D Palac
email: dpalacin@ub.edu

F O Wagner
email: wagner@math.univ-lyon1.fr

ELIMINATION OF HYPERIMAGINARIES AND STABLE INDEPENDENCE IN SIMPLE CM-TRIVIAL THEORIES

In a simple CM-trivial theory every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries. Moreover, such a theory eliminates hyperimaginaries whenever it eliminates finitary hyperimaginaries. In a supersimple CM-trivial theory, the independence relation is stable.

Introduction

An important notion introduced by Shelah for a first-order theory is that of an imaginary element: the class of a finite tuple by a ∅-definable equivalence relation. The construction obtained by adding all imaginary elements to a structure does not change its basic model-theoretic properties, but introduces a convenient context and language to talk about quotients (by definable equivalence relations) and canonical parameters of definable sets. In the context of a stable theory it also ensures the existence of canonical bases for arbitrary complete types, generalizing the notion of a field of definition of an algebraic variety.

The generalization of stability theory to the wider class of simple theories necessitated the introduction of hyperimaginaries, classes of countable tuples modulo ∅-type-definable equivalence relations. Although the relevant model-theory for hyperimaginaries has been reasonably well understood [START_REF] Hart | Coordinatisation and canonical bases in simple theories[END_REF], they cannot simply be added as extra sorts to the underlying structure, since inequality of two hyperimaginaries amounts to non-equivalence, and thus a priori is an open, but not a closed condition. While hyperimaginary elements are needed for the The first author was partially supported by research project MTM 2008-01545 of the Spanish government and research project 2009SGR 00187 of the Catalan government. The second author was partially supported by the research projct ANR-09-BLAN-0047 of the Agence Nationale de la Recherche. This work was partially done while the first author was visiting the Institut Camille Jordan of Lyon. The first author wishes to express his gratitude to the members of Lyon logic group for their hospitality. He also would like to thank Enrique Casanovas for his valuable comments.

general theory, all known examples of a simple theory eliminate them in the sense that they are interdefinable (or at least interbounded) with a sequence of ordinary imaginaries; the latter condition is called weak elimination. It has thus been asked (and even been conjectured):

Question. Do all simple theories eliminate hyperimaginaries?

The answer is positive for stable theories [START_REF] Pillay | Pas d'imaginaires dans l'infini![END_REF], and for supersimple theories [START_REF] Buechler | Supersimple theories[END_REF]. Among non-simple theories, the relation of being infinitely close in a non-standard real-closed field gives rise to non-eliminable hyperimaginaries; Casanovas and the second author have constructed non-eliminable hyperimaginaries in a theory without the strict order property [START_REF] Casanovas | The free roots of the complete graph[END_REF].

A hyperimaginary is finitary if it is the class of a finite tuple modulo a type-definable equivalence relation. Kim [START_REF] Kim | A note on Lascar strong types in simple theories[END_REF] has shown that small theories eliminate finitary hyperimaginaries, and a result of Lascar and Pillay [START_REF] Lascar | Hyperimaginaries and automorphism groups[END_REF] states that bounded hyperimaginaries can be eliminated in favour of finitary bounded ones. We shall show that in a CM-trivial simple theory all hyperimaginaries are interbounded with sequences of finitary hyperimaginaries. We shall deduce that in such a theory hyperimaginaries can be eliminated in favour of finitary ones. In particular, a small CM-trivial simple theory eliminates hyperimaginaries. However, even the question whether all one-based simple theories eliminate hyperimaginaries is still open.

Elimination of hyperimaginaries is closely related to another question, the stable forking conjecture:

Question. In a simple theory, if a | ⌣B M for some model M containing B, is there a stable formula in tp(a/M) which forks over B ? If we do not require M to be a model, nor to contain B, this is called strong stable forking. Every known simple theory has stable forking; Kim [START_REF] Kim | Simplicity, and stability in there[END_REF] has shown that one-based simple theories with elimination of hyperimaginaries have stable forking. Kim and Pillay [START_REF] Kim | Around stable forking[END_REF] have strengthened this to show that one-based simple theories with weak elimination of imaginaries hyperimaginaries have strong stable forking; on the other hand pseudofinite fields (which are supersimple of SU-rank 1) do not. Conversely, stable forking implies weak elimination of hyperimaginaries (Adler).

