Siberian Mathematical Journal, Vol. 43, No. 5, pp. 858–867, 2002 c 2002 Kuda˘ıbergenov K. Zh. Original Russian Text Copyright

HOMOGENEOUS MODELS AND STABLE DIAGRAMS K. Zh. Kuda˘ıbergenov

UDC 510.67

Abstract: Some fragment is studied of stability theory in the category of D-sets. Conditions are given for existence of D-homogeneous models of however large power. A categoricity theorem is proven for the class of (D, λ)-homogeneous models. Keywords: homogeneous model, stability theory, categoricity theorem

Introduction In this article we study some fragment of stability theory in the category of D-sets, i.e., subsets of models of some fixed theory T in which only the types of a given family D are realizable. Such an activity was initiated and deeply developed in the articles by S. Shelah [1, 2]. In the articles [3–7], the author studied (D, λ)-homogeneous models of stable theories, exploiting the methods and results of [1, 2]. Stability of a theory was needed only for utilizing tools from stability theory such as ranks and forking. However, already after the article [1] it has became clear that if similar tools are available within the category of D-sets then the stability of a theory is redundant. In 1994 the author tried to restate (or, more precisely, extend to a new situation) some part of the article [3] by means of a variant of Morley’s rank without going beyond the scope of the category of D-sets. The resultant theory repeated almost word for word (for a more general case) the classical theory of total transcendence (see [8]) and the content of [3], so that the author postponed his drafts to an indefinite time. However, recently B. I. Zil0 ber, to whom the author is sincerely grateful, communicated to the author about the increased interest in homogeneous models and a large amount of new papers on stability theory for classes of submodels of a sufficiently large homogeneous model. Having recognized the risk of his results being rediscovered, the author decided to publish his results obtained by this time. In § 1 we collect preliminary information on the category in which we work: definitions of D-sets, Dmodel, D-type, etc. In § 2 we give the definition of rank of a D-type and list some of its simplest properties. In the subsequent sections we assume that the rank of every D-type is less than ∞. In § 5 and § 6 we extend some results of [3] to the new situation: we exhibit conditions for existence of D-homogeneous models of however large power and prove a categoricity theorem for the class of (D, λ)-homogeneous models. In § 3 and § 4 we prove the needed technical assertions on the existence and properties of indiscernible sets in D-models as well as assertions about the existence of (D, λ)-prime models. Note that here we do not presume the existence of a large D-homogeneous model. Our exposition is self-contained, and all assertions are given together with their complete proofs. § 1. Preliminaries

Let T be a complete theory of a language L which has infinite models. We will work in some sufficiently saturated model of T . All elements (sets) are elements (sets) of this model. Models are denoted by M and N . We do not discriminate between the notation of a model and its universe. We denote by L(A) the enrichment of the language L with the names of elements in a set A. We adhere to the notations of [1]. Let D(A) be the set of all types over ∅ realizable by finite sequences of elements of a set A and let D be a set of the form D(M ), where M is a model. We call a set A a D-set if D(A) ⊆ D and call a model N a D-model if D(N ) = D. The set D is referred to as Almaty. Translated from Sibirski˘ı Matematicheski˘ı Zhurnal, Vol. 43, No. 5, pp. 1064–1076, September–October, 2002. Original article submitted February 23, 2001.

858

c 2002 Plenum Publishing Corporation 0037-4466/02/4305–0858 $27.00

a finite diagram; for short, we call it a diagram. The diagram of an ω-saturated model of T is denoted by D(T ). We consider only complete types. We say that p is a type over A and write dom(p) = A if p is a maximal consistent set of formulas with parameters in A. We denote by p|B the restriction of p to B, i.e., the type over B that lies in p. A type p over A is called a D-type over A if for some (equivalently, any) element a realizing p the set A ∪ {a} is a D-set. Elementary mappings are referred to as morphisms (they are really morphisms in the category of D-sets). Morphisms act naturally on the set of D-types. The following obvious lemma is very important. Lemma 1.0. (1) The family of all D-types is closed under subtypes and unions of chains. (2) A type p is a D-type if and only if the type p|B is a D-type for every finite B ⊆ dom(p). We denote the set of all D-types over A by SD (A) and denote the power of a set A by |A|. Definition. (i) A diagram D is called λ-stable if |SD (A)| ≤ λ for every A of power |A| ≤ λ. (ii) A diagram D is called stable if it is λ-stable for some λ. We recall the definition of a homogeneous model. Definition. (1) A model M is called λ-homogeneous if, for every A ⊆ M of power |A| < λ and every a ∈ M , each morphism from A into M extends to a morphism from A ∪ {a} into M . (2) A model M is called homogeneous if it is |M |-homogeneous. (3) A model M is called (D, λ)-homogeneous if D(M ) = D and M is λ-homogeneous. (4) A model M is called D-homogeneous if D(M ) = D and M is homogeneous. The following result by Keisler and Morley [9] plays a fundamental role in the study of homogeneous models. Lemma 1.1. If a model M is (D, λ)-homogeneous and D-sets A ⊆ B are such that |A| < λ and |B| ≤ λ, then each morphism from A into M extends to a morphism from B into M . This yields the next Lemma 1.2. A model M is (D, λ)-homogeneous if and only if D(M ) ⊆ D and, for every A ⊆ M of power |A| < λ, each D-type over A is realizable in M . § 2. Rank

