, g : P × R → Q be functions such that if S(b, a), then f(b, a) = a_{1} and g(b, a) = a_{2}; otherwise, f(b, a) = a_{2} and g(b, a) = a_{1}. It can be shown that ⟨P, Q, R, f, g⟩ is a simple structure (notice we have ‘forgotten’ S). Consider now tp(aa_{1}/b), where e.g. S(b, a); clearly f(b, a) = a_{1} and a_{1} ∈ dcl(ab). But Cb(aa_{1}/b) = ∅, and obviously a_{1} ∉ dcl(a).

Secondly, in the simple case, there is no hope for a property similar to (3) above, see 2.3.4 which expresses this in a more specific language.

To overcome the first (in fact nonessential) difficulty, we consider multifunctions instead of functions, and to deal with the second, we need to complete after each composition.

Material in the first three sections is almost all from [BY], with a few modifications and simplifications (e.g. we consider completions and reductions in any of the variables, thus replacing Ben-Yaacov’s ‘strongness on the left and right’). For Section 2.4, I had a proof scheme in the simple case (in particular I had recognised the importance of what in the present language is called ‘π with π^{−1} ∘ π generic’) but was lacking the right definition of germs. However, upon seeing Ben-Yaacov’s definition, I managed to reprove the results of that section independently.

2.1. Partial generic multiactions

From now onwards, we shall be working in a simple theory.

Definition 2.1.1. Let π_{i}(x_{i}), i < α be partial types (in hyperimaginary sorts) over a hyperimaginary parameter e. Their product, ⨂_{i<α} π_{i}(⟨x_{i} : i < α⟩) is the partial type, if one exists, which is true of ⟨a_{i} : i < α⟩ if and only if for every i, π(a_{i}), and ⟨a_{i} : i < α⟩ is independent over e.

Definition 2.1.2. A partial type π over e has definable independence if for every π′ over e, π ⊗_{e} π′ exists.

The following is straightforward from the definition.

Proposition 2.1.3. (1) Every complete type has definable independence over its

domain.

(2) If π_{i} has definable independence for all i < α, then ⨂_{i<α} π_{i} exists and has definable independence.

(3) If π has definable independence and π′ ⊢ π, then π′ has it too.

(4) If π has definable independence over e, ∼ is an e-hyperdefinable equivalence relation, π is ∼-invariant, then π/∼ has definable independence over e.

Definition 2.1.4. Let π(x, y, z) be a partial type in three hyperimaginary sorts over ∅. We say that π defines a partial generic multiaction if:

(1) π ↾_{x}, π ↾_{y} and π ↾_{z} have definable independence;

(2) π(x, y, z) implies that x, y, z are pairwise independent;

(3) for any f, a, there are at most boundedly many b such that π(f, a, b), and in that case we write b ∈ f(a) or b ≈ f(a).

We shall use the following notation: Fun(π) = π ↾_{x}, Arg(π) = π ↾_{y}, Val(π) = π ↾_{z}, and if f ∈ Fun(π), Γ(f)(y, z) := π(f, y, z) and usually we will identify f with its graph, writing ‘xy ⊧ f’ in place of ‘xy ⊧ Γ(f)’.

In the above, it is enough to require that just Fun(π) and Arg(π) have definable independence, since for b ∈ Val(π) and any e, b | ⌣ e if and only if there are f ∈ Fun(π) and a ∈ Arg(π) with fa | ⌣ e.

Definition 2.1.5. Let π be a partial generic multiaction.

(1) π is invertible, if for every b there is at most boundedly many a’s with π(f, a, b), i.e. if π^{−1} is a generic action, where π^{−1}(f, b, a) if and only if π(f, a, b);

(2) π is complete in x if for any f ∈ Fun(π), Γ(f) is a Lascar strong type (or an amalgamation base) over f; similarly we define completeness in y or z; if we just say ‘complete’ without mentioning a variable, it will mean ‘complete in x’;

(3) π is reduced (in x) if it is complete (in x) and whenever there is a | ⌣ fg with f(a) ≈ g(a), then f = g; it is reduced in y, if it is complete in y and whenever there is f | ⌣ aa′ with f(a) ≈ f(a′), then a = a′; if π is invertible and complete in z, we say it is reduced in z if whenever there are f | ⌣ b, b′ and a such that b, b′ ∈ f(a), then b = b′.

(4) π is trivial, if π(a, b, c) implies that {a, b, c} is independent.

Definition 2.1.6. Two partial generic actions π(x, y, z) and π′(x′, y, z) are isomorphic if there is a hyperdefinable bijection φ : Fun(π) → Fun(π′) such that for every f ∈ Fun(π), Γ(f) = Γ(φ(f)).

Two constructions, yielding examples of complete and reduced multiactions, are discussed in the next section.

2.2. Completion and reduction

Definition 2.2.1. Let π(x, y, z) be a partial generic multiaction. Let \underline{x} be the sort (yz, x)/LS, where LS(yz, x, y′z′, x′) is the hyperdefinable equivalence relation from 1.2.17 saying that x = x′ and yz ≡_{x}^{L} y′z′. An element of this sort can be identified with a pair f_{p}, where f is of the x sort, and p is a Lascar strong type over f in the yz sort. Let \underline{π}((y′z′, x)/LS, y, z) = π(x, y, z) ∧ LS(yz, x, y′z′, x), i.e. \underline{π}(f_{p}, a, b) if and only if b ∈ f(a) and lstp(ab/f) = p. This \underline{π} we shall call the completion of π (in x).

For f ∈ Fun(π), we define \underline{f} := \{(ab, f)_{LS} : b ∈ f(a)\}, which is obviously a bounded hyperdefinable set. In the same way, we can complete with respect to y and z.

Proposition 2.2.2. (1) The completion of a partial generic multiaction (in any of the variables) is a complete partial generic multiaction (in the respective variable).

(2) If a multiaction π(x, y, z) is complete in some variable, (e.g. x), the completion with respect to some other variable (say y) will still be complete in x.

Proof. Part (1) is easy. For (2), let π(x, y, z) be complete in x, and let \underline{π}_{y} be the completion with respect to y. We need to show that \underline{π}_{y} is still complete in x. Suppose \underline{π}_{y}(f, a_{p}, b) and \underline{π}_{y}(f, a′_{p′}, b′); then π(f, a, b), lstp(fb/a) = p, π(f, a′, b′) and lstp(fb′/a′) = p′. By assumption of completeness in x, let φ : ab ≡_{f}^{L} a′b′. But then, by invariance of Lascar strong types, φ(p) = p′, so in fact φ : a_{p}b ≡_{f}^{L} a′_{p′}b′. □

Definition 2.2.3. Let π(x, y, z) be a complete multiaction, f, g ∈ Fun(π). Let f ∼_{1} g (or sometimes we write ∼_{1}^{x}) if there is a | ⌣ fg such that f(a) ≈ g(a), and let ∼ be the transitive closure of ∼_{1}. It is clear by the definable independence of Fun(π) that ∼_{1} is hyperdefinable, as well as reflexive and symmetric, and we will show below that ∼ is also hyperdefinable. The class of a function f, denoted f̄, we call the germ of f, and we define the reduction of π, π̄(x̄, y, z) (where x̄ = x/∼, the sort of germs) as the partial type such that π̄(f̄, a, b) if and only if there is f ∈ f̄ with π(f, a, b). The set of all germs of π is denoted Germ(π). In a similar way, we can reduce with respect to y and z, by dividing by the corresponding relations ∼^{y}, ∼^{z} (a ∼_{1}^{y} a′ if there is f | ⌣ aa′, f(a) ≈ f(a′), b ∼_{1}^{z} b′ if there is f | ⌣ bb′ and a | ⌣ f such that b, b′ ∈ f(a)).

Lemma 2.2.4. Relations ∼_{1}^{x}, and ∼_{1}^{y} are generically transitive. If π is invertible, so is ∼_{1}^{z} and we have that ∼^{y} and ∼^{z} are such that any two elements which are related are in fact interbounded. Thus, ∼^{x} and ∼^{y} are hyperdefinable, and so is ∼^{z} when π is invertible.

Proof. First of all, let us show that f ∼_{1}^{x} g if and only if Γ(f) and Γ(g) have a common nonforking extension. If a | ⌣ fg with b ∈ f(a)∩g(a), then obviously ab | ⌣_{f} g and ab | ⌣_{g} f so ab realises a common nonforking extension of Γ(f) and Γ(g). The converse is also straightforward.

Assume now f ∼_{1} g, g ∼_{1} h and f | ⌣_{g} h. In other words, Γ(f) and Γ(g) have a common nonforking extension, the same as Γ(g) and Γ(h). By completeness, Γ(g) is a Lascar strong type. Thus, since f | ⌣_{g} h, we can apply the Independence Theorem and get that Γ(f) and Γ(h) have a common nonforking extension and thus ∼_{1}^{x} is generically transitive.

To see the same for e.g. ∼_{1}^{y} when π is complete in y, let a ∼_{1}^{y} a′, a ∼_{1}^{y} a′′, a′ | ⌣_{a} a′′. There will be some f | ⌣ aa′ and b, f′ | ⌣ aa′′ and b′ such that b ∈ f(a)∩f(a′), b′ ∈ f′(a) ∩ f′(a′′). Then fb ≡_{a}^{L} f′b′, fb | ⌣_{a} a′, f′b′ | ⌣_{a} a′′ and a′ | ⌣_{a} a′′, so by the Independence Theorem we may assume fb = f′b′ | ⌣_{a} a′a′′ and thus a′ ∼_{1}^{y} a′′.

When π is invertible, if e.g. a ∼_{1}^{y} a′, let f | ⌣ aa′ and b such that b ∈ f(a) ∩ f(a′). Then in particular f | ⌣_{a} a′, and, since a′ ∈ bdd(fa) (a′ ∈ f^{−1}(b), b ∈ f(a)), we get a′ | ⌣_{a} a′ so a′ ∈ bdd(a).

Finally, use 1.2.20, to get that ∼^{?} is hyperdefinable, for ? ∈ {x, y, z}. □

Definition 2.2.5. In view of the above considerations, if π is not necessarily complete, for f ∈ Fun(π), we let f̂ be the set of germs of all completions of Γ(f) to a Lascar strong type.

Remark 2.2.6. It is clear that for complete π, f ∈ Fun(π), f̄ = Cb(Γ(f)) by [HKP].

Proposition 2.2.7. If π is a complete partial generic multiaction (in x), then π̄ is a reduced partial generic multiaction (and similarly in y and z). Moreover, if e.g. π was complete in x and y, and we reduce with respect to x, the result is still complete in y. If π is nontrivial, so is the completion.

Proof. Fun(π̄) has definable independence as a quotient of Fun(π). If π̄(f̄, a, b), there is f ∈ f̄ with π(f, a, b), so {f, a, b} is pairwise independent and therefore {f̄, a, b} as well. By the previous remark, ab | ⌣f̄f, so b | ⌣_{af̄}af, and, as b ∈ bdd(af), b | ⌣_{af̄}b, which implies b ∈ bdd(af̄).

The ‘moreover’ part is easy. □

Definition 2.2.8. Two partial generic multiactions π and π′ are equivalent, written π ≈ π′, if their reductions are isomorphic.

2.3. Composing germs

Definition 2.3.1. Let π(u, x, y) and π′(v, y, z) be partial generic multiactions. Define π′ ∘ π((v, u), x, z) to be the partial type such that π′ ∘ π((g, f), a, c) if:

(1) {g, f, a} is independent;

(2) c ∈ g ∘ f(a), i.e. there is b ∈ f(a) such that c ∈ g(b).

Proposition 2.3.2. (1) π′ ∘ π always exists and it is a generic action; (2) (π′ ∘ π)^{−1} = π^{−1} ∘ π′^{−1}.

Proof.

(1) Since Fun(π) and Fun(π′) have definable independence, so does Fun(π) × Fun(π′), and π′∘π exists. As Fun(π′∘π) ⊢ Fun(π)×Fun(π′), it also has definable independence. It is easy to check the definable independence of Arg(π′ ∘ π) and the boundedness property. Suppose π′ ∘ π(f, g, a, c), so b ∈ f(a), c ∈ g(b), and {f, g, a} is independent. In particular, a | ⌣ fg, so a | ⌣_{f} g, and, as b ∈ bdd(fa), b | ⌣_{f} g. Since b | ⌣ f, we get that b | ⌣ fg. Continuing, b | ⌣_{g} f, so, by c ∈ bdd(gb) and c | ⌣ g, we conclude c | ⌣ fg, as required.

(2) is trivial. □

Proposition 2.3.3. Let π and π′ be partial generic multiactions with Arg(π′) = Val(π) being Lascar strong types. Then, for any f ∈ Fun(π), g ∈ Fun(π′) with f | ⌣ g, g ∘ f ∈ Fun(π′ ∘ π). If π′ is nontrivial, so is the composition.

Proof. Let f | ⌣ g and let ab ⊧ f, b′c ⊧ g. By assumption, b ≡^{L} b′, b | ⌣ f, b′ | ⌣ g, f | ⌣ g, so by the Independence Theorem, we may assume b = b′ | ⌣ fg. We may also assume a | ⌣_{bf} g, and so g | ⌣ abf and a | ⌣ fg, so g ∘ f is defined on a. □

Remark 2.3.4. It is clear that in a general simple theory, a composition of two complete multiactions need not be complete any more. Thus, if we want that the composition of two germs lives on germs again, we need to observe all the possible completions of the composition and their germs. This is how multivaluedness appears in simple theories, in some sense a perfectly natural consequence of the crucial difference between simplicity and stability, simple theories not having stationarity which helps to determine objects uniquely.

The following obvious but important lemmas allow us to formalise these considerations.

Lemma 2.3.5. Let f ∈ Fun(π), g ∈ Fun(π′), f | ⌣ g and h ∈ Germ(π′ ∘ π). We say that

abc witness that h ∈ { \atop g ∘ f} and write abc ⊧ h ∈ { \atop g ∘ f} if a | ⌣ fgh and ab ⊧ f, bc ⊧ g,

ac ⊧ h. With this definition, h ∈ { \atop g ∘ f} if and only if there are witnesses for it.

Lemma 2.3.6. Let π′ ∘ π be defined, f ∈ Germ(π), g ∈ Germ(π′), h ∈ Germ(π′ ∘ π), f | ⌣ g.

(1) If π is invertible and f | ⌣ h, then abc ⊧ h ∈ { \atop g ∘ f} if and only if bac ⊧ g ∈ { \atop h ∘ f^{−1}}.

(2) If π′ is invertible and g | ⌣ h, then abc ⊧ h ∈ { \atop g ∘ f} if and only if acb ⊧ f ∈ { \atop g^{−1} ∘ h}.

A desirable property of germs is that reduction and composition commute. Indeed, we have:

Lemma 2.3.7. Let f ∈ Fun(π), g ∈ Fun(π′), f | ⌣ g. Then { \atop g ∘ f} = ⋃_{f̄∈f̂,ḡ∈ĝ} { \atop ḡ ∘ f̄}.

Proof. Let abc ⊧ h ∈ { \atop g ∘ f}, let f̄:= Cb(ab/f), ḡ := Cb(bc/g). Then clearly h ∈ { \atop ḡ ∘ f̄}.

On the other hand, let abc ⊧ h ∈ { \atop ḡ ∘ f̄}. We can rechoose them such that

abc | ⌣_{f̄ḡh} f, or even abc | ⌣f̄ ; ḡ f, as h ∈ bdd(f̄ḡ). Then, as f | ⌣ ḡ, f | ⌣f̄abcḡh. If

a′b′ ⊧ f with f̄ = Cb(a′b′/f), since a′b′ | ⌣f̄f, ab | ⌣f̄ ḡh and f | ⌣f̄ ḡh, by the Independence Theorem we may assume that ab ⊧ f and ab | ⌣f̄fḡh. Moreover, we may assume that c | ⌣_{abf̄ḡ} f and also abc | ⌣_{fḡh} g, yielding g | ⌣ḡ abcfh. As above for f, using the Independence Theorem, we may assume that bc ⊧ g and bc | ⌣ḡ fgh, but

then abc ⊧ h ∈ { \atop g ∘ f}. □

Corollary 2.3.8. π′ ∘ π ≈ π̄′ ∘ π̄. Moreover, if π_{i} ≈ π′_{i} then π_{1} ∘ π_{0} ≈ π′_{1} ∘ π′_{0}.

Definition 2.3.9. We say that the composition π′ ∘ π is generic if for any f ∈ Germ(π), g ∈ Germ(π′) with f | ⌣ g and h ∈ { \atop g ∘ f}, we have h | ⌣ f and h | ⌣ g.

There is another potential obstacle to successfully defining a multioperation: a composition of two, three or more germs might not be of the same sort. Luckily, the following solves the problem.

Theorem 2.3.10. Let π(x, y, z) be an invertible partial generic multiaction, which is also complete in y, Arg(π) be a Lascar strong type and the composition π^{−1} ∘ π generic.

(0) If we denote π̂ = \overline{π^{−1} ∘ π}, we have that π̂ ∘ π̂ ≈ π̂.

This yields a multioperation (with boundedly many values) ∗ : P ⊗ P → ℙ^{bdd}(P), for P = Germ π̂, given by g ∗ f := { \atop g ∘ f}, for f | ⌣ g such that (P, ∗) is in fact a hyperdefinable polygroup chunk, i.e.

(1) (Generic independence) for every f | ⌣ g, if h ∈ f ∗ g, then h | ⌣ f and h | ⌣ g;

(2) (Generic associativity) for {f, g, h} independent, (f ∗ g) ∗ h = f ∗ (g ∗ h) (as sets); (3) (Generic surjectivity) for any f | ⌣ g, there is h such that g ∈ f ∗ h; in fact,

g ∈ f ∗ h if and only if h ∈ f^{−1} ∗ g.

Proof.

(0) Let abc ⊧ h ∈ { \atop h_{1} ∘ h_{0}}, for h_{0} | ⌣ h_{1} ∈ Fun(π̂). There are f_{0}, g_{0} ∈ Fun(π) and

d_{0} such that ad_{0}b ⊧ h_{0} ∈ { \atop f_{0}^{−1} ∘ g_{0}} and f_{1}, g_{1} ∈ Fun(π) such that bd_{1}c ⊧ h_{1} ∈

{ \atop g_{1}^{−1} ∘ f_{1}}. By completeness in y, f_{0}d_{0} ≡_{b}^{L} f_{1}d_{1}. Also, f_{0}d_{0} | ⌣_{b} ah_{0} (by genericity

of π^{−1} ∘ π, h_{0} | ⌣ f_{0}, so abd_{0} ⊧ g_{0} ∈ { \atop f_{0} ∘ h_{0}}, so in particular b | ⌣ f_{0}h_{0}, and thus

f_{0} | ⌣_{b} h_{0}, so f_{0}d_{0} | ⌣_{b} ah_{0}) and similarly f_{1}d_{1} | ⌣_{b} ch_{1}, and ah_{0} | ⌣_{b} ch_{1}. Thus, by the Independence Theorem, we can assume that f_{1}d_{1} = f_{0}d_{0} =: fd and ah_{0} | ⌣_{bdf} ch_{1} and also ag_{0}h_{0} | ⌣_{bdf} cg_{1}h_{1}. In particular, ag_{0} | ⌣_{bdf} g_{1}, and by assumptions above, ag_{0} | ⌣_{d} bf, so ag_{0} | ⌣_{d} fg_{0}, ag_{0} | ⌣ fg_{1}, a | ⌣ fg_{0}g_{1} and finally a | ⌣ fg_{0}g_{1}h_{0}h_{1}h, so adc ⊧

h ∈ { \atop g_{1}^{−1} ∘ g_{0}}.

Conversely, let adc ⊧ h ∈ { \atop g_{1}^{−1} ∘ g_{0}}, g_{0}, g_{1} ∈ Fun(π), g_{0} | ⌣ g_{1}. Since d ∈ Val(π), we can find f ∈ Fun(π) and b with bd ⊧ f and we may assume that bf | ⌣_{d} acg_{0}g_{1}h, so ag_{0} | ⌣_{bdf} cg_{1}. Now, f | ⌣ g_{0}g_{1}, a | ⌣ fg_{0}, b | ⌣ fg_{1}, so we may take h_{0} := Cb(ab/fg_{0}) ∈

{ \atop f^{−1} ∘ g_{0}} and h_{1} := Cb(bc/fg_{1}) ∈ { \atop g_{1}^{−1} ∘ f}, and we have that abc ⊧ h ∈ { \atop h_{1} ∘ h_{0}}.

(1) follows from genericity of π^{−1} ∘ π.

(2) Let l ∈ (f ∗ g) ∗ h; so there are k, bcd ⊧ k ∈ f ∗ g and ab′d′ ⊧ l ∈ k ∗ h. Therefore, we have bd ≡_{k}^{L} b′d′, bd | ⌣_{k} fg, b′d′ | ⌣_{k} lh, fg | ⌣_{k} lh, so by the Independence Theorem we may assume that b′d′ = bd | ⌣_{k} fglh. Then, abc ⊧ m := Cb(ac/gh) ∈ g ∗ h and abd ⊧ l ∈ f ∗ m.

(3) is easy by 2.3.6. □

Remark 2.3.11. In the previous theorem, requiring that π(x, y, z) be complete in y on top of the assumption of genericity of π^{−1} ∘ π is not too strong, since if, for example, we had π such that π^{−1} ∘ π is generic, and if π_{y} is the completion in y, then π_{y}^{−1} ∘ π_{y} is also generic.

Proof. Let a_{p}bc_{q} ⊧ h ∈ { \atop g^{−1} ∘ f} ∈ Germ(π_{x}^{−1} ∘ π_{x}). In particular, xyz ⊧ h′ := Cb(xz/fg) ∈ { \atop g^{−1} ∘ f} ∈ Germ(π^{−1} ∘ π), so by the assumption, h′ | ⌣ f and h′ | ⌣ g. However, as x_{p}z_{q} ∈ bdd(xz), xz | ⌣_{h′} fg implies that x_{p}z_{q} | ⌣_{h′} fg, so h ∈ bdd(h′) and thus h | ⌣ f, h | ⌣ g, as required. □

2.4. Group configuration

Definition 2.4.1. An algebraic quadrangle is a diagram (a, b, c, x, y, z), where: · any pair and any non-collinear triple is independent; · bdd(ab) = bdd(ac) = bdd(bc); · bdd(xa) = bdd(ya), · bdd(zc) = bdd(xc), · bdd(yb) = bdd(zb); · b is interbounded with Cb(yz/b), · a is interbounded with Cb(xy/a), · c is interbounded with Cb(xz/c).

a b c x z y

Clearly, replacing any element in an algebraic quadrangle by an interbounded element gives another algebraic quadrangle, and we call such quadrangles algebraically equivalent.

Although it is possible, we do not attempt to find, given an algebraic quadrangle, an equivalent one where each point is actually definable over the other two on the same line, since this property is not preserved by reduction anyway, in view of 2.2.

Theorem 2.4.2. Let (a, b, c, x, y, z) be an algebraic quadrangle. We may assume (by considering an equivalent quadrangle with y and z replaced by bdd(y) and bdd(z)) that y and z are boundedly closed. If we let π(ξ, η, ζ) = lstp(byz), then π is invertible partial generic multiaction, complete in η and ζ, with π^{−1} ∘ π generic.

Proof. Let b_{1} | ⌣ b_{2} ⊧ tp(b) and let tut_{1} ⊧ h ∈ { \atop b_{1}^{−1} ∘ b_{2}}. We need to show that h | ⌣ b_{1} and h | ⌣ b_{2}. Without loss of generality, b_{2} = b. We have btu ⊧ π, b_{1}t_{1}u ⊧ π, so let φ ∈ LAut_{b} such that tu ↦ yz and b_{1}t_{1}u ↦ b′_{1}y_{1}z. Then φ(lstp(tt_{1}/bb_{1})) = lstp(yy_{1}/bb′_{1}) and it’s enough to show φ(h) | ⌣ b and φ(h) | ⌣ b′_{1}.

So, it boils down to: starting with h = Cb(yy_{1}/bb_{1}) for some {b, b_{1}, y} independent with b_{1}y_{1}z ≡ byz, we need to show h | ⌣ b and h | ⌣ b_{1}.

Since b_{1}y_{1} ≡_{z} by, b_{1}y_{1} | ⌣_{z} by and by | ⌣_{z} cx, the Independence Theorem implies (as z = bdd(z) so in fact b_{1}y_{1} ≡_{z}^{L} by) there is b_{2}y_{2} | ⌣_{z} bcxy with b_{2}y_{2} ≡_{byz} b_{1}y_{1} (*) and b_{2}y_{2} ≡_{cxz} by (**). Let a_{2} such that a_{2}b_{2}cxy_{2}z ≡ abcxyz. Let h_{2} := Cb(yy_{2}/bb_{2}); then, if we manage to show h_{2} | ⌣ b and h_{2} | ⌣ b_{2} we’ll be done by (*). We may also assume b_{2}y_{2} | ⌣_{z} abcxy (***). From here on, we proceed as in [P3], 5.4.7:

(i). aa_{2}bb_{2} | ⌣ y. From (***), (**) and b | ⌣ z, we get b_{2} | ⌣ abcy, which, together with y | ⌣ abc gives y | ⌣ abb_{2}c, but a_{2} ∈ bdd(b_{2}c) so y | ⌣ aa_{2}bb_{2}.