While we shall not attack the stable forking conjecture as such, we shall show in the last section that the independence relation x | ⌣y 1 y 2 is stable, meaning that it cannot order an infinite indiscernible sequence.

Preliminaries

As usual, we shall work in the monster model C of a complete firstorder theory (with infinite models), and all sets of parameters and all sequences of elements will live in C eq . Given any sequences a, b and any set of parameters A, we write a ≡ A b whenever a and b have the same type over A. We shall write a ≡ Proof. Let a be a finite tuple with e ∈ dcl heq (a), and h a finitary hyperimaginary over e. So there is a finite tuple b with h ∈ dcl heq (eb) ⊆ dcl heq (ab). Then there is a type-definable equivalence relation E over ∅ such that e and a E are interdefinable, and a type-definable equivalence relation F a over a such that h and b Fa are interdefinable. Moreover, F a only depends on the

E-class of a, that is, if a ′ Ea, then F a ′ = F a .
Type-define an equivalence relation by

xy Ēuv ⇔ xEu ∧ yF x v.
It is easy to see that h is interdefinable with (ab) Ē over e. Moreover, (ab) Ē is clearly finitary, and hence eliminable in T . So h is eliminable in T (e).

The following fact appears in [8, Proof of Proposition 2.2], but was first stated as such in [1, Lemma 2.17].

Fact 2.5. Let h be a hyperimaginary and let a be a sequence of imaginaries such that a ∈ bdd(h) and h ∈ dcl heq (a). Then, h is eliminable.

Fact 2.6. [1, Lemma 2.18] Let h, e be hyperimaginaries with h ∈ bdd(e). Then the set of e-conjugates of h is interdefinable with a hyperimaginary h ′ . Fact 2.7. [8, Theorem 4.15] A bounded hyperimaginary is interdefinable with a sequence of finitary hyperimaginaries.

Proposition 2.8. If T eliminates finitary hyperimaginaries, then T eliminates quasi-finitary hyperimaginaries.

Proof. Let h be a quasi-finitary hyperimaginary and let a be a finite tuple of imaginaries such that h ∈ bdd(a). Consider a ′ ≡ h a with bdd(a) ∩ bdd(a ′ ) = bdd(h). Let h ′ be the hyperimaginary corresponding to the set of aa ′ -conjugates of h. Then h ′ is aa ′ -invariant, and hence finitary. It is thus interdefinable with a sequence e of imaginaries.

On the other hand, h ∈ bdd(a)∩bdd(a ′ ), as are all its aa ′ -conjugates. Thus h ′ ∈ bdd(a) ∩ bdd(a ′ ) = bdd(h). Hence e ∈ acl eq (h) and h ∈ bdd(h ′ ) = bdd(e). By Fact 2.7, there is a sequence h ′′ of finitary hyperimaginaries interdefinable with h over e. By Lemma 2.4 and elimination of finitary hyperimaginaries we see that h ′′ is interdefinable over e with a sequence e ′ of imaginaries. So h ∈ dcl heq (ee ′ ) and e ′ ∈ dcl eq (eh). Moreover, ee ′ ∈ acl eq (h) since e ∈ acl eq (h). Hence h is eliminable by Fact 2.5.

The following remarks and lemmata will need G-compactness.

Remark 2.9. Let T be G-compact over a set A. The following are equivalent:

(1) a ≡ Ls A b iff a ≡ s A b for all sequences a, b. (2) Aut(C/bdd(A)) = Aut(C/acl eq (A)).

(3) bdd(A) = dcl heq (acl eq (A)).

Proof. Easy exercise.

Remark 2.10. Let T be a G-compact theory and assume further that a ≡ Ls A b ⇔ a ≡ s A b for all sequences a, b and for any set A. Let now h be a hyperimaginary and let e be a sequence of imaginaries such that h and e are interbounded. Then h is eliminable.

Proof. It follows from Remark 2.9 that bdd(e) = dcl heq (acl eq (e)). Fix an enumeration ē of acl eq (e) and observe that h ∈ dcl heq (ē) and ē ∈ bdd(h). Then apply Fact 2.5 to eliminate h.

