THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES

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1 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES BYUNGHAN KIM Abstract. This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries, and make some new observations on the strong types over those. In particular we show that in a simple theory, Ltp stp in real context implies that for hyperimaginary context. The Lascar group introduced by Lascar [8], and related subjects have been studied by many authors ([9][1][7][6][3] and more). Notably in [9][1], a new look on the Lascar group is given, and using compact Lie group theory Lascar and Pillay proved that any bounded hyperimaginary is interdefinable with a sequence of finitary bounded hyperimaginaries. Good summaries on the Lascar group are written in [10][11]. While this is another short expository note stating known results in [8][9][1][7], we supply a couple of new observations. We study the Lascar group in slightly more general context namely over hyperimaginaries. The notion of strong types over hyperimaginaries is somewhat subtler even at the level of the definition (see Example 3.5). As a by-product we show that in a simple theory if Ltp(a/A) stp(a/a) for real tuples a and A, then the same holds for hyperimaginaries. A question remains whether this holds in any theory. We work with an arbitrary complete theory T in L, and a fixed large saturated model M = T of size κ, as usual. We recall some of definitions. Unless said otherwise a tuple can have an infinite size (< κ). By a hyperimaginary we mean an equivalence class of a typedefinable equivalence relation over. So a hyperimaginary has the form a/e = a E where a is a tuple from M and E(x, y) is the -type-definable equivalence relation on M x. We call a E an E-hyperimaginary. We say the hyperimaginary is finitary if a is a finite tuple. In general we put a E := a. In the note arity means an arity of a real tuple. From now on a, b, c,..., A, B,... denote hyperimaginaries, but M, N,.. denote elementary small submodels of M. Clearly any tuple from This is a note submitted for the proceedings of the conference: Recent developments in model theory, Oléron, France, June 2011, where the author gave a talk on different subjects in [4],[5]. This work is supported by an NRF grant

2 2 BYUNGHAN KIM M or M eq is also a hyperimaginary. We call such a tuple real or imaginary, respectively. Given a hyperimaginary c, Aut c (M) denotes the set of all automorphisms of M fixing c (i.e., fixing the equivalence class setwise). A relation is said to be over c or c-invariant if it is Aut c (M)-invariant. A hyperimaginary a is said to be bounded over b (written a bdd(b)) if a has only boundedly many automorphic conjugates over b, i.e., {f(a) f Aut b (M)} < κ. Say a is bounded if a bdd( ). Similarly we say a is definable (algebraic, resp.) over b written a dcl(b) (a acl(b) resp.) if {f(a) f Aut b (M)} is a singleton (finite, resp.). Two hyperimaginaries a, b are said to be interdefinable or equivalent if a dcl(b) and b dcl(a). We use nonstandard notation a b to denote a dcl(b). We put acl eq (b) = (acl(b) M eq ) {b}. Notice the difference between acl(b) and acl eq (b); both are somewhat newly introduced when b is a hyperimaginary. In Corollary 3.4 we shall see that acl(b) and acl eq (b) are interdefinable. As is known the type of a over b, tp(a/b), makes sense, and p S E (b) means p is a type of some E-hyperimaginary over b. As usual a b c means c = tp(a/b). Of course when we say a b c, both a, c must be hyperimaginaries for a common E, so in general E-(complete) types are computed in M/E. For two partial E-types p(x), q(x) we write q p if p = q and q = p. For the additional introduction to hyperimaginaries the reader may see [11]. 1. Lascar group We restate definitions from [8][9][7]. Throughout we fix a hyperimaginary A. Recall that Autf A (M)(= a subgroup of Aut A (M) generated by {f Aut A (M) f Aut M (M) for some model M A}) is a normal subgroup of Aut A (M). We recall a fact on Lascar (strong) types. Definition 1.1. Let a, b be hyperimaginaries such that tp(a/a) = tp(b/a). We define d A (a, b) to be the least natural number n( 1) such that: there are sequences I 1, I 2,..., I n and hyperimaginaries a = a 0, a 1,..., a n = b such that a i 1I i and a i I i are both A-indiscernible for each 1 i n. (If there is no such n < ω, then we write d A (a, b) =.) The following is proved in [7] when a, b, A are real. The same proof works for hyperimaginaries. Fact 1.2. The following are equivalent (a, b are E-hyperimaginaries).