Definition. We define the rank of a D-type p over a finite set as follows: (1) RD (p) ≥ 0 for every such p; (2) RD (p) ≥ δ, where δ is a limit ordinal, if RD (p) ≥ α for all α < δ; (3) RD (p) ≥ α + 1 if there exist inconsistent D-types p0 and p1 over finite sets such that RD (pi ) ≥ α and p ⊆ pi , i < 2. If RD (p) ≥ α holds, whereas RD (p) ≥ α + 1 fails, then we write RD (p) = α. If RD (p) ≥ α for all α then we write RD (p) = ∞. In the general case we put RD (p) = min{RD (p|B) : B ⊆ dom(p), |B| < ω}. Lemma 2.1. (i) RD (p) = 0 ⇐⇒ in every D-model the type p is realizable by a unique element. (ii) If p ⊆ q then RD (q) ≤ RD (p). (iii) If there exists a type of rank α and β < α then there exists a type of rank β. (iv) If p ∈ SD (A) and f is a morphism with dom(f ) = A then RD (p) = RD (f (p)). (v) If p ⊆ q ⊆ p0 and RD (p) = RD (p0 ) then RD (p) = RD (q). 859

(vi) There exists an ordinal α such that RD (p) ≥ α implies RD (p) = ∞ for every D-type p. (The least such ordinal α is denoted by αD .) Proof follows easily from definitions. Definition. A type p ∈ SD (A), where |A| < ω, is called (D, λ)-stable if the relation |{q ∈ SD (B) : p ⊆ q}| ≤ λ holds for every B ⊇ dom(p) of power |B| ≤ λ. It is clear that a diagram D is λ-stable if and only if every D-type over a finite set is (D, λ)-stable. Proposition 2.2. Suppose that p ∈ SD (A) and |A| < ω. (1) Assume that the type p is (D, µ)-stable, µ < 2ω , and either of the following conditions is satisfied: (a) if A ⊆ B ⊆ C, D(C) ⊆ D, |C| ≤ ω, and p ⊆ q ∈ SD (B) then q lies in a D-type over C; (b) if A ⊆ B ⊆ C, D(C) ⊆ D, |C| < ω, p ⊆ q ∈ SD (B), and RD (q) = ∞ then q lies in a D-type over C of rank ∞. Then RD (p) < ∞. (2) If RD (p) < ∞ then the type p is (D, µ)-stable for every µ ≥ |D|. Proof. (1) Assume the contrary. Construct D-types ps , s ∈ ω> 2, over finite sets such that (i) if s ⊆ t then ps ⊆ pt ; (ii) ps0 and ps1 are inconsistent; (iii) RD (ps ) = ∞. Put p∅ = p. Assume that a type ps has been defined and RD (p) ≥ αD + 1. Then by definition there exist inconsistent types ps0 and ps1 such that ps ⊆ psi and RD (psi ) ≥ αD , i < 2. (If condition (b) holds then we may assume S that dom(ps0 ) = dom(ps1 ).) Setting pη = n<ω pη|n , η ∈ ω 2 (and enlarging pη in the case of condition (a)), we obtain 2ω D-types over the countable set ∪{dom(ps ) : s ∈ ω> 2} which extend p. This contradicts µ-stability. (2) Given q ∈ SD (B) of rank < ∞, choose a finite Bq ⊆ B such that RD (q) = RD (q|Bq ). By Lemma 2.3 the mapping q 7→ q|Bq is injective. Therefore, |{q ∈ SD (B) : p ⊆ q}| ≤ | ∪ {SD (Bq ) : RD (q) < ∞}| ≤ |D| · |B|, whence the sought assertion ensues. The proof of Proposition 2.2 is over. Lemma 2.3. Let q be a D-type of rank α. If p0 , p1 ∈ SD (A) are extensions of q of the same rank then p0 = p1 . In particular, if q ⊆ pi ∈ SD (Ai ) and RD (q) = RD (pi ), i < 2, and if f : A0 → A1 is a bijective morphism that is the identity on dom(q), then f (p0 ) = p1 . Proof. Suppose the contrary. We may assume that the set dom(q) is finite. Then it lies in a finite B ⊆ A such that p0 |B 6= p1 |B. Since q ⊆ pi |B ⊆ pi , by Lemma 2.1(v) RD (q) = RD (pi |B) = α, whence by definition RD (q) ≥ α + 1; a contradiction. The proof of Lemma 2.3 is over. Definition. Call a diagram D almost totally transcendental if RD (p) < ∞ for every D-type p. If D = D(T ) then the theory T can be called almost totally transcendental, too. Note that if T is the theory of countably many independent unary predicates then T is almost totally transcendental but not totally transcendental. It is easy to construct an example of a nonstable theory T such that some diagram D ⊆ D(T ) is almost totally transcendental. Corollary 2.4. (1) If a diagram D is µ-stable, µ < 2ω , and either of conditions (a) and (b) of Proposition 2.2 holds for each D-type p over a finite set, then D is almost totally transcendental. (2) If a diagram D is almost totally transcendental then D is µ-stable for every µ ≥ |D|. Convention. Henceforth D is an almost totally transcendental diagram. 860