It turns out for G-compact theories that elimination of hyperimaginaries can be decomposed as weak elimination of hyperimaginaries plus the equality between Lascar strong types and strong types over parameter sets. Proof. The proof in [START_REF] Casanovas | Simple theories and hyperimaginaries[END_REF] is nice and intuitive; however, we will give another one using Remark 2.10. If T eliminates bounded hyperimaginaries, then Aut(C/bdd(∅)) = Aut(C/acl eq (∅)). By Remark 2.9 we get Lstp = stp. For the other direction, let e ∈ bdd(∅) and let ā be an enumeration of acl eq (∅). It is clear that e and ā are interbounded. By Remark 2.10, e is eliminable. Lemma 2.12. Suppose T is G-compact and assume further that T eliminates finitary hyperimaginaries. Then a ≡ Ls A b iff a ≡ s A b for all sequences a, b and for any set A.

Proof. Since T is G-compact, it is enough to check the condition for finite A. But then T (A) eliminates finitary hyperimaginaries by Remark 2.4, and hence all bounded hyperimaginaries by Fact 2.7. Now applying Fact 2.11 we obtain that a ≡ Ls b iff a ≡ s b in T (A).

Elimination of hyperimaginaries in simple theories

In this section T will be a simple theory. Recall that the canonical base of a over b, denoted Cb(a/b), is the smallest definably closed subset C of bdd(b) such that a | ⌣C b and tp(a/C) is Lascar strong. Lemma 3.1. For any a and any h ∈ bdd(c) we have Cb(a/h) ⊆ dcl(ac) ∩ bdd(h). Therefore, the canonical base of the type of an imaginary finite tuple over a quasi-finitary hyperimaginary is finitary. Furthermore, if b ∈ Cb(a/c) then dcl(ab) ∩ bdd(b) ⊆ Cb(a/c). In particular, if c ∈ dcl(a) then Cb(a/c) = dcl(a) ∩ bdd(c).

Proof. Since h ∈ bdd(c), equality of Lascar strong types over c refines equality of Lascar strong types over h, and the class of a modulo the former is clearly in dcl(ac). So the class of a modulo the latter is in dcl(ac), and Cb(a/h) ∈ dcl(ac) ∩ bdd(h). As a consequence, if a is a finite tuple and h is a quasi-finitary hyperimaginary bounded over some finite tuple c, then Cb(a/h) is definable over the finite tuple ac. We can now state (and prove) the main result.

Theorem 3.5. Let T be a simple CM-trivial theory. Then every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries.

Proof. By Lemma 3.1 every hyperimaginary is interbounded with a canonical base. Since Cb(A/B) is interdefinable with {Cb(ā/B) : ā ∈ A finite}, it is enough to show that canonical bases of types of finite tuples are interbounded with sequences of finitary hyperimaginaries. This is precisely Proposition 3.4.

Corollary 3.6. A simple CM-trivial theory eliminates hyperimaginaries whenever it eliminates finitary ones.

Proof. By Theorem 3.5 every hyperimaginary is interbounded with a sequence of finitary hyperimaginaries and so with a sequence of imaginaries. Since T is simple, it is G-compact, whence Lstp = stp over any set by Lemma 2.12. We conclude that every hyperimaginary is eliminable by Remark 2.10.

Corollary 3.7. Every small simple CM-trivial theory eliminates hyperimaginaries.

Proof. A small simple theory eliminates finitary hyperimaginaries by [START_REF] Kim | A note on Lascar strong types in simple theories[END_REF]. Now apply Corollary 3.6.

Stable independence for CM-trivial theories

Recall that an ∅-invariant relation R(x, y) is stable if there is no infinite indiscernible sequence (a i , b i : i < ω) such that R(a i , b j ) holds if and only if i < j. In this section, we shall show that independence is a stable relation, even with varying base set. We hope that this will help elucidate the stable forking problem. Theorem 4.1. In a supersimple CM-trivial theory, the relation R(x; y 1 y 2 ) given by x | ⌣y 1 y 2 is stable.

Proof. Suppose not. Then there is an indiscernible sequence I = (a i : i ∈ Q) and tuples b, c such that

• I + = (a i : i > 0) is indiscernible over I -bc,
• I -= (a i : i < 0) is indiscernible over I + bc, and • a i | ⌣c b if and only if i > 0. We consider limit types with respect to the cut at 0. Put p = lim(I/I), p + = lim(I + /Ibc) and p -= lim(I -/Ibc).