(ii). y_{2} ∈ bdd(aa_{2}y). This is easy, and so is the following claim:

(iii). y_{2} ∈ bdd(bb_{2}y). From the above, forking calculus yields:

(iv). aa_{2} | ⌣ b. (v). aa_{2} | ⌣ b_{2}.

Now, by (i), (ii), (iii), lstp(yy_{2}/aa_{2}bb_{2}) does not fork over both aa_{2}, bb_{2} and thus

h_{2} = Cb(yy_{2}/bb_{2}) = Cb(yy_{2}/aa_{2}) ∈ bdd(aa_{2}).

By Claims (iv) and (v), it is independent from each b and b_{2}, as required.

□

CHAPTER 3 Almost hyperimaginaries

Classes of finite tuples for definable equivalence relations in a monster model ℭ of a first order theory, called imaginaries, were organised into the multi-sorted first order structure called ℭ^{eq} by Shelah, to allow a smoother development of stability theory. This turned out particularly important in geometric stability theory, since canonical bases of types exist in ℭ^{eq}.

On the other hand, the construction of canonical bases in simple theories [HKP] seems to require quotienting by type-definable equivalence relations, which lead to the introduction of hyperimaginaries as classes of (possibly infinite, but small) tuples with respect to type-definable equivalence relations.

Alas, not even the hyperimaginary universe is general enough, since in our construction of the group from the polygroup chunk, certain relations appear, which, at the moment, we cannot guarantee to be type-definable. They belong, however, to a very well behaved class of invariant relations and it is possible to develop some model theory for quotients over such relations, called graded almost hyperimaginaries, as shown in this chapter.

In sections 3.1 and 3.2 we define ultraimaginaries and graded almost hyperimaginaries and develop a satisfactory ‘forking calculus’, or a theory of independence for them, while in Section 3.3, we study the gradedly almost hyperdefinable polygroups and polyspaces in simple theories. The core relation needed to identify elements that happen to be in the intersection of independent enough products is discussed in Section 3.4, and we prove the Hrushovski-Weil style group (and space) chunk theorem for the gradedly almost hyperdefinable case in Section 3.5. One might pose a question whether it is necessary to consider graded almost hyperimaginaries, since if we observe the hyperdefinable polygroup chunk obtained in 2.3.10, the core relation on it has bounded classes (any two related elements are interbounded), and it would be much easier to develop the theory for quotients over such equivalence relations. The reason for choosing such generality is that in order to apply the group chunk theorem, we might be forced to leave the realm of relations with bounded classes. An additional bonus for choosing to work in the category of gradedly almost hyperdefinable groups in simple theories is that our subsequent results will show it ‘closed under group configuration’.

The fact that ultraimaginaries might be of importance in the construction of a group from the polygroup chunk and that the proof of the group chunk theorem will go through was already noted in [To], where the group I obtained lived on ultraimaginaries, but the relation I was forced to use was not as well-behaved as the present core relation and I lacked control over the dimension of such objects. A possibility of studying hyperdefinable polygroups and properties of generic elements in the supersimple case was first recognised in [To1], then improved by Ben-Yaacov to the framework of gradedly almost hyperdefinable polygroups in general simple theories. For the presentation here we adopt a different definition of stratified ranks and correct, by distinguishing ‘full’ structures, some inaccuracies present in a working version of Ben-Yaacov’s notes. The group chunk theorem in the simple (hyperdefinable) case is by Wagner, see [W], 4.7.1. For the gradedly almost hyperdefinable case, it is given in [To] for ultraimaginaries, and Ben-Yaacov noted that that it preserves almost hyperdefinability. The full details presented here were not written down before. I have not seen the space chunk theorem 3.5.3 appear anywhere in the literature.

It should be remarked here that the framework of graded almost hyperimaginaries might turn out to be unnecessary generality because it is indeed our hope, since most of the construction can be done hyperdefinably, that the relation in question can be made type-definable as well. Nevertheless, at least for now, it is required.

3.1. Ultraimaginaries and almost-hyperimaginaries

To clarify some of the notation, if R is an equivalence relation, the R-class of a is denoted by a_{R} or a/R. When R is just reflexive and symmetric, a_{R} := {x : xRa}, ‘a ∈_{R} A’ means there is a′ ∈ A with aRa′, ‘A ≈_{R} B’ means there are a ∈ A, b ∈ B with aRb, and ‘A =_{R} B’ means that for every a ∈ A there is b ∈ B with aRb and vice versa. If R_{i} comes from some grading, then ∈_{i}, ≈_{i} and =_{i} will stand for ∈_{R_{i}}, ≈_{R_{i}} and =_{R_{i}}, respectively.

Definition 3.1.1. Let (I, ≤) be a directed partial order.

(1) An I-graded equivalence relation R is a direct limit of reflexive symmetric type-definable relations R_{i} such that:

(a) if i ≤ j, then R_{i} ⊆ R_{j};

(b) for every i, j, there is k (i ≤ k, j ≤ k) such that xR_{i}yR_{j}z implies xR_{k}z;

(2) An invariant equivalence relation R is almost type-definable if there is a reflexive symmetric type-definable relation R′ finer than R such that each R-class can be covered by boundedly many R′-classes. It is I-gradedly almost typedefinable if it is I-graded and almost type-definable.

(3) A class of an invariant equivalence relation is called an ultraimaginary. A class of a (gradedly) almost type-definable relation is called a (graded) almost hyperimaginary.

In the end, the objects we will be dealing with will live on graded almost hyperimaginaries. Our approach is partly inspired by the development of hyperimaginaries in [HKP].

Convention 3.1.2. From this point onwards, in order to save precious letters of the alphabet, we abuse the notation somewhat, and write a_{R} and b_{R} even when a and b are not even of the same sort, since it will always be clear from the context which relation is involved in a_{R} and which in b_{R}.

Furthermore, we might have defined (gradedly) almost hyperdefinable equivalence relations on hyperimaginaries as the ones which can be covered by boundedly many classes of hyperdefinable reflexive and symmetric relations, but this approach gives the same objects as quotienting tuples of real elements by (gradedly) almost type-definable equivalence relations from 3.1.1, so we use these terms interchangeably.

Definition 3.1.3. Two ultraimaginaries a_{R} and b_{R} have the same type over a hyperimaginary c, denoted a_{R} ≡_{c} b_{R}, if there are a′ ∈ a_{R} and b′ ∈ b_{R} such that a′ ≡_{c} b′. They have the same Lascar strong type, denoted a_{R} ≡_{c}^{L} b_{R} if there are a′ ∈ a_{R} and b′ ∈ b_{R} with a′ ≡_{c}^{L} b′.

Lemma 3.1.4. For ultraimaginaries a_{R} and b_{R} and a hyperimaginary c, the following statements are equivalent:

(1) a_{R} ≡_{c} b_{R};

(2) there is an automorphism fixing c and taking a_{R} to b_{R};

(3) for every a′ ∈ a_{R} there is b′ ∈ b_{R} with a′ ≡_{c} b′.

Similarly, for Lascar strong types, the following are equivalent:

(1) a_{R} ≡_{c}^{L} b_{R};

(2) a_{R} and b_{R} are equivalent modulo any bounded c-invariant equivalence relation;

(3) for every a′ ∈ a_{R} there is b′ ∈ b_{R} with a′ ≡_{c}^{L} b′.

Definition 3.1.5. An infinite sequence ⟨a_{i}_{R} : i ∈ I⟩ of ultraimaginaries is indiscernible over a hyperimaginary set A if there is an A-indiscernible sequence ⟨c_{i} : i ∈ I⟩ such that c_{i} ∈ a_{i}_{R}.

3.2. Dividing

Definition 3.2.1. We say that two ultraimaginaries a_{R} and b_{Q} are independent over a hyperimaginary c, if there are a′ ∈ a_{R} and b′ ∈ b_{Q} such that a′ | ⌣_{c} b′.

We exercise particular care not to consider independence over an ultraimaginary, or between infinite tuples of ultraimaginaries, due to complications caused by the lack of compactness. Naturally, a finite set of ultraimaginaries can be considered as a single ultraimaginary by juxtaposing the equivalence relations.

Proposition 3.2.2. Independence (defined between finite tuples of ultraimaginaries over a hyperimaginary set) has the following properties:

(1) (Bounded elements) If a_{R} | ⌣_{c} b_{R}, and a′_{R} ∈ bdd(a_{R}c), then a′_{R} | ⌣_{c} b_{R}. Conversely, if R is graded and a_{R} | ⌣_{b′} a_{R} for every b′ ∈ b_{R}, then a_{R} ∈ bdd(b_{R}).

(2) (Extension) for any a_{R}, A, b_{R} there is an a′_{R} ≡_{A} a_{R} such that a′_{R} | ⌣_{A} b_{R};

(3) (Symmetry) a_{R} | ⌣_{A} b_{R} if and only if b_{R} | ⌣_{A} a_{R};

(4) (Transitivity) If a_{R} | ⌣_{A} b_{R}c_{R} and b_{R} | ⌣_{A} c_{R} then a_{R}b_{R} | ⌣_{A} c_{R}. a_{R} | ⌣_{A} bc_{R} if

and only if a_{R} | ⌣_{A} b and a_{R} | ⌣_{Ab} c_{R}.

(5) (Local Character). For every a_{R} and A, there is A_{0} ⊆ A with |A_{0}| ≤ |T | such

that a_{R} | ⌣_{A_{0}} A.

(6) (Independence Theorem) If a_{0}_{R} ≡_{A}^{L} a_{1}_{R}, a_{i}_{R} | ⌣_{A} b_{i}_{R} for i < 2, b_{0}_{R} | ⌣_{A} b_{1}_{R},

then there is a_{R} with a_{R}b_{i}_{R} ≡_{A} a_{i}_{R}b_{i}_{R} and a_{R} | ⌣_{A} b_{0}_{R}b_{1}_{R}.

Proof.

(1) We may assume a | ⌣_{c} b and let ā′ be any enumeration of representatives of all a_{R}c-conjugates of a′_{R}. We may assume ā′ | ⌣_{ac} b, which implies aā′ | ⌣_{c} b. Of course, there will be i such that a′Ra′_{i} and then a′_{i} | ⌣_{c} b implies that a′_{R} | ⌣_{c} b_{R}.

For the converse, suppose a_{R} is not bounded over b_{R}; there will be a b-indiscernible sequence a_{α}b_{α} such that α ≠ β implies a_{α}_{R} and a_{β}_{R} are distinct, and b_{α}Rb. Moreover, by an automorphism, we may assume it is b′-indiscernible and a_{0}b_{0} = ab, for some b′Rb. The existence of such a sequence implies a′′ |̸ ⌣_{b′} a′ for any a′Ra and a′′Ra: suppose a′R_{i}a, a′′R_{i}a, hence a′′R_{j}a′ for some j and let a′_{α} be such that a′_{α}a_{α} ≡_{b′} a′a; it is still true that α ≠ β implies a′_{α} is not R-related to a′_{β}. We may in fact assume that a′_{α} is b′-indiscernible with a′_{0} = a′. Let k such that R_{j}(x, y)∧R_{j}(x, z) → R_{k}(y, z); by indiscernibility of a′_{α}, we can find ψ(y, z) ∈ R_{k} such that α ≠ β implies ¬ψ(a′_{α}, a′_{β}). By compactness, find a formula φ(x, y) ∈ R_{j} such that φ(x, y)∧φ(x, z) → ψ(y, z). But then φ(x, a′) ∈ tp(a′′/b′a′), and φ(x, a′) 2-divides over b′ since if e.g. φ(c, a′_{α})∧φ(c, a′_{β}), then ψ(a′_{α}, a′_{β}), which is a contradiction. (2), (3) and (5) are easy.

(4) If a_{R} | ⌣_{A} b_{R}c_{R} and b_{R} | ⌣_{A} c_{R}, we may assume that a | ⌣_{A} bc, and let b′Rb, c′Rc with b′ | ⌣_{A} c′. We may assume b′c′ | ⌣_{Abc} a, so a | ⌣_{A} bb′cc′ and we are done. The rest is similar.

(6) Let a_{0}_{R} ≡_{A}^{L} a_{1}_{R}, a_{i}_{R} | ⌣_{A} b_{i}_{R}, b_{0}_{R} | ⌣_{A} b_{1}_{R}. We may assume a_{0} ≡_{A}^{L} a_{1} and b_{0} | ⌣_{A} b_{1}. Let a′_{i}Ra_{i} and b′_{i}Rb_{i} such that a′_{i} | ⌣_{A} b′_{i}. In fact, it is possible to choose b′_{i} such that b′_{i} ≡_{A}^{L} b_{i} (let b′′_{i} realise the nonforking extension of lstp(b_{i}/Ab′_{i}) to Ab′_{i}a′_{i}; then b′′_{i} | ⌣_{Ab′_{i}} a′_{i} and by above a′_{i} | ⌣_{A} b′_{i}b′′_{i}). So, let a′′_{i} be such that a′_{i}b′_{i} ≡_{A}^{L} a′′_{i}b_{i}. Now, since

a′′_{i} ≡_{A}^{L} a′_{i}, a′_{i}Ra_{i}, and a_{0} ≡_{A}^{L} a_{1}, we have a′′_{0}_{R} ≡_{A}^{L} a′′_{1}_{R}. Thus, there will be some a′′′_{i}Ra′′_{i} such that a′′′_{0} ≡_{A}^{L} a′′′_{1} and we may assume a′′′_{i} | ⌣_{A} b_{i} (let a′′′′_{i} realise the nonforking extension of lstp(a′′′_{i}/Aa′′_{i}) to Aa′′_{i}b_{i}; then a′′′′_{i} | ⌣_{Aa′′_{i}} b_{i}, but b_{i} | ⌣_{A} a′′_{i} implies b_{i} | ⌣_{A} a′′′′_{i}). Now just apply the (hyperimaginary) Independence Theorem to this situation. □

Let us show now the special properties of independence for almost hyperimaginaries.

Lemma 3.2.3. Let R be almost type-definable, witnessed by R′, a, b, c ∈ ℭ. Then, if a_{R} | ⌣_{c} b, there is a′R′a such that a′ | ⌣_{c} b.

Proof. Let a′′Ra such that a′′ | ⌣_{c} b, and let ā be such that a/R = ⋃ a_{i}/R′. We may assume ā | ⌣_{a′′} cb, which implies ā | ⌣_{c} b. Now, there is i such that a ∈ a_{i}/R′, so we may take a′ := a_{i}. □

As a corollary, we get the first order characterisation of independence for almost hyperimaginaries:

Corollary 3.2.4. Let R be almost type-definable, witnessed by R′. Then a_{R} | ⌣_{c} b_{R} if and only if there are a′R′a, b′R′b such that a′ | ⌣_{c} b′.

Proof. If a_{R} | ⌣_{c} b_{R}, let b_{0}Rb such that a_{R} | ⌣_{c} b_{0}. Then, by the lemma above, there is a′R′a with a′ | ⌣_{c} b_{0}, and, in particular, a′ | ⌣_{c} b_{R}. By the lemma again, there is b′R′b with a′ | ⌣_{c} b′. The converse is trivial. □

Remark 3.2.5. It is clear by the above corollary that we can make sense of independence between infinite (small) tuples of almost hyperimaginaries and in that case we will have the Finite Character of independence as well.

3.3. Almost hyperdefinable polygroups

Let us describe the category of gradedly almost hyperdefinable (multi)structures and maps we shall be working in.

Definition 3.3.1. Assume S = S_{0}/R, S′ = S′_{0}/R′, where S_{0} and S′_{0} are type-definable sets, R, R′ are I and J-graded almost type-definable relations and f(x, y) a typedefinable relation on S_{0} × S′_{0}. Then f defines a gradedly almost type-definable partial multimap S_{0}/R → S′_{0}/R′ if:

(1) there exists j ∈ J such that for every a ∈ S, f(a) can be covered by a bounded set of R′_{j}-classes (i.e. the fact that f(a)/R is bounded is witnessed by some R′_{j});

(2) for every R_{i}, there is R′_{j} such that for every a ∈ S, f(a/R_{i}) ⊆ f(a)/R′_{j}.

If in (1), at most a single R′_{j}-class is needed, then f induces a partial map. If f(a) ≠ ∅ for every a ∈ S, we say the (multi)map is total.

An equational multistructure is a structure in a language consisting of symbols for multivalued functions (operations), ‘∈’ and ‘=’, where axioms are universal closures of formulas of the form ⋀_{n} x_{n} ∈ τ_{n} → ⋁_{m} y_{m} ∈ σ_{m}, where τ_{n} and σ_{m} are terms.

A gradedly almost type definable equational (multi)structure S in some theory is given as S = S_{0}/R, where S_{0} is a type-definable set, R is a gradedly almost hyperdefinable equivalence relation and each n-ary (multi)operation f on S is a gradedly definable (multi)map f : S_{0}^{n}/R^{n} → S_{0}/R such that, for every axiom of the form

⋀_{n} x_{k_{n}} ∈ τ_{n}(x_{0}, …, x_{N−1}) → ⋁_{m} x_{l_{m}} ∈ σ_{m}(x_{0}, …, x_{N−1}),

for every i ∈ I there is j ∈ I such that for every a_{k}/R, k < N, if the premise holds up to R_{i} (for every n, there are a′_{k}R_{i}a_{k}, k < N with a′_{k_{n}} ∈ τ_{n}(a′_{0}, …, a′_{N−1})), one of the conclusions holds up to R_{j}.

Convention 3.3.2. In view of 3.1.2, even if our structure is multi-sorted, with sorts S_{i} = S_{i}^{0}/R_{i}, we shall denote all R_{i}’s by the same letter, e.g. R, since knowing that x ∈ S_{i}^{0} clearly indicates that x_{R} should be read as x_{R_{i}}. For example, a gradedly almost hyperdefinable polyspace (P_{0}/R, X_{0}/Q) will be denoted as (P_{0}/R, X_{0}/R).

Let us explore the extent of the above definition on an example of a gradedly almost definable polygroup. It is possible to formulate a similar statement at a ‘metalevel’ for an arbitrary equational multistructure, but we wish to be extra precise here for reasons that will emerge in 3.3.4.

Lemma 3.3.3. Let (P = P_{0}/R, ∗) be a gradedly almost hyperdefinable polygroup. Then, for every i and j, there is k such that:

(1) a ∈_{i} b′ ∗ c′, b′R_{j}b, c′R_{j}c implies a ∈_{k} b ∗ c.

(2) d ∈_{i} (a′ ∗ b′) ∗ c′, a′R_{j}a, b′R_{j}b, c′R_{j}c implies d ∈_{k} a ∗ (b ∗ c).

(3) c ∈_{i} a′ ∗ b′, a′R_{j}a, b′R_{j}b implies b ∈_{k} a^{−1} ∗ c, and symmetrically for inverses from the right.

Proof.

(1) If aR_{i}a′ ∈ b′ ∗ c′, b′R_{j}b, c′R_{j}c, by the fact that ∗ induces a graded multimap, a′ ∈_{j′} b ∗ c for some j′, and then a ∈_{k} b ∗ c for some k.

(2) Assume the situation is as in the statement. Let dR_{i}d′ ∈ (a′ ∗b′)∗c′, so the premise of the axiom t ∈ (x ∗ y) ∗ z → t ∈ x ∗ (y ∗ z) is satisfied up to R_{max{i,j}}, so there is l such that the conclusion holds up to l, i.e. there are d′′R_{l}d, a′′R_{l}a, b′′R_{l}b, c′′R_{l}c with d′′ ∈ a′′ ∗(b′′ ∗c′′), so let e′′ ∈ b′′ ∗c′′ such that d′′ ∈ a′′ ∗e′′. By (1), there will be l′ such that e′′R_{l′}e for some e ∈ b ∗ c, and again by (1), there will be l′′ such that d′′ ∈_{l′′} a ∗ e, or, in other words, dR_{l}d′′R_{l′′}a ∗ (b ∗ c).

(3) Follows in a similar fashion to (2), using that ^{−1} induces a graded map. □

Remark 3.3.4. There is a complication which is not apparent when dealing with hyperdefinable objects, but presents itself here. It concerns the ‘right’ definition of a graded map. The above definition is minimal such that the stratified ranks for gradedly almost hyperdefinable polygroups, to be defined below, have sensible properties like e.g. translation invariance. However, when trying to do the same for gradedly almost hyperdefinable polyspaces, this definition is not rich enough. Namely, the following lemma may not hold for polyspaces, and in case we want a uniform treatment for polygroups and polyspaces, we have to require the conclusion of the lemma in the definition of graded maps and call such maps and corresponding structures full.

In this exposition, we choose the minimal definition, as we have reservations about the naturality of the alternative one, but then we have to prove the existence of generics for polygroups using stratified ranks first, and then obtain the existence of generics for polyspaces as a corollary.

Lemma 3.3.5. Let (P_{0}/R, ∗) be a gradedly almost hyperdefinable polygroup and let R_{0} be such that z ∈ x ∗ y implies y ∈_{0} x^{−1} ∗ z and the same from the right. Then, for every i there is j such that if c ∈_{i} a ∗ b, there is a′R_{j}a with c ∈_{0} a′ ∗ b.

Proof. Let cR_{i}c′ ∈ a ∗ b. Then a ∈_{0} c′ ∗ b^{−1}, and by (1) of the previous lemma, a ∈_{j} c ∗ b^{−1}, so let aR_{j}a′ ∈ c ∗ b^{−1}, but then c ∈_{0} a′ ∗ b. □

Remark 3.3.6. I believe it should be possible to handle even more general objects, where e.g. S is a direct limit of S_{i}/R_{i} for i ∈ I, and a map f from S to S′ = lim_{j∈J} S′_{j}/R′_{j} consists of an increasing index map ι : I → J and type-definable maps f_{i} : S_{i} → S′_{ι(i)}, where we would have a similar requirement for uniformity of satisfaction of axioms. This actually seems like a more natural setup from a classical mathematical perspective, but we shall not need such generality in the applications that follow.

Definition 3.3.7. If (P, X) = (P_{0}/R, X_{0}/R) is a gradedly almost hyperdefinable polyspace, g ∈ P_{0} and π is a partial type extending X_{0}, we denote by g ∗ π(x) the partial type which is true of x if there is y ⊧ π such that x ∈ g ∗ y. If Q is some type-definable reflexive and symmetric relation, by (π/Q)(x) we denote the partial type satisfied by x if there is y ⊧ π with xQy.

For the rest of this chapter we shall assume we are in a simple theory, even though some of the definitions would make sense in an arbitrary theory as well.

Definition 3.3.8. Let (P, X) = (P_{0}/R, X_{0}/R) be a gradedly almost hyperdefinable polyspace in a simple theory. We say that x_{R} is a generic element of X over a set A if for every g | ⌣_{A} x_{R}, for every y ∈ g · x, y_{R} | ⌣ Ag. Since every polygroup acts on itself by left and right translation, we can define both left and right generics in a polygroup.

In polygroups, the usual properties of generics follow. In particular, left and right generics coincide and we just call them generics:

Lemma 3.3.9. Let P = P_{0}/R be a gradedly almost hyperdefinable polygroup. (1) If a_{R} is left generic in P, g_{R} | ⌣ a_{R}, b_{R} ∈ g_{R} ∗ a_{R}, then b_{R} is left generic;

(2) if g_{R} is left generic, then g_{R}^{−1} is left generic;

(3) if g_{R} is left generic, it is right generic, too.

Proof.

(1) is a particular case of 3.3.13.

(2) Let g_{R} be left generic, and let g′ | ⌣ g realise tp(g). Let h_{R} ∈ g′_{R} ∗ g_{R} and we may assume h | ⌣ g′. Then g_{R}^{−1} ∈ h_{R}^{−1} ∗ g′_{R} is generic by the previous claim.

(3) follows easily from (2). □

From now on, let R_{0} witness the almost type-definability of R, and let it be high enough in the grading such that y ∈ x ∗ z^{−1} implies x ∈_{0} y ∗ z, and let R_{1} be coarser than R_{0}^{2} and such that x ∈_{0} y ∗ z and yR_{0}y′ imply x ∈_{1} y′ ∗ z.