3 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES3 (1) Ltp(a/A) = Ltp(b/A) (or write a L A b), i.e. there is f Autf A (M) such that f(a) = b. (2) d A (a, b) < ω. (3) = F (a, b) for any A-invariant bounded equivalence relation F coarser than E. In conclusion x E L A y E is an invariant bounded equivalence relation over A coarser than E, and is the finest among those. Definition 1.3. (Lascar group) Gal L (M, A) := Aut A (M)/ Autf A (M). In the rest of the paper, when there is no risk of confusion we may omit A for notational convenience, so A M for any model mentioned below, and Aut(M), Autf(M) below indeed mean Aut A (M), Autf A (M), respectively. As said in the introduction, sections 1 and 2 form a summary of known results from [1][9][10] when A is real. Most of the arguments there go through even when A is a hyperimaginary, and we will repeat some arguments for the sake of completion. Remark 1.4. We argue that the Lascar group depends only on T and A, and we write Gal L (T, A) for Gal L (M, A). We have Gal L (T, A) 2 T + A. (1) Let M be a (small) elementary submodel of M. For f, g Aut(M), if f(m) M g(m) then f. Autf(M) = g. Autf(M) in Gal L (M): There is h in Autf(M) which fixes M pointwise and sends f(m) to g(m). Then since s := f 1.h 1.g fixes M too, s Autf(M). Thus the claim follows. (2) In (1), we can choose M having size T + A. Since there are at most 2 T + A -many types over M, we have Gal L (M, A) 2 T + A. (3) Let M M be saturated and M > M. There is a canonical isomorphism from Gal L (M) to Gal L (M ): Let f Aut(M). Any two automorphisms of M extending f are in the same coset in Gal L (M ). This induces a homomorphism α : Aut(M) Gal L (M ). We claim that ker(α) = Autf(M): If f Autf(M) then clearly any extension of f in Aut(M ) is in Autf(M ). Conversely, let f Autf(M ) extend f Aut(M). Then for a small model M M, by 1.2, M L f (M) = f(m) in M. Hence there is g Autf(M) such that f(m) = g(m). Then g 1.f Autf(M) since it fixes M, so f Autf(M) too. Let us also write α : Gal L (M) Gal L (M ) for the induced injection. We claim that α is surjective: Let g Aut(M ). For a small model M M, there is M M such that g(m) M

4 4 BYUNGHAN KIM M. Now there is f Aut(M) sending M to M. Then by (1), α(f) = g. Autf(M ). In the rest of this section, for f, g Aut(M), we write f g if f. Autf(M) = g. Autf(M). We shall endow Gal L (T ) with a quotient topology to make it a compact (but not necessarily Hausdorff) topological group. Let π : Aut(M) Gal L (= Gal L (T, A)) be the canonical projection. Fix a model M M and let S M (M) := {tp(f(m)/m) f Aut(M)}, equipped with its Stone topology. Then by 1.4(1), π factors through the surjection µ : Aut(M) S M (M) sending f to tp(f(m)/m), i.e. there is a canonical surjection ν = ν M : S M (M) Gal L such that π = ν.µ. We use these maps. We give Gal L the quotient topology under the map ν. This topology is independent from the choice of M: It suffices to show this for a model N( M), an elementary extension of M (since any two models have a common extension). Now the map ν N : S N (N) Gal L factors through the restriction map S N (N) S M (M) sending tp(f(n)/n) to tp(f(m)/m). Since the restriction map is continuous and both S N (N), S M (M) are compact Hausdorff, ν M, ν N induce the same quotient topology. Lascar originally introduced the topology on Gal L in terms of ultrafilters. It is known that his and the quotient topology coincide. Contrary to what the reader might expect the proof that Gal L is a topological group is quite subtle. The only known complete and correct proof can be found in [10], and the proof goes through when working over some hyperimaginary A: Proposition 1.5. Gal L (T, A) is a compact topological group. 2. Quotient groups of the Lascar group We introduce two canonical subgroups of Gal L (T, A). Note that {id} is a closed normal subgroup of Gal L (T, A). The connected component of Gal L (T, A) containing id is denoted by Gal 0 L(T, A); it is also closed and normal. Definition 2.1. (1) Autf KP (M, A) := π 1 ({id}). (2) Autf S (T, A) := π 1 (Gal 0 L(T, A)). (3) Gal KP (T, A) := Gal L (T, A)/{id} = Aut A (M)/ Autf KP (M, A). (4) Gal S (T, A) := Gal L (T, A)/ Gal 0 L(T, A) = Aut A (M)/ Autf S (M, A). (5) We say that T is G-compact over A if Gal L (T, A) is Hausdorff.

