Some remarks on indiscernible sequences
Some remarks on indiscernible sequences Enrique Casanovas∗ Departamento de L´ogica Universidad de Barcelona Baldiri Reixac s/n 08028 Barcelona, Spain March 28, 2002. Revised September 9, 2002
Abstract We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs. Mathematics Subject Classification: 03C45 Keywords: indiscernibles, heirs and coheirs, simple theories, independence theorem
1
Introduction
The Independence Theorem for Lascar strong types in a simple theory was first established by Kim and Pillay in [3] and [1]. It was obtained as a corollary of the Independence Theorem over a model. Shami gave in [6] a more direct proof. The results we present in this paper arose of our wanting to simplify even more the proof of Shami, and trying to see clearly the path leading from the fact that in a simple theory if a, b start an infinite ∗
Work partially supported by grant PB98-1231 of the Spanish Government. The author wishes to thank Daniel Lascar and Anand Pillay for helpful suggestions improving the results in section 3
1
indiscernible sequence and ϕ(x, a) does not fork over the empty set, then ϕ(x, a) ∧ ϕ(x, b) does not fork over the empty set, to the fact that in a simple theory if ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set, b, b0 start an infinite indiscernible sequence and a ^ | bb0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. We discovered that the only ingredient necessary to prove one fact from the other one is what we call generic homogeneity of the indiscernibility relation. This basically means that in a simple theory it is possible to do a back-and-forth argument adding independent tuples to tuples starting infinite indiscernible sequences. This is explained and proven in section 2. The proofs only require very basic facts about forking in simple theories. In particular the Independence Theorem is not used at all. In section 3 we turn our attention to coheirs. It is well known that coheirs are a useful tool to obtain indiscernible sequences. In a simple theory they provide Morley sequences. The theory of heirs and coheirs was developed by Lascar and Poizat for stable theories in [5] and was adapted to the simple setting by Lascar and Pillay in [4]. One novelty of our treatment is that we work in a more general context. Proposition 3.1 works for any theory and Proposition 3.2 is stated in such a way that it can be applied both to simple and to o-minimal theories. These two propositions give reformulations of the properties of starting an infinite indiscernible sequence and of starting a Morley sequence in terms of coheirs. In particular we prove that in these circumstances it is always possible to reach a situation where both heir and coheir independence hold. This does not seem to have been noticed before. Our notation is standard. We work in a monster model of a complete theory T and we use a, b, c for sequences, possibly of infinite length. By a ≡A b we mean that tp(a/A) = tp(b/A).
2
Generic homogeneity
Definition 1 We say that a, b are an indiscernible pair if there is an infinite indiscernible sequence starting with a, b. The indiscernibility relation is the relation R such that R(a, b) if and only if a, b are an indiscernible pair.
2
Definition 2 Let R be a symmetric binary relation such that whenever R(a, b) then a and b are sequences of the same length. We say that R is homogeneous if for all a, b, c such that R(a, b) there is some d such that R(ac, bd). If the ambient theory is simple, we say that R is generically homogeneous if for all a, b, c such that c ^ | a b there is some d such that R(ac, bd). Proposition 2.1 In a simple theory the indiscernibility relation is generically homogeneous. Proof. Let c ^ | a b and assume I = (ai : i < ω) is an infinite indiscernible sequence with a = a0 and b = a1 . Since (an : n ≥ 1) is a-indiscernible and c ^ | a b, there is an ac-indiscernible sequence (a0n : n ≥ 1) such that (an : n ≥ 1) ≡ab (a0n : n ≥ 1). Thus we may assume that an = a0n for all n ≥ 1. Let c0 = c and choose for n ≥ 1 some cn such that ca0 a1 . . . ≡ cn an an+1 . . . Since (an : n ≥ 1) is ac-indiscernible, cab ≡ caam . Hence cab ≡ cn an an+m , i.e., in the sequence (cn an : n < ω) all triangles cn an an+m have the same type p(x, y, z) = tp(cab). By Ramsey’s Theorem there is an indiscernible sequence (dn bn : n < ω) where all triangles dn bn bn+m satisfy p(x, y, z). Clearly we may assume that c = d0 , a = b0 and b = b1 . Then d = d1 witnesses the generic homogeneity for a, b, c. 2 Remarks 2.2 1. The homogeneity of the indiscernibility relation fails dramatically in any theory: if a0 , a1 , . . . , is an infinite indiscernible sequence and b = a1 it does not exist c such that a0 b, a1 c start an infinite indiscernible sequence. The reason is that since b = a1 then, by indiscernibility, b = a2 and hence a1 = a2 and, again by indiscernibility, ai = aj for all i, j. 2. In a stable theory, if I is an indiscernible sequence starting with a, b and c ^ | ab we may assume that I ^ | c and from this it follows that I is c-indiscernible. Hence ac, bc start an indiscernible sequence. Thus 3
in this particular case of a stable theory and c ^ | ab, d = c witnesses the generic homogeneity of the indiscernibility relation for a, b, c. Fact 2.3 (Kim) Let T be simple. Assume that ϕ(x, a) does not fork over the empty set and a, b are an indiscernible pair. Then ϕ(x, a) ∧ ϕ(x, b) does not fork over the empty set. Proof. See Corollary 3.16 in [1] or Corollary 3.4 in [2].