Definition 3.3.10. Let D(·, φ, ϵ, λ, k) ≥ α, for a formula φ(x, z), ϵ(x, y) ∈ R_{1}, cardinal λ and k ∈ ω be the least ordinal-valued function on partial types π satisfying:

· D(π, φ, ϵ, λ, k) ≥ 0 if π is consistent;

· D(π, φ, ϵ, λ, k) ≥ α for α limit, if D(π, φ, ϵ, λ, k) ≥ β for all β < α;
· D(π, φ, ϵ, λ, k) ≥ α + 1 if there is f ∈ P, and ⟨c_{i} : i < λ⟩ such that

(1) {φ(x, c_{i}) : i < λ} is ϵ-k-inconsistent, i.e. ⋀_{i∈I} ∃y[ϵ(x, y) ∧ φ(y, b_{i})] is not realised for any I ⊂ ω, |I| = k;

(2) for every i < λ, D((f ∗ π)/R_{1}(x) ∪ φ(x, c_{i}), φ, ϵ, λ, k) ≥ α;

Remark 3.3.11. For any φ, ϵ and k, the rank D(·, φ, ϵ, ω, k) is closed and continuous: D(π, φ, ϵ, ω, k) = min{D(ψ, φ, ϵ, ω, k) : π ⊢ ψ}. Furthermore, if π(x) = π(x, A) is a partial type with parameters from A, then for every n < ω there is a partial type ν_{n}^{π}(X) (equivalent to the consistency of a certain tree) such that ⊧ ν_{n}^{π}(A′) if and only if D(π(x, A′), φ, ϵ, ω, k) ≥ n. By compactness then, it is clear that if e.g. f and c_{i}, i < ω witness that D(π, φ, ϵ, ω, k) ≥ n + 1, we can in fact find d_{i}, i < λ for any λ, with the same property as c_{i}, and thus it follows that D(π, φ, ϵ, λ, k) = D(π, φ, ϵ, ω, k) for any λ ≥ ω, so from now on we may omit λ. Also, if φ, ϵ and k are clear from the context, we shall write D(π) for shorthand.

Proof. By induction on n we show that D(ψ, φ, ϵ, k) ≥ n for all ψ provable from π implies D(π, φ, ϵ, k) ≥ n, and there is a partial type ν_{n}^{π} such that ⊧ ν_{n}^{π}(A′) if and only if D(π(x, A′)) ≥ n. Clearly this holds for n = 0, where ν_{0}^{π}(X) is the partial type ∃x π(x, X). Assume the statement holds for n and D(ψ) ≥ n+1 for all ψ(x, A) implied by π(x, A). Thus, for every such ψ there are b^{ψ} and ⟨b_{i}^{ψ} : i < ω⟩ such that for all i < ω, D((b^{ψ} ∗ ψ(x, A))/R_{1} ∪ φ(x, b_{i}^{ψ})) ≥ n and {φ(x, b_{i}^{ψ}) : i < ω} is ϵ-k-inconsistent. By inductive assumption there is ν_{n}^{ψ}(X, x, y) true of (A, b^{ψ}, b_{i}^{ψ}) for every i < ω such that D((b′ ∗ ψ(x, A′))/R_{1} ∪ φ(x, b′′)) ≥ n if and only if ⊧ ν_{n}^{ψ}(A′, b′, b′′). Clearly, if ψ ⊢ ψ′, then ν_{n}^{ψ} ⊢ ν_{n}^{ψ′}. By compactness, there are ⟨b′, b′_{i} : i < ω⟩ such that ν_{n}^{ψ}(A, b′, b′_{i}) for all ψ implied by π and all i < ω, and {φ(x, b′_{i}) : i < ω} is ϵ-k-inconsistent. By induction hypothesis D((b′ ∗ π)/R_{1} ∪ φ(x, b′_{i})) ≥ n for all i < ω so D(π) ≥ n + 1 by definition. We may define ν_{n+1}^{π}(X) as

∃x∃⟨y_{i} : i < ω⟩ [ ⋀_{i<ω,π⊢ψ} ν_{n}^{ψ}(X, x, y_{i}) ∧ ⋀_{I⊂ω,|I|=k} ¬∃z ⋀_{s=1}^{k} ∃y [ϵ(z, y) ∧ φ(y, y_{s})]] .

□

Even though the definition of stratified ranks makes sense for arbitrary polyspaces, the lemma below needs fullness.

Lemma 3.3.12. Stratified ranks on graded almost definable polygroups have the following properties:

(1) Ultrametric Property. If π ⊢ ⋁_{i<β} π_{i}, for π, π_{i} type-definable, then D(π) ≤ sup{D(π_{i}) : i < β}.

(2) Translation Invariance. D(π) = D((g ∗ π)/R_{j}) for every g ∈ P and R_{j}.

(3) Finiteness. For every π, φ, ϵ, k, D(π, φ, ϵ, k) < ω.

(4) Witnessing Dividing. Suppose q ⊇ p are complete types over B ⊇ A. Then q/R_{1} doesn’t divide over A if and only if D(p, φ, ϵ, k) = D(q, φ, ϵ, k), for every φ, ϵ and k.

Proof.

(1) Let |β| < λ. We proceed by induction. Assume D(π) ≥ α + 1; let g and c_{i}, i < λ be such that for every i, D((g ∗ π)/R_{1}(x) ∪ φ(x, c_{i})) ≥ α and {φ(x, c_{i}) : i < λ} is ϵ-k-contradictory. Then, since (g ∗ π)/R_{1}(x) ∧ φ(x, c_{i}) ⊢ ⋁_{j<β}(g ∗ π_{j})/R_{1} ∧ φ(x, c_{i}), by induction hypothesis, there is j = j(i) such that D((g ∗ π_{j})/R_{1} ∪ φ(x, c_{i})) ≥ α. By the pigeonhole principle applied to i ↦ j(i), there is j_{0} < β such that for λ-many c_{i}’s, D((g ∗ π_{j_{0}})/R_{1} ∪ φ(x, c_{i})) ≥ α and thus D(π_{j_{0}}) ≥ α + 1.

(2) Suppose D((g ∗ π)/R_{j}) ≥ α + 1; there are f ∈ P and c_{i} : i < λ such that for every i < λ, D((f ∗(g∗π)/R_{j})/R_{1}∪φ(x, c_{i})) ≥ α and φ(x, c_{i}) is ϵ-k-inconsistent. Consider an x realising the above type; there are z ⊧ π, yR_{j}y′ ∈ g ∗ z, x ∈_{1} f ∗ y. This means that x ∈_{j′} f ∗ y′ ⊆ f ∗ (g ∗ z) for some higher j′, and in fact x ∈_{k} (f ∗ g) ∗ z for some higher k. By 3.3.5, there is h′R_{l}f ∗ g such that x ∈_{0} h′ ∗ z. Let {h_{β} : β < μ} be boundedly many elements such that (f ∗g)/R_{l} ⊆ ⋃_{β<μ} h_{β}/R_{0}. Thus, there must be a β such that hR_{0}h_{β}. By assumption on R_{1}, as x ∈_{0} h ∗ z, we get that x ∈_{1} h_{β} ∗ z. In other words, (f ∗ (g ∗ π)/R_{j})/R_{1} ⊢ ⋁_{β<μ}(h_{β} ∗ π)/R_{1}, so, by the ultrametric property, there is h_{β} such that D((h_{β} ∗ π)/R_{1} ∪ φ(x, c_{i})) ≥ α. However, we might have started with λ > μ, so even though the choice of h_{β} depends on c_{i}, we can apply the pigeonhole principle to get a fixed h such that there are still λ many i’s with D((h ∗ π)/R_{1} ∪ φ(x, c_{i})) ≥ α, witnessing D(π) ≥ α + 1.

(3) If some D(π, φ, ϵ, k) ≥ ω, the formula φ would have the tree property, contradicting simplicity.

(4) Suppose q/R_{1} divides over A; let φ_{0}(x, b_{0}) ∈ q/R_{1} be a dividing formula, and let ⟨b_{i} : i < ω⟩ be an indiscernible sequence in tp(b_{0}/A) such that {φ_{0}(x, b_{i}) : i < ω} is inconsistent. But then, there will be an ϵ ∈ R_{1} and φ ∈ q such that ∃y ϵ(x, y) ∧ φ(y, b_{0}) ⊢ φ_{0}(x, b_{0}). Then, as q ⊢ p ∪ φ(x, b_{0}), we have that for every i < ω, D(p ∪ φ(x, b_{i})) ≥ D(q), and as {φ(x, b_{i}) : i < ω} are ϵ-k-inconsistent, we have that D(p) ≥ D(q) + 1.

For the converse, let us show by induction on n that if a_{R} | ⌣_{A} B and D(a/A) ≥ n, then D(a/B) ≥ n. This is clear for n = 0, so suppose D(a/A) ≥ n + 1. There are b and ⟨b_{i} : i < ω⟩ such that for every i < ω, D((b ∗ tp(a/A))/R_{1} ∧ φ(x, b_{i})) ≥ n and {φ(x, b_{i}) : i < ω} is ϵ-k-contradictory. By continuity of ranks, we can extend (b ∗ tp(a/A))/R_{1} ∧ φ(x, b_{i}) to a complete type p over Abb_{0} of the same rank and we may assume that some c ∈_{R} b ∗ a realises it. Let a′R_{0}a with a′ | ⌣_{A} B and we may assume bb_{0} | ⌣_{Aa′} B, implying B | ⌣_{A} a′bb_{0} and c_{R} | ⌣_{Abb_{0}} B. By induction hypothesis, D(c/Abb_{0}) ≥ n implies D(c/Bbb_{0}) ≥ n. As b_{0} | ⌣_{Ab} B, we may assume that ⟨b_{i} : i < ω⟩ is indiscernible over Bb, so ⟨b_{i} : i < ω⟩ witness that D(c/Bb) ≥ n + 1. Thus, by translation invariance, D(a/B) ≥ D(a/Bb) = D(c/Bb) ≥ n + 1. □

Proposition 3.3.13. (1) There exist generic elements in gradedly almost hyperde-

finable polygroups.

(2) If (P, X) = (P_{0}/R, X_{0}/R) is a gradedly almost hyperdefinable polyspace x_{R} ∈ X is generic if and only if there is a generic g_{R} ∈ P and x_{0}_{R} ∈ X with g_{R} | ⌣ x_{0}_{R} such that x_{R} ∈ g_{R} ∗ x_{0}_{R}.

(3) Generic elements exist in every orbit of a gradedly almost hyperdefinable polyspace.

Proof.

(1) Let (P_{0}/R, ∗) be the polygroup in question. Enumerate all the possible triplets (φ, ϵ, k) and choose a type p ⊢ x ∈ P_{0} with maximal D(·, φ, ϵ, k)-rank in the lexicographic order with respect to the above enumeration. Let a ⊧ p and let g | ⌣ a_{R}, b ∈ g∗a. Then, we know that, since a_{R} | ⌣ g (implying tp(a/g)/R_{1} does not divide over ∅), D(a, φ, ϵ, k) = D(a/g, φ, ϵ, k) = D(b/g, φ, ϵ, k) ≤ D(b, φ, ϵ, k). By the choice of a, we have equality and b_{R} | ⌣ g (the equality D(a/g) = D(b/g) is obtained as follows: from b_{R} ∈ g_{R}∗a_{R} we can deduce that b ∈_{j} g∗a for some R_{j}, so tp(b/g) ⊢ (g∗tp(a/g))/R_{j} and by translation invariance we get that D(b/g) ≤ D(a/g); similarly, from a_{R} ∈ g_{R}^{−1} ∗ b_{R}, we get D(a/g) ≤ D(b/g)).

(2) Let x_{0}_{R} ∈ X, g_{R} ∈ P generic with g_{R} | ⌣ x_{0}_{R} and let x_{R} ∈ g_{R} ∗ x_{0}_{R}. Without loss of generality, g | ⌣ x_{0}. Take any f | ⌣ x_{R}. We may assume that f | ⌣ x and furthermore that gx_{0} | ⌣_{x} f, implying gx_{0}x | ⌣ f. Let y_{R} ∈ f_{R}∗x_{R} ⊆ f_{R}∗(g_{R}∗x_{0}_{R}) = (f_{R}∗g_{R})∗x_{0}_{R}, so there is h_{R} ∈ f_{R} ∗ g_{R} with y_{R} ∈ h_{R} ∗ x_{0}_{R}. As g_{R} is generic, h_{R} | ⌣ f. Now, we have f | ⌣ gx_{0}, which implies x_{0} | ⌣_{f} g and x_{0} | ⌣_{f} h_{R} (since h_{R} ∈ bdd(fg)), so x_{0}f | ⌣ h_{R}, f | ⌣ x_{0}h_{R} and subsequently, as y_{R} ∈ bdd(x_{0}h_{R}), we get the required f | ⌣ y_{R}.

On the other hand, if x_{R} ∈ X is generic, find a generic g_{R} ∈ G such that x | ⌣ g and let x_{0} ∈ g^{−1} ∗ x. Then g_{R} | ⌣ x_{0}_{R} by genericity of x_{R}.

(3) is a trivial corollary of (1) and (2). □

Definition 3.3.14. Let S_{0} be a type-definable set, R a gradedly almost type-definable equivalence relation on S_{0}, ∗ : S_{0}/R ⊗ S_{0}/R → S_{0}/R a gradedly type-definable multimap, and a gradedly defined map ^{−1} : S_{0}/R → S_{0}/R such that whenever a_{R} | ⌣ b_{R}, a ∗ b is defined (in particular, we require that for every i, there is j such that if a_{R} | ⌣ b_{R}, a/R_{i} ∗ b/R_{i} ⊆ (a ∗ b)/R_{j}). We say that S = (S_{0}/R, ∗) is a gradedly almost hyperdefinable polygroup chunk, if there is some R_{0} in the grading of R such that:

(1) Generic independence: If a | ⌣ b, c ∈ a ∗ b, then c_{R} | ⌣ a and c_{R} | ⌣ b.

(2) Generic associativity: If {a, b, c} is independent, (a ∗ b) ∗ c =_{0} a ∗ (b ∗ c).

(3) Generic inverse: If b_{R} | ⌣ c_{R} and a ∈ b∗c, then b ∈_{0} a∗c^{−1}, and, if b ∈ a∗c^{−1},

then a ∈_{0} b ∗ c; similarly from the left. If furthermore X_{0}/R is a gradedly almost hyperdefinable set, and ∗ : S_{0}/R ⊗ X_{0}/R → X_{0}/R a gradedly type-definable multimap (i.e. for every a_{R} | ⌣ x_{R}, a ∗ x is defined), we say that (S, X, ∗) = (S_{0}/R, X_{0}/R, ∗) forms a polyspace chunk, if the R_{0} from above can be coarsened such that:

(1’) If a | ⌣ x, y ∈ a ∗ b, then y_{R} | ⌣ a.

(2’) If {a, b, x} is independent, (a ∗ b) ∗ x =_{0} a ∗ (b ∗ x).
(3’) If a_{R} | ⌣ x_{R} and y ∈ a ∗ x, then x ∈_{0} a^{−1} ∗ y.

Proposition 3.3.15. Let P = P_{0}/R be a gradedly almost hyperdefinable (poly)group. Then the generic elements of P form a gradedly almost hyperdefinable (poly)group chunk.

Proof. By continuity of ranks, it follows that an element a_{R} is generic if and only if D(a, φ, ϵ, k) = D(P_{0}, φ, ϵ, k) for all φ, ϵ and k, which is a type-definable condition S_{0}, and S_{0}/R is the required almost hyperdefinable (poly)group chunk. □

A dual problem is whether from a (poly)group chunk we can reconstruct a (poly)group. An answer in the case of a group chunk is provided by 3.5.1. As for polygroup chunks, in Chapter 4 we will present a construction of a gradedly almost hyperdefinable group chunk, given a gradedly almost hyperdefinable polygroup chunk, coreless in the sense below.

3.4. The core relation

Definition 3.4.1. Let S = S_{0}/R be an I-gradedly almost hyperdefinable polygroup chunk.

(1) For a, a′ ∈ S_{0} we say that a ∼_{i1} a′ if there is x_{R} | ⌣ a_{R}a′_{R} such that a ∗ x ≈_{i} a′ ∗x. Let ∼_{in} be the n-th iterate of ∼_{i1}, and let ∼ be the I ×ω-graded direct limit of those, called the core relation.

(2) S is coreless if ∼ is the same as R, i.e. for every (i, n) ∈ I × ω, there is j such that R_{j} is coarser than ∼_{in}.

Lemma 3.4.2. (1) For every i there is j such that if a ∼_{i1} a′, there is x | ⌣ aa′

with a ∗ x ≈_{j} a′ ∗ x.

(2) ∼ is an I × ω-graded almost type-definable equivalence relation on P such that every ∼-class contains boundedly many R-classes (i.e. if a_{R} ∼ a′_{R}, then bdd(a_{R}) = bdd(a′_{R}) as almost-hyperimaginaries).

(3) For every i there is j such that a ∼_{i1} a′ if and only if there are c, b, c_{R} | ⌣ a_{R}a′_{R} with a, a′ ∈_{j} b ∗ c (and the other way around).

(4) ∼ is (gradedly and generically) regular: for every (i, n) ∈ I × ω there is (j, m) ∈ I × ω such that whenever a ∼_{in} a′ and b_{R} | ⌣ a_{R} (and thus also b_{R} | ⌣ a′_{R}), for every c ∈ a ∗ b there is c′ ∈ a′ ∗ b with c ∼_{jm} c′.

(5) By regularity, P/∼ is again a graded almost hyperdefinable polygroup chunk and it is coreless.

Proof.

(1) Let x_{R} | ⌣ a_{R}a′_{R} such that a ∗ x ≈_{i} a′ ∗ x. There are x_{0}R_{0}x, a_{0}R_{0}a, a′_{0}R_{0}a′ such that x_{0} | ⌣ a_{0}a′_{0}. By gradedness of ∗, a_{0} ∗ x_{0} ≈_{k} a′_{0} ∗ x_{0} for some k and we may assume that x_{0} | ⌣_{a_{0}a′_{0}} aa′. Thus, x_{0} | ⌣ aa′ and by gradedness of ∗ again, a ∗ x_{0} ≈_{j} a′ ∗ x_{0} for some j.

(2) If a ∼_{i1} a′, then there is j and x | ⌣ aa′ with a ∗ x ≈_{j} a′ ∗ x. Then, x | ⌣_{a} a′, and since a′_{R} ∈ bdd(xa) (a′_{R} ∈ (a_{R} ∗x_{R})∗x_{R}^{−1}), we get that a′_{R} | ⌣_{a} a′_{R}. Of course, we could have done the same for any a_{0}Ra, since if e.g. a_{0}R_{i}a ∼_{i1} a′, then clearly a_{0} ∼_{k1} a′ for some k. Therefore, a′_{R} ∈ bdd(a_{R}).

Thus, if R_{0} witnesses the almost type-definability of R, ∼_{01} will witness the almost type-definability of ∼.

(3) Let x_{R} | ⌣ a_{R}a′_{R} with a ∗ x ≈_{i} a′ ∗ x, and let y be in this intersection (up to R_{i}). Then, there is j such that a, a′ ∈_{j} y ∗ x^{−1}, and these are as required.

(4) Suppose a ∼_{i1} a′, and let (by previous claims) x, y and i′ be such that x | ⌣ aa′ and a, a′ ∈_{i′} x ∗ y. We may additionally choose xy | ⌣_{aa′} b. If c ∈ a ∗ b, there is i′′ such that c ∈_{i′′} (x ∗ y) ∗ b =_{0} x ∗ (y ∗ b), so there will be a z ∈ y ∗ b with c ∈_{i′′} x ∗ z. On the other hand, by associativity (from x ∗ (y ∗ b) =_{0} (x ∗ y) ∗ b), we get that x ∗ z ≈_{j′} a′ ∗ b, so pick c′ inside up to R_{j′}. Thus, we may assume (by coarsening R_{j′}) that c, c′ ∈_{j′} x ∗ z, and since x | ⌣ aa′b, we get that c ∼_{j1} c′ for some higher j.

(5) To see corelessness, let g_{∼} | ⌣ a_{∼}b_{∼} such that a, b ∈_{jn} g ∗ h. Let a′ ∼_{jn} a and b′ ∼_{jn} b with a′, b′ ∈ g ∗ h. Then, since each ∼-class is covered by boundedly many R-classes, g_{R} | ⌣ a′_{R}b′_{R}, so we have a ∼_{jn} a′ ∼_{01} b′ ∼_{jn} b and a ∼_{j,2n+1} b. □

It is possible to define a similar relation on a gradedly almost hyperdefinable polygroup, and it will have all the analogous properties to the above, and even more: the class of the scalar identity element will form a subpolygroup, and the quotient by the relation will be the quotient by the subpolygroup. We skip the details as we will not need it later. The definition is as follows.

Definition 3.4.3. Let P = P_{0}/R be an I-graded almost hyperdefinable polygroup. We will say a ∼_{i1} a′ if there is a generic element x_{R} with x_{R} | ⌣ a_{R}a′_{R} such that a∗x ≈_{i} a′∗x. Let ∼_{in} be the n-closure of ∼_{i1}, and let ∼ be the direct limit of all of those.

3.5. Group chunk theorem

Theorem 3.5.1. Let P = (P_{0}/R, ∗) be a graded almost hyperdefinable group chunk in a simple theory. There is a gradedly almost hyperdefinable group G and a graded bijection σ between P and the set of generics of G, such that generically ∗ is mapped to the group operation. Moreover, the construction of G = P^{2} has the following universal property: for any polygroup Z and a graded homomorphism φ : P → Z into generics of Z (i.e. φ(a ∗ b) ∈ φ(a) ∗ φ(b) for a | ⌣ b, and φ(a^{−1}) = φ(a)^{−1}), there is a unique graded φ^{2} : G → Z such that φ = φ^{2} ∘ σ. In particular, such G is unique up to a gradedly almost type-definable isomorphism.

\begin{matrix}P^{2} \\ P & Z \end{matrix}σ φ φ^{2}

Proof. It was already noted in [To] that the proof of the group chunk theorem by Wagner ([W], 4.7.1) is quite robust and can be done at the level of ultraimaginaries however bad the relation we are quotienting is. The point is that the independence conditions can be restricted to representatives.

In what follows, we assume that R_{0} is coarse enough to witness the graded almost hyperdefinability of R and to ‘absorb’ all the algebraic operations (e.g. if z ∈ x ∗ y, then x ∈_{0} z ∗ y^{−1}). In particular, if x | ⌣ yA, we can find z ∈_{0} x ∗ y with z | ⌣ yA (x | ⌣ yA implies x | ⌣_{y} A, so for z′ ∈ x ∗ y, z′_{R} ∈ bdd(xy) and z′_{R} | ⌣ y yield z′_{R} | ⌣ yA and there is zR_{0}z′ with z | ⌣ ya and z ∈_{0} x ∗ y).

Let us write (a, b)Q_{i}(a′, b′) for pairs from P_{0} if there are x, y ∈ P_{0} such that x | ⌣ aba′b′, y | ⌣ aba′b′ and a ∗ x ≈_{i} a′ ∗ y, b ∗ x ≈_{i} b′ ∗ y. Let Q be the direct limit of Q_{i} and let [a, b] := (a, b)/Q. Claim: Q is gradedly almost type-definable equivalence relation.

Clearly Q_{i}’s are reflexive and symmetric type-definable relations. For almost typedefinability, let R_{0} witness the almost type-definability of R. Fix some (a, b) and suppose e.g. (a′, b′) ∈ (a, b)/Q_{i}; there are x and y, x | ⌣ aba′b′, y | ⌣ aba′b′ with a ∗ x ≈_{i} a′ ∗ y and b ∗ x ≈_{i} b′ ∗ y. Let a/R ⊆ ⋃_{α<μ} a_{α}/R_{0} and b/R ⊆ ⋃_{α<μ} b_{α}/R_{0} (for small μ) and without loss of generality xy | ⌣_{aba′b′} āb̄.

By 3.3.5, we can find a′′R_{j}a and b′′R_{j}b with a′′ ∗ x ≈_{1} a′ ∗ y and b′′ ∗ x ≈_{1} b′ ∗ y for some fixed R_{1}. There will be a_{α} with a′′R_{0}a_{α} and b_{β} with b′′R_{0}b_{β}, so for a fixed 2 ∈ I, we will have a′ ∗ y ≈_{2} a_{α} ∗ x, b′ ∗ y ≈_{2} b_{β} ∗ x, and thus (a′, b′)Q_{2}(a_{l}, b_{k}), meaning that (a, b)/Q can be covered by the bounded union ⋃_{α<μ,β<μ}(a_{α}, b_{β})/Q_{2}.

Let us show the graded transitivity. Choose x, y witnessing (a, b)Q_{i}(a′, b′) with xy | ⌣_{aba′b′} a′′b′′, and x′, y′ witnessing (a, b)Q_{j}(a′′, b′′) with x′y′ | ⌣_{aba′′b′′} a′b′xy. In particular, this implies that each of x, y, x′, y′ is independent of aba′b′a′′b′′. Take u ∈ P_{0} independent of everything. Then, for some t ∈_{0} x ∗ u, t | ⌣ aba′b′a′′b′′xx′yy′ by generic independence. By generic surjectivity, let u′ ∈_{0} x^{′−1}∗t with u′ | ⌣ aba′b′a′′b′′x′y′. Then, by generic associativity (as all the triples below are independent):

\begin{aligned}a′ ∗ (y ∗ u) & ≈_{0} (a′ ∗ y) ∗ u ≈_{i′} (a′ ∗ x) ∗ u ≈_{0} a ∗ (x ∗ u) \\ & ≈_{1} a ∗ (x′ ∗ u′) ≈_{0} (a ∗ x′) ∗ u′ ≈_{j′} (a′′ ∗ y′) ∗ u′ ≈_{0} a′′ ∗ (y′ ∗ u′), \end{aligned}

for some constant R_{1}, i′ depending only on i and j′ depending on j, and thus a′ ∗ (y ∗ u) ≈_{k} a′′ ∗ (y′ ∗ u′) for some k depending only on i and j. Similarly we get b′ ∗ (y ∗ u) ≈_{k} b′′ ∗ (y′ ∗ u′). Now, since ∗ defines a (single-valued) product, we may assume that y ∗ u is covered by a single R_{0}-class, so we only may need to coarsen k a little to choose v ∈_{0} y ∗ u and v′ ∈_{0} y′ ∗ u′ with a′ ∗ v ≈_{k} a′′ ∗ v′ and b′ ∗ v ≈_{k} b′′ ∗ v′, showing (a′, b′)Q_{k}(a′′, b′′). Claim: if a ∗ x ≈_{i} b ∗ x for some x with x | ⌣ a, x | ⌣ b, there is j such that for every y with y | ⌣ a, y | ⌣ b, a ∗ y ≈_{j} b ∗ y.