By finite satisfiability, p + and p -are both non-forking extensions of p, which is Lascar-strong. Let

A = Cb(p) = Cb(p + ) = Cb(p -) ∈ bdd(I + ) ∩ bdd(I -).
As p + and p -do not fork over A, we have

a i | ⌣ A I + bc for all i < 0, and a i | ⌣ A I -bc for all i > 0.
We consider first e 0 = bdd(a 1 c) ∩ bdd(Ac). Then bdd(a 1 e 0 ) ∩ bdd(Ae 0 ) = e 0 .

Put A 0 = Cb(a 1 /e 0 ). By CM-triviality By compactness and Ramsey we may suppose in addition that Ī = (ā i : i ∈ Q) is B-indiscernible, Ī+ = (ā i : i > 0) is indiscernible over Bb Īand Ī-= (ā i : i < 0) is indiscernible over Bb Ī+ . We shall add B to the parameters and suppress it from the notation. We may further assume that ā′ Let w P (.) denote the weight with respect to that non-orthogonality class P of regular types. Then āi is P-semi-regular; since ā-1 | ⌣ b we obtain w P (ā -1 ) > w P (ā -1 /b) by [START_REF] Pillay | Geometric Stability Theory[END_REF]Lemma 7.1.14] (the proof works just as well for the simple case).

A 0 ∈ e 0 ∩ bdd(Cb(a 1 /Ae 0 )) ⊆ bdd(a 1 c) ∩ bdd(A), since a 1 | ⌣A e 0 implies Cb(a 1 /Ae 0 ) ⊆ bdd(A). Note that a 1 | ⌣c b yields a 1 | ⌣A 0 c b. Moreover c ∈ e 0 , so a 1 | ⌣A 0 e 0 implies a 1 | ⌣A 0 c,
We again consider limit types with respect to the cut at 0. Put Let A 1 = Cb(ā -1 /e 1 ). By CM-triviality

A 1 ∈ e 1 ∩ bdd(Cb(ā -1 / Āe 1 )) ⊆ bdd(ā -1 b) ∩ bdd( Ā),
w P (ā -1 /A 1 b) = w P (ā -1 /A 1 ) = w P (ā 1 /A 1 ). Recall that A 1 ⊆ bdd(ā -1 b). Then w P (ā -1 /b) = w P (ā -1 A 1 /b) = w P (ā -1 /A 1 b) + w P (A 1 /b) = w P (ā 1 /A 1 ) + w P (A 1 /b) ≥ w P (ā 1 /A 1 b) + w P (A 1 /b) = w P (ā 1 A 1 /b) ≥ w P (ā 1 /b) = w P (a 1 ) = w P (a -1 ) > w P (ā -1 /b).
This final contradiction proves the theorem.

Remark 4.3. Note that the proof only uses the conclusion of Fact 4.2. The theorem thus still holds for simple CM-trivial theories with finite weights (strongly simple theories) and enough regular types, for instance CM-trivial simple theories without dense forking chains.

Question. By [START_REF] Palacín | Ample Thoughts[END_REF]Theorem 4.20] it is sufficient to assume that every regular type is CM-trivial, as this implies global CM-triviality. However, for a regular type p a more general notion of CM-triviality is often more appropriate, namely

cl p (aA) ∩ cl p (B) = cl p (A) ⇒ Cb(a/cl p (A)) ⊆ cl p (Cb(a/cl p (B))).
If this holds for all regular types p, is independence still stable? given by x | ⌣cl Σ (y 1 ) y 2 is stable.

Proof. Suppose not. Then there is an indiscernible sequence I = (a i : i ∈ Q) and tuples b, c such that Remark 4.9. If the theory is supersimple, we can take P to be the family of all non one-based regular types.

• I + = (a i : i > 0) is indiscernible over I -bc, • I -= (a i : i < 0) is indiscernible over I + bc,
Remark 4.10. Theorem 4.8 generalises the fact that independence is stable in a one-based theory. For a true generalisation of Theorem 4.1 to arbitrary theories, one should take P to be the family of all 2-ample types. This is work in progress.

Fact 2 .

 2 11. [2, Proposition 18.27] Assume that T is G-compact. Then T eliminates all bounded hyperimaginaries iff a ≡ Ls b ⇔ a ≡ s b for all sequences a, b.

Remark 3 . 3 .Proposition 3 . 4 .