2
Proposition 2.4 Let T be simple and assume that ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. If b, b0 are an indiscernible pair and a ^ | b b0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. Proof. Apply Proposition 2.1 finding c such that ba, b0 c are an indiscernible pair. By Fact 2.3 ϕ(x, a) ∧ ψ(x, b) ∧ ϕ(x, c) ∧ ψ(x, b0 ) does not fork over the empty set. In particular ϕ(x, a)∧ψ(x, b0 ) does not fork over the empty set. 2 Proposition 2.4 is slightly more general than Lemma 1 from [6] (if we pay attention only to the case of indiscernible pairs) in that we require | bb0 . The general case of equality of Lascar strong a^ | b b0 in place of a ^ types and the Independence Theorem for Lascar strong types follow easily as shown in [6]. For completeness we give now these two short proofs. It can be easily generalized to types instead of formulas. Recall that equality of Lascar strong types is the transitive closure of the indiscernibility relation. Corollary 2.5 Let T be simple and assume that ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. If Lstp(b) = Lstp(b0 ) and a ^ | bb0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. Proof. Find b1 , . . . , bn such that b = b1 , b0 = bn and bi , bi+1 are an indiscernible pair for all i. Let a0 be such that a0 ≡bb0 a and a0 ^ | bb0 b1 , . . . bn . By 0 Proposition 2.4 we see that ϕ(x, a ) ∧ ψ(x, bi ) does not fork over the empty set for all i ≤ n. Hence ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set . 2
4
Corollary 2.6 (Independence Theorem) Let T be simple and a ^ | b. If there are c, d such that |= ϕ(c, a), c ^ | a, |= ψ(d, b), d ^ | b, and Lstp(c) = Lstp(d), then ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. Proof. Choose b0 ^ | c ab such that Lstp(cb0 ) = Lstp(db). Then |= ϕ(c, a) ∧ ψ(c, b0 ) and c ^ | ab0 . Therefore ϕ(x, a)∧ψ(x, b0 ) does not fork over the empty set. Since a ^ | bb0 by Corollary 2.5, ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set . 2
3
Coheirs
Definition 3 We say that a, b are in coheir position over M if tp(a/M b) is a coheir of tp(a/M ). Similarly, a, b are in heir position over M if tp(a/M b) is a heir of tp(a/M ). Finally, we say that a, b are in nice position over M if they are both in heir and coheir position over M , i.e., if a, b are in coheir position over M and also b, a are in coheir position over M . Proposition 3.1 The following are equivalent. 1. a, b are an indiscernible pair. 2. For some infinite sequence I, I a b is indiscernible and I a a is indiscernible over b. 3. a, b are in nice position over some model M such that a ≡M b. 4. a, b are in coheir (heir) position over some model M such that a ≡M b. Proof. It is clear that 1 implies 2 and that 3 implies 4. It is well known that if a, b are in coheir position over some model M , then they start an infinite M -indiscernible sequence. Hence 1 follows from 4. We now prove 3 assuming 2. Let U be a non principal ultrafilter over ω and let I = (ai : i < ω). The limit type modulo U of the elements ai over some set C is lim tp(ai /C) = {ϕ(x, c) : c ∈ C, {i < ω :|= ϕ(ai , c)} ∈ U }. U
5
Then tp(b/I) = limU tp(ai /I) = tp(a/I). Let N ⊇ I be a model. Choose a realization b0 of limU tp(ai /N ). Since b0 ≡I b, there is a model N 0 such b0 N ≡I bN 0 . Then tp(b/N 0 ) = limU tp(ai /N 0 ). Let a0 be a realization of limU tp(ai /N 0 b). Then a0 ≡N 0 b and a0 , b are in coheir position over N 0 . Notice that tp(a/Ib) = limU tp(ai /Ib) = tp(a0 /Ib). Since a0 ≡Ib a we find finally a model M such that a0 N 0 ≡Ib aM . It follows that a ≡M b and a, b are in coheir position over M . As indicated before this implies that a, b belong to an infinite M -indiscernible sequence. Thus we may iterate this step and we can construct an elementary chain (Mi : i < ω) such that a ≡Mi b for all i < ω, ai , bi are in coheir position over Mi for even i and bi , ai are in coheir position over Mi for odd i. Let Mω be the union of the chain. Clearly a ≡Mω b and a, b are in a nice position over Mω . 