Suppose we have z with z | ⌣ ax and z | ⌣ bx. Let c ∈_{0} z^{−1} ∗ x, with c | ⌣ x. We may assume c | ⌣_{xz} ab, which will imply a | ⌣ cxz and b | ⌣ cxz and {a, c, x}, {b, c, x} will be independent triples. Thus,

a ∗ z ≈_{1} a ∗ (x ∗ c) ≈_{0} (a ∗ x) ∗ x ≈_{i′} (b ∗ x) ∗ c ≈_{0} b ∗ (x ∗ c) ≈_{1} b ∗ z,

for some constant R_{1}, and i′ depending only on i, so there will be k depending only on i such that a ∗ z ≈_{k} b ∗ z. Now, if we let z | ⌣ abxy, by above we will have that a ∗ z ≈_{k} b ∗ z, but, since also y | ⌣ az and y | ⌣ bz, by the proof above, there will be j such that a ∗ y ≈_{j} b ∗ y. Claim: If x is such that x | ⌣ ab, then for every i, there is j such that if a′ ∈_{i} a ∗ x, b′ ∈_{i} b ∗ x, (a, b)Q_{j}(a′, b′). If c | ⌣ ab, there is k and there are d, d′ with d | ⌣ ab and d′ | ⌣ ab such that (a, b)Q_{k}(c, d) and (a, b)Q_{k}(d′, c).

Let y | ⌣ abx. As a′_{R} ∈ bdd(ax) and b′_{R} ∈ bdd(bx), there will be a′′R_{0}a′ and b′′R_{0}b′ such that y | ⌣ aba′′b′′x. Let z ∈_{0} x∗y such that z | ⌣ aba′′b′′. Then, a∗z ≈_{1} a∗(x∗y) ≈_{0} (a ∗ x) ∗ y ≈_{i′} a′′ ∗ y and b ∗ z ≈_{1} b ∗ (x ∗ y) ≈_{0} (b ∗ x) ∗ y ≈_{i′} b′′ ∗ y for some R_{1} and i′ depending only on i. We may assume yz | ⌣_{aba′′b′′} a′b′, so y | ⌣ aba′b′, z | ⌣ aba′b′ and a ∗ z ≈_{i′′} a′′ ∗ y ≈_{1} a′ ∗ y by gradedness, as a′R_{0}a′′, so a ∗ z ≈_{j} b′ ∗ y and similarly a ∗ z ≈_{j} b′ ∗ y for some j depending only on i, showing (a, b)Q_{j}(a′, b′).

If c | ⌣ ab, let x ∈_{0} a^{−1} ∗ c with x | ⌣ ab. Then, c ∈_{1} a ∗ x for some R_{1}, pick any d ∈_{1} b ∗ x and by the previous paragraph there is R_{k} such that (c, d)Q_{k}(a, b).

Now, for (a_{0}, b_{0}) and (b_{1}, c_{0}), we can pick b | ⌣ a_{0}b_{0}b_{1}c_{0} so by the previous claim there will be (a, b)Q_{k}(a_{0}, b_{0}) and (b, c)Q_{k}(b_{1}, c_{0}). Thus, it makes sense to define (a_{0}, b_{0}) ∘ (b_{1}, c_{0}) = {(a, c) : ∃b, (a, b)Q_{k}(a_{0}, b_{0}) ∧ (b, c)Q_{k}(b_{1}, c_{0})}, this relation clearly being type-definable. Claim: The relation ∘ gradedly defines a single-valued operation on R-classes.

To see this, it is enough to show that for every i, there is j such that if (a′, b′)Q_{i}(a, b) and (b′, c′)Q_{i}(b, c), then (a′, c′)Q_{j}(a, c). Let x | ⌣ aba′b′ and y | ⌣ aba′b′ with a ∗ x ≈_{i} a′ ∗ y, b ∗ x ≈_{i} b′ ∗ y, x′ | ⌣ bcb′c′ and y′ | ⌣ bcb′c′ with b ∗ x ≈_{i} b′ ∗ y, c ∗ x ≈_{i} c′ ∗ y. We may choose x′y′ | ⌣_{bcb′c′} aa′xy, so in particular x′ | ⌣ x. By generic surjectivity, let z ∈_{0} x^{−1} ∗ x′ with z | ⌣ aba′b′xy (since x′ | ⌣ abca′b′c′xy), and the line below makes sense (where 1 ∈ I depends only on 0):

b′ ∗ y′ ≈_{i} b ∗ x′ ≈_{1} b ∗ (x ∗ z) ≈_{0} (b ∗ x) ∗ z ≈_{i′} (b′ ∗ y) ∗ z ≈_{0} b′ ∗ (y ∗ z).

Let t ∈_{0} y ∗ z with t | ⌣ y (implying t | ⌣ b′) such that b′ ∗ y′ ≈_{k} b′ ∗ t, k depending only on i. As b′ | ⌣ y′ and b′ | ⌣ t, by one of the previous claims we get that there is k′ such that for every u with u | ⌣ y′, u | ⌣ t, u ∗ y′ ≈_{k′} u ∗ t. In particular,

a′ ∗ y′ ≈_{k′} a′ ∗ t ≈_{1} a′ ∗ (y ∗ z) ≈_{0} (a′ ∗ y) ∗ z ≈_{i′} (a ∗ x) ∗ z ≈_{0} a ∗ (x ∗ z) ≈_{1} a ∗ x′.

Thus, for some j depending only on i, a′ ∗ y′ ≈_{j} a ∗ x′, and similarly c′ ∗ y ≈_{j} c ∗ x′, so (a′, c′)Q_{j}(a, c). Claim: There is j such that for every a, b, (a, a)Q_{j}(b, b).

Let c | ⌣ ab, x ∈_{0} a^{−1} ∗ c and y ∈_{0} b^{−1} ∗ c such that x | ⌣ ab and y | ⌣ ab. By the previous claim, there is some i with (a, a)Q_{i}(c, c)Q_{i}(b, b) so (a, a)Q_{j}(b, b) for some fixed j.

Clearly, [a, a] will act as a unit for multiplication, and any class [a, b] has an inverse [b, a]. Claim: Multiplication is (gradedly) associative, i.e. there is k such that ((a, b)∘(a′, b′))∘ (a′′, b′′) =_{k} (a, b) ∘ ((a′, b′) ∘ (a′′, b′′)).

Let x | ⌣ aba′b′a′′b′′ and find a_{0} ∈_{0} a′ ∗ x and b_{0} ∈_{0} b′ ∗ x with a_{0} | ⌣ aba′′b′′ and b_{0} | ⌣ aba′′b′′. By the previous claims, there is i such that we can find c and c′ with (a, b)Q_{i}(c, a_{0}) and (a′′, b′′)Q_{i}(b_{0}, c′). By gradedness of ∘, there will be j with ((a, b) ∘ (a′, b′))∘(a′′, b′′) =_{j} ((c, a_{0})∘(a_{0}, b_{0}))∘(b_{0}, c′) =_{j} (c, c′), but the same result is obtained (up to Q_{j}) from (a, b) ∘ ((a′, b′) ∘ (a′′, b′′)), so the existence of k is clear. Thus, the set of Q-classes forms an almost hyperdefinable group G = (P_{0} × P_{0})/Q with ∘ inducing the operation. Claim: For every i there is k such that for any x | ⌣ a and y | ⌣ a, any a′ ∈_{i} a ∗ x, a′′ ∈_{i} a ∗ y, (a′, x)Q_{k}(a′′, y).

Let u | ⌣ axy and z ∈_{0} x^{−1} ∗ u with z | ⌣ axy, z′ ∈_{0} y^{−1} ∗ u with z′ | ⌣ axy. Since a′ ∗ z =_{i′} (a ∗ x) ∗ z =_{0} a ∗ (x ∗ z) =_{1} a ∗ (y ∗ z′) =_{0} (a ∗ y) ∗ z′ =_{i′} a′′ ∗ z′, pick t ∈_{j} a′ ∗ z and t ∈_{j} a′′ ∗ z′ and by previous claims (a′, x)Q_{j′}(t, u)Q_{j′}(a′′, y), so (a′, x)Q_{k}(a′′, y) for some k.

The above claim gives rise to a gradedly almost type-definable map σ : P → G. Since the reader should be accustomed to extending the classical arguments to ac-

commodate almost-hyperimaginaries by now, we shall only run through the remaining intermediate claims needed for the proof at the ultraimaginary level. Claim: G = σ(P)^{2}.

If a, b ∈ P_{0}, pick x | ⌣ ab. Let y ∈_{0} a ∗ x^{−1} with y | ⌣ ab and let y′ ∈_{0} x ∗ b^{−1} with y′ | ⌣ ab. Then [a, b] = [a, x] ∘ [x, b] = [y ∗ x, x] ∘ [y′ ∗ b, b] ∈ σ(P)^{2}. Claim: σ is injective.

Suppose σ(a) = σ(b) for a, b ∈ P_{0}. If x | ⌣ ab, [a ∗ x, x] = [b ∗ x, x], so [a, b] = [a ∗ x, b ∗ x] = [a ∗ x, x] ∘ [b ∗ x, x]^{−1} = 1_{G} = [a, a]. Thus, there are x | ⌣ ab and y | ⌣ ab with a ∗ x =_{R} a ∗ y =_{R} b ∗ x; therefore a ∗ z =_{R} b ∗ z for all z with z | ⌣ a, z | ⌣ b. Take c | ⌣ ab and let a_{1} =_{R} a ∗ c^{−1} with a_{1} | ⌣ ab, and b_{1} =_{R} b ∗ c^{−1} with b_{1} | ⌣ ab. For any y | ⌣ aa_{1}bb_{1}c the triples {a_{1}, c, y} and {b_{1}, c, y} are independent, so:

a_{1} ∗ (c ∗ y) =_{R} (a_{1} ∗ c) ∗ y =_{R} a ∗ y =_{R} b ∗ y =_{R} (b_{1} ∗ c) ∗ y =_{R} b_{1} ∗ (c ∗ y).

Therefore a_{1} ∗ x =_{R} b_{a} ∗ x for all x with x | ⌣ a_{1} and x | ⌣ b_{1}. Thus a =_{R} a_{1} ∗ x =_{R} b_{1} ∗ x =_{R} b. Claim: σ generically preserves multiplication and maps P onto the generics of G.

Let a | ⌣ b and take x | ⌣ ab. Then

σ(a ∗ b) = [a ∗ b ∗ x, x] = [a ∗ b ∗ x, b ∗ x] ∘ [b ∗ x, x] = σ(a) ∘ σ(b).

Let c ∈ P_{0} and consider a and b with ab | ⌣ c and let c′Rc ∗ a with c′ | ⌣ ab. Then σ(c) = [c′, a] and σ(c) ∘ [a, b] = [c′, b]. As c′ | ⌣ ab, let dRc′ ∗ b^{−1} with d | ⌣ ab. So σ(c) ∘ [a, b] = [d ∗ b, b] = σ(d). Hence σ(c) ∘ [a, b] | ⌣[a, b] and σ(c) is generic.

Suppose now we have a graded homomorphism φ : P → Z. We may define φ^{2}([a, b]) := φ(b) ∗ φ(a)^{−1} ∩ φ(b′) ∗ φ(a′)^{−1}, for some [a′, b′] = [a, b] with a′ | ⌣ ab (such (a′, b′) can be found by translating by x | ⌣ ab). Clearly such a map is welldefined and single-valued, seen as follows. If [a, b] = [a′, b′], let t | ⌣ aba′b′ such that b ∗ (a^{−1} ∗ t) = b ∗ (a′^{−1} ∗ t), so φ(b) ∗ φ(a^{−1}) ∗ φ(t) ≈ φ(b′) ∗ φ(a′^{−1}) ∗ φ(t); pick z in the intersection. Then φ(b) ∗ φ(a)^{−1} ≈ z ∗ φ(t)^{−1} ≈ φ(b′) ∗ φ(a′)^{−1}, and choose h in the first and k in the second intersection. Since φ(t) | ⌣ hk, they are core-related and the map has nonempty value. It is also single-valued as each two potential values are core-related. The reader can check that this can be done gradedly.

The uniqueness part follows from the universal property. □

Remark 3.5.2. If in the previous theorem the map φ happens to be (generically) onto of type 3 (i.e. if φ(C) ∈ a ∗ b for a | ⌣ b, there are A and B with φ(A) = a, φ(B) = b and C = A ∗ B), the induced map φ^{2} is onto of type 3.

Proof. Let us show that φ^{2} is of type 3. Assume φ^{2}([E, F]) = z ∈ u ∗ v. We may assume E | ⌣ uv and F | ⌣ uv by translation, and let X | ⌣ EF uv and E′ := E ∗ X, F′ := F ∗ X, so by definition of φ^{2}, z = f ∗ e^{−1} ∩ f′ ∗ e′^{−1}, where φ(E) = e, φ(F) = f, φ(E′) = e′ and φ(F′) = f′. Since φ is a homomorphism, φ(X) =: x ∈ e^{−1}∗e′∩f^{−1}∗f′. Now, since z ∈ f′ ∗ e′^{−1} ∩ u ∗ v, by transposition we can find b′ ∈ u^{−1} ∗ f′ ∩ v ∗ e′. By the second form of associativity, from u^{−1} ∗ (f′ ∗ x^{−1}) = (u^{−1} ∗ f′) ∗ x^{−1} we get that u^{−1} ∗f ≈ b′ ∗x^{−1} and similarly v ∗e ≈ b′ ∗x^{−1}. Since x | ⌣ efuv, any two elements from the two intersections are core-equivalent and we can choose b ∈ u^{−1} ∗f ∩v ∗e∩b′ ∗x^{−1}. By the fact that x, b and b′ are generic and φ is generically onto of type 3, let B and B′ be such that B′ = B ∗ X, φ(B) = b and φ(B′) = b′. Now, it is easily checked that [E, B] = [E′, B′], [B, F] = [B′, F′] and they witness that u = φ^{2}([E, B]), v = φ^{2}([B, F]) and [E, F] = [E, B] ∘ [B, F]. □

Corollary 3.5.3. If (P, Y, ∗) (∗ standing for both the group chunk operation and the generic action) is a gradedly almost hyperdefinable space chunk in a simple theory, then there is a gradedly almost hyperdefinable space (G, X, ∘) and a graded bijection σ between (P, Y) and the generics of (G, X) such that generically ∗ is mapped onto the group operation and action ∘.

Proof. We shall not worry about the graded almost hyperdefinability of objects we construct in this proof, assuming the reader can rewrite the material in the style of the previous proof. Alternatively, the reader can imagine we are working in the hyperdefinable category.

We need to consider the dual equivalence relation between the pairs from P to the one given in the proof above, since we wish to deal here with the generic action of P on X from the left: we say that (a, b)L(a′b′) if there are u | ⌣ aba′b′, v | ⌣ aba′b′ with u ∗ a = v ∗ a′ and u ∗ b = v ∗ b′. Then it is clear that G := (P × P)/L is a group with the usual ∘. Note that [a, b] = [a′, b′] if and only if there is t | ⌣ aba′b′ with (t ∗ a^{−1}) ∗ b = (t ∗ a′^{−1}) ∗ b′.

Let X := (P × P × Y)/L′, where (a, b, x)L′(a′, b′, x′) if there is an u | ⌣ abxa′b′x′ such that ((u ∗ a^{−1}) ∗ b) ∗ x = ((u ∗ a′^{−1}) ∗ b′) ∗ x′. It is easily shown that this is an equivalence relation and we denote the class of (a, b, x) by [a, b, x].

Notice, if [a, b] = [a′, b′], then trivially [a, b, x] = [a′, b′, x]. Thus, for any [a′, b′] ∈ G and [b′′, c′, x], by results for L-classes, we can find [a, b] = [a′, b′] and [b, c, x] = [b′′, c′, x′], and we can define the action of G on X by [a, b] ∘ [b, c, x] := [a, c, x]. To see that it is well-defined, let [a′, b′] = [a, b] and [b′, c′, x′] = [b, c, x]. Let u | ⌣ aba′b′ such that (u ∗ a^{−1}) ∗ b = (u ∗ a′^{−1}) ∗ b′ and we can pick v | ⌣ bcxb′c′x′u such that ((v ∗ b^{−1}) ∗ c) ∗ x = ((v ∗ b′^{−1}) ∗ c) ∗ x. Choose any z | ⌣ abcxa′b′c′x′uv and let z′ be such that z^{−1} = z′ ∗ v (in particular, z′ is again independent of everything). Now, since {z, u ∗ a^{−1}, b} is an independent triple (and similarly with dashes), by generic associativity we get that (u ∗ a^{−1}) ∗ (b ∗ z) = (u ∗ a′^{−1}) ∗ (b′ ∗ z) =: t^{−1} (where t is again independent of everything). Also, since {z′, (v ∗ b^{−1}) ∗ c, x}, {z′, v ∗ b^{−1}, c} and {z′, v, b} are independent triples,

\begin{aligned}z′ ∗ (((v ∗ b^{−1}) ∗ c) ∗ x) & = (z′ ∗ ((v ∗ b^{−1}) ∗ c)) ∗ x = ((z′ ∗ (v ∗ b^{−1})) ∗ c) ∗ x \\ & = (((z′ ∗ v) ∗ b^{−1}) ∗ c) ∗ x = ((z^{−1} ∗ b^{−1}) ∗ c) ∗ x. \end{aligned}

Similarly we get that z′ ∗ (((v ∗ b′^{−1}) ∗ c′) ∗ x′) = ((z^{−1} ∗ b′^{−1}) ∗ c′) ∗ x′. Now, since (b ∗ z)^{−1} = t ∗ (u ∗ a^{−1}) = (t ∗ u) ∗ a^{−1} and (b′ ∗ z)^{−1} = (t ∗ u) ∗ a′^{−1}, we get that (((t∗u)∗a^{−1})∗c)∗x = (((t∗u)∗a′^{−1})∗c′)∗x′ so t∗u witnesses that [a, c, x] = [a′, c′, x′].

It is clear that (P, Y) embeds into (G, X) via x ↦ [a, a, x] for x ∈ X and any a ∈ G and, a ↦ [b, b ∗ a] for any a ∈ G and b | ⌣ a. To see e.g. that for a | ⌣ x, σ(a ∗ x) = σ(a) ∘ σ(x), it is enough to notice that if whenever a | ⌣ b and a | ⌣ x, [b, b, a ∗ x] = [b, b ∗ a, x]. By the previous theorem, it is clear that P is in bijection with generics of G. Suppose now that x_{0} is generic in X. Then, by 3.3.13, there are g ∈ G generic and y ∈ X such that g | ⌣ y and x_{0} = g ∘ y. Let y = [b, c, x] such that g | ⌣ bcx and g = [a, a ∗ g] for a | ⌣ gbcx. Now, since a ∗ g ∗ b^{−1} | ⌣ bc, g ∗ b^{−1} ∗ c | ⌣ a and g ∗ b^{−1} ∗ c | ⌣ x,

\begin{matrix}g ∘ y = & [a, a ∗ g] ∘ [b, c, x] = [a, a ∗ g] ∘ [a ∗ g, (a ∗ g ∗ b^{−1}) ∗ c, x] \\ = & [a, a ∗ (g ∗ b^{−1} ∗ c), x] = [a, a, (g ∗ b^{−1} ∗ c) ∗ x], \end{matrix}

showing that x_{0} is in the image of σ. □

3.6. Stabilizers

In this section, we work in the hyperdefinable category, assuming the interested reader can extend the results to gradedly almost hyperdefinable objects. If π is a partial type over a set A, we say it is generic if it is contained in a generic type.

Lemma 3.6.1. The following are equivalent in a hyperdefinable polygroup P:

(1) π is generic over A;

(2) for every b ∈ P, b ∗ π does not fork over ∅;

(3) for every b ∈ P, b ∗ π does not fork over A.

Proof.

(1) ⇒ (2). Since π is generic, let a ⊧ π be generic over A with a | ⌣_{A} b. By genericity, if c ∈ b ∗ a, c | ⌣ Ab, but c ⊧ b ∗ π so b ∗ π does not fork over ∅.

(2) ⇒ (3) is trivial.

(3) ⇒ (1). Let b be generic over A. Then b ∗ π does not fork over A so there are a ⊧ π and c ∈ b ∗ a with c | ⌣_{A} b. Therefore b and b^{−1} are generic over Ac, and so is a ∈ b^{−1} ∗ c. □

Definition 3.6.2. Let P be a (hyperdefinable) polygroup, and let p be a type in P over A. Let S(p) := {g ∈ P : g ∗ p ∪ p doesn’t fork over A}, and let stab(p) be the subpolygroup of P generated by S(p).

Lemma 3.6.3. (1) g ∈ S(p) if and only if there are u, v ⊧ p, u | ⌣_{A} g, v | ⌣_{A} g,

u ∈ g ∗ v;

(2) g ∈ S(p) if and only if g^{−1} ∈ S(p);
(3) if g | ⌣_{A} g′ ∈ S(p) and p is an amalgamation base, then g ∗ g′ ≈ S(p).

Proof.

(1) By definition, if g ∈ S(p), there are u, v ⊧ p such that u ∈ g ∗ v and u | ⌣_{A} g. Then, for every D(·) = D(·, φ, ϵ, k),

D(p) = D(v/A) ≥ D(v/Ag) = D(u/Ag) = D(u/A) = D(p).

Thus we have equality all the way through and v | ⌣_{A} g.

(2) is a trivial consequence of (1).

(3) Let u, v ⊧ p, u | ⌣_{A} g, v | ⌣_{A} g, u ∈ g∗v and v′, w ⊧ p, v′ | ⌣_{A} g′, w | ⌣_{A} g′, u ∈ g′∗v. Now, by the Independence Theorem, we may assume that v′ = v | ⌣_{A} gg′. Thus, we have u ∈ g ∗ (g′ ∗ w) = (g ∗ g′) ∗ w, so there is g′′ ∈ g ∗ g′ with u ∈ g′′ ∗ w, and obviously u | ⌣_{A} g′′, so g′′ ∈ S(p). □

Remark 3.6.4. Clearly S(p) is a hyperdefinable set (over A). Unfortunately, the condition proved above, that g | ⌣_{A} g′ ∈ S(p) implies g ∗ g′ ≈ S(P) is too weak to show (without invoking the later techniques used in blowing-up) that stab(p) is generated in finitely many steps from S(P) (if we had g ∗ g′ ⊆ S(p), then it would be generated in two steps like in the case of groups). Nevertheless, it will be an invariant (over A) object.

Lemma 3.6.5. If X is a generic hyperdefinable (over A) set, the subpolygroup H generated by X has a bounded index in P.

Proof. Suppose H has an unbounded index in P. Then, we can find an A-indiscernible sequence ⟨g_{i} : i < ω⟩, such that {g_{i} ∗H : i < ω} are distinct cosets. Then, {g_{i} ∗X : i < ω} is inconsistent, implying that g_{0} ∗X forks over A, so, by 3.6.1, X is not generic. □

Lemma 3.6.6. A type p (over ∅, say) in a hyperdefinable polygroup is generic if and only if S(p) is generic.

Proof. If p is generic, let u | ⌣ v ⊧ p and let g ∈ u ∗ v^{−1}. By genericity of u and v, u | ⌣ g, v | ⌣ g, and u ∈ g ∗ v, so g ∈ S(p) is generic.

The converse if similarly straightforward. □

CHAPTER 4 Blowup

Our original aim was to obtain a group from a group configuration in a simple theory. In Chapter 2, we managed to extract a polygroup chunk (P, ∗) from this information, and it is perfectly natural to hope that some variant of Hrushovski-Weil group chunk theorem (see 3.5.1) will yield a group, or at least a polygroup. The technical difficulties imposed by multivaluedness, however, are overwhelming, one of the reasons being that this polygroup chunk, in spite of some good properties like generalised associativity to be discussed below, has some serious shortcomings. For example, even if y, y′ ∈ f(x), for f ∈ P, x | ⌣ f, it can a priori happen that y | ⌣ y′. Also, if h, h′ ∈ f ∗ g for f | ⌣ g ∈ P, we might have h | ⌣ h′.

Generalised associativity properties allow us to blow up the original polygroup chunk and obtain an improved polygroup chunk which actually becomes a group chunk modulo the core relation from 3.4, where 3.5.1 is going to be applicable. The construction has a familiar algebraic-geometric flavour which, together with the universal property it satisfies, should hopefully provide a good enough justification for its name.