§ 3. Indiscernible Sets

Lemma 3.1. If I is an infinite indiscernible D-sequence over A then I is an indiscernible set over A. Proof. Assume the contrary. Then using Mal0 tsev’s compactness theorem we find a densely ordered indiscernible D-sequence of large power which has still more cuts extendible to pairwise distinct D-types. This contradicts stability of the diagram D. The proof of Lemma 3.1 is over. Definition. Suppose that p ∈ SD (A). A sequence I = {ai : i < α} is called a Morley sequence of the type p over A if for every i < α the type pi = tp(ai /A ∪ {aj : j < i}) is an extension of p of the same rank. Proposition 3.2. If I = {ai : i < α} is an infinite Morley sequence over A then I is an indiscernible set over A. Proof. In view of Lemma 3.1, it suffices to prove that (aik : k ≤ n) and (ajk : k ≤ n) realize the same type over A for all i0 < · · · < in < α and j0 < · · · < jn < α. We do this by induction on n. For n = 0 this is trivial. Next, by the induction hypothesis there exists a morphism f : A∪{aik : k < n} → A∪{ajk : k < n} that is the identity on A and that takes aik into ajk . Put qsn = tp(asn /A ∪ {ask : k < n}), s ∈ {i, j}. It suffices to show that f (qin ) = qjn . Since p0 ⊆ qsn ⊆ psn and I is a Morley sequence, by Lemma 2.1(v) RD (qsn ) = RD (p0 ), s ∈ {i, j}. Then by Lemma 2.3 f (qin ) = qjn . The proof of Proposition 3.2 is over. Proposition 3.3. Let I be an indiscernible set over A, and let ¯b be a finite collection of elements such that tp(¯b/A ∪ I) is a D-type. Then there exists a finite J ⊆ I such that I − J is indiscernible over A ∪ ¯b ∪ J. Proof. Choose a finite J ⊆ I such that the rank of the type pJ = tp(¯b/A∪J) equals that of the type pI = tp(¯b/A∪I). Let I0 and I1 be finite subsets of the same power in I −J. Since I is indiscernible over A, there exists a morphism f : A ∪ J ∪ I0 → A ∪ J ∪ I1 that is the identity on A ∪ J. Put pi = tp(¯b/A ∪ J ∪ Ii ), i < 2. It suffices to show that f (p0 ) = p1 . Since pJ ⊆ pi ⊆ pI , by the choice of J and Lemma 2.1(v) RD (pi ) = RD (pJ ). Then by Lemma 2.3 f (p0 ) = p1 . The proof of Proposition 3.3 is over. Definition. If I is an infinite indiscernible set then define Av(I/A) = {ϕ(x, a ¯) ∈ L(A) : |¬ϕ(I, a ¯)| < ω}. Corollary 3.4. Av(I/A) ∈ SD (A) for every infinite indiscernible set I and every A such that I ∪ A is a D-set. Proof. By Proposition 3.3, for every finite B ⊆ A almost all elements of I realize Av(I/B). Hence, Av(I/B) ∈ SD (B). The proof of Corollary 3.4 is over. Proposition 3.5. Let M be a (D, ω)-homogeneous model and p ∈ SD (M ). Then for every D-set B ⊇ M the type p has an extension in SD (B) of the same rank. Proof. Choose a finite A ⊆ M such that RD (p|A) = RD (p). Take ¯b ∈ B. By Lemma 1.1 there exists a morphism f : A ∪ ¯b → M that is the identity on A. Put p¯b = f −1 (p|A ∪ f (¯b)). By Lemma 2.1(v),(iv) RD (p¯b ) = RD (p). If ¯c ∈ B then p¯b , pc¯ ⊆ p¯b¯c , because p¯b and p¯b¯c |A ∪ ¯b coincide as extensions of p|A of the same rank (Lemma 2.3). Therefore, ∪{p¯b : ¯b ∈ B} is a sought extension of p of the same rank. The proof of Proposition 3.5 is over. 861