2 In the next proposition we will consider a theory having an independence relation ^ | with some basic properties, in a similar situation as described in [3]. Among these properties we will have invariance, transitivity, symmetry, and extension (defined as in [3]). But we replace local character and finite character respectively for a weaker and a stronger property. In place of local character we only postulate that a complete type does not fork (in the sense of ^ | ) over its own domain, i.e., c ^ | A A. This weak form of the local character in fact follows directly from extension and backwards transitivity, so we do not need to state it separately. In place of finite character we demand that if a complete type forks (again, in the sense of our independence relation ^ | ) over a set, it contains a formula which forks over that set, i.e., {tp(c/B) : c ^ | A B} is closed in S(B) for any A ⊆ B. This properties are sufficient to show that if a type tp(c/B) is finitely satisfiable in A ⊆ B then c^ | A B. We do not assume the independence relation satisfies the Independence Theorem over a model nor the Independence Theorem for Lascar strong types. Simple theories and also o-minimal theories satisfy all the conditions, so the next proposition applies to these cases. In this context by a Morley sequence over A we understand an ordered sequence (ai : i ∈ I) which is indiscernible over A and independent over A. 6
This last condition means that every ai is independent of its predecessors {aj : j < i} over A or, equivalently, that every ai is independent of the remaining elements {aj : j 6= i} over A. Proposition 3.2 Let T be a theory with an independence relation ^ | with the properties of invariance, transitivity, symmetry, extension and such that {tp(c/B) : c ^ | A B} is closed in S(B) for any A ⊆ B. Then the following are equivalent. 1. a, b start an infinite Morley sequence. 2. For some infinite sequence I ^ | ab, I a b is indiscernible and I a a is indiscernible over b. 3. a, b are in nice position over some model M ^ | ab such that a ≡M b. 4. a, b are in coheir (heir) position over some model M ^ | a such that a ≡M b. Proof. As remarked above, our assumptions on the independence relation imply that if tp(c/B) is finitely satisfiable in A ⊆ B, then c ^ | A B. In particular coheirs are independent extensions. The rest is basically the same argument as in the proof of Proposition 3.1. In the proof of 3 from 2, since tp(b/N 0 ) = limU tp(ai /N 0 ) is finitely satisfiable in I, it follows that b ^ | I N 0. | N 0 and ab ^ | M . Thus we can We may assume that N 0 ^ | Ib a. Hence ab ^ construct the elementary chain (Mi : i < ω) with ab ^ | Mi for all i < ω. Then ab ^ | Mω . In the proof of 1 from 4 and also in the iteration done in the proof of 3 from 2 we use the fact that coheir extensions are independent extensions in order to construct an infinite Morley sequence over M starting with a, b. In the proof of 1, since a ^ | M , it is also a Morley sequence over the empty set. 2 Remarks 3.3 1. If T is simple one can also add in Proposition 3.2 the condition that a ^ | b and a, b are an indiscernible pair. To prove that a, b start an infinite Morley sequence under this hypothesis one needs the Independence Theorem. 7
2. As pointed out to us by Pillay, the proof of 3 from 1 in Proposition 3.2 for the particular case of T simple can also be done directly using the Independence Theorem and the results in [4]. In this article Lascar and Pillay show (Corollary 4) that for simple T any type p ∈ S(A) has a non-forking extension q over some model M ⊇ A such that every non-forking extension of q is a heir of q. If we start with p(x) = Lstp(a) = Lstp(b) and we assume that a ^ | b, it is easy to show, as indicated in the proof of Theorem 7 in [4], that we may assume that q = tp(a/M ) = tp(b/M ) and that ab ^ | M . This is the point where the Independence Theorem is used. Now it is clear that a, b are in a nice position over M .