However, there is an even more important historical reference to be made here. Model theoretic group configuration originates in the classical reconstruction of a division ring from a projective geometry by von Staudt, Hilbert, Veblen-Young, Artin and von Neumann. A blowup procedure extracting a group (resp. space) chunk out of the polygroup (resp. polyspace) chunk is written in Section 4.2 in a way which is, at the notational level, completely parallel to some parts of von Neumann’s proof from [vN]. Also, the analogy to linear algebra (e-numbers being matrices, e-tuples vectors, multiplication of numbers corresponding to matrix multiplication, action of a number on a tuple corresponding to action of a matrix on a vector) we find rather amusing. This is not at all surprising given that, at a very abstract level, the axioms of a polygroup correspond to the axioms of a modular lattice.

Of course, it is desirable to keep the construction hyperdefinable as long as we can, because our hope is that the resulting group can be obtained hyperdefinable. That is why in Section 4.1 we present a blowup construction which produces a (hyperdefinable) partial generic multiaction which in fact is the improved polygroup chunk. One drawback of this construction is that it is even more localised than the previous one.

In order to obtain a construction of a global character, in Section 4.3 we discuss a sheaf-like (or manifold-like, depending on the reader’s mathematical taste) blowup, constructed by pasting together the local blowups. It satisfies the universal property normally associated with blowing-up in algebraic geometry.

The first rather naive attempt to obtain a group from the polygroup chunk just by quotienting described in [To], in the particular example of a double coset space G // H corresponds to descending to the group G/N, where N is the normal closure of H. This was unsatisfactory since we couldn’t guarantee the nontriviality of the resulting group. Wagner suggested that, since we require G // H to be a type-definable polygroup, one of the sufficient conditions for boundedness of products is that the family of conjugates of H be uniformly commensurable. But then, Schlichting’s theorem ([W], 4.2.4) implies the existence of a normal subgroup N uniformly commensurable with the family of conjugates of H and G/N is thus a nontrivial group. This suggested that it would be better to look for the group ‘interpreted’ in the polygroup chunk, which resulted in a variety of blowup procedures by Ben-Yaacov and myself. I present three of mine here. It is interesting to notice a posteriori that a construction of a character similar to our blowup was used by Pillay and Kowalski in [KoP] to obtain the group configuration in the special case of ACFA, where the ‘blowup map’ is roughly a ↦ ⟨σ^{i}(a) : i ∈ ℤ⟩.

4.1. Hyperdefinable blowup

Let (P, ∗) be the polygroup chunk of germs obtained from a suitable partial generic multiaction π, as in 2.3.10. We present here a construction of an improved polygroup chunk where whenever y, y′ ∈ f(x), bdd(y) = bdd(y′) and if h, h′ ∈ f ∗ g, bdd(h) = bdd(h′) and thus quotienting by the core relation gives an almost hyperdefinable group chunk. Intuitively, we force each point into an intersection of ‘independent enough’ products, which suffices to determine it better (e.g. in an algebraic quadrangle, each point is determined up to interboundedness, or in the example 1.4.4, a point is uniquely determined in the intersection of two lines).

Lemma 4.1.1. There are e and f in P and x_{0}, x, y_{0}, y such that x | ⌣ ef, x_{0}x ⊧ e, y_{0}y ⊧ e, xy ⊧ f.

Proof. Start with some x′_{0}x ⊧ e′, xy′ ⊧ f′. Now, x ≡^{L} y′, x | ⌣ e′, y′ | ⌣ x, e′ | ⌣ x, so by the Independence Theorem, there is y | ⌣ xe′, φ : x ≡_{e′} y, ψ : y′ ≡_{x} y, and now x′_{0}x ⊧ e′, φ(x′_{0})y ⊧ e′, xy ⊧ ψ(f′). Now y | ⌣ xe′ implies e′ | ⌣ xy, so let e realise the nonforking extension of tp(e′/xy) to xyf, where f = ψ(f′). Thus, for some x_{0} and y_{0}, x_{0}x ⊧ e, y_{0}y ⊧ e, xy ⊧ f and x | ⌣ ef. □

From now on, fix an e as in the lemma.

Definition 4.1.2. Let x̃ stand for x_{0}x, f̃ for ⟨f,_{0}f, f_{0}⟩. We define a partial generic multiaction π^{2} as follows: π^{2}(f̃, x̃, ỹ) if f | ⌣ e, x_{0}xy ⊧ f_{0} ∈ f ∗ e, xyy_{0} ⊧ _{0}f ∈ e^{−1} ∗ f, and in such a case we shall write x̃ỹ ⊧ f̃.

We can complete and reduce π^{2} to get π̄^{2}, although the reduction doesn’t change the completion \underline{π}^{2}: if x̃ | ⌣_{e} f̃_{p}g̃_{q} with ỹ ∈ f̃_{p}(x̃) ∩ g̃_{q}(x̃), then x | ⌣ fg, y ∈ f(x) ∩ g(x), so f = g since they are both germs. Similarly, x_{0}y witness f_{0} = g_{0} and xy_{0} witness 0f = _{0}g. Now, we claim that, in the language from Chapter 2 (we abuse the notation a little in what follows, as we omit ‘_{p}’ from f̃_{p} ∈ Fun(π̄^{2}), but we bear in mind that lstp(x̃ỹ/f̃) for x̃ỹ ⊧ f̃ is fixed):

Theorem 4.1.3. The pair (P^{2}, ∗), where P^{2} = Fun(π̄^{2}) and multioperation ∗ : P^{2} ⊗_{e}

P^{2} → ℙ^{bdd}(P^{2}) is given by g̃ ∗ f̃ := { \atop g̃ ∘ f̃} (for f̃ | ⌣_{e} g̃), forms a (hyperdefinable) polygroup chunk with:

· for all ỹ, ỹ′ ∈ f̃(x̃), bdd(ỹ) = bdd(ỹ′);

· for all h̃, h̃′ ∈ f̃∗ g̃, bdd(h̃) = bdd(h̃′) (in fact, h̃ and h̃′ are core-related).

Proof. For the reader who refuses to believe the notation as a ‘black box’, here is a detailed proof that for any f̃ | ⌣_{e} g̃, f̃ ∗ g̃ ≠ ∅ (the reason essentially being that Arg(π^{2}) = Γ(e) is an amalgamation base).

Let x̃ỹ ⊧ f̃, ỹ′z̃ ⊧ g̃. As ỹ ≡_{e}^{L} ỹ′, ỹ | ⌣_{e} f̃, ỹ′ | ⌣_{e} g̃, by the Independence Theorem, we may assume that ỹ′ = ỹ | ⌣_{e} f̃g̃ and we can take h := Cb(xz/fg) ∈ f ∗ g ∩ g_{0} ∗_{0}f, h_{0} := Cb(x_{0}z/gf_{0}) ∈ h ∗ e ∩ g ∗ f_{0}, _{0}h := Cb(xz_{0}/_{0}gf) ∈ e^{−1} ∗ h ∩_{0}g ∗ f. In the case as above we write x̃ỹz̃ ⊧ h̃ ∈ g̃ ∗ f̃.

It is easy to see this product is e.g. associative: let {f̃, g̃, h̃} be e-independent, let x̃ỹz̃ ⊧ k̃ ∈ f̃∗ g̃ and ũx̃′z̃′ ⊧ l̃ ∈ k̃ ∗ h̃. Since we are in a complete situation, x̃z̃ ≡L ; k̃ x̃′z̃′, x̃z̃ | ⌣k̃ f̃g̃, x̃′z̃′ | ⌣k̃ l̃h̃, and f̃g̃ | ⌣k̃ l̃h̃, so by the Independence Theorem, we may assume that x̃′z̃′ = x̃z̃ | ⌣k̃ f̃g̃h̃l̃, and if we let m := Cb(uy/gh), m_{0} := Cb(u_{0}y/gh_{0}), 0m := Cb(uy_{0}/_{0}gh), we have l̃∈ f̃∗ m̃ and m̃ ∈ g̃ ∗ h̃.

If ỹ, ỹ′ ∈ f̃(x̃), then y, y′ ∈ f(x) ∩ f_{0}(x_{0}), but as fx | ⌣_{y} f_{0}x_{0} (x_{0}xy ⊧ f_{0} ∈ f ∗ e, so x_{0} | ⌣ ff_{0} and as f_{0} | ⌣ f we get f | ⌣ f_{0}x_{0} and the required independency follows from y ∈ bdd(f_{0}x_{0}) and x ∈ bdd(fy)), this implies y′ | ⌣_{y} y′ so y′ ∈ bdd(y). Similarly one sees the interboundedness of y_{0} and y′_{0}.

If h̃, h̃′ ∈ f̃∗ g̃, then h, h′ ∈ f ∗ g ∩ g_{0} ∗_{0}f, and f | ⌣_{0}gf_{0} (ef | ⌣ g implies ff_{0} | ⌣ g_{0} which gives the required) so h and h′ are core-related and thus interbounded. The interboundedness of h_{0} and h′_{0}, as well as _{0}h and _{0}h′ follows in a similar way. □

The following proposition states that our polygroup chunk has ‘generalised associativity’ for each α.

Proposition 4.1.4. Let {g_{i} : i < α} be independent germs. Then, for any choice of {_{0}g_{i} ∈ g_{0}^{−1} ∗ g_{i} : 0 < i < α}, there are {_{i}g_{j} ∈ g_{i}^{−1} ∗ g_{j} : i ≠ j < α} such that for all {i, j, k}, _{i}g_{j} ∈ _{i}g_{k} ∗_{k}g_{j}, and _{i}g_{j}^{−1} = _{j}g_{i}. Moreover, for any f | ⌣{g_{i},_{i}g_{j} : i ≠ j < α}, we can find f_{i} ∈ f ∗ g_{i} with f_{j} ∈ f_{i} ∗_{i}g_{j} for every i ≠ j.

We call any such ⟨g_{i},_{i}g_{j} : i ≠ j < α⟩ an α-frame.

Proof. Let a_{i}b_{i}c_{i} witness _{0}g_{i} ∈ g_{0}^{−1} ∗ g_{i}, i.e. a_{i}b_{i} ⊧ Γ(g_{i}), c_{i}b_{i} ⊧ Γ(g_{0}), a_{i} | ⌣ g_{0}g_{i}_{0}g_{i}. By induction on k, it is possible to assume: (‡) there are bc with bc | ⌣_{g_{0}}{g_{i},_{0}g_{i} :

0 < i ≤ k} such that a_{i}bc ⊧ _{0}g_{i} ∈ g_{0}^{−1} ∗ g_{i}, for 0 < i ≤ k. For k = 1 there’s nothing to prove. For the successor case, suppose (‡) holds for k and some bc. By hypotheses of the proposition, c_{k+1}b_{k+1} ≡_{g_{0}}^{L} cb, cb | ⌣_{g_{0}}{g_{i},_{0}g_{i} : i ≤ k}, c_{k+1}b_{k+1} | ⌣_{g_{0}} g_{k+1} and g_{k+1} | ⌣_{g_{0}}{g_{i},_{0}g_{i} : i ≤ k} so by the Independence Theorem we may assume c_{k+1}b_{k+1} = cb | ⌣_{g_{0}}{g_{i},_{0}g_{i} : i ≤ k+1}. If l is a limit and (‡) holds for all k < l, then by compactness we can find bc which are ‘good’ for all k < l and we continue by the Independence Theorem as in the successor case. This induction finishes at α; rename cb as a_{0}b.

Now, for i ≠ j > 0, let _{i}g_{j} := Cb(a_{j}a_{i}/g_{i}g_{j}). From the above independencies, it is clear that a_{j}ba_{i} witness _{i}g_{j} ∈ g_{i}^{−1} ∗ g_{j}, and a_{j}a_{k}a_{i} witness _{i}g_{j} ∈ _{i}g_{k} ∗_{k}g_{j}.

Now let f | ⌣{g_{i},_{i}g_{j} : i ≠ j < α}, and let e.g. ed ⊧ Γ(f). By above, we have a_{i}b ⊧ Γ(g_{i}) for all i. Since Arg(π) is a Lascar strong type, b ≡^{L} d. Also, b | ⌣{g_{i},_{i}g_{j} : i ≠ j < α} and d | ⌣ f, so by the Independence Theorem we may assume that b = d | ⌣{f} ∪ {g_{i},_{i}g_{j} : i ≠ j < α} and for f_{i} we may take Cb(a_{i}e/g_{i}f). □

Remark 4.1.5. The polygroup chunk from 4.1.3 will obviously give a gradedly almost hyperdefinable group chunk after quotienting by the core relation. Moreover, it is possible to generalise the construction as follows, with the idea of getting the group chunk in a hyperdefinable way.

Denote by C(f) the intersection of all products containing f. If there is a frame e = ⟨e_{i},_{i}e_{j} : i ≠ j < α⟩ such that for every g, h with {g, h, e} independent we have that if f ∈ ⋂_{i<α} g_{i} ∗_{i}h for g_{i} ∈ g ∗ e_{i} and g_{j} ∈ g_{i} ∗_{i}e_{j}, _{i}h ∈ e_{i}^{−1} ∗ h and _{i}h ∈ _{i}e_{j} ∗_{j}h, then C(f) = ⋂_{i<α} g_{i} ∗ h_{i} (i.e. the intersection on the right hand side stabilises), then f ∼^{α} f′ if C(f) = C(f′) will become a type-definable equivalence relation, and we can do the blowup construction in such a way that whenever h̃, h̃′ ∈ f̃∗g̃ we have h̃ ∼^{α} h̃′, thus yielding a hyperdefinable group chunk. This property is not so unreasonable to ask for, since in the motivating example of reconstructing a group from a double coset space mentioned in the introduction to this chapter, it is possible to find an n-frame such that the intersection of any n products as above is a singleton (after some initial normalizations). Also, as mentioned in the introduction to this section, in the example of reconstructing a group from a projective space, a 2-frame suffices.

The construction of π^{α} goes by finding a suitable frame ē = ⟨e_{i},_{i}e_{j} : i ≠ j < α⟩ with x_{i}x ⊧ e_{i}, i < α, x | ⌣ ē and x_{i}xx_{j} ⊧ _{j}e_{i} ∈ e_{j}^{−1} ∗ e_{i}. Denote by x̃ := ⟨x_{i} : i < α⟩ˆ x, and we fix p := lstp(x̃/ē). Then, if f̃:= fˆ ⟨_{i}f, f_{i},_{i}f_{j} : i ≠ j < α⟩, we let π^{α}(f̃, x̃, ỹ) if f | ⌣ ē, x̃, ỹ ⊧ p, x_{i}xy ⊧ f_{i} ∈ f ∗ e_{i}, xyy_{i} ⊧ _{i}f ∈ e_{i}^{−1} ∗ f, x_{i}yy_{j} ⊧ _{j}f_{i} ∈ e_{j}^{−1} ∗ f_{i}, x_{i}xy_{j} ⊧ _{j}f_{i} ∈ _{j}f ∗ e_{i} and in such a case we write x̃ỹ ⊧ f̃. The construction continues by completing, with the product of f̃_{q} | ⌣ē g̃_{r} given as (now forgetting the types q and r and witnesses) h̃ such that h ∈ f ∗ g ∩ ⋂_{i} f_{i} ∗_{i}g, _{i}h ∈ e_{i}^{−1} ∗ h ∩_{i}f ∗ g ∩ ⋂_{j} _{i}f_{j} ∗_{j}g, h_{i} ∈ h ∗ e_{i} ∩ f ∗ g_{i} ∩ ⋂_{j} f_{j} ∗_{j}g_{i}, _{i}h_{j} ∈ _{i}h ∗ e_{j} ∩ e_{i}^{−1} ∗ h_{j} ∩_{i}f ∗ g_{j} ∩ ⋂_{k} _{i}f_{k} ∗_{k}g_{j}, thus determining the product up to ∼^{α}.

Remark 4.1.6. If we consider what the generalised associativity means for the polygroup associated to a projective geometry from 1.4.4, the Pasch axiom is equivalent to associativity (i.e. generalised associativity for α = 3), and e.g. the Desargues axiom corresponds to generalised associativity for α = 4, as shown in the figure below. Hence, both in our and the classical case, we can ‘coordinatise’ only in the presence of a ‘higher associativity’ condition.

g_{0}

g_{2}
g_{3} _{0}g_{3}

0g_{2}

0g_{1}

1g_{3}

2g_{3}
1g_{2}

g_{1}

Remark 4.1.7. A negative feature of this construction is that for an arbitrary f | ⌣ e, it may be impossible to find witnesses x_{0}xy_{0}y as in 4.1.1, so not every f | ⌣ e can be blown up. This will be rectified in the next section at the expense of losing hyperdefinability. Nevertheless, the rank situation is still good, i.e. SU(P^{2}) = SU(P), since if f | ⌣ e are from 4.1.1, at least every f′ ⊧ tp(f/e) can be blown up.

4.2. The ‘classical’ blowup construction

Let (S, X) = (S_{0}/R, X_{0}/Q) be a coreless gradedly almost hyperdefinable polyspace chunk, where R is I-graded and Q is J-graded. Assume R_{0} is like in 3.3.14. In what follows, we often claim without a proof the existence of certain R_{ι}’s such that some algebraic manipulations give a result up to R_{ι}, and in each such case it will follow from almost hyperdefinability and gradedness of the polyspace chunk, e.g. if {a_{R}, b_{R}, c_{R}} is independent, we can find a′R_{0}a, b′R_{0}b, c′R_{0}c such that {a′, b′, c′} is independent, so (a′ ∗ b′) ∗ c′ =_{0} a′ ∗ (b′ ∗ c′), and, by gradedness of ∗, (a ∗ b) ∗ c =_{ι} a ∗ (b ∗ c) for some ι.

Definition 4.2.1. A tuple e = ⟨e_{i}, e_{ij} : i ≠ j < n⟩ from S_{0}, we shall call an n-frame (up to R_{ϵ}), or sometimes just a frame, if:

(1) {e_{i}_{R} : i < n} is independent;

(2) for every i ≠ j < n, e_{ij} ∈_{ϵ} e_{i}^{−1} ∗ e_{j};
(3) for every i ≠ j < n, e_{ij} =_{ϵ} e_{ji}^{−1};
(4) for every i, j, k < n, e_{ij} ∈_{ϵ} e_{ik} ∗ e_{kj}.

Given an n-frame e, a tuple (a matrix?) a = ⟨a_{ij} : i ≠ j < n⟩ from S_{0} is called an e-number (up to R_{ι}), if there exists an a ∈ S_{0}, a_{R} | ⌣ e such that:

(1) for every i ≠ j < n, a_{ij} ∈_{ι} e_{i}^{−1} ∗ a ∗ e_{j};

(2) for every i, j, k, a_{ij} ∈_{ι} a_{ik} ∗ e_{kj} and a_{ij} ∈_{ι} e_{ik} ∗ a_{kj}.

A tuple ⟨_{i}x : i < n⟩ in X_{0} is called a left e-tuple (up to some Q_{ξ}), provided there exists an x ∈ X_{0} such that:

(1) for every i < n, _{i}x ∈ e_{i}^{−1} ∗ x;

(2) for every i ≠ j < n, _{i}x ∈_{ξ} e_{ij} ∗_{j}x.

We define right e-tuples dually.

Lemma 4.2.2. There is an R_{ϵ} such that, given {e_{i}_{R} : i < n} independent, for any choice of e_{0i} ∈ e_{0}^{−1} ∗ e_{i}, 0 < i < n, we can find e_{ij}, 0 < i < n, j < n such that ⟨e_{i}, e_{ij} : i ≠ j < n⟩ is a frame (up to R_{ϵ}).

Proof. Let e_{0i} ∈ e_{0}^{−1} ∗ e_{i} be given, and put e_{i0} := e_{0i}^{−1}. Now, e_{0}^{−1} ∈ e_{0i} ∗ e_{i}^{−1} ∩ e_{0j} ∗ e_{j}^{−1} (up to some R_{ϵ}), and therefore e_{i0} ∗ e_{0j} ≈_{ϵ} e_{i}^{−1} ∗ e_{j}, (for some coarser R_{ϵ}), and we can pick an element e_{ij} in both.

We claim that this will form a frame. By the second form of associativity, since e.g. (e_{13} ∗ e_{3}^{−1}) ∗ e_{2} =_{R} e_{13} ∗ (e_{3}^{−1} ∗ e_{2}), we get that e_{1}^{−1} ∗ e_{2} ≈_{ϵ} e_{13} ∗ e_{32} (for some R_{ϵ}), so pick h in the intersection.

So, we have obtained e_{12}, h ∈_{ϵ} e_{1}^{−1} ∗ e_{2}, and e_{1}_{R} | ⌣ e_{12}_{R}h_{R} as it is easy to check that e_{1}_{R} | ⌣ e_{01}_{R}e_{02}_{R}e_{03}_{R}, which implies e_{1}_{R} | ⌣ e_{01}_{R}e_{02}_{R}e_{13}_{R}e_{32}_{R}. Therefore, by corelessness, e_{12} ∈_{ϵ} e_{13} ∗ e_{32} for some higher R_{ϵ}. We can check the other relations in a similar fashion. □

From now on, fix an n-frame e up to some R_{ϵ}, and let X_{e}^{0} := {x ∈ X_{0} : x_{Q} | ⌣ e} and S_{e}^{0} := {a ∈ S_{0} : a_{R} | ⌣ e} and denote by (S_{e}, X_{e}) = (S_{e}^{0}/R, X_{e}^{0}/Q) the resulting almost hyperdefinable polyspace chunk over e.

Lemma 4.2.3. There is Q_{ξ} such that, given x ∈ X_{e}^{0}, for any choice of _{0}x ∈ e_{0}^{−1} ∗ x, we can find (uniquely up to Q) _{i}x, 0 < i < n such that ⟨_{i}x : i < n⟩ is an e-tuple (up to Q_{ξ}).

Proof. Let _{0}x ∈ e_{0}^{−1} ∗ x be given. Since (e_{10} ∗ e_{0}^{−1}) ∗ x =_{Q} e_{10} ∗ (e_{0}^{−1} ∗ x), by the second form of associativity, we get that e_{1}^{−1} ∗ x ≈_{ξ} e_{10} ∗_{0}x (for some Q_{ξ}), so we can pick _{1}x inside.

Similarly, we get that e_{20} ∗ _{0}x ≈_{ξ} e_{2}^{−1} ∗ x ≈_{ξ} e_{21} ∗ _{1}x; pick h in the first, k in the second intersection. Thus, since h, k ∈_{ξ} e_{2}^{−1} ∗ x, and e_{2}_{R} | ⌣ h_{Q}k_{Q} (the last independency follows from e_{2}_{R} | ⌣ e_{01}_{R}e_{02}_{R}_{0}x_{Q}, which is easily verified), h and k are core-related and thus, by corelessness, we can define _{2}x up to some higher Q_{ξ}. We proceed analogously. □

Lemma 4.2.4. There is a certain R_{ι} higher in the grading than R_{ϵ} such that, given any a ∈ S_{e}^{0} we can pick ⟨a_{ij} : i ≠ j < n⟩ forming an e-number up to R_{ι}. In other words, e-numbers are plentiful.

Proof. Let a_{R} | ⌣ e. By 4.2.3 there are a_{i} ∈_{ι} a ∗ e_{i} with a_{j} ∈_{ι} a_{i} ∗ e_{ij} and _{i}a ∈_{ι} e_{i}^{−1} ∗ a with _{i}a ∈_{ι} e_{ij} ∗_{j}a, for some R_{ι}.

Since (e_{i}^{−1} ∗ a) ∗ e_{j} =_{R} e_{i}^{−1} ∗ (a ∗ e_{j}), _{i}a ∗ e_{j} ≈_{ι} e_{i}^{−1} ∗ a_{j} (for a higher R_{ι}), so take a_{ij} in the intersection.

By (e_{i}^{−1} ∗ a_{k}) ∗ e_{kj} =_{R} e_{i}^{−1} ∗ (a_{k} ∗ e_{kj}), a_{ik} ∗ e_{kj} ≈_{ι} e_{i}^{−1} ∗ a_{j} ∋_{ι} a_{ij} (for some R_{ι}); take h in the intersection. If we can show that a_{j}_{R} | ⌣ a_{ij}_{R}a_{ik}_{R}e_{kj}_{R}, we will have that h and a_{ij} are core-related and thus equal modulo some coarser R_{ι}.

Since a_{ij} ∈_{R} _{i}a ∗ e_{j}, a_{ik} ∈_{R} _{i}a ∗ e_{k}, e_{kj} ∈_{R} e_{k}^{−1} ∗ e_{j}, it is enough to check that a_{j}_{R} | ⌣_{i}a_{R}e_{j}_{R}e_{k}_{R}. By assumption, e_{i}_{R} | ⌣ a_{R}e_{j}_{R}e_{k}_{R}, so _{i}a_{R} | ⌣ e_{j}_{R}e_{k}_{R}a_{j}_{R}, but a_{j}_{R} | ⌣ e_{j}_{R}e_{k}_{R} so everything is independent and a_{j}_{R} | ⌣_{i}a_{R}e_{j}_{R}e_{k}_{R}.

The fact that a_{ij} ∈_{R} e_{ik} ∗ a_{kj} is proved symmetrically. □

Here is a criterion allowing us to recognise an e-number intrinsically.