Proposition 3.6. Let M be a (D, ω)-homogeneous model and let a type p ∈ SD (M ) be not realizable in M . Then p = Av(I/M ) for some infinite indiscernible set I ⊆ M . Proof. Choose a finite A ⊆ M such that RD (p) = RD (p|A). Inducting on n < ω, find pairwise distinct elements an ∈ M such that an realizes p|A ∪ {ai : i < n}. Then I = {an : n < ω} is a Morley sequence over A. By Proposition 3.2 I is an indiscernible set over A. To complete the proof, it suffices to show that for every finite B ⊆ M almost all elements of I realize p|B. So, let B ⊆ M be finite. Using Proposition 3.5, extend I to a Morley sequence I ∗ = {an : n < ω1 } over A such that I ∗ − I is a Morley sequence of the type p. By Proposition 3.2 I ∗ is an indiscernible set. By Proposition 3.3 there exists a finite J ⊆ I ∗ such that I ∗ − J is indiscernible over B. Take a ∈ I − J and a∗ ∈ (I ∗ − I) − J. Then tp(a/B) = tp(a∗ /B) = p|B. The proof of Proposition 3.6 is over. Proposition 3.7. Let A be a D-set of regular power λ, Z ⊆ A, and |D| + |Z| < λ. Then there exists a set I ⊆ A of power λ which is indiscernible over Z. Proof. Assume Y ⊆ A, |Y | < λ. Call a type p ∈ SD (Y ) maximal if the set p(A) of elements in A realizing p has power λ. Such a type exists, because the cardinal λ is regular and by Corollary 2.4 |SD (Y )| ≤ |Y | + |D| < λ. In the set ∪{SD (Y ) : Z ⊆ Y, |Y | < λ} choose a maximal type p∗ of least rank. Let p∗ ∈ SD (Y ∗ ). If Y ∗ ⊆ Y ⊆ A and |Y | < λ then p∗ has a maximal extension q ∈ SD (Y ), since p∗ is maximal while λ is regular and |SD (Y )| < λ. Once p∗ ⊆ q, it follows that RD (q) ≤ RD (p∗ ). Therefore, by the choice of p∗ we have RD (p∗ ) = RD (q). This fact and Lemma 2.3 imply that p∗ has a unique maximal extension in SD (Y ). Inducting on i < λ, define elements ai ∈ A and types pi ∈ SD (Yi ), Yi = Y ∗ ∪ {aj : j < i}, such that (i) p0 = p∗ ; (ii) ai realizes pi ; (iii) pi is a maximal extension of the type p0 ; (iv) pi ⊆S pi+1 ; (v) pi = j
Definition. (1) A model M is called (D, λ)-prime over a set A ⊆ M if M is (D, λ)-homogeneous and for every (D, λ)-homogeneous model N each morphism from A into N extends to a morphism from M into N . (2) A type p ∈ SD (A) is called (D, λ)-isolated if there exists B ⊆ A of power |B| < λ such that the type p|B has a unique extension in SD (A) (in this event we say that the type p is D-isolated over B). (3) Call a D-set B a (D, λ)-construction over A if B = A ∪ {ai : i < δ}, where the type tp(ai /A ∪ {aj : j < i}) is (D, λ)-isolated for all i < δ. (4) A model M is called (D, λ, 1)-prime over A if M is (D, λ)-homogeneous and its universe is a (D, λ)-construction over A. Lemma 4.1. If B is a (D, λ)-construction over A then for every (D, λ)-homogeneous model N each morphism from A into N extends to a morphism from B into N . In particular, if a model M is (D, λ, 1)-prime over A then M is (D, λ)-prime over A. Proof. Obvious. 862