References [1] B. Kim, Simple first order theories, Ph. D. Thesis, University of Notre Dame 1996. [2] B. Kim, Forking in simple unstable theories, Journal of the London Mathematical Society 57 (1998), 257-267. [3] B. Kim and A. Pillay, Simple theories, Annals of Pure and Applied Logic 88 (1997), 149-164. [4] D. Lascar and A. Pillay Forking and fundamental order in simple theories, The Journal of Symbolic Logic 64 (1999), 1155-1158. [5] D. Lascar and B. Poizat, An introduction to forking, The Journal of Symbolic Logic 44 (1979), 330-350. [6] Z. Shami, Definability in low simple theories, The Journal of Symbolic Logic 65 (2000), 1481-1490.
8
Abstract We prove a property of generic homogeneity of tuples starting an infinite indiscernible sequence in a simple theory and we use it to give a shorter proof of the Independence Theorem for Lascar strong types. We also characterize the relation of starting an infinite indiscernible sequence in terms of coheirs. Mathematics Subject Classification: 03C45 Keywords: indiscernibles, heirs and coheirs, simple theories, independence theorem
1
Introduction
The Independence Theorem for Lascar strong types in a simple theory was first established by Kim and Pillay in [3] and [1]. It was obtained as a corollary of the Independence Theorem over a model. Shami gave in [6] a more direct proof. The results we present in this paper arose of our wanting to simplify even more the proof of Shami, and trying to see clearly the path leading from the fact that in a simple theory if a, b start an infinite ∗
Work partially supported by grant PB98-1231 of the Spanish Government. The author wishes to thank Daniel Lascar and Anand Pillay for helpful suggestions improving the results in section 3
1
indiscernible sequence and ϕ(x, a) does not fork over the empty set, then ϕ(x, a) ∧ ϕ(x, b) does not fork over the empty set, to the fact that in a simple theory if ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set, b, b0 start an infinite indiscernible sequence and a ^ | bb0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. We discovered that the only ingredient necessary to prove one fact from the other one is what we call generic homogeneity of the indiscernibility relation. This basically means that in a simple theory it is possible to do a back-and-forth argument adding independent tuples to tuples starting infinite indiscernible sequences. This is explained and proven in section 2. The proofs only require very basic facts about forking in simple theories. In particular the Independence Theorem is not used at all. In section 3 we turn our attention to coheirs. It is well known that coheirs are a useful tool to obtain indiscernible sequences. In a simple theory they provide Morley sequences. The theory of heirs and coheirs was developed by Lascar and Poizat for stable theories in [5] and was adapted to the simple setting by Lascar and Pillay in [4]. One novelty of our treatment is that we work in a more general context. Proposition 3.1 works for any theory and Proposition 3.2 is stated in such a way that it can be applied both to simple and to o-minimal theories. These two propositions give reformulations of the properties of starting an infinite indiscernible sequence and of starting a Morley sequence in terms of coheirs. In particular we prove that in these circumstances it is always possible to reach a situation where both heir and coheir independence hold. This does not seem to have been noticed before. Our notation is standard. We work in a monster model of a complete theory T and we use a, b, c for sequences, possibly of infinite length. By a ≡A b we mean that tp(a/A) = tp(b/A).