Lemma 4.2.5. Let n ≥ 4. If a tuple ã = ⟨a_{ij} : i ≠ j < n⟩ satisfies (1) for every i ≠ j < n, a_{ij}_{R} | ⌣ e;

(2) for every i < n, {a_{ij}_{R} : j ≠ i} (and dually, {a_{ji}_{R} : j ≠ i}) is independent;

(3) for every i, j, k, a_{ij} ∈_{ι} a_{ik} ∗ e_{kj} and a_{ij} ∈_{ι} e_{ik} ∗ a_{kj},

(for some R_{ι}), there is an a ∈ S_{0}, a_{R} | ⌣ e, witnessing that ã is an e-number. Furthermore, such a is unique up to some R_{κ} and we obtain a graded map π : ã ↦ a.

Proof. We just give a quick sketch of a proof. It is possible to find _{i}a ∈_{κ} ⋂_{j} a_{ij} ∗ e_{j}^{−1} and a_{j} ∈_{κ} ⋂_{i} e_{i} ∗ a_{ij} for some big enough κ. Since (e_{i} ∗ a_{ij}) ∗ e_{j}^{−1} =_{R} e_{i} ∗ (a_{ij} ∗ e_{j}^{−1}), a_{j} ∗ e_{j}^{−1} ≈_{κ} e_{i} ∗_{i}a for every i, j (and some κ). Finally, by given independencies, all things in such intersections are R_{κ}-equivalent, for high enough κ, and that gives the sought for a; for example, observe a_{1} ∗ e_{1}^{−1} ≈_{κ} e_{0} ∗ _{0}a ≈_{κ} a_{2} ∗ e_{2}^{−1}, and let h be in the first, k in the second intersection. Thus, h, k ∈_{κ} e_{0} ∗_{0}a, and e_{0}_{R} | ⌣ h_{R}k_{R} (since e_{0}_{R} | ⌣ a_{31}_{R}e_{1}_{R}e_{2}_{R}e_{3}_{R}). □

An analogous characterization of e-tuples is even easier, so we omit the proof.

Lemma 4.2.6. Let n ≥ 2. If x̃ is a tuple in X_{0} such that (1) for every i < n, _{i}x_{Q} | ⌣ e;

(2) {_{i}x_{Q} : i < n} is independent;
(3) for every i ≠ j < n, _{i}x ∈_{ξ} e_{ij} ∗_{j}x,

(for some Q_{ξ}), there is an x ∈ X_{0}, x_{Q} | ⌣ e witnessing that x̃ is an e-tuple. This x is unique up to Q and we obtain a graded map π : x̃ ↦ x.

The previous two lemmas show that the blowup of our original polyspace is actually a bounded covering of it.

Proposition 4.2.7. Let n ≥ 4. Let ã and b̃ be two e-numbers (up to some R_{ι}), a_{R} | ⌣_{e} b_{R}. There is a unique e-number c̃ (up to some coarser R_{κ} depending only on R_{ι}) such that c_{ij} ∈_{κ} ⋂_{k≠i,j} a_{ik} ∗ b_{kj}. Furthermore, π(c̃) ∈_{κ} π(ã) ∗ π(b̃).

Proof. We give the proof for n = 4, which is the only case we need, and at some point, we choose particular values for indices to enhance readability.

Firstly, the product is unique, since if e.g. c_{10}, c′_{10} ∈_{κ} a_{12} ∗ b_{20} ∩ a_{13} ∗ b_{30}, then they are core-related because a_{12}_{R} | ⌣ a_{13}_{R}b_{30}_{R}.

For the existence, by associativity, it’s easily seen that ⋂_{k≠i,j} a_{ik} ∗ b_{kj} ≠_{ι} ∅, so we can pick c_{ij} inside. We claim that this c is an e-number up to some R_{κ}. Since (a_{01} ∗ b_{12}) ∗ e_{23} =_{R} a_{01} ∗ (b_{12} ∗ e_{23}), c_{02} ∗ e_{23} ≈_{κ} a_{01} ∗ b_{13} ∋ c_{03} (for some κ). Take h in the intersection; now h, c_{03} ∈_{κ} a_{01} ∗ b_{13} and a_{01}_{R} | ⌣ h_{R}c_{03}_{R}, so c_{03} is core-related to h, i.e. they are equal with respect to some coarser R_{κ}.

To see the last independence, it is enough to check that a_{01}_{R} | ⌣ c_{02}_{R}e_{23}_{R}c_{03}_{R}, or a_{01}_{R} | ⌣ a_{02}_{R}a_{03}_{R}b_{23}_{R}b_{32}_{R}. If we let a =_{R} π(ã) and b =_{R} π(b̃) the above follows from e_{1}_{R} | ⌣ a_{R}b_{R}e_{0}_{R}e_{2}_{R}e_{3}_{R}.

For the ‘furthermore’ part, let c =_{κ} π(c̃) (for high enough R_{κ}), so c_{01} ∈_{κ} e_{0}^{−1} ∗c∗e_{1} (some κ); let _{0}c be as in the proof of 4.2.5 such that c_{01} ∈_{κ} _{0}c ∗ e_{1}. Since also c_{01} ∈_{κ} a_{02} ∗ b_{21}, we can find _{k}b such that _{0}c ∗ e_{1} ≈_{κ} a_{02} ∗_{2}b ∗ e_{1}, and this implies, since e_{1}_{R} | ⌣_{0}c_{R}a_{02}_{R}_{2}b_{R} (follows from e_{1}_{R} | ⌣ a_{R}b_{R}e_{0}_{R}e_{2}_{R}e_{3}_{R}), that _{0}c ∈_{κ} a_{02} ∗_{2}b (for some higher κ). But then e_{0}^{−1} ∗ c ≈_{κ} e_{0}^{−1} ∗ a_{2} ∗_{2}b, and since e_{i}_{R} | ⌣ c_{R}a_{2}_{R}_{2}b_{R} (follows from e_{0}_{R} | ⌣ a_{R}b_{R}e_{1}_{R}e_{2}_{R}e_{3}_{R}), c ∈_{κ} a_{2} ∗_{2}b, or c ∗ b^{−1} ∗ e_{2} ≈_{κ} a ∗ e_{2}, and since e_{2} | ⌣ a_{R}b_{R}c_{R} (follows from e_{2}_{R} | ⌣ a_{R}b_{R}e_{0}_{R}e_{1}_{R}e_{3}_{R}), c ∈_{κ} a ∗ b (for some κ). □

Proposition 4.2.8. If a is an e-number (up to R_{ι}), x an e-tuple (up to Q_{ξ}) and a_{R} | ⌣_{e} x_{Q}, there is a unique e-tuple y (up to some Q_{ζ} depending only on R_{ι} and Q_{ξ}) such that y_{i} ∈_{ζ} a_{ij} ∗ x_{j}.

Proof. Suppose y_{0}, y′_{0} ∈_{ζ} a_{01} ∗x_{1} ∩a_{02} ∗x_{2}. Since a_{01}_{R} | ⌣ a_{02}_{R}x_{2}_{Q}, we have that y_{0}Qy′_{0} so we have uniqueness.

For the existence, since (a_{01}∗e_{12})∗x_{2} =_{Q} a_{01}∗(e_{12}∗x_{2}), we get that a_{02}∗x_{2} ≈_{ζ} a_{01}∗ x_{1}, (for a high enough Q_{ζ}) and take h in the intersection. Similarly, we can find k ∈_{ζ} a_{03} ∗x_{3} ∩a_{01} ∗x_{1}. By all the independencies, {a_{03}_{R}, x_{3}_{Q}, e_{32}_{R}, e_{31}_{R}} is independent, so e_{31}_{R} | ⌣ a_{03}_{R}x_{3}_{Q}e_{32}_{R} and thus a_{01}_{R} | ⌣ a_{03}_{R}x_{3}_{Q}e_{32}_{R}, yielding a_{01}_{R} | ⌣ a_{02}_{R}x_{2}_{Q}a_{03}_{R}x_{3}_{Q} and therefore hQ_{ζ}k for course enough Q_{ζ} and we can define y_{0}. Similarly for the remaining y_{i}’s. □

The reader should recall the definition of type 3 morphisms between polygroups from 1.4.12 before proceeding to the next result.

Theorem 4.2.9. Let (S, X) = (S_{0}/R, X_{0}/Q) be a gradedly almost hyperdefinable polyspace chunk with a fixed n-frame e for n ≥ 4. The set of e-numbers S̃_{e} and the set of etuples X̃_{e} with the multiplication of numbers and action of numbers on tuples is a gradedly almost hyperdefinable space chunk (over e) and the map π : (S̃_{e}, X̃_{e}) → (S_{e}, X_{e}) is a gradedly almost hyperdefinable generic epimorphism of type 3 with bounded fibres.

Proof. We will show just the group chunk part, since checking the properties of the action is completely analogous. Let R_{0} and R_{1} be such that e-numbers exist up to R_{0}, and the product of two numbers up to R_{0} is a number up to R_{1}. From the construction of numbers, it is clear that there is a fixed R_{2} such that for every number a up to R_{1}, there is a number a′R_{2}a which is a number up to R_{0}. Let S_{1} be the type-definable set of e-numbers up to R_{1}. Let · be the type-definable relation on S_{1}^{3} given by: for a, b ∈ S_{1}, c ∈ a · b if there are a′R_{2}a, b′R_{2}b with c in the product of a and b as in 4.2.7 (this is clearly nonempty for a_{R} | ⌣_{e} b_{R} by the choice of R_{0}, R_{1} and R_{2}, and all the c’s in the product of a and b are R_{3}-related for some R_{3}). The gradedness of the induced map is clear.

For generic independence, let a | ⌣_{e} b and c ∈ a·b. In particular, a_{ij} | ⌣ be, and since c_{ik} ∈_{R} a_{ij} ∗ b_{jk}, by generic independence of the original polygroup chunk, c_{ik}_{R} | ⌣ be, so c_{ik}_{R} | ⌣_{e} b, and as c_{R} ∈ bdd(c_{ik}_{R}e), c_{R} | ⌣_{e} b. Similarly we get c_{R} | ⌣_{e} a.

For generic associativity, take {a_{R}, b_{R}, c_{R}} e-independent e-numbers, let x_{R} = a_{R} · b_{R}, z_{R} = b_{R} · c_{R}, u_{R} = x_{R} · c_{R} and v_{R} = a_{R} · z_{R}. We need to show u_{R} = v_{R}.

Let us start with e.g. (a_{10} ∗ b_{02}) ∗ c_{23} =_{R} a_{10} ∗ (b_{02} ∗ c_{23}); it implies that x_{12} ∗ c_{23} ≈_{i} a_{10}∗z_{03} (for some R_{i}), so take h in both. Now, we have h, v_{13} ∈_{i} a_{10}∗z_{03} and we would like to show that h and v_{13} are core-related. This will hold if a_{10}_{R} | ⌣ h_{R}v_{13}_{R}, but since v_{13} ∈_{i} a_{12} ∗ z_{23} as well, it is enough to show a_{10}_{R} | ⌣ a_{13}_{R}b_{32}_{R}c_{23}_{R}a_{12}_{R}b_{21}_{R}c_{13}_{R}, which follows from a_{10}_{R} | ⌣ a_{12}_{R}b_{21}_{R}c_{13}_{R}e_{12}_{R}e_{23}_{R}. As {a_{12}_{R}, b_{21}_{R}, c_{13}_{R}, e_{R}} is independent, and {e_{20}_{R}, e_{12}_{R}, e_{23}_{R}} is independent, we have e_{20}_{R} | ⌣ a_{12}_{R}b_{21}_{R}c_{13}_{R}e_{12}_{R}e_{23}_{R}, which clearly implies what is required. In a similar way, we can get that u_{13} and h are core-equivalent and thus also u_{13} and v_{13}.

Generic invertibility follows directly from uniqueness and the corresponding property of S, and the fact that π is a generic homomorphism is shown above.

To see that π is generically of type 3, it is enough to show that if c ∈_{R} a ∗ b, for some a_{R} | ⌣_{e} b_{R} ∈ S_{e} and π(c̃) =_{R} c, we can find ã and b̃ with π(ã) =_{R} a, π(b̃) =_{R} b and c̃ =_{R} ã · b̃. We will work modulo R, and it is just a technicality to show that everything can be done gradedly. Thus, let c̃ with π(c̃) = c and c ∈_{R} a ∗ b be given. Reconstruct c_{i} and _{i}c as in 4.2.5 such that _{i}c ∈_{R} e_{i}^{−1} ∗ c, c_{i} ∈_{R} c ∗ e_{i} and c_{ij} ∈_{R} _{i}c ∗ e_{j} ∩ e_{i}^{−1} ∗ c_{j} (this can be done uniquely). By the second form of associativity applied to e_{i}^{−1} ∗ c ∗ b^{−1}, find _{i}a ∈_{R} e_{i}^{−1} ∗ a ∩ _{i}c ∗ b^{−1}. We claim that ia ∈_{R} e_{ij} ∗ _{j}a. Again by the second form of associativity applied to e_{i}^{−1} ∗ e_{j} ∗ _{j}a, we get _{i}a ∈_{R} e_{i}^{−1} ∗ a ≈_{R} e_{ij} ∗ _{j}a, and let h be in the intersection. To show that ia and h are core-related, it is enough to check that e_{i}_{R} | ⌣_{i}a_{R}_{j}a_{R}e_{ij}_{R}, or, since ia ∈_{R} _{i}c ∗ b^{−1}, that e_{i}_{R} | ⌣_{i}c_{R}_{j}c_{R}b_{R}e_{ij}_{R}. In fact, as _{j}c ∈_{R} e_{ji} ∗ _{i}c, it is enough to see that e_{i}_{R} | ⌣_{i}c_{R}b_{R}e_{ij}_{R}, but that follows from independence of {b_{R}, c_{R}, e_{i}_{R}, e_{j}_{R}}. Similarly (by observing a^{−1} ∗ c ∗ e_{j}) we can choose b_{j} ∈ b ∗ e_{j} ∩ a^{−1} ∗ c_{j}. Also, if we choose any a_{0} ∈ a ∗ e_{0}, there will be unique (up to R) _{0}b ∈ e_{0}^{−1} ∗ b and a_{i} ∈ a ∗ e_{i}, ib ∈ e_{i}^{−1} ∗ b such that c ∈_{R} a_{i} ∗ _{i}b and a_{i} ∈_{R} a_{j} ∗ e_{ji} and _{i}b ∈_{R} e_{ij} ∗ _{j}b. From this data, as in 4.2.4, we can get a_{ij} and b_{ij} such that they form e-numbers ã and b̃. To check that c̃ =_{R} ã ∗ b̃, let us first show that c_{ij} ∈_{R} _{i}a ∗ b_{j}. By associativity, from e_{i}^{−1} ∗ a ∗ b_{j} and the construction above, where c_{j} ∈_{R} a ∗ b_{j}, we get that _{i}a ∗ b_{j} ≈ e_{i}^{−1}∗c_{j} ∋ c_{ij}, so pick h in the intersection. We claim that c_{ij}Rh. From independence of {a_{R}, b_{R}, e_{i}_{R}, e_{j}_{R}}, we have that e_{i}_{R} | ⌣ a_{R}b_{R}c_{R}e_{j}_{R}, so e_{i}_{R} | ⌣ a_{R}c_{j}_{R}b_{j}_{R}, and it is easily seen that a_{R} | ⌣ c_{j}_{R}b_{j}_{R} so a_{R} | ⌣ e_{i}_{R}c_{j}_{R}b_{j}_{R}, a_{R} | ⌣ c_{ij}_{R}b_{j}_{R}e_{i}_{R} and a_{i}_{R} | ⌣ c_{ij}_{R}b_{j}_{R}e_{i}_{R}. But, as e_{i}_{R} | ⌣ c_{ij}_{R}b_{j}_{R}, we have that e_{i}_{R} | ⌣ c_{ij}_{R}_{i}a_{R}b_{j}_{R}, implying e_{i}_{R} | ⌣ c_{ij}_{R}h_{R} so c_{ij}Rh by corelessness. The fact that c_{ij} ∈_{R} a_{ik} ∗ b_{kj} is shown in a similar way using the independence of {a_{R}, b_{R}, e_{i}_{R}, e_{j}_{R}, e_{k}_{R}}. □

Remark 4.2.10. When dealing with the coreless polygroups and polygroup chunks, we did not need to discuss the generalised associativity properties (unlike in the hyperdefinable case in the previous section) as they are trivially satisfied in this context.

4.3. Universal blowup

It is much easier to develop the mini-blowup described in Section 4.1 modulo R and get the group chunk of similar characteristics as in the previous section. Let us shortly explain how it should go at the level of ultraimaginaries (the full details for the gradedly almost hyperdefinable version are given in [BTW]), without worrying about the graded almost hyperdefinability issues, since our aim in this section is much higher, to construct a blowup with the universal property.

Let (P, ∗) be the coreless polygroup chunk, and pick an e ∈ P. Let P_{e} := {a ∈ P : a | ⌣ e}. The group chunk (P̃_{e}, ∗) consists of tuples ã = ⟨a,_{e}a, a_{e}⟩, where a ∈ P_{e}, ea ∈ e^{−1} ∗ a, a_{e} ∈ a ∗ e, and the operation ∗ is defined as follows: if ã | ⌣_{e} b̃ (which implies {a, b, e} is independent), let c ∈ a∗b∩a_{e}∗_{e}b, _{e}c ∈ e^{−1}∗c∩_{e}a∗b, c_{e} ∈ c∗e∩a∗b_{e}. It is easy to check that these values are unique (up to the core relation) and we say that c̃ = ã ∗ b̃. The map π : P̃_{e} → P_{e} given by ⟨a,_{e}a, a_{e}⟩ ↦ a is a bounded covering (a generic epimorphism of type 3 with bounded fibres).

Let e_{i}, i < λ be a long Morley sequence and let e_{ij} form a frame (as in the previous section), i.e. e_{ij} ∈ e_{i}^{−1} ∗ e_{j} and the usual e_{ik} ∈ e_{ij} ∗ e_{jk} and e_{ji} = e_{ij}^{−1}.

Let P_{i} denote {a ∈ P : a | ⌣ e_{i}}. Then obviously P = ⋃_{i} P_{i}. Observe mini-blowups P̃_{i} and let P^{†} = ⋃_{i} P̃_{i}. The projection π can clearly be extended to the whole of P^{†}. We have transition maps f_{ij} between some parts of P̃_{i} and P̃_{j}: if ã = ⟨a,_{i}a, a_{i}⟩ ∈ P̃_{i}, and ã | ⌣ e_{j} (which amounts to a | ⌣ e_{i}e_{j}), then we can uniquely find a_{j} ∈ a∗e_{j} ∩a_{i} ∗e_{ij} (the intersection is nonempty by the second form of associativity applied to a∗e_{i} ∗e_{ij}) and similarly _{j}a ∈ e_{j}^{−1} ∗ a ∩ e_{ji} ∗_{i}a, so let f_{ij} : (a,_{i}a, a_{i}) ↦ (a,_{j}a, a_{j}).

Lemma 4.3.1. Transition maps are partial generic homomorphisms:

(1) If ã ∈ P̃_{i} is independent of e_{j}e_{k}, then f_{jk}(f_{ij}(ã)) = f_{ik}(ã).

(2) For e_{j} | ⌣ ã ∈ P̃_{i}, f_{ji}(f_{ij}(ã)) = ã.
(3) For ã | ⌣_{e_{i}} b̃ ∈ P̃_{i}, if e_{k} | ⌣ ãb̃, then f_{ik}(ã ∗ b̃) = f_{ik}(ã) ∗ f_{ik}(b̃).

Proof.

(1) For example, let a_{j} = a ∗ e_{j} ∩ a_{i} ∗ e_{ij} and a_{k} = a ∗ e_{k} ∩ a_{j} ∗ e_{jk}, and, on the other hand, a′_{k} = a ∗ e_{k} ∩ a_{i} ∗ e_{ik}. Since a | ⌣ e_{i}e_{j}e_{k}, we have that a_{k} | ⌣ e_{ij}e_{ik}e_{k}, but, as e_{k} | ⌣ e_{ij}e_{ik}, a_{k} | ⌣ a_{k}e_{ij}e_{ik}. Furthermore, as e_{jk} ∈ e_{ij}^{−1} ∗ e_{ik}, a_{j} ∈ a_{k} ∗ e_{kj}, a_{i} ∈ a_{j} ∗ e_{ij}, we get that e_{k} | ⌣ a_{i}e_{ik}a_{j}e_{jk} and thus e_{k} | ⌣ a′_{k}a_{k}, so we can conclude that a_{k} and a′_{k} are core-related.

(2) is easy.

(3) The assumption that ã | ⌣_{e_{i}} b̃ and e_{k} | ⌣ ãb̃ amounts to the independence of {a, b, e_{i}, e_{k}}.

Let c̃ = ã ∗ b̃; so in particular c_{i} ∈ c ∗ e_{i} ∩ a ∗ b_{i}. Suppose c′_{k} is the third coordinate of f_{ik}(c̃); c′_{k} ∈ c ∗ e_{k} ∩ c_{i} ∗ e_{ik}. On the other hand, b_{k} ∈ b ∗ e_{k} ∩ b_{i} ∗ e_{ik}, and the third coordinate of f_{ik}(ã) ∗ f_{ik}(b̃) is c_{k} ∈ c ∗ e_{k} ∩ a ∗ b_{k}. As both c_{k}, c′_{k} ∈ a ∗ e_{k}, if we can show that e_{k} | ⌣ c_{i}e_{ik}ab_{k}, we would have c_{k} = c′_{k} by corelessness. It clearly suffices to show, as c_{i} ∈ a ∗ b_{i} that e_{k} | ⌣ ab_{i}e_{ik}ab_{k}, but since b_{k} ∈ b_{i} ∗ e_{ik}, it is enough to check e_{k} | ⌣ ab_{i}e_{ik}, which easily follows from the independence of {a, b, e_{i}, e_{k}} and generic independence. □

It is much clearer now what we need to do in order to get a generic operation:

Definition 4.3.2. We shall say that ã ∈ P̃_{i} is ∼-related to ã′ ∈ P̃_{j} if there is e_{k}, ã | ⌣ e_{k}, ã′ | ⌣ e_{k} such that f_{ik}(ã) = f_{jk}(ã′).

Lemma 4.3.3. We have ã ∼ ã′, for ã ∈ P̃_{i} and ã′ ∈ P̃_{j} if and only if for every e_{l} with ã | ⌣ e_{l}, ã′ | ⌣ e_{l}, f_{il}(ã) = f_{jl}(ã′).

Proof. Let e_{k} witness that ã ∼ ã′, and consider any e_{l} | ⌣ ã, e_{l} | ⌣ ã′.

If we take any e_{m} with e_{m} | ⌣ e_{k}ã and e_{m} | ⌣ e_{k}ã′, in particular we have that e_{m} | ⌣ e_{k}e_{i}a and e_{m} | ⌣ e_{k}e_{j}a, and thus, since a | ⌣ e_{i}e_{k} and a | ⌣ e_{j}e_{k}, we have a | ⌣ e_{i}e_{k}e_{m} and a | ⌣ e_{j}e_{k}e_{m}, so, by 4.3.1, f_{km}(f_{ik}(ã)) = f_{km}(f_{jk}(ã)) implies that f_{im}(ã) = f_{jm}(ã).

Now, we can choose e_{m} | ⌣ e_{k}e_{l}ãã′. By the previous paragraph, we get f_{im}(ã) = f_{jm}(ã′). Furthermore, e_{l} | ⌣ e_{m}ã and e_{l} | ⌣ e_{m}ã′, so again by the previous paragraph, f_{il}(ã) = f_{jl}(ã′). □

Corollary 4.3.4. The relation ∼ is an equivalence relation.

Let us denote the ∼-class of ã by [ã].

Remark 4.3.5. Even though we said at the beginning of this section that we shall not worry about the almost hyperdefinability of our objects, we feel obliged to provide the reader with a hint how to show graded almost hyperdefinability of ∼. We claim that P̃_{i} ∋ ã ∼ ã′ ∈ P̃_{j} if and only if there is e^{†} | ⌣ ãã′ such that f_{i†}(ã) = f_{j†}(ã′) (where, of course, the fact that {e_{i}, e_{j}, e^{†}} is independent implies that it can be completed to a frame e′, provided we choose e′_{ij} := e_{ij} and any e′_{i†} ∈ e_{i}^{−1} ∗ e^{†}; this gives a meaning to f_{i†} and f_{j†}). The latter relation is clearly a good candidate for becoming almost hyperdefinable in view of our considerations around the core relation.

Suppose that e^{†} is as above, and find k_{0} such that {e_{i}, e_{j}, e^{†}, e_{k} : k ≥ k_{0}} is independent. After choosing e′′_{ij} := e_{ij}, e′′_{ik} := e_{ik} for k ≥ k_{0} and e′′_{i†} := e′_{i†}, we can complete this to a frame e′′. However, since e′′_{kl} ∈ e′′_{ki} ∗e′′_{il} ∩e_{k}^{−1} ∗e_{l} = e_{ki} ∗e_{il} ∩e_{k}^{−1} ∗e_{l} ∋ e_{kl}, by all the independencies, e′′_{kl} = e_{kl} (core related) for k, l ≥ k_{0} or k ≥ k_{0}, l = j. Then, since e^{†} is in e′′, by the previous lemma, f′′_{ik}(ã) = f′′_{jk}(ã′) for some k ≥ k_{0}, but as f′′_{ik} = f_{ik} and f′′_{jk} = f_{jk}, ã ∼ ã′ with respect to the original frame e. The converse is trivial by 4.3.3.