Theorem 4.2. If there exists a (D, λ)-homogeneous model then for every D-set A there exists a (D, λ, 1)-prime model over A. In particular, for every D-set A there exists a model which is (D, λ)prime over A. Proof. Let N be a (D, λ)-homogeneous model. Let M be a maximal (D, λ)-construction over A. It suffices to demonstrate that M is a (D, λ)-homogeneous model. Suppose that B ⊆ M , |B| < λ, and p ∈ SD (B). We demonstrate that p is realizable by an element of M . In view of Lemmas 1.1 and 4.1, we may assume that M ⊆ N . By Lemma 1.2 some element a ∈ N realizes the type p. Therefore, p lies in a D-type over M (for example, in tp(a/M )). Take such D-type q of least rank. Choose a finite set C ⊆ M such that RD (q) = RD (q|C). If q|C ∪ B ⊆ q 0 ∈ SD (M ) then RD (q 0 ) ≤ RD (q|C ∪ B) ≤ RD (q|C) = RD (q). Therefore, RD (q 0 ) = RD (q) by the choice of q, whence by Lemma 2.3 q 0 = q. Hence, the type q is (D, λ)-isolated. By maximality of M , this implies that q (and in consequence p) is realizable by an element of M . It remains to show that M is a model. Suppose that a formula ϕ(x, a ¯) in L(M ) is realizable by an element b ∈ N . Then tp(b/¯ a) is a D-type. By the above, tp(b/¯ a), as well as the formula ϕ(x, a ¯), is then realizable by an element of M . By the Tarski–Vaught test, M is a model. The proof of Theorem 4.2 is over. Corollary 4.3. Let A ⊆ B be D-sets, |A| < λ, and p ∈ SD (A). Suppose that there exists a (D, λ)homogeneous model. Then p extends to a D-type over B. Proof. By Theorem 4.2 there exists a model M which is (D, λ)-prime over B. By Lemma 1.2 some element a ∈ M realizes p. Then tp(a/B) is a sought extension of the type p. The proof of Corollary 4.3 is over. Corollary 4.4. Let λ ≥ |D| + |T |. Suppose that there exist a D-set A of power λ and a (D, ω)homogeneous model N . Then there exists a D-model of power λ which includes A. In particular, if N includes an infinite indiscernible set then there exists a D-model of power λ. Proof. If there is an infinite indiscernible D-set then by the compactness theorem there exists such a set of power λ. By Theorem 4.2 there exists a model M which is (D, ω)-prime over A. It remains to take an elementary submodel in M of power λ which includes A. The proof of Corollary 4.4 is over. § 5. Homogeneous Models

Proposition 5.1. The following are equivalent: (1) the model M is (D, λ)-homogeneous; (2) the model M is (D, ω)-homogeneous and I ≥ λ for every maximal infinite indiscernible set I ⊆ M . Proof. (1)⇒(2): Assume the contrary. Let I be a counterexample, a ∈ I, and f : I − {a} → I a bijection. Since the set I is indiscernible while the model M is λ-homogeneous, f extends to a morphism g : I → M . Then g(I) is an indiscernible set in M which properly extends I. This contradicts the maximality of I. (2)⇒(1): Let A ⊆ M , |A| < λ, a ∈ M , p = tp(a/A), and let f : A → M be a morphism. We have to prove that f (p) is realizable in M . By Theorem 4.2 there exists a model N which is (D, ω)-prime over A. In view of (D, ω)-primeness, we may assume that N ≺ M and f extends to a morphism h : N → M . If a ∈ N then h(a) realizes f (p). Suppose that a ∈ / N . Then the type q = tp(a/N ) ∈ SD (N ) is not realizable in the (D, ω)homogeneous model N . By Proposition 3.6, q = Av(I/N ) for some infinite indiscernible set I ⊆ N . Then h(q) = Av(h(I)/h(N )). Extend h(I) to a maximal indiscernible set I1 ⊆ M . In view of (2), |I1 | ≥ λ. By Proposition 3.3 there exists I2 ⊆ I1 such that |I2 | ≤ |h(A)| < λ and I1 − I2 is indiscernible over h(A). Since Av(I1 − I2 /h(A)) = Av(I1 /h(A)) = Av(h(I)/h(A)) = h(q|A) = f (p), each element of I1 − I2 realizes f (p). The proof of Proposition 5.1 is over. 863

Theorem 5.2. The following are equivalent: (1) for every λ ≥ |D| + |T | there exists a D-homogeneous model of power λ; (2) there exists a (D, ω)-homogeneous model of power greater than |D|; (3) there exists a (D, ω)-homogeneous model including an infinite indiscernible set; (4) there exist D-models M ≺ N such that M 6= N and M is (D, ω)-homogeneous; (5) there exist a D-set of power greater than |D| and a (D, ω)-homogeneous model. Proof. (1)⇒(2): Obvious. (2)⇒(3): Follows from Proposition 3.7. (1)⇒(4): Let M and N be D-homogeneous models and |M | < |N |. Then M 6= N and, in view of Lemma 1.1, we can consider M ≺ N . (4)⇒(3): Take a ∈ N − M . Then the type tp(a/M ) ∈ SD (M ) is omitted in M . By Proposition 3.6 the model M includes an infinite indiscernible set. (3)⇒(1): Assume that λ ≥ |D| + |T |. Inducting on i, construct an elementary chain {Mi : i ≤ λ} such that (i) Mi is a model of power λ; (ii) all types S in SD (Mi ) are realizable in Mi+1 ; (iii) Mi = j
Proof. Let I be a maximal infinite indiscernible set in M . In view of Proposition 5.1, it suffices to prove that |I| ≥ λ. Assume that |I| < λ. Let p = Av(I/M ). By Corollary 3.4 p ∈ SD (M ). Choose a finite A ⊆ M such that RD (p) = RD (p|A). Then A ⊆ Mn for some n < δ. By (D, λ)-homogeneity, we can construct a Morley sequence J = {bi : i < λ} of the type p|A in the model Mn . By Proposition 3.2 J is an indiscernible set. It follows from the proof of Proposition 3.6 that p = Av(J/M ). By Proposition 3.3 there exists J 0 ⊆ J such that |J 0 | ≤ |I| < λ and J − J 0 is indiscernible over I. Take a ∈ J − J 0 . Then tp(a/I) = Av(J − J 0 /I) = Av(J/I) = p|I = Av(I/I). Hence, I ∪ {a} is an indiscernible set in M properly including I, which contradicts the maximality of I. The proof of Theorem 5.4 is over. § 6. Categoricity