2
Generic homogeneity
Definition 1 We say that a, b are an indiscernible pair if there is an infinite indiscernible sequence starting with a, b. The indiscernibility relation is the relation R such that R(a, b) if and only if a, b are an indiscernible pair.
2
Definition 2 Let R be a symmetric binary relation such that whenever R(a, b) then a and b are sequences of the same length. We say that R is homogeneous if for all a, b, c such that R(a, b) there is some d such that R(ac, bd). If the ambient theory is simple, we say that R is generically homogeneous if for all a, b, c such that c ^ | a b there is some d such that R(ac, bd). Proposition 2.1 In a simple theory the indiscernibility relation is generically homogeneous. Proof. Let c ^ | a b and assume I = (ai : i < ω) is an infinite indiscernible sequence with a = a0 and b = a1 . Since (an : n ≥ 1) is a-indiscernible and c ^ | a b, there is an ac-indiscernible sequence (a0n : n ≥ 1) such that (an : n ≥ 1) ≡ab (a0n : n ≥ 1). Thus we may assume that an = a0n for all n ≥ 1. Let c0 = c and choose for n ≥ 1 some cn such that ca0 a1 . . . ≡ cn an an+1 . . . Since (an : n ≥ 1) is ac-indiscernible, cab ≡ caam . Hence cab ≡ cn an an+m , i.e., in the sequence (cn an : n < ω) all triangles cn an an+m have the same type p(x, y, z) = tp(cab). By Ramsey’s Theorem there is an indiscernible sequence (dn bn : n < ω) where all triangles dn bn bn+m satisfy p(x, y, z). Clearly we may assume that c = d0 , a = b0 and b = b1 . Then d = d1 witnesses the generic homogeneity for a, b, c. 2 Remarks 2.2 1. The homogeneity of the indiscernibility relation fails dramatically in any theory: if a0 , a1 , . . . , is an infinite indiscernible sequence and b = a1 it does not exist c such that a0 b, a1 c start an infinite indiscernible sequence. The reason is that since b = a1 then, by indiscernibility, b = a2 and hence a1 = a2 and, again by indiscernibility, ai = aj for all i, j. 2. In a stable theory, if I is an indiscernible sequence starting with a, b and c ^ | ab we may assume that I ^ | c and from this it follows that I is c-indiscernible. Hence ac, bc start an indiscernible sequence. Thus 3
in this particular case of a stable theory and c ^ | ab, d = c witnesses the generic homogeneity of the indiscernibility relation for a, b, c. Fact 2.3 (Kim) Let T be simple. Assume that ϕ(x, a) does not fork over the empty set and a, b are an indiscernible pair. Then ϕ(x, a) ∧ ϕ(x, b) does not fork over the empty set. Proof. See Corollary 3.16 in [1] or Corollary 3.4 in [2].