Definition 4.3.6. Let P̃ := P^{†}/∼, and let [ã] | ⌣[b̃] ∈ P̃. Without loss of generality, P̃_{i} ∋ ã | ⌣ b̃ ∈ P̃_{j}. We define [ã]∗[b̃] := [f_{ij}(ã)∗b̃](= [ã∗f_{ji}(b̃)]). Also, let π([ã]) := π(ã). We call P̃ the universal blowup.

Theorem 4.3.7 (Universal blowup). Let P̃ →^{π} P be the universal blowup of a coreless polygroup chunk P with respect to some frame e = ⟨e_{i}, e_{ij}⟩. Then, (P̃, ∗) is a group chunk and π is a (generic) epimorphism of type 3 with bounded fibres.

Furthermore, for every group chunk Z with a frame f and every epimorphism φ : Z → P such that φ(f) = e (i.e. φ(z ∗ t) ∈ φ(z) ∗ φ(t) for z | ⌣ t and φ(z^{−1}) = φ(z)^{−1}), there is a unique φ̃ : Z → P̃ such that φ = π ∘ φ̃.

\begin{aligned}Z & P̃ \\ & P \end{aligned}φ π φ̃

Proof. The multiplication is well-defined by 4.3.3 and 4.3.1: let us say we have P̃_{i} ∋ ã | ⌣ b̃ ∈ P̃_{j}, and let ã ∼ ã′ ∈ P̃_{k}, witnessed by some e_{l} | ⌣ ãã′b̃, i.e. f_{il}(ã) = f_{kl}(ã′), and suppose ã′ | ⌣ b̃. Let c̃:= f_{ij}(ã) ∗ b̃, and let c̃′ := f_{kj}(ã′) ∗ b̃. Now, applying f_{jl} to the last equation, we get that

f_{jl}(c̃′) = f_{jl}(f_{kj}(ã′)) ∗ f_{jl}(b̃) = f_{kl}(ã′) ∗ f_{jl}(b̃) = f_{il}(ã) ∗ f_{jl}(b̃).

Applying f_{lj} to both sides yields c̃′ = f_{ij}(ã) ∗ b̃ = c̃.

For generic independence, let P̃_{i} ∋ ã | ⌣ b̃ ∈ P̃_{j} and let c̃:= f_{ij}(ã)∗b̃. We have that ae_{i} | ⌣ be_{j}, so b | ⌣ ae_{i}e_{j} and by generic independence of P, since c ∈ a∗b, c | ⌣ ae_{i}e_{j}, so ce_{j} | ⌣ ae_{i} and c̃ | ⌣ ã. To see that [ã] · [b̃] | ⌣ b̃, we can check in a similar way as above that ã ∗ f_{ji}(b̃) | ⌣ b̃. Associativity, invertibility and the fact that π is of type 3 follow from the corresponding properties on each P̃_{i}.

For the ‘furthermore’ part, let z ∈ Z and let z_{i} := z ∗ f_{i}, _{i}z := f_{i}^{−1} ∗ z and put φ̃(z) := [⟨φ(z), φ(_{i}z), φ(z_{i})⟩] for big enough i. It is clear that φ̃ : Z → P̃ since φ(z_{i}) = φ(z ∗ f_{i}) ∈ φ(z) ∗ e_{i} and similarly φ(_{i}z) ∈ e_{i}^{−1} ∗ φ(z). □

Remark 4.3.8. In the statement of the theorem, if we require that φ be a generic epimorphism of type 3, we can omit the assumption that there is a frame f in Z with φ(f) = e because e can be pulled back via φ to give such a frame.

Proof. Let the frame e = ⟨e_{i}, e_{ij}⟩ be given and pick an arbitrary f_{0} such that φ(f_{0}) = e_{0}. Since for every i, e_{0} ∈ e_{i} ∗ e_{i0}, using the fact that φ is generically of type 3, we can find f_{i} and f_{i0} such that f_{0} = f_{i} ∗ f_{i0} and φ(f_{i}) = e_{i}, φ(f_{i0}) = e_{i0}. As Z is the group chunk, this extends uniquely to a frame by f_{ji} = f_{ij}^{−1} and f_{ij} = f_{i0} ∗ f_{0j}. □

CHAPTER 5 Applications and geometric simplicity

The aim of this chapter is to demonstrate how the group configuration theorem obtained through the previous chapters gives rise to ‘geometric simplicity theory’, i.e. the set of methods commonly referred to as geometric stability can be partially extended to simple theories.

The fact that the group configuration at the moment produces an almost hyperdefinable and not hyperdefinable groups creates some difficulties, and our hope is that they are not significant. Much more serious obstacles come from the lack of knowledge on groups in simple theories; for example, whether an SU-rank 1 group hyperdefinable in a simple theory has a ‘large’ abelian part.

In Section 5.1, we solve what might be considered a fundamental problem in the classical theory of hypergroups (see the extensive literature in [Co]) for the class of almost hyperdefinable polygroups in simple theories: given a polygroup of that class, we find a group in the same class which is ‘closely’ related to it. We were quite surprised that the classical literature lacks the ability to (nontrivially) answer such questions and our hope is that this result will initiate some development. This problem, posed by myself, was first approached by Wagner and myself using stabilisers and the idea of using the universal property of the group chunk theorem is by Ben-Yaacov.

In Section 5.2 we finally decide to recover the space corresponding to the original partial generic multiaction and not just the group, and use it in Section 5.5 to prove that pseudolinearity implies one-basedness in an ω-categorical simple theory. The action was originally defined by myself in a similar fashion for the case of a polygroup, but there I was lacking the generic faithfulness. The present state of these two sections comes from collaboration with Wagner.

In Section 5.3 we show that at least in an ω-categorical theory, our group configuration theorem gives an interpretable group. It is due to myself.

Section 5.4 shows that an (ω-categorical) one-based, SU-rank 1 structure interprets a vector space over a (finite field) division ring as a stable reduct, thus partially proving the stable forking conjecture mentioned in Section 1.2. Also, we argue that this is the first (two) case(s) of a Zil’ber-type trichotomy for simple theories. I presented a proof of this in [To], but the proof had an omission which was later patched up with help of Wagner. Independently, the same result was obtained by de Piro and Kim in [dPK] by an ad hoc method and partially by Vassiliev in [Va] using generic pairs.

5.1. Getting a group from a polygroup

Definition 5.1.1. Two polygroups P_{1} and P_{2} are said to be polyisogenous, if there is a subpolygroup H < P_{1} × P_{2} such that the first projection of H has a bounded index in P_{1} and the second projection of H has a bounded index in P_{2}, and kernels of both projections are bounded.

Intuitively, H induces a bounded-to-bounded correspondence (almost a homomorphism) between big parts of P_{1} and P_{2}.

Remark 5.1.2. In a simple theory, with every gradedly almost hyperdefinable polygroup we can associate a gradely almost hyperdefinable group which is polyisogenous to it.

Let P be the polygroup in question, coreless without loss of generality, and observe its generic part X, which is a polygroup chunk by 3.3.15. We can blow it up, obtain a group chunk and use it to generate a group G. In particular, there will be a generic element g′ ∈ G interbounded with a generic element g ∈ P (over an independent parameter which we suppress for the moment).

Look at H := stab(gg′) < P × G. Since stab(g) × stab(g′) < H, it is clear (by 3.6.5 and 3.6.6) that the first projection of H is of bounded index in P, and the second is of bounded index in G and the fact that the kernels of both projections are bounded should follow from the fact that g and g′ are interbounded, so H will realise a polyisogeny between P and G.

Unfortunately, the polyisogeny constructed above need not be gradedly almost hyperdefinable, but we do not attempt to refine the result since a much more explicit correspondence can be obtained using the universal property of the group chunk as follows.

Theorem 5.1.3. Let P be a coreless gradedly almost hyperdefinable polygroup in a simple theory. There is a gradedly almost hyperdefinable group G which is a bounded covering of P, i.e. with an epimorphism π : G → P of type 3 with bounded fibres.

Proof. We show how to get the group using e.g. the blowup from Section 4.2 (it would be more ‘universal’ to use the blowup from 4.3, but then the group would be ‘manifoldlike’); let X be the generic part of P and let π : X̃_{e} → X_{e} be a blowup with respect to some frame e. By the universal property from 3.5.1 and 3.5.2, if G_{e} is the group generated by X̃_{e}, π lifts to π^{2} : G_{e} → P of type 3.

Let us show that π^{2} : G_{e} → P is surjective (we work at the level of ultraimaginaries). Let a ∈ P and let g | ⌣ ae, g′ | ⌣ age be generics of P (in particular, g, g′ ∈ X_{e}). Let h ∈ g^{−1} ∗ a (so h | ⌣ ae) and h′ ∈ g^{′−1} ∗ a (so h′ | ⌣ aghe). Clearly h, h′ ∈ X_{e}, too. Then, a ∈ g ∗ h ∩ g′ ∗ h′, so by the transposition property (associativity), there is d ∈ g^{−1} ∗ g′ ∩ h ∗ h^{′−1}, and d ∈ X_{e}. By surjectivity of π, there is d̃∈ X̃_{e} with π(d̃) = d, and, since π is generically of type 3, we can find g̃, g̃′, h̃, h̃′ ∈ X̃_{e} with π(g̃) = g, π(g̃′) = g′, π(h̃) = h, π(h̃′) = h′ and d̃ = g̃^{−1} · g̃′ = h̃ · h̃′^{−1}. Let ã := g̃ · h̃ = g̃′ · h̃′. Then, π^{2}(ã) ∈ π^{2}(g̃) ∗ π^{2}(h̃) ∩ π^{2}(g̃′) ∗ π^{2}(h̃′) = g ∗ h ∩ g′ ∗ h′, and therefore must be core-equivalent to a.

For boundedness of fibres, let ã ∈ G_{e}, and pick a generic g̃ | ⌣_{e} ã. Let π^{2}(ã) = a; if ⟨ã_{i} : i < κ⟩ is an unbounded sequence of ae-conjugates of ã, we may refine it and get an ag̃e-indiscernible sequence ⟨ã_{i} : i < ω⟩ such that π^{2}(ã_{i}) = a (when we speak about indiscernible sequences of ultraimaginaries over ultraimaginaries, it is understood that it holds for some representatives). Then, writing g̃′_{i} := g̃ ∗ ã_{i}, we have that ⟨g̃′_{i} : i < ω⟩ is ag̃e-indiscernible, but since a ∈ π(g̃)^{−1} ∗ π(g̃′_{0}), and g̃ is bounded over π(g̃), we get that g̃′_{0} is bounded over ag̃e and the sequence must be constant. □

Remark 5.1.4. (Added in proof). In the above, since π is onto of type 3, we get a (weak form) of a structure theorem for gradedly almost hyperdefinable coreless polygroups in simple theories: P ≅ G // π, see [Cr] for further analysis of quotients of polygroups by homomorphisms of type 3. By analysing the structure of the blowup further, Ben-Yaacov in [BY1] obtains much more: P ≅ G // H for some H < G.

5.2. Reconstructing the action

Thus far, we have always constructed a group starting with a good enough partial generic multiaction π(x, y, z), where π(f, a, b) intuitively means that f acts on a’s and b’s. However, we have ‘forgotten’ about reconstructing this original action, even though we have proved the space chunk theorem and have formulated some blowup constructions for polyspace chunks. There is a good reason for postponing the discussion of the action part until now: it is hard to do at the level of the polygroup chunk (we cannot control ranks of objects involved, and cannot obtain the action generically faithful), so we choose to do it after the blowup procedure has been done.

Let us start with a hyperdefinable polygroup chunk obtained in 4.1, i.e. (after adding the parameters to the language), we will be working with (P_{0}, ∗), P_{0} = Germ(π), such that y, y′ ∈ f(x), for f ∈ P_{0}, x ∈ Arg(π), f | ⌣ x, we have that bdd(y) = bdd(y′), and if h, h′ ∈ f ∗ g, f | ⌣ g ∈ P_{0}, bdd(h) = bdd(h′). Thus, P_{0} is a group chunk modulo the core relation denoted by R in this section.

Definition 5.2.1. Let X_{0} := {(g, x) : g ∈ P_{0}, x ∈ Arg(π), g | ⌣ x}. We shall write (g, x) ∼_{i} (g′, x′) if there are h, h′ ∈ P_{0}, each independent of gxg′x′, with h(x) ≈ h′(x′) and g ∗ h^{−1}R_{i}g′ ∗ h′^{−1}. It is clear that every ∼_{i} is symmetric, and for every g | ⌣ x we can find h | ⌣ gx such that x ∈ dom(h), so ∼_{i} is reflexive as well. Let ∼ be the direct limit of ∼_{i}’s.

Lemma 5.2.2. The relation ∼ is a gradedly almost hyperdefinable equivalence relation.

Proof. To see almost hyperdefinability, let R_{0} witness the almost-hyperdefinability of R, and let g/R ⊆ ⋃_{α<μ} g_{α}/R_{0}. Suppose (g′, x′) ∼_{i} (g, x); there will be h | ⌣ gxg′x′ and h′ | ⌣ gxg′x′ with h(x) ≈ h′(x′) and g ∗ h^{−1}R_{i}g′ ∗ h′^{−1}. We may assume that hh′ | ⌣_{gxg′x′} ḡ. Let g′′R_{j}g such that g′′ ∗ h^{−1}R_{1}g′ ∗ h′^{−1} for some fixed R_{1}, and let

g_{α}R_{0}g′′. Then g_{α}∗h^{−1}R_{2}g′∗h′^{−1} and h and h′ are independent of g_{α}xg′x′, so (g_{α}, x) ∼_{2} (g′, x′) and therefore ∼_{2} witnesses the almost hyperdefinability of ∼.

For graded transitivity, if (g, x) ∼_{i} (g′, x′) and (g, x) ∼_{j} (g′′, x′′), let h | ⌣ gxg′x′, h′ | ⌣ gxg′x′ with y ∈ h(x)∩h′(x′) and g∗h^{−1}R_{i}g′∗h′^{−1}, and h_{1} | ⌣ gxg′′x′′, h_{2} | ⌣ gxg′′x′′ with y′ ∈ h_{1}(x)∩h_{2}(x′′) and g ∗h_{1}^{−1}R_{j}g′′ ∗h_{2}^{−1}. We may assume yhh′ | ⌣_{gxg′x′} g′′x′′ and y′h_{1}h_{2} | ⌣_{gxg′′x′′} g′x′hh′y. Since {x, h, h_{1}} is independent, y′xy ⊧ h^{†} := Cb(y′y/hh_{1}) ∈ h ∗ h_{1}^{−1}. Also, x′′y′y ⊧ h′′ := Cb(x′′y/h^{†}h_{2}) ∈ h^{†} ∗ h_{2}, since y′ | ⌣ h^{†}h_{2} (from above, we can get h | ⌣ y′h_{1}h_{2}, so by generic independence and h^{†} ∈ h ∗ h_{1}^{−1}, h^{†} | ⌣ y′h_{2}). Then, y ∈ h′(x′) ∩ h′′(x′′) and

\begin{aligned}g′ ∗ h′^{−1} & =_{i} g ∗ h^{−1} =_{1} g ∗ (h_{1}^{−1} ∗ h^{†}^{−1}) =_{0} (g ∗ h_{1}^{−1}) ∗ h^{†}^{−1} \\ & =_{j′} (g′′ ∗ h_{2}^{−1}) ∗ h^{†}^{−1} =_{0} g′′ ∗ (h_{2}^{−1} ∗ h^{†}^{−1}) =_{1} g′′ ∗ h′′^{−1}, \end{aligned}

for some constant 1 and j′ depending only on j, so (g′, x′) ∼_{k} (g′′, x′′) for some k depending only on i and j. □

We denote the ∼-class of (g, x) ∈ X_{0} by [g, x].

Lemma 5.2.3. The hyperdefinable relation, given by (k, x) ∈ f ∗ (g, x) if and only if f | ⌣ gx and k ∈ f ∗ g, induces a gradedly almost hyperdefinable generic action ∗ : P_{0}/R ⊗ X_{0}/∼ → X_{0}/∼.

Proof. It is enough to check that if e.g. (g, x) ∼_{i} (g′, x′) and f | ⌣ gx, f | ⌣ g′x′, then there is j such that for every k ∈ f ∗g and k′ ∈ f ∗g′, (k, x) ∼_{j} (k′, x′). Let h | ⌣ gxg′x′ and h′ | ⌣ gxg′x′ with h(x) ≈ h′(x′) and g ∗ h^{−1}R_{i}g′ ∗ h′^{−1} and let k ∈ f ∗ g, k′ ∈ f ∗ g′. We may assume that hh′ | ⌣_{gxg′x′} f, which implies that h | ⌣ kxk′x′ and h′ | ⌣ kxk′x′

and f | ⌣ gh, f | ⌣ g′h′. Thus, by regularity of R, f ∗ (g ∗ h^{−1})R_{i′}f ∗ (g′ ∗ h′^{−1}) for some i′ and k ∗ h^{−1}R_{j}k′ ∗ h′^{−1} for some higher j, showing what was required. □

Lemma 5.2.4. If xy ⊧ f, x′y ⊧ f′, then [f, x] = [f′, x′].

Proof. Find g | ⌣ ff′xx′y such that yt ⊧ g for some t. Then, xyt ⊧ h := Cb(xt/fg) ∈ g∗f and x′yt ⊧ h′ := Cb(x′t/f′g) ∈ g∗f′. Thus, t ∈ h(x)∩h′(x′) and f ∗h^{−1}Rg^{−1}Rf′∗ h′^{−1}, witnessing (f, x) ∼ (f′, x′). □

Proposition 5.2.5. (P, X) := (P_{0}/R, X_{0}/∼, ∗) is a gradedly almost hyperdefinable space chunk where the action is generically faithful and generically transitive.

Proof. Generic associativity and invertibility of the action follow easily. For generic faithfulness, let [g, x] | ⌣ f_{R}f′_{R} such that f_{R} ∗ [g, x] = f′_{R} ∗ [g, x]. We may assume gx | ⌣ ff′ and let k ∈ f ∗ g, k′ ∈ f′ ∗ g such that e.g. [k, x] = [k′, x]. Let h | ⌣ kxk′ and h′ | ⌣ kxk′ such that h(x) ≈ h′(x) and k ∗ h^{−1}Rk′ ∗ h′^{−1}. Since x | ⌣ g, it follows that x | ⌣ gff′ and x | ⌣ kk′, which, together with h | ⌣ kxk′ implies x | ⌣ hkk′, and, as h ∈ bdd(hkk′), x | ⌣ hh′ and therefore by h(x) ≈ h′(x) and reducedness, we conclude h = h′. Then, k ∗ h^{−1}Rk′ ∗ h^{−1} implies that kRk′ and finally fRf′.

In order to check generic transitivity, let x | ⌣ z ⊧ Arg(π). We can find xy ⊧ a and zy′ ⊧ b for some a, b ∈ P. Since Arg(π) is a Lascar strong type, y ≡^{L} y′, y | ⌣ x, y′ | ⌣ z and x | ⌣ z so by the Independence Theorem we may assume (renaming a and b) that y′ = y | ⌣ xz and xy ⊧ a, zy ⊧ b. By 5.2.4, it follows that [a, x] = [b, z], which clearly implies generic transitivity. □

Lemma 5.2.6. For any g | ⌣ x, [g, x] and x are interbounded over the independent parameter g.

Proof. Claim: If xy ⊧ f, then [f, x] ∈ bdd(y). Let f′x′ realise the nonforking extension of tp(fx/y) to fxy. Thus, f′x′ | ⌣_{y} fx and x′y ⊧ f′ so by 5.2.4, [f, x] | ⌣_{y}[f, x] implying [f, x] ∈ bdd(y). Claim: For any g and x, [g, x] | ⌣ g. For any [g, x] it is possible to find f | ⌣ gx such that xy ⊧ f for some y ∈ Arg(π). Then, for k ∈ f ∗ g^{−1}, k | ⌣ gx and [f, x] ∈ k ∗ [g, x]. Notice that if xy ⊧ f, then [f, x] | ⌣ f since [f, x] ∈ bdd(y). But this will again be true for any [g, x]: from x | ⌣_{f} g we get that [f, x] | ⌣ fgk as [f, x] | ⌣ f, and then [g, x] | ⌣ g as [g, x] | ⌣ k.

Now we will be able to obtain x ∈ bdd([g, x], g): if x′ ≡_{[g,x]g} x, then [g, x] = [g, x′], so there are h and h′, each independent from gxx′ with h(x) ≈ h′(x′) and g ∗ h^{−1}Rg ∗ h′^{−1}. Therefore hRh′, implying that h and h′ are interbounded and eventually bdd(x) = bdd(x′). In fact, xQx′, the (I × ω)-graded relation Q being the direct limit of relations Q_{i,n}, where Q_{i,n} is the n-th transitive closure of Q_{i,1} and xQ_{i,1}x′ if there are hR_{i}h′, hh′ | ⌣ xx′ with h(x) ≈ h′(x′). So x_{Q} ∈ dcl([g, x], g) and thus [g, x] and x are interbounded over the independent parameter g. □

Applying the space chunk theorem 3.5.3 yields:

Corollary 5.2.7. Starting with a partial generic multiaction π(x, y, z) as in 4.1, we can get a gradedly almost hyperdefinable space (G, X) where the action is faithful and transitive. Furthermore, over some independent parameters a generic element of G is interbounded with an element of Germ(π) and a generic element of X is interbounded with an element of Arg(π).

5.3. Group configuration in ω-categorical theories

Since our result in general is not completely satisfactory (the group living on almost-hyperimaginaries and not necessarily hyperimaginaries), we might attempt to improve it under the assumption of ω-categoricity. Extra care is required, however, to stay within a finite power of ℭ^{eq}, as expounded below.

My aim here is to show that it is indeed possible, starting from a good enough configuration on finite tuples, to get an interpretable group.

Remark 5.3.1. Recall that in an ω-categorical theory, if a, b are finite tuples in ℭ^{eq}, then Cb(a/b) will be finite, and also it is possible to replace b by an (interalgebraic) finite tuple b_{0} such that tp(a/b_{0}) is equivalent to lstp(a/b). Furthermore, this finiteness is uniform: if a’s belong to a sort S_{1} and b’s belong to a sort S_{2}, since there is only finitely many types tp(ab), all the possible canonical bases of the form Cb(a/b) are contained in finitely many sorts.

Proof. The relation R(a, a′, b) iff a ≡_{b}^{L} a′ (for some finite tuples a, a′, b), is an invariant relation so it is definable by ω-categoricity. Then the definable set R(x, a, b) is clearly equivalent to lstp(a/b) and for b_{0} we can take the name for it. □

We will profit from this remark in the ω-categorical case for the following reason: in order to use ω-categoricity, we need to keep working over (uniformly) finite tuples of imaginaries. In the general case, however, to achieve completeness in y of some generic action π(x, y, z) = lstp(fab), needed in 2.3.10, we typically replace a by bdd(a), which would take us into considering infinite tuples even if f, a, b were finite. But in ωcategorical case, it is possible to replace a by a finite tuple a_{0} (by 5.3.1) depending on tp(fb/a) so the completion in y will still live on finite tuples. We formalise the above discussion as follows.

Lemma 5.3.2. Completion with respect to any of the variables is a finitary operation, i.e. if π(x, y, z) is a type-definable partial generic multiaction on finite tuples of imaginaries, the completion in any of the variables is on finite tuples of imaginaries. Furthermore, reduction with respect to any of the variables is a finitary operation as well.

Proof. Let us consider e.g. the completion in x. The relation LS(yz, x, y′z′, x′), true if x = x′ and yz ≡_{x}^{L} y′z′ is definable by ω-categoricity. Then, the first variable in the completion \underline{π} (compare to 2.2.1) is of the sort (yz, x)/LS and thus imaginary.

For reduction, if e.g. π(x, y, z) is complete in x, observe the transitive closure ∼ of f ∼_{1} f′ if there is x | ⌣ ff′ with f(x) ≈ f′(x). This is a definable relation by ω-categoricity and therefore the first variable in the reduction π̄ is of the imaginary sort x/∼. □

Theorem 5.3.3. An ω-categorical simple theory with an algebraic quadrangle on finite tuples of imaginaries interprets an infinite group.

Proof. Let us start with an algebraic quadrangle (a, b, c, x, y, z) on finite tuples. By the above lemma, we may assume the generic partial multiaction π := lstp(byz) is complete in all the variables without destroying finiteness. Now, as in 2.4.2, it follows that π^{−1} ∘ π is generic and we can apply 2.3.10 to it and obtain a polygroup chunk (P, ∗).