5 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES5 KP stands for Kim-Pillay and S stands for Shelah or strong. Again, below we may omit the subscript A. Remark 2.2. (1) Recall that for topological groups H G, the quotient group G/H is Hausdorff iff H is closed in G. Hence both Gal KP (T ), Gal S (T ) are compact Hausdorff. Moreover Gal S (T ) is totally disconnected, so a profinite group. Now T is G-compact iff {id} is closed iff Autf(M) = Autf KP (M) (then Gal KP (T ) and Gal L (T ) are canonically isomorphic). (2) For the rest of this section we endow Aut(M) with a topology having basic open sets of the form {f Aut(M) f(a) = b} for some real n-tuples a, b M. (However the topologies of Gal S, Gal KP are always the quotient topologies obtained from Gal L.) (3) Let Γ be a subgroup of Aut A (M). Fix F an -type-definable equivalence relation on M α (α : an arity). We write E F Γ to denote an equivalence relation such that for F -hyperimaginaries c, d, we have E F Γ (c, d) iff d = f(c) for some f Γ. So E F Γ is an equivalence relation on M α /F, or equivalently an equivalence relation on M α coarser than F. When we write E = Γ (x, y), = means the finest real equivalence relation x = y. We omit F if F is clear from context. Note that if Γ is normal then E Γ is A- invariant, but in general it need not be. When Γ = Autf(M, A) we know that E Γ is L A, and E Autf KP (M,A) is denoted by KP A. Notice that Γ is closed iff Γ = Γ = {f Aut A (M) for each finite real tuple a M, E = Γ (a, f(a)), i.e. f stabilizes all E = Γ -classes of finite arities}. Lemma 2.3. (1) Let H be a closed normal subgroup of Gal L, and let H = π 1 (H ). Then given F, E F H is a type-definable bounded equivalence relation (over A). (2) The restriction map π is continuous, so both Autf S (M), Autf KP (M) are closed in Aut(M). Proof. (1) Let a = u F = p(x) S F (A) where u is a real tuple of arity α. Fix a model (u )M = M p M. Since H is closed, there is Φ(x, M) over M type-defining ν 1 (H ) i.e. f H iff Φ(f(M), M) holds. Note that H being a group implies that Φ(x, y ) type-defines an equivalence relation on tp(m). Moreover x L A M Φ(x, M). Thus Φ(x, y ) is a bounded equivalence relation on tp(m/a). Note also that since H is normal it easily follows that Φ(x, y ) E = H(x, y ) on tp(m/a). Then by taking existential quantifiers to Φ(x, y ), clearly E H = (x, y) is type-definable on q(x) = tp(u/a) too. Hence EH F (x, y) is type-defined

6 6 BYUNGHAN KIM by Ψ p (x, y) zw(e = H(z, w) q(z) q(w) F (z, x) F (w, y)) on tp(a/a) = p(x). We put Ψ p(x, y) (Ψ p (x, y) p(x) p(y)) x A y. Therefore clearly E F H(x, y) {Ψ p p(x) S F (A)}. (2) Let f Aut(M), and let U be an open subset of Gal L (T ) containing π(f) = f. Autf(M). Since ν is continuous, ν 1 (U) S M (M) contains a basic open neighborhood V φ(x) = {tp(g(m)/m) φ(x) : g Aut(M)} of µ(f) = tp(f(m)/m), where φ(x) is some formula over M. a M be a finite tuple corresponding to x. Then simply µ 1 (V φ(x) ) = {g Aut(M) : g(a) = φ(x)}. Since f µ 1 (V φ(x) ), in particular = φ(f(a)) holds. Therefore a basic open neighborhood {h Aut(M) h(a) = f(a)} of f is contained in µ 1 (V φ(x) ). Hence π is continuous. Now fix a closed normal subgroup H Gal L and let H = π 