We recall that a class of models is called µ-categorical if every two models of power µ of this class are isomorphic. Denote by H (D, λ) the class of all (D, λ)-homogeneous models. We will prove a categoricity theorem for this class. Definition. (1) Fix a D-type of least possible rank which is of the form p∗ = Av(I ∗ /A∗ ), where is an infinite indiscernible set and |A∗ | < ω. Say that a type p is D-minimal if p = f (p∗ ) for some morphism f . (2) If a type p ∈ SD (A) is D-minimal and A ⊆ M then define a p-basis for a model M to be a maximal Morley sequence of the type p in M . I∗

Lemma 6.1. If a model M is (D, λ)-homogeneous then there exist a finite set A ⊆ M and a Dminimal type p ∈ SD (A). Proof. Follows from Lemma 1.1. Definition. Say that a type p is (D, λ)-non-two-cardinal if for arbitrary (D, λ)-homogeneous models M ≺ N , N 6= M , |M | > |D| + |T |, the condition dom(p) ⊆ M implies realizability of p in N − M . Definition. Say that a model M is (D, λ)-minimal over a set A ⊆ M if the conditions A ⊆ N ≺ M and N is (D, λ)-homogeneous imply that M = N . Proposition 6.3. Suppose that M is a (D, λ)-homogeneous model of power |M | > |D| + |T |, a type p ∈ SD (A) is D-minimal and (D, λ)-non-two-cardinal, A ⊆ M , and I is a p-basis for M . Then the model M is (D, λ)-prime and (D, λ)-minimal over A ∪ I. Proof. Let A ∪ I ⊆ N ≺ M and let a model N be (D, λ)-homogeneous. If N 6= M then, since the type p is non-two-cardinal, p is realizable by some element a ∈ N − M . By Proposition 3.6 tp(a/M ) = Av(J/M ) for some infinite indiscernible set J ⊆ M . Then RD (tp(a/M )) = RD (p) by D-minimality of the type p. Hence, I ∪ {a} is a Morley sequence of the type p, which contradicts the maximality of I. Therefore, N = M and the model M is (D, λ)-minimal over A ∪ I. By Theorem 4.2 there exists a model N ≺ M which is (D, λ)-prime over A ∪ I. By (D, λ)-minimality, we have N = M ; i.e., the model M is (D, λ)-prime over A ∪ I. The proof of Proposition 6.3 is over. Corollary 6.4. Suppose that a D-minimal type p∗ is (D, λ)-non-two-cardinal. Assume that κ > |D| + |T |. Then every (D, λ)-homogeneous model M0 of power κ is homogeneous and the class H (D, λ) is κ-categorical. Proof. By Theorem 5.2 there exists a D-homogeneous model M1 of power κ. Let pi ∈ SD (Ai ) be a D-minimal type, Ai ⊆ Mi , and let Ii be a pi -basis for the model Mi , i < 2. By Corollary 5.3 and Proposition 6.3 |Ii | = κ, and therefore there exists a surjective morphism f : A0 ∪ I0 → A1 ∪ I1 . Extending f to a morphism g : M0 → M1 , from Proposition 6.3 we obtain g(M0 ) = M1 . 865