Since P = Germ(π^{−1} ∘ π) is obtained by composing two multiactions, completing the result in the first variable and then reducing it, and all these operations are finitary, we obtain an interpretable polygroup chunk (P, ∗). The core relation on P, being invariant, will become definable by ω-categoricity, hence the blowup (choose e.g. the classical blowup of 4.2 which only requires finitely many parameters), and later the construction of the group from the group chunk can be done definably. □

Using the results from Section 5.2 and arguing in a similar way, we get:

Corollary 5.3.4. An ω-categorical simple theory with an algebraic quadrangle on finite tuples of imaginaries interprets an infinite space.

5.4. One-based case

Lemma 5.4.1. In a one-based theory, for every partial generic multiaction π, π^{−1} ∘ π is generic.

Proof. Clearly, if f ∈ Germ(π) and ab ⊧ f, then f = Cb(ab/f) ∈ bdd(ab) by onebasedness (1.2.22). Now, let abc ⊧ h ∈ { \atop g ∘ f}, so a | ⌣ fgh. In particular, since f | ⌣ g, we get f | ⌣_{a} g and thus b | ⌣_{a} c. Then, as above, h ∈ bdd(ac) and f ∈ bdd(ab), so h | ⌣_{a} f and h | ⌣ f. Similarly h | ⌣ g. □

We will make significant use of the following result ([W], 4.8.18), which relies on the fact that every group in a one-based theory is bounded-by-abelian ([W], 4.8.4):

Theorem 5.4.2. Let G be a hyperdefinable connected group in a simple theory, and suppose that a generic type p of G is locally modular and regular. If R denotes the division ring of p-endogenies of G, then a tuple (g_{0}, …, g_{n}) in G is dependent if and only if there are r_{i} ∈ R, not all zero, with ∑_{i=0}^{n} r_{i}(g_{i}) ⊂ cl_{p}(∅).

Theorem 5.4.3. Let T be an ω-categorical, SU-rank 1, one-based nontrivial. Then T interprets a vector space over a finite field (as a stable ‘reduct’).

Proof. Since T is one-based nontrivial, we can get a pairwise independent, nonindependent triple {a, b′, c} such that each element is bounded over the other two, as in [BH]. By ω-categoricity, find a finite tuple b such that tp(ac/b) is equivalent to lstp(ac/b′).

Then, π := stp(abc) is a generic action, invertible, complete in the second variable, Arg(π) is a (Lascar) strong type and, by 5.4.1, π^{−1} ∘ π is generic. Using the group configuration machinery (noting that all the germs will be (uniformly) finite tuples by choice of π and ω-categoricity as in 5.3) we get a group G (interpretable over finitely many parameters) with SU(G) = 1. Thus, G^{0} := G_{∅}^{0} is a connected group of SU-rank 1 and by 5.4.2, the ring of endogenies is a division ring and induces a vector space structure on G^{0}.

In what follows, we will show that endogenies have boundedly finite order, so by Wedderburn’s theorem the ring of endogenies will actually be a finite field. Since every endogeny is induced by a definable subgroup of G^{0} × G^{0}, we can identify it with an imaginary element (the code of the subgroup). So let r be an endogeny, and let v ∈ G^{0} be a nonzero vector. By ω-categoricity, there are only finitely many types among {tp(r^{i}v/rv) : i ∈ ω}. But, since all r^{i} are defined over r, we get that there are m, n such that r^{m} = r^{n} and so r is of finite order. Pick now any two endogenies r and s. If σ : rv ≡_{v} sv, then σ(r)v = sv, so since we’re in a vector space, σ(r) = s and it follows that r and s are of the same order. By ω-categoricity again, there can be only finitely many orders of endogenies, as required. □

Remark 5.4.4. Notice that ω-categoricity is not essential in the above. Reproving 5.4.2 on 1-based groups for 1-based gradedly almost hyperdefinable groups gives directly a gradedly almost hyperdefinable vector space over a division ring.

We can thus view this result as a first (nontrivial locally modular) case of a Zil’bertype trichotomy for simple theories. The remaining non-modular case would involve studying a generalisation of Zariski structure framework from [HZ] to simple structures. I have already done some preliminary investigation in the direction of pseudo-Zariski structures, aimed at characterising Zariski closed sets over pseudofinite (or bounded PAC) fields.

Furthermore, the result can also be viewed as a first satisfactory step towards the stable forking conjecture, since inside a simple structure we have found a stable one where the independence is clearly governed by the stable structure (independence coincides with linear independence).

Remark 5.4.5. A careful reader might have noticed that we have not used the full potential of 5.4.2 in the above, since the group we obtained was of SU-rank 1, so its generic type was also of SU-rank 1 and thus regular, but the theorem will work for any group with a regular generic.

Let p be nontrivial regular locally modular type in a (ω-categorical) simple theory. Then, rewriting all of the group configuration techniques from this thesis in the language of regular types as in [P3], Chapter 7, will give a (definable) group G with a regular generic type, which will again yield a vector space by 5.4.2. The details of this will appear in [TW].

Remark 5.4.6. The purpose of this remark is to persuade the model-theoretic community that, at least in an ω-categorical case, references to the classical results about pregeometries, e.g.

· ([Ar]) a projective geometry of dimension not less than 4, in which each closed set of dimension 2 contains at least 3 elements, is isomorphic to a projective geometry over some division ring; · ([DH]) a locally projective (i.e. non-trivial and locally modular), locally finite geometry of dimension greater than 4, in which all closed sets of dimension 2 have the same size, is an affine or projective geometry over a finite field;

can be replaced by a model-theoretic construction. For example, if D is a solution set of an SU-rank 1 Lascar strong type, and it is nontrivial, then [dPK] have shown that for each independent a, b ∈ D, cl(a) ∪ cl(b) is properly contained in cl(ab), so it makes sense to define a ∗ b := cl(ab) − (cl(a) ∪ cl(b)). If D is modular, it is easy to check that ∗ is generically associative, and Steinitz exchange implies generic invertibility (where a^{−1} = a), so (D, ∗) will be a definable polygroup chunk by ω-categoricity. We can make it coreless by dividing out by the core relation and then apply the blowup procedure to obtain a group chunk, and eventually a group and even a division ring.

The construction from this section is an adaptation of two famous classical constructions: firstly, the group configuration corresponds to the (Hilbert), Veblen-Young and von Neumann construction of the division ring from a projective geometry, and then, once an abelian group has been obtained, one might study the nearring of endomorphisms of it; if one is lucky (typically in some linear-like framework), this will be a ring or even a field. In this case, however, we weren’t lucky, we had one-basedness.

5.5. Pseudolinearity implies linearity

We will need this technical result:

Lemma 5.5.1. If x and x′ are interbounded, and e := Cb(x/B), e′ := Cb(x′/B), then e and e′ are interbounded.

Proof. We have x | ⌣_{e} B and x′ | ⌣_{e′} B, but since bdd(x) = bdd(x′), from the first we get x′ | ⌣_{e} B which implies e′ ∈ bdd(e) and from the second x | ⌣_{e′} B which implies e ∈ bdd(e′). □

Lemma 5.5.2. If D is k-linear, then G(D) (as defined in 1.3.8) is k-linear.

We include this lemma just for completeness, as we will not need the full power of it because we will only deal with plane curves of the form lstp(ab′/C), where b′ ∈ ℭ^{eq} is actually interalgebraic with some real b, so the fact that SU(Cb(ab′/C)) ≤ k will follow from the previous lemma and k-linearity.

Proof. Let x, y ∈ G(D), B ⊆ G(D) such that SU(xy/B) = 1, x | ⌣ y. Without any loss of generality, B ⊆ D. For xy, find an independent finite set F ∪ {a, b} ⊆ D such that xy is interalgebraic with ab over F, xy | ⌣ F. Take F′ realising a nonforking extension of tp(F/xy) to xyabBF; then F′ | ⌣ xyabBF. Since F ≡_{xy} F′, let a′b′ such that abF ≡_{xy} a′b′F′. Now acl^{eq}(xyF′) = acl^{eq}(a′b′F′). Claim: SU(a′b′/F′B) = 1. To prove the claim, note first that a′b′ | ⌣_{F′} B; otherwise, we would get that xy | ⌣ B which would be a contradiction. If SU(a′b′/F′B) = 0, then x, y ∈ acl(F′B) and subsequently x, y ∈ acl(B), again a contradiction.

Denote by e := Cb(xy/B) = Cb(xy/BF′) (so e ∈ bdd(B)), and e′ := Cb(a′b′/F′B). By the above claim and linearity of D, SU(e′) ≤ k. By an argument similar to 5.5.1, since a′b′ | ⌣_{e′} F′B, in particular a′b′ | ⌣_{e′F′} B so by interboundedness over F′, xy | ⌣_{e′F} B so e ∈ bdd(e′F′). Therefore, SU(e/F′) ≤ k, but since e ∈ bdd(B) and B | ⌣ F′, SU(e) = SU(e/F′) ≤ k and we are done. □

Let us state some results of that shall be used significantly below, allowing us to find large abelian parts in our group.

Theorem 5.5.3.

(1) ([Mp], [W], 6.2.35). An ω-categorical simple group is nilpotent-by-finite.

(2) ([EW], [W], 6.2.32). An ω-categorical supersimple group is finite-by-abelian-by-finite.

Theorem 5.5.4. A pseudolinear set D in an ω-categorical simple theory is linear.

Proof. Let D be k-linear, and pick a plane curve q = lstp(bc/a) (a, b, c finite) such that for a_{0} := Cb(q), SU(a_{0}) = k. By ω-categoricity, find b′ such that tp(ac/b′) is equivalent to lstp(ac/b). Then, by 5.5.1, SU(Cb(b′c/a)) is still k, and it easily follows that π := lstp(a, b′, c) is a partial generic multiaction, invertible, complete in the second variable, π^{−1} ∘π generic, Arg(π) is a Lascar strong type, the only possibly nontrivial claim being that π^{−1} ∘ π is generic, shown as follows: pick a_{1} | ⌣ a_{2} ⊧ Germ(π)

and a_{3} ∈ { \atop a_{1}^{−1} ∘ a_{2}}. Then SU(a_{3}) ≤ k by 5.5.2, as a_{3} is again a canonical base of a plane curve in G(D). From rank considerations, we get a_{3} | ⌣ a_{1} and a_{3} | ⌣ a_{2}.

Thus, it gives rise to an interpretable group (similar remarks as before about getting this interpretable on finite tuples apply); moreover, using Section 5.2 and the space chunk theorem 3.5.3, we can get a group G of SU-rank k acting on a set X of SU-rank at most 1; in fact, below we will only need the generic part of that action and the fact that it can be obtained generically faithful.

By 5.5.3(2), G is finite-by-abelian-by-finite, and in fact we may assume, after dividing by a finite normal subgroup that G is actually abelian-by-finite, so in particular Z(G^{0}) = G^{0} (G^{0} := G_{∅}^{0}) is infinite. We continue as in [P3], after Lemma 2.4.21. Claim: If f ∈ G^{0} and x ∈ X, x | ⌣ f, then f ∈ dcl(x, f · x). Let g ∈ G^{0} be such that tp(g/x, f · x) = tp(f/x, f · x). In particular g · x = f · x. Let f′ ∈ Z(G^{0}) such that SU(f′) ≥ 1 (as Z(G^{0}) is infinite) and f′ | ⌣ fgx.

Let y := f′ · x. We claim that f′ |̸ ⌣ xy. Suppose otherwise; then f′ | ⌣ xy implies f′ | ⌣_{x} y so y | ⌣_{x} y and thus y ∈ bdd(x). Similarly we get x ∈ bdd(y). Now pick f′′ realising the nonforking extension of lstp(f′/x) to xf′. Since y ∈ bdd(x), f′′ · x = y too, and thus (f′^{−1} · f′′) · x = x = 1 · x, and x | ⌣ f′f′′ so by generic faithfulness of the action, f′^{−1} · f′′ = 1, implying f′ ∈ bdd(∅) which contradicts SU(f′) ≥ 1.

From this, it easily follows that f′ · x | ⌣ fg and x | ⌣ ff′, x | ⌣ gf′. Then

\begin{aligned}f · y = f · (f′ · x) & = (f · f′) · x = (f′ · f) · x = f′ · (f · x) \\ & = f′ · (g · x) = (f′ · g) · x = (g · f′) · x = g · (f′ · x) = g · y. \end{aligned}

Thus, by generic faithfulness, f = g, proving the claim.

In the above, let f ∈ G^{0} with SU(f) = k. By the claim and f | ⌣ x, we have SU(f) = SU(f/x) ≤ SU(f · x/x) ≤ 1, which implies k = 1. □

Using the fact that linearity is equivalent to one-basedness (1.3.10), we obtain:

Corollary 5.5.5. A pseudolinear set D in an ω-categorical simple theory is one-based.

Remark 5.5.6. Again, if we are willing to rewrite the group configuration techniques in the language of regular types of [P3], Chapter 7, we can rephrase the above result as follows.

Let p be a regular type (over ∅) in an ω-categorical simple theory, k-linear in the sense that whenever q(x, y) is a p-minimal extension of p(x) ∪ p(y) with SU_{p}(q) = 1 (plane curve), then SU_{p}(Cb(q)) ≤ k, and weight k is achieved for some plane curve. Then, an analogous proof as above in the framework of regular types would yield k = 1 and p is locally modular (where the appeal to 5.5.3(2) is replaced by the fact that if p is a finitely based regular type, a p-connected group of finite SU_{p}-rank is (central bounded-by-abelian)-by-bounded; it is conceivable that, with some additional work, even 5.5.3(1) should suffice). This will be expounded in [TW].

Bibliography

[Ar] E. Artin. Geometric Algebra. Interscience Publishers, New York, 1957.

[Ba] John T. Baldwin. An almost strongly minimal non-Desarguesian projective plane. Trans. AMS, 342 (1994), no. 2.

[BY] Itay Ben-Yaacov. Group configurations and germs in simple theories. Preprint, 2000.

[BY1] Itay Ben-Yaacov. More on the polygroup blow-up. Preprint, 2001.

[BTW] Itay Ben-Yaacov, Ivan Tomašić and Frank Wagner. Constructing an almost hyperdefinable group. In preparation, 2001.

[BH] Elisabeth Bouscaren and Ehud Hrushovski. On one-based theories. Journal of Symbolic Logic, 59(2), 1994.

[Bu] Steven Buechler. Lascar strong types in some simple theories. To appear in Journal of Symbolic Logic.

[BL] Steven Buechler and Olivier Lessmann. Simple homogeneous models. Preprint, 2000.

[BPW] Steven Buechler, Anand Pillay and Frank O. Wagner. Elimination of hyperimaginaries for supersimple theories. Preprint, 1998.

[ChH] Zoe Chatzidakis and Ehud Hrushovski. The model theory of difference fields. Trans. AMS, 351(8), 1999.

[ChMv] Zoe Chatzidakis, L. van den Dries and A. J. Macintyre. Definable sets over finite fields. J. reine angew. Math. 427, (1992), 107-135.

[ChP] Zoe Chatzidakis and Anand Pillay. Generic structures and simple theories. Annals of Pure and Applied Logic 95, (1998), 71–92.

[CHL] G. Cherlin, L. Harrington and A. H. Lachlan. ω-categorical, ω-stable structures. Annals of Pure and Applied Logic 28, (1985), 103–135.

[CH] G. Cherlin and E. Hrushovski. Finite structures with few types. In preparation.

[Cr] Stephen D. Comer. Polygroups derived from cogroups. Journal of Algebra 89, (1984), 397–405.

[Co] Piergiulio Corsini. Prolegomena of hypergroup theory. Supplement to Riv. Mat. Pura Appl. Aviani Editore, Tricesimo, 1993.

[dPK] Tristram de Piro and Byunghan Kim. The geometry of 1-based minimal types. Preprint, MIT, 2001.

[DH] J. Doyen and X. Hubaut. Finite regular locally projective spaces. Mathematische Zeitschrift 119, 1971.

[Du] J.-L. Duret. Les corps faiblement algébriquement clos non séparablement clos ont la propriété d’indépendance. In Model Theory of Algebra and Arithmetic, Lecture Notes Math. 834, Springer, Berlin, 1980, 136–162.

[E] David Evans. ℵ_{0}-categorical structures with a predimension. To appear in Annals of Pure and Applied Logic.

[EW] David Evans and Frank O. Wagner. Supersimple ω-categorical groups. Journal of Symbolic Logic, to appear.

[GKTV] Rami Grossberg, Alexei Kolesnikov, Ivan Tomašić, Monica VanDieren. The equality D = S = R. Preprint, Carnegie Mellon University, 1999.

[GIL] Rami Grossberg, José Iovino, Olivier Lessmann. A primer of simple theories. Preprint, Carnegie Mellon University, 1999.

[HH] Victor Harnik and Leo Harrington. Fundamentals of forking. Ann. Pure Appl. Logic, 26, 245– 286, 1984.

[HKP] Bradd Hart, Byunghan Kim and Anand Pillay. Coordinatization and canonical bases in simple theories. Journal of Symbolic Logic, to appear.

[Hi] David Hilbert. Grundlagen der Geometrie. Teubner, Leipzig und Berlin. 9. Auflage, Stuttgart 1962, herausgeg. von P. Bernays. (first publ. 1899.)

[H0] Ehud Hrushovski. Contributions to Stable Model Theory. PhD thesis, University of California at Berkeley, Berkeley, 1986.

[H1] Ehud Hrushovski. Locally modular regular types. In John T. Baldwin, editor, Classification Theory: Proceedings of the US-Israel Binational Workshop on Model Theory in Mathematical Logic, Chicago ’85, LNM 1292. Springer-Verlag, Berlin, 1987.

[H2] Ehud Hrushovski. Finitely based theories. Journal of Symbolic Logic, 54, 1989.

[H3] Ehud Hrushovski. Pseudofinite fields and related structures. Preprint, 1991.

[H4] Ehud Hrushovski. Strongly minimal expansions of algebraically closed fields. Israel Journal of Mathematics, 79, 1992.

[H5] Ehud Hrushovski. A new strongly minimal set. Annals of Pure and Applied Logic, 62, 1993.

[H6] Ehud Hrushovski. The Mordell-Lang conjecture for function fields. Journal of the American Mathematical Society, 9, 1996.

[H7] Ehud Hrushovski. Simplicity and the Lascar group. Preprint, Hebrew University of Jerusalem, 1998.

[H8] Ehud Hrushovski. The Manin-Mumford conjecture and the model theory of difference fields. Annals of Pure and Applied Logic, to appear.

[HP1] Ehud Hrushovski and Anand Pillay. Groups definable in local fields and pseudo-finite fields. Israel J. of Mathematics, 85, 203–262, 1994.

[HP2] Ehud Hrushovski and Anand Pillay. Definable groups of algebraic groups over finite fields. Crelle’s Journal, 462, 69–91, 1995.

[HZ] Ehud Hrushovski and Boris Zil’ber. Zariski geometries. Journal of the American Mathematical Society., 9, no. 1, 1996.

[Ja] James Jantosciak. Transposition hypergroups: noncommutative Join Spaces. Journal of Algebra, 187, 1997.

[Ki] Byunghan Kim. Simple first order theories. Ph. D. thesis, University of Notre Dame, 1996.

[Ki1] Byunghan Kim. Forking in simple unstable theories. J. of London Math. Society, (2) 57 (1998) 257–267.

[KP] Byunghan Kim and Anand Pillay. Simple theories, Annals of Pure and Applied Logic, 88 (1997) 149-164.

[KP1] Byunghan Kim and Anand Pillay. From stability to simplicity. Bulletin of Symbolic Logic (4) 1 (1998), 17–36

[KP2] Byunghan Kim. Around stable forking. Preprint, 2000.

[KoP] Piotr Kowalski and Anand Pillay. A note on groups definable in difference fields. Preprint, 2000.

[L1] Daniel Lascar. Types définissables et produit de types. C. R. Acad. Sci. Paris Sir. A-B, 272, A1233–A1236, 1971.

[L2] Daniel Lascar. Ranks and definability in superstable theories. Israel J. Math., 23, 53–87, 1976.

[L3] Daniel Lascar. On the category of models of a complete theory. Journal of Symbolic logic, 82, 249–266, 1982.

[L4] Daniel Lascar. Stability in Model Theory. John Wiley & Sons, Inc., New York, 1987.

[L5] Daniel Lascar. The group of automorphisms of a relational saturated structure. Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, Kluwer Acad. Publ., Dordrecht, 1993.

[L6] Daniel Lascar. Recovering the action of an automorphism group. Logic: from foundations to applications (Staffordshire, 1993), 313–326, Oxford Sci. Publ., Oxford Univ. Press, New York, 1996.

[LP] Daniel Lascar and Anand Pillay. Hyperimaginaries ans automorphism groups. Preprint, 1998.

[LPo] Daniel Lascar and Bruno Poizat. Fundamentals of forking. J. Symbolic Logic, 1977.

[Ma] Angus J. Macintyre. Generic automorphisms of fields. Joint AILA-KGS Model Theory Meeting (Florence, 1995). Ann. Pure Appl. Logic 88, no. 2–3, 1997.

[Mp] H. Dugald Macpherson. Absolutely ubiquitous structures and ℵ_{0}-categorical groups. Oxford Quarterly Journal of Mathematics (2), 483–500, 1998.

[Mk] Michael Makkai. A survey of basic stability theory, with particular emphasis on orthogonality and regular types. Israel J. Math., 49, 181–238, 1984.

[Mar] F. Marty. Sur une généralisation de la notion de group. IV^{∘} Congress des Mathematiciens Scandinave, Stockholm, 1934.

[vN] John von Neumann. Continuous geometry. Princeton University Press, New Jersey, 1960.

[P1] Anand Pillay. An Introduction to Stability Theory. Clarendon Press, Oxford University Press, New York, 1983.

[P2] Anand Pillay. Forking, normalization and canonical bases. Annals of Pure and Applied Logic 32 (1986), 61–82.

[P3] Anand Pillay. Geometric Stability Theory. Oxford Logic Guides 32, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.

[P4] Anand Pillay. Forking in the category of existentially closed structures. Preprint, University of Ilinois at Urbana-Champaign, 1998.

[P5] Anand Pillay. Definability and definable groups in simple theories. Preprint, 1998.

[PP] Anand Pillay et Bruno Poizat. Pas d’imaginaires dans l’infini! J. Symbolic Logic, Vol. 52, No. 2, 1987.

[PSW] A. Pillay, T. Scanlon and F. O. Wagner. Supersimple fields and division rings. Math. Res. Lett. 5 (1998), no. 4, 473-483.

[Pou] Massoud Pourmahdian. Model theory of simple theories. PhD Thesis, Oxford, 2000.

[Sha] Z. Shami. A natural finite equivalence relation definable in low theories and a criterion for lstp = stp. Preprint, 1997.

[Sha1] Ziv Shami. Internality and type-definablity of the automorphism group in simple theories. Preprint, Notre Dame, 2000.

[S] Saharon Shelah. Classification Theory and the Number of Nonisomorphic Models. North-Holland Publishing Co., Amsterdam, 1978.

[S93] Saharon Shelah. Simple unstable theories. Ann. Math. Logic, 19, 177–203, 1980.

[vS] K. G. C. von Staudt. Beiträge zur Geometrie der Lage. Heft 2, 1857.

[To] Ivan Tomašić. Group configuration in simple theories. Preprint, University of Edinburgh, 2000.

[To1] Ivan Tomašić. Hypergroups definable in simple theories. Preprint, University of Edinburgh, 2000.

[TW] Ivan Tomašić and Frank O. Wagner. Applications of the group configuration theorem in simple theories. Preprint, 2001.

[Va] Evgueni V. Vassiliev. Expansions of rank 1 structures. PhD thesis, University of Notre Dame, Notre Dame, 2001.

[VY] O. Veblen and J. W. Young. Projective geometry I & II. Ginn & Co., Boston, 1916.

[W1] Frank O. Wagner. Relational structures and dimensions. Automorphisms of first-order structures, 153–180, Oxford Sci. Publ., Oxford Univ. Press, New York, 1994.

[W] Frank O. Wagner. Simple theories. Kluwer Academic Publishers, 2000.

[W2] Frank O. Wagner. On almost orthogonality in simple theories. Preprint, 2001.

[Z1] Boris Zil’ber. Groups and rings whose theory is categorical (in Russian). Fundamenta Mathematicae, 95, 1977.

[Z2] Boris Zil’ber. Strongly minimal countably categorical theories (in Russian). Sibirskiǐ Matematicheskiǐ Zhurnal, 21, no. 2, 1980.

[Z3] Boris Zil’ber. Totally categorical theories; structural properties and the non-finite axiomatizability. In L. Pacholski, J. Wierzejevski, and A. Wilkie, editors, Model theory of Algebra and Arithmetic, (LNM 834). Springer-Verlag, Berlin, 1980.

[Z4] Boris Zil’ber. Strongly minimal countably categorical theories II (in Russian). Sibirskiǐ Matematicheskiǐ Zhurnal, 25, no. 3, 1984.

[Z5] Boris Zil’ber. Strongly minimal countably categorical theories III (in Russian). Sibirskiǐ Matematicheskiǐ Zhurnal, 25, no. 4, 1984.

[Z6] Boris Zil’ber. Uncountably categorical theories. AMS Translations of Mathematical Monographs, Vol. 117, 1993.