 3334 For the second assertion put b ′ = dcl(ab) ∩ bdd(b). Since b ′ ∈ dcl(ab) there is an equivalence relation E on tp(a/b) type-definable over b such that b ′ is interdefinable over b with a E . As b ′ ∈ bdd(b) and b ∈ Cb(a/c), the E-class of a is bounded over Cb(a/c); as tp(a/Cb(a/c)) is Lascarstrong, a E ∈ Cb(a/c). The "in particular" clause is essentially [1, Remark 3.8]: If c ∈ dcl(a) then clearly c ∈ Cb(a/c); the assertion follows. Recall the definition of CM-triviality. Definition 3.2. A simple theory T is CM-trivial if for every tuple a and for any sets A ⊆ B with bdd(aA) ∩ bdd(B) = bdd(A) we have Cb(a/A) ⊆ bdd(Cb(a/B)). As in [10, Corollary 2.5], in the definition of CM-triviality we may take A ⊆ B to be models of the ambient theory and a to be a tuple from the home sort. Therefore, it makes no difference in the definition of CM-triviality whether we consider hyperimaginaries or just imaginaries. Now we characterize canonical bases in simple CM-trivial theories in terms of finitary hyperimaginaries. Assume the theory is simple CM-trivial. If a is a finite imaginary tuple, then bdd(Cb(a/B)) = bdd(Cb(a/b) : b ∈ X), where X is the set of all finitary b ∈ bdd(Cb(a/B)). Proof. Since Cb(a/b) ⊆ bdd(b) ⊆ bdd(Cb(a/B)) for b ∈ X, we have bdd(Cb(a/b) : b ∈ X) ⊆ bdd(Cb(a/B)). For the reverse inclusion, for every b ∈ X let b be a real tuple with Cb(a/b) ∈ dcl( b); we choose them such that ( b : b ∈ X) | ⌣ (Cb(a/b):b∈X) aB, whence ( b : b ∈ X) | ⌣B a. Now, if a | ⌣( b:b∈X) B then there is a finite tuple b ′ ∈ B ∪ { b : b ∈ X} and a formula ϕ(x, b ′ ) ∈ tp(a/B, b : b ∈ X) which divides over ( b : b ∈ X). Put b = bdd(ab ′ ) ∩ bdd(B, b : b ∈ X). Then b is a quasi-finitary hyperimaginary, and by CM-triviality Cb(a/ b) ⊆ bdd(Cb(a/B, b : b ∈ X)) = bdd(Cb(a/B)). Since Cb(a/ b) is finitary by Lemma 3.1, it belongs to X. Note that b ′ ∈ b; but a | ⌣Cb(a/ b) b, so ϕ(x, b ′ ) cannot divide over Cb(a/ b), and even less over ( b : b ∈ X) as this contains Cb(a/ b). Thus, a | ⌣( b:b∈X) B, whence a | ⌣(Cb(a/b):b∈X) B by transitivity. Therefore Cb(a/B) ⊆ bdd(Cb(a/b) : b ∈ X). Question. The same proof will work without assuming CM-triviality if for every finite tuple b ∈ B there is some quasi-finitary hyperimaginary b ∈ bdd(B) with b ∈ dcl( b) such that Cb(a/ b) ⊆ bdd(Cb(a/B)). Is this true in general?

whence a 1 |

 1 ⌣A 0 cb by transitivity. On the other hand, suppose bc | ⌣A 0 a -1 . Then b | ⌣A 0 c a -1 ; as b | ⌣c a 1 implies b | ⌣c A 0 , we obtain b | ⌣c A 0 a -1 , contradicting a -1 | ⌣c b. Therefore bc | ⌣A 0 a -1 . Since I remains indiscernible over A 0 , and both I + and I -remain indiscernible over A 0 bc, we may add A 0 to the parameters and suppose c = ∅ (replacing b by bc).

Fact 4 . 2 .