1 (H ). We write x H y if E = H(x, y) holds, which on any arity, due to 2.3(1), is a bounded type-definable equivalence relation (over A). Proposition 2.4. (1) H = {f Aut(M) f stabilizes all the H - classes of any arities} = {f Aut(M) f stabilizes all the H - classes of finite real arities}. (2) The following are equivalent. (a) c 0 H c 1 for real tuples. (b) c 0 H c 1 for each corresponding finite subtuple c i of c i (i = 0, 1). Proof. (1) Due to 2.3(2), H is closed in Aut(M). Hence it comes from 2.2(3). (2) Clearly (a) implies (b). Assume (b) holds. In this proof all the tuples are real. Note that there is h H such that h (c 0) = c 1. Hence for any finite d 0, there is d 1 = h (d 0 ) with c 0d 0 H c 1d 1. Thus by compactness there is a sufficiently saturated M i containing c i such that for each finite corresponding b i M i, b 0 H b 1 (*). In particular tp(m 0 ) = tp(m 1 ) and there is h Aut(M) sending M 0 to M 1. Let d/ H with arbitrary finite d be given. We claim h fixes d/ H, so by (1), h H, and (a) follows: Due to the saturation of M 0 and the boundedness of H, there is d M 0 such that tp(d) = tp(d ) and d H d holds. Then by (*), h clearly fixes d / H = d/ H. Let

7 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES7 In [9], using compact Lie group theory, Lascar and Pillay showed that any bounded hyperimaginary e is equivalent to a sequence of finitary bounded hyperimaginaries. As a corollary of Proposition 2.4, the result can be directly obtained when Aut e (M) is a normal subgroup of Aut(M). (This is observed by Casanovas and Potier [2].) Corollary 2.5. Let e = c F with real c be a hyperimaginary bounded over A. Assume additionally Aut ea (M) Aut A (M). Then there are finitary hyperimaginaries e i (i I) such that e and (e i i I) are interdefinable over A. Proof. In the proof we again omit A. Let p(x) = tp(e). We may reset F as (p(x) p(y) F (x, y)) x y, so that F as a whole, a typedefinable bounded equivalence relation (over A). Choose a model M containing c. Note that F type-defines a bounded equivalence relation on M M too. Moreover Aut e (M) = {f Aut(M) : = F (f(c), c)}. Hence π(aut e (M)) is a closed subgroup of Gal L. By the assumption it is normal as well. Hence our corollary follows from 2.4(2). Corollary 2.6. (1) For real x, y, x KP A y is the finest bounded type-definable equivalence relation over A. Precisely, for real u i (i = 0, 1) the following are equivalent. (a) u 0 KP A u 1 holds. (b) u 0 A u 1 ; and E (u 0, u 1 ) holds for any -type-definable equivalence relation E such that u 0 /E bdd(a). (c) u 0 KP A u 1 for each corresponding finite subtuple u i of u i. (2) Autf KP (M, A) = {f Aut(M, A) f stabilizes all the bounded A- type-definable equivalence classes} = {f Aut(M, A) f stabilizes all the bounded A- type-definable equivalence classes of finite arities}. (3) The following are equivalent. (a) a KP A b where a = u F, b = v F with real u, v. (b) FA KP (u, v) holds where FA KP (x, y) zz (z KP A z F (z, x) F (z, y)) z(f (z, y) z KP A x), (z s A x is of course equality of KP-types over A of real tuples). FA KP (x, y) is a bounded type-definable equivalence relation over A coarser than F, and is the finest among those. (c) a bdd(a) b.