Proposition 6.5. Suppose that κ ≥ |D| + |T | + λ, N is a (D, λ)-homogeneous model of power |N | > κ including an infinite set indiscernible over A, A ⊆ N , and p ∈ SD (A). If a type p is omitted in N then there exists a (D, λ)-homogeneous model M ≺ N of power κ such that the type p|A ∩ M is omitted in M and M includes an infinite set indiscernible over A ∩ M . Proof. Inducting on i, construct an elementary chain {Mi : i ≤ ω} as follows. Choose a D-set B ⊆ N of power κ which includes an infinite set indiscernible over A. By Theorem 4.2 there exists a model M0 ≺ N which (D, λ)-prime over B. By Corollary 5.3 |M0 | = κ. Suppose that a model Mi has been constructed, Mi ≺ N , |Mi | = κ. Since p is omitted in N , for every element a ∈ Mi there is a formula ϕa (x, ¯ba ) ∈ p such that |= ¬ϕ(a, ¯ba ). By Theorem 4.2 there exists a model Mi+1 ≺ N which is (D, λ)-prime over Mi ∪ {¯ba : a ∈ Mi }. By Corollary 5.3 |Mi+1 | = κ. It is clear that no element of Mi realizes p|A ∩ Mi+1 . Put M = Mω . By Theorem 5.4 the model M is (D, λ)-homogeneous. Clearly, |M | = κ. No element a ∈ M realizes p|A ∩ M , since a ∈ Mi for some i and by construction a does not realize p|A ∩ Mi+1 . The proof of Proposition 6.5 is over. In the next proposition (which is Theorem 9.6 of [3]) D is an arbitrary (not necessarily stable) diagram. Proposition 6.6. Suppose that p ∈ SD (A), I is an indiscernible sequence over A, I0 ⊆ I, |I0 | ≥ λ, a model M is (D, λ)-homogeneous, A ∪ I0 ⊆ M , and a model N is (D, λ, 1)-prime over A ∪ I. If p is realizable in N then p is also realizable in M . Proof. Let N = A ∪ I ∪ {ai : i < δ} be a (D, λ)-construction and pi = tp(ai /A ∪ I ∪ {aj : j < i}). Suppose that an element a ∈ N realizes p. It is clear that if a ∈ A ∪ I then p is realizable in M . Assume that a = ai0 for some i0 < δ. Inducting on n < ω, construct sets An ⊆ N as follows. Choose A0 ⊆ dom(pi0 ) such that |A0 | < λ and the type pi0 is D-isolated over A0 . Suppose that An has been i i defined and |An | ≤ λ. For every a i ∈ An , choose An ⊆ dom(pi ) such that An < λ and the type pi is i . Put A i :a ∈A D-isolated over AS = ∪ A n+1 i n . Clearly, |An+1 n n S| ≤ λ. Let I1 = I ∩ n<ω An and {as(i) : i < δ1 } = {ai : i < δ} ∩ n<ω An , where s(i) < s(j) < i0 for all i < j < δ1 . Put s(δ1 ) = i0 . By construction, for every i ≤ δ1 the type qi = tp(as(i) /A ∪ I ∪ {as(j) : j < i}) is (D, λ)-isolated. Since I is an indiscernible sequence over A and |I1 | ≤ λ ≤ |I0 |, there exists a morphism f : A ∪ I1 → A ∪ I0 that is the identity on A. Since the type qi is (D, λ)-isolated for every i ≤ δ1 and the model M is (D, λ)-homogeneous, inducting on i ≤ δ1 + 1 we can define morphisms fi : A ∪ I1 ∪ {as(j) : j < i} → M such that f0 = f and fj ⊆ fi for all j < i. Since the element as(δ1 ) = ai0 realizes p, it follows that fδ1 +1 (as(δ1 ) ) realizes p in M . The proof of Proposition 6.6 is over. Theorem 6.7. The following are equivalent: (1) the class H (D, λ) is κ-categorical for every κ > |D| + |T |; (2) the class H (D, λ) is κ-categorical for some κ > |D| + |T | + λ; (3) for all κ > |D| + |T | every (D, λ)-homogeneous model of power κ is homogeneous; (4) for some κ > |D| + |T | + λ every (D, λ)-homogeneous model of power κ is homogeneous; (5) if M and N are (D, λ)-homogeneous models, |M | > |D| + |T |, M ≺ N , M 6= N , and p = Av(I/A) for some A ⊆ M of power |A| < |M | and an infinite indiscernible set I ⊆ M , then p is realizable in N − M; (6) there exists a (D, λ)-non-two-cardinal D-minimal type. Proof. (1)⇒(2): Obvious. (2)⇒(4): Follows from Theorem 5.2. (4)⇒(3): Assume the contrary. Let N be a counterexample to (3). Then there exist A ⊆ N , |A| < |N |, and a type p ∈ SD (A) which is omitted in N . By Proposition 3.7 there is an infinite set in N which is indiscernible over A. By Proposition 6.5 there exists a (D, λ)-homogeneous model M ≺ N of power |D| + |T | + λ such that the type p|B is omitted in M and M includes an infinite set I0 indiscernible 866