 42 [START_REF] Frank | Simple theories[END_REF] Theorem 5.2.18] In a supersimple theory, for any finitary a there are some B | ⌣ a and a hyperimaginary finite tuple ā of independent realizations of regular types over B, such that ā is dominationequivalent with a over B.ByFact 4.2 there are B | ⌣ a 1 and an independent tuple ā1 of realizations of regular types over B such that ā1 is domination-equivalent with a 1 over B. Since B | ⌣ a 1 and I is indiscernible, we may assume by [13, Theorem 2.5.4] that Ba i ≡ Ba 1 for all i ∈ Q, and B | ⌣ I. So there are āi for i ∈ Q with Ba i āi ≡ Ba 1 ā1 . We can also assume B | ⌣I b, whence B | ⌣ Ib. In particular b | ⌣a i B, so for i > 0 we obtain b | ⌣ Ba i and thus b | ⌣B a i , while for i < 0 we have b | ⌣ a i B and b | ⌣ B, whence b | ⌣B a i . By domination-equivalence, āi | ⌣B b for i > 0 whereas āi | ⌣B b for i < 0.

- 1 |

 1 ⌣ b for any proper subtuple ā′ -1 ⊆ ā-1 . Claim. All the regular types in āi are non-orthogonal. Proof of Claim: Consider c, c ′ ∈ ā-1 and put c = ā-1 \ {c, c ′ }. Then cc | ⌣ b and cc ′ | ⌣ b by minimality, whence c | ⌣ bc and c ′ | ⌣ bc, as ā-1 = ccc ′ is an independent tuple. Suppose c | ⌣bc c ′ . Then c | ⌣ bcc ′ , whence c | ⌣cc ′ b and finally b | ⌣ ccc ′ , contradicting b | ⌣ ā-1 . So tp(c/bc) and tp(c ′ /bc) are non-orthogonal; as they do not fork over ∅ we get tp(c) non-orthogonal to tp(c ′ ). The claim now follows, as all āi have the same type over ∅.

  p = lim( Ī/ Ī), p+ = lim( Ī+ / Īb) and p-= lim( Ī-/ Īb). Once more, p is Lascar-strong and p+ and p-are non-forking extensions of p by finite satisfiability; let Ā = Cb(p) = Cb(p + ) = Cb(p -) ∈ bdd( Ī+ ) ∩ bdd( Ī-). As before, āi | ⌣ Ā Ī+ b for all i < 0, and āi | ⌣ Ā Īb for all i > 0. Put e 1 = bdd(ā -1 b) ∩ bdd( Āb). Then bdd(ā -1 e 1 ) ∩ bdd( Āe 1 ) = e 1 .

since ā- 1 |

 1 ⌣ Ā e 1 implies Cb(a -1 / Āe 1 ) ⊆ bdd( Ā). As b ∈ e 1 and ā-1 | ⌣A 1 e 1 we obtain ā-1 | ⌣A 1 b. Moreover ā1 ≡ A 1 ā-1 , since A 1 ⊆ bdd( Ā) and Ī remains indiscernible over Ā. Therefore

Corollary 4 . 4 .

 44 An ω-categorical supersimple CM-trivial theory has stable forking. Proof. Suppose A | ⌣B C. Then there are finite tuples ā ∈ A and c ∈ C with ā | ⌣B c. By supersimplicity, there is a finite b ∈ B with āc | ⌣ b B. Thus ā | ⌣ b c. By ω-categoricity there is a formula ϕ(x, ȳ1 ȳ2 ) which holds if and only if x | ⌣ ȳ1 ȳ2 . Then ϕ is stable by Theorem 4.1, and ϕ(x, bc) ∈ tp(ā/ bc). Let Σ be an ∅-invariant family of types. Recall the definition of Σ-closure: cl Σ (A) = {a : tp(a/A) is Σ-analysable}.

Fact 4 . 5 .Corollary 4 . 6 .

 4546 [START_REF] Frank | Simple theories[END_REF] Lemma 3.5.3 and 3.5.5] If dcl(AB) ∩ cl Σ (A) ⊆ bdd(A), then B | ⌣A cl Σ (A). If A | ⌣B C, then A | ⌣cl Σ (B) C.In a supersimple CM-trivial theory the relation R(x; y 1 y 1 )

  and • a i | ⌣cl Σ (c) b if and only if i > 0. Put c ′ = dcl(bc) ∩ cl Σ (c). By Fact 4.5 we have b | ⌣c ′ cl Σ (c), so by transitivity a i | ⌣c ′ b for i > 0. Suppose a i | ⌣c ′ b for i < 0. Since cl Σ (c ′ ) = cl Σ (c), Fact 4.5 yields a i | ⌣cl Σ (c) b, a contradiction. Thus a i | ⌣c ′ b if and only if i > 0, contradicting Theorem 4.1.To conclude the paper we prove a version of Corollary 4.6 without the assumption of CM-triviality, but for a particular ∅-invariant family, namely the family P of all non one-based types.