8 8 BYUNGHAN KIM Proof. (1)(a) (b) This comes from the argument in the proof of 2.3(1). Note that for a type Ψ(x, M) type-defining ν 1 ({id}), Ψ(x, y) must be the finest bounded type-definable equivalence relation on tp(m/a). The equivalence of (1)(c) to others, and (2) are due to 2.4. (3)(a) (b) Again this is from the argument in the proof of 2.3(1). Note that if u KP A v and F (u, u ), F (v, v ) with real u, v so that there is f Autf KP (M, A) with v = f(u ), then f(u) KP A u and F (f(u), v) holds. Hence one can easily verify that Ψ p there with H = Autf KP (M, A) is equivalent to FA KP above on p(x) = tp(a/a). Note also that FA KP is coarser than both s and F. It remains to show FA KP is the finest one as stated. There clearly is a finest one; let it be F. If FA KP (u, v ) holds, then there is u such that F (u, u ) u KP A v. Hence F (u, u ) and F (u, v ); so F (u, v ) must hold. (c) (b) Obvious. Note only that due to (1)(b), u/ KP A is equivalent over A to a hyperimaginary. (b) (c) Assume (b). Let a 0 bdd(a). By compactness it suffices to show c 0 Aa0 c 1. Suppose {a i i I} is the set of all A-conjugates of a 0. Write p(x, a 0 ) = p(x F, a 0 ) = tp(c 0 /Aa 0 ) and p(x, a i ) is the conjugate of p(x, a 0 ) over A. Now there is a maximal subset J I containing 0 such that {p(x, a i ) i J} is realized by c 0. Put a J = a i i J, and let p (xz) = tp(c 0 a J /A); q(z) = tp(a J /A). Consider Ē(x, y) z(p (x, z) p (y, z) q(z)) x F A y F. Due to the maximality of J, Ē is an A-type-definable equivalence relation, bounded, and coarser than F. Thus by (b), Ē(c 0, c 1 ) holds. Note that c 0 = i J p(x, a i), so c 1 = p(x, a 0 ), and c 0 Aa0 c 1 as desired. Proposition 2.7. The following are equivalent. (1) T is G-compact over A. (2) Autf(M, A) is closed in Aut A (M), and for each finite arity, x L A y is type-definable over A. (3) For any arity, x L A y iff x KP A y. (4) For any arity, x L A y is type-definable. (5) For any F, x F L A y F iff x F KP A y F. Proof. (1) (2),(3) By 2.3, 2.6. (2) (1) Let f Autf KP (M). Then by (2) and 2.6, f fixes all L - classes of finite arities. Then since Autf(M) is closed, from 2.2(3), f Autf(M). (3) (1) Let f Autf KP (M). Due to (3), for a model M, we have M L f(m), and there is g Autf(M) such that f(m) = g(m). Thus by 1.4(1), f Autf(M). (3) (4), (1) (5) (3) Clear.

9 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES9 As is well-known, any simple theory T is G-compact over an arbitrary hyperimaginary. 3. Strong types of hyperimaginaries Now we talk about strong types in the hyperimaginary context which is somewhat more subtle, even the definition, than in the real case. Recall that a finite equivalence relation means an equivalence relation having finitely many classes. Definition 3.1. We say two hyperimaginaries a, b have the same strong or Shelah type over A, written a s A b or stp(a/a) = stp(b/a), if E AutfS (M,A)(a, b) holds. Proposition 3.2. (1) Gal 0 L(T, A) is the intersection of all closed (normal) subgroups of Gal L (T, A) having finite index. (2) Autf S (M, A) = {f Aut A (M) f stabilizes all strong types over A of finite arities}. Proof. (1) We recall 2.2(1). Clearly Gal 0 L is contained in any closed subgroup of Gal L of finite index. Moreover Gal S is a profinite group. In a profinite