over B, where B = A ∩ M . Since N is (D, λ)-homogeneous, |I0 | ≥ λ. By the compactness theorem there exists a D-set I ⊇ I0 of power κ which is indiscernible over B. By Theorem 4.2 there exists a model N0 which is (D, λ, 1)-prime over B ⊆ I. By Corollary 5.3 |N0 | = κ. Since |B| ≤ |M | < |N0 | and the D-type p|B is omitted in N0 by Proposition 6.6, it follows that N0 is not homogeneous, contradicting (4). (3)⇒(1): Obvious. (3)⇒(5): Let M , N , p, A, and I be the same as in (5). We have to prove that p is realizable in N − M . Let κ = |M |+ . Inducting on i, construct an elementary chain {Mi : i ≤ κ} as follows. Put M0 = M , a ∈ N − M . By Theorem 4.2 there exists a model M1 ≺ N which is (D, λ)-prime over M ∪ {a}. By Lemma 5.3 |M0 | = |M |. In view of (3), the models M and M1 are isomorphic. Suppose that a model Mi , 1 ≤ i < κ, has been constructed and is isomorphic to M . Since the model M is homogeneous and |A| < |M |, there exists an isomorphism fi of M onto Mi which is the identity on A. Extend fi to a morphism gi with domain M1 . Put Mi+1 = gi (M1 ). Then Mi ≺ Mi+1 and Mi 6= Mi+1 . S For a limit δ ≤ κ, put Mδ = i<δ Mi . By the induction hypothesis, the model Mi is isomorphic to M for every i < δ. Therefore, for δ < κ the model Mδ is (D, λ)-homogeneous by Theorem 5.4 and |Mδ | = |M |. In view of (3), M and Mδ are isomorphic models. Extend I to a maximal indiscernible set I0 ⊆ Mκ . By (3), the model Mκ is homogeneous and |Mκ | ≥ κ (since Mi 6= Mi+1 for any i < κ). Thereby |I0 | ≥ κ. By Proposition 3.3 there exists I1 ⊆ I0 such that |I1 | ≤ ω +|A| < κ and I0 −I1 is indiscernible over A. We have Av(I0 −I1 /A) = Av(I0 /A) = Av(I/A) = p. Since |I0 − I1 | ≥ κ > |M0 |, there exist i < κ and an element b ∈ Mi+1 − Mi such that b belongs to I0 − I1 and hence realizes p. Since gi is a morphism that is the identity on A, it follows that gi−1 (b) realizes p. Since gi (M ) = Mi and gi (M1 ) = Mi+1 , we have gi−1 (b) ∈ M1 − M ⊆ N − M . (5)⇒(6): Obvious. (6)⇒(1): Follows from Corollary 6.4. The proof of Theorem 6.7 is over. Remark. The author has made incorrect abridgements in a right proof of the implication (3)⇒(1) of Theorem 5.2 in proofreading; in consequence, there appear gaps in the above proof. We can fill them as follows: The models Mi must be (D, ω)-homogeneous. Put M0 = M ω , where {M i : i ≤ ω} is an elementary chain of D-models of power λ; M 0 is chosen arbitrarily; take unions at limit steps; and in M i+1 all D-types over finite subsets of the model M i are realized (this is done by analogy to the construction of Mi+1 on using Corollary 4.3). If the model Mi is available then we arrange the D-model 0 Mi+1 of power λ, as described in the article, wherein all types of SD (Mi ) are realized; afterwards we 0 arrange the (D, ω)-homogeneous model Mi+1  Mi+1 of power λ by analogy to the construction of M0 . We proceed then as in the body of the article. The author is grateful to B. I. Zil0 ber for his interest in the paper and fruitful discussions. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Shelah S., “Finite diagrams stable in power,” Ann. Math. Logic, 2, 69–118 (1970). Shelah S., Classification Theory and the Number of Non-Isomorphic Models, North-Holland, Amsterdam (1978). Kudaibergenov K. Zh., “Homogeneous models of stable theories,” Siberian Adv. Math., 3, No. 3, 1–33 (1993). Kuda˘ıbergenov K. Zh., “On homogeneous models of unidimensional theories,” Algebra i Logika, 34, No. 1, 61–78 (1995). Kuda˘ıbergenov K. Zh., “On homogeneous models of some weakly minimal theories,” Sibirsk. Mat. Zh., 40, No. 6, 1253–1259 (1999). Kuda˘ıbergenov K. Zh., “On homogeneous models of locally modular theories of finite rank,” Mat. Trudy, 5, No. 1, (2002). Kuda˘ıbergenov K. Zh., “On homogeneous models of totally transcendental nonmultidimensional theories,” Algebra i Logika, 30, No. 1, 48–73 (1991). Sacks G. E., Saturated Model Theory [Russian translation], Mir, Moscow (1976). Keisler H. J. and Morley M. D., “On the number of homogeneous models of a given power,” Israel J. Math., 5, No. 2, 73–78 (1967).

867