Fact 4 . 7 .

 47 [START_REF] Palacín | Ample Thoughts[END_REF] Corollary 5.2] In a simple theory a | ⌣cl P (a)∩bdd(b) b for all tuples a and b, where P is the family of all non one-based types.

Theorem 4 . 8 .

 48 In a simple theory, the relation R(x; y 1 y 2 ) given by x | ⌣cl P (y 1 ) y 2 is stable, where P is the family of all non one-based types.Proof. Suppose not. Then there is an indiscernible sequence I = (a i : i ∈ Q) and tuples b, c such that• I + = (a i : i > 0) is indiscernible over I -bc, • I -= (a i : i < 0) is indiscernible over I + bc,and • a i | ⌣cl P (c) b if and only if i > 0. As before, we consider limit types with respect to the cut at 0. Let p = lim(I/I), p + = lim(I + /Ib) and p -= lim(I -/Ib). By finite satisfiability, p + and p -are both non-forking extensions of p, which is Lascar-strong. Let A = Cb(p) = Cb(p + ) = Cb(p -) ∈ bdd(I + ) ∩ bdd(I -). As in the proof of Theorem 4.1 we have a i | ⌣ A I + bc for all i < 0, and a i | ⌣ A I -bc for all i > 0. We consider first e = cl P (a 1 )∩bdd(A). Then a 1 | ⌣e A by Fact 4.7; since I remains indiscernible over bdd(A) we have a -1 ≡ bdd(A) a 1 , whence e = cl P (a -1 ) ∩ bdd(A) and a -1 | ⌣e A. On the other hand, since e ∈ bdd(A) and a i | ⌣A bc for i ∈ Q we obtain a 1 | ⌣ e bc and a -1 | ⌣ e bc. Now put c ′ = dcl(bc) ∩ cl P (c); note that cl P (c ′ ) = cl P (c). Then b | ⌣c ′ cl P (c) by Fact 4.5. Moreover, a 1 | ⌣cl P (c) b yields cl P (a 1 ) | ⌣cl P (c) b by Fact 4.5, whence cl P (a 1 ) | ⌣c ′ b. Thus e | ⌣c ′ b and hence e | ⌣c ′ bc since c ⊆ c ′ . But now c ′ ⊆ dcl(bc) and a -1 | ⌣e bc imply that a -1 | ⌣c ′ bc. Hence a -1 | ⌣cl P (c) b by Fact 4.5, as cl P (c ′ ) = cl P (c). This contradiction finishes the proof.

  s A b if in addition a and b lie in the same class modulo all A-definable finite equivalence relations (i.e. if a and b have the same strong type over A), and a ≡ Ls A b if they lie in the same class modulo all A-invariant bounded equivalence relations (i.e. if a and b have the same Lascar strong type over A). Recall that a theory is G-compact over a set A iff ≡ LsA is type-definable over A (in which case it is the finest bounded equivalence relation type-definable over A). A theory T is G-compact whenever it is G-compact over any A. In particular, simple theories are G-compact[START_REF] Kim | A note on Lascar strong types in simple theories[END_REF].Definition 2.1. A hyperimaginary h is finitary if h ∈ dcl heq (a) for some finite tuple a of imaginaries, and quasi-finitary if h ∈ bdd(a) for some finite tuple a of imaginaries. Definition 2.2. A hyperimaginary h is eliminable if it is interdefinable with a sequence e = (e i : i ∈ I) of imaginaries, i.e. if there is such a sequence e with dcl heq (e) = dcl heq (h). A theory T eliminates (finitary/quasi-finitary) hyperimaginaries if all (finitary/quasi-finitary) hyperimaginaries are eliminable in all models of T . Let e be a finitary hyperimaginary. If T eliminates finitary hyperimaginaries, then T (e) eliminates finitary hyperimaginaries.

	Remark 2.3. [8, Corollary 1.5] If h ∈ dcl heq (a), then there is a type-
	definable equivalence relation E on tp(a) such that h and the class a E
	of a modulo E are interdefinable.
	Lemma 2.4.