group, the identity is the intersection of all normal closed subgroups of finite index. Hence (1) follows. (2) follows from 2.2(3), 2.3(2). Due to Lemma 2.3, x s A y is bounded type-definable over A. We have a more precise collection of formulas type-defining it. Proposition 3.3. Let u, v be real tuples. The following are equivalent. (1) u A v; and for any -definable equivalence relation E, if u/e acl(a), then E(u, v) holds. (2) u s A v. (3) For any A-type-definable equivalence relation E, if u/e has finitely many conjugates over A, then E(u, v) holds. (4) u acl(a) v. (5) u acl eq (A) v. (6) u 0 s A u 1 for each corresponding finite subtuple u i of c i (i = 0, 1). Proof. (1) (2) By 3.2(1), Gal 0 L(T, A) = i H i where H i is a closed normal subgroup of Gal L (T, A) having finite index, so H i is open as well. Due to the similar argument as in the proof of 2.3(1) there is a formula F i (x, y ) over defining an equivalence relation on tp(u) hence on M u by compactness, such that u/f i acl(a). Since H i is normal it again follows that on p(x) = tp(u/a), F i (x, y ) defines E Hi (x, y),

10 10 BYUNGHAN KIM in particular p(x) = F i (x, u ) E Hi (x, u) (the corresponding finite u u). Therefore (1) (2) follows. (2) (3) Let E(x, y) type-define an equivalence relation over A such that e = u/e acl(a). Then Aut ea (M) is a subgroup of Aut A (M) containing Autf A (M). Now E(x, u) clearly defines E AuteA (M)(x, u) on tp(u/a). Hence π(aut ea (M)) is closed (may not be normal) and it has finite index in Gal 0 L(T, A) as automorphisms in a coset send e to a different E-class. Therefore Gal 0 L(T, A) π(aut ea (M)), and (2) (3) follows. (3) (1); and (4) (5) (1) Obvious. (3) (4) The proof is the same as that of Corollary 2.6(3)(b) (c). This time there J I are finite, and q is algebraic. (1) (6) follows by compactness. Corollary 3.4. acl(a) and acl eq (A) are interdefinable. Proof. Let u/e acl(a) (u real). It suffices to show u/e is definable over acl eq (A). Due to 3.3, u E dcl(u/ s ) and u/ s is definable over acl eq (A). Example 3.5. We give a couple of examples related to Proposition 3.3. In (1) that u A v is essential. Let L = {=}, and let u v be singletons in the infinite M. By quantifier elimination, the 2nd clause of (1) holds with A = v, but u v v much less u s v v. Also if A is real as well then u s A v iff for any A-definable finite equivalence relation E, E(u, v) holds. This no longer holds in hyperimaginary context. Even the right hand side need not imply u A v. The difference comes from that in real context any formula in tp(u/v) is v-invariant, but not in general if v is a hyperimaginary: Consider a typical example where Ltp stp. That is a model (C; {U n (x, y)} 0<n ω ) where C is the unit circle, and U n (a, b) holds for a, b C iff the length of the shorter arc from a to b is n 1. Let F be an equivalence relation on C type-defined by {U n (x, y) 0 < n ω}. Choose u, v C such that u F v F. Then for any v F -invariant finite definable equivalence relation E (there almost no such except trivial one), E(u, v) holds. But not even u vf v holds. In the same manner the properties of strong types of hyperimaginaries follow. Proposition 3.6. Let a = u F, b = v F with real u, v be F -hyperimaginaries. The following are equivalent. (1) a s A b.

11 THE LASCAR GROUP, AND THE STRONG TYPES OF HYPERIMAGINARIES11 (2) FA s(u, v) holds where F A s (x, y) is a bounded type-definable equivalence relation over A coarser than F such that F s A(x, y) zz (z s A z F (z, x) F (z, y)) z(f (z, y) z s A x), (z s A x is equality of strong types over A of real tuples). (3) a acl(a) b. Proof. (1) (2) By the similar reason as in the proof of 2.6(3)(a) (b). (1) (3) comes from 3.3. Proposition 3.7. The following are equivalent. (1) Autf KP (M, A) = Autf S (M, A). (2) Gal KP (T, A) is compact Hausdroff and totally disconnected. (3) KP A is equivalent to s A for any arity. (4) KP A is equivalent to s A for finite arities. (5) KP A is equivalent to s A for any hyperimaginary variables. (6) acl(a) and bdd(a) are interdefinable. In Example 3.5, Autf KP Autf S. Some non G-compact examples (so Autf Autf KP ) are constructed in [1]. In an example L is different from KP for a finite tuple; while they can be equal for all finite arities in another non G-compact example. In [10], given any compact Hausdorff topological group G a corresponding theory T G is constructed so that Gal L (T G ) = G. That Ltp stp in real (hyperimaginary, resp.) context means for any tuple c and a set A both real (hyperimaginaries, resp.), it holds that Ltp(c/A) stp(c/a). Proposition 3.8. Assume T is simple. Then Ltp stp in real context implies that in hyperimaginary context. (In particular both low theories and supersimple theories have Ltp stp in hyperimaginary context.) Proof. Assume Ltp stp in real context. Let u E be a hyperimaginary and let v, v be real. It suffices to show v s u E v implies v L u E v. Choose a model M containing u. Now E clearly type-defines an equivalence relation on M as well, and u/e and M/E are interdefinable. Hence there is no harm to suppose that u is some enumeration of the model. Now let a hyperimaginary e := Cb(u/u E ), and let F (x, y) be E(x, y) x L y. Then u F is hyperdefined by x L u E u. As known u F = Cb(u/u E ): Since F is finer than E, clearly u E dcl(u F ). Hence u u F u E. Also u F bdd(u E ). Thus tp(u/u F ) is a Lascar type, and e dcl(u F ). Conversely, note now u e u E. Since u E dcl(u), we have u E bdd(e), so u F bdd(e) too. Then tp(u/e) tp(u/ bdd(e)) = tp(u/u F ) = F (x, u). Therefore u F dcl(e). So we can put e = u F.

12 12 BYUNGHAN KIM We claim that v s e v implies v L e v : Suppose that v s e v. We can clearly assume u e v, Take v such that v L e v and v e v. Now there is u such that u v s e uv. Hence by type-amalgamation there is u = tp(u/ev) tp(u /ev ). Then u v u v, and as u ( e) is a model, v L e v so v L e v, as desired. By the claim and 3.7, bdd(u E ) = bdd(u F ) = acl(u F ). Moreover by the definition of F and our assumption, u F is definable over acl(u E ). Hence bdd(u E ) = acl(u E ), and by 3.7 again, Ltp stp over u E, as wanted. Question Is Proposition 3.8 true for all T? The following is a comment given by an anonymous referee. We express our thanks for that: Another approach to defining the strong type over a hyperminaginary would be to consider the classical theory as a theory in continuous logic with the discrete metric. In this case, there is no distinction between hyperimaginaries and imaginaries and one could rework material on the Lascar group in this context. References [1] E. Casanovas, D. Lascar, A.Pillay, and M. Ziegler, Galois groups of first order theories, Journal of Math. Logic 1 (2001) [2] E. Casanovas and J. Potier. Normal hyperimaginaries, arxiv: v2. [3] J. Gismatullin and L. Newelski, G-compactness and groups, Arch. Math. Logic 47 (2008) [4] J. Goodrick, B. Kim, and A. Kolesnikov, Homology of types in model theory I: Basic concepts and connections with type amalgamation, submitted. [5] J. Goodrick, B. Kim, and A. Kolesnikov, Homology of types in model theory II: A Hurewicz theorem for stable theories, submitted. [6] E. Hrushovski, Simplicity and the Lascar group, unpublished note. [7] B. Kim, A note on Lascar strong types in simple theories, J. of Symbolic Logic 63 (1998) [8] D. Lascar, On the category of models of a complete theory, J. of Symbolic Logic 47 (1982) [9] D. Lascar and A. Pillay, Hyperimaginaries and automorphism groups, J. of Symbolic Logic 66 (2001) [10] M. Ziegler, Introduction to the Lascar group, in Tits buildings and the model theory of groups, London Math. Lecture Note Series 291, Cambridge Univ. Press (2002) [11] F. O. Wagner, Simple theories, Kluwer Academic Publishers, Dordrecht (2000). Department of Mathematics, Yonsei University, 134 Shinchon-dong, Seodaemun-gu, Seoul , South Korea address: bkim@yonsei.ac.kr