2
Proposition 2.4 Let T be simple and assume that ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. If b, b0 are an indiscernible pair and a ^ | b b0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. Proof. Apply Proposition 2.1 finding c such that ba, b0 c are an indiscernible pair. By Fact 2.3 ϕ(x, a) ∧ ψ(x, b) ∧ ϕ(x, c) ∧ ψ(x, b0 ) does not fork over the empty set. In particular ϕ(x, a)∧ψ(x, b0 ) does not fork over the empty set. 2 Proposition 2.4 is slightly more general than Lemma 1 from [6] (if we pay attention only to the case of indiscernible pairs) in that we require | bb0 . The general case of equality of Lascar strong a^ | b b0 in place of a ^ types and the Independence Theorem for Lascar strong types follow easily as shown in [6]. For completeness we give now these two short proofs. It can be easily generalized to types instead of formulas. Recall that equality of Lascar strong types is the transitive closure of the indiscernibility relation. Corollary 2.5 Let T be simple and assume that ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. If Lstp(b) = Lstp(b0 ) and a ^ | bb0 , then ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set. Proof. Find b1 , . . . , bn such that b = b1 , b0 = bn and bi , bi+1 are an indiscernible pair for all i. Let a0 be such that a0 ≡bb0 a and a0 ^ | bb0 b1 , . . . bn . By 0 Proposition 2.4 we see that ϕ(x, a ) ∧ ψ(x, bi ) does not fork over the empty set for all i ≤ n. Hence ϕ(x, a) ∧ ψ(x, b0 ) does not fork over the empty set . 2
4
Corollary 2.6 (Independence Theorem) Let T be simple and a ^ | b. If there are c, d such that |= ϕ(c, a), c ^ | a, |= ψ(d, b), d ^ | b, and Lstp(c) = Lstp(d), then ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set. Proof. Choose b0 ^ | c ab such that Lstp(cb0 ) = Lstp(db). Then |= ϕ(c, a) ∧ ψ(c, b0 ) and c ^ | ab0 . Therefore ϕ(x, a)∧ψ(x, b0 ) does not fork over the empty set. Since a ^ | bb0 by Corollary 2.5, ϕ(x, a) ∧ ψ(x, b) does not fork over the empty set . 2
3
Coheirs
Definition 3 We say that a, b are in coheir position over M if tp(a/M b) is a coheir of tp(a/M ). Similarly, a, b are in heir position over M if tp(a/M b) is a heir of tp(a/M ). Finally, we say that a, b are in nice position over M if they are both in heir and coheir position over M , i.e., if a, b are in coheir position over M and also b, a are in coheir position over M . Proposition 3.1 The following are equivalent. 1. a, b are an indiscernible pair. 2. For some infinite sequence I, I a b is indiscernible and I a a is indiscernible over b. 3. a, b are in nice position over some model M such that a ≡M b. 4. a, b are in coheir (heir) position over some model M such that a ≡M b. Proof. It is clear that 1 implies 2 and that 3 implies 4. It is well known that if a, b are in coheir position over some model M , then they start an infinite M -indiscernible sequence. Hence 1 follows from 4. We now prove 3 assuming 2. Let U be a non principal ultrafilter over ω and let I = (ai : i < ω). The limit type modulo U of the elements ai over some set C is lim tp(ai /C) = {ϕ(x, c) : c ∈ C, {i < ω :|= ϕ(ai , c)} ∈ U }. U
5
Then tp(b/I) = limU tp(ai /I) = tp(a/I). Let N ⊇ I be a model. Choose a realization b0 of limU tp(ai /N ). Since b0 ≡I b, there is a model N 0 such b0 N ≡I bN 0 . Then tp(b/N 0 ) = limU tp(ai /N 0 ). Let a0 be a realization of limU tp(ai /N 0 b). Then a0 ≡N 0 b and a0 , b are in coheir position over N 0 . Notice that tp(a/Ib) = limU tp(ai /Ib) = tp(a0 /Ib). Since a0 ≡Ib a we find finally a model M such that a0 N 0 ≡Ib aM . It follows that a ≡M b and a, b are in coheir position over M . As indicated before this implies that a, b belong to an infinite M -indiscernible sequence. Thus we may iterate this step and we can construct an elementary chain (Mi : i < ω) such that a ≡Mi b for all i < ω, ai , bi are in coheir position over Mi for even i and bi , ai are in coheir position over Mi for odd i. Let Mω be the union of the chain. Clearly a ≡Mω b and a, b are in a nice position over Mω . 2 In the next proposition we will consider a theory having an independence relation ^ | with some basic properties, in a similar situation as described in [3]. Among these properties we will have invariance, transitivity, symmetry, and extension (defined as in [3]). But we replace local character and finite character respectively for a weaker and a stronger property. In place of local character we only postulate that a complete type does not fork (in the sense of ^ | ) over its own domain, i.e., c ^ | A A. This weak form of the local character in fact follows directly from extension and backwards transitivity, so we do not need to state it separately. In place of finite character we demand that if a complete type forks (again, in the sense of our independence relation ^ | ) over a set, it contains a formula which forks over that set, i.e., {tp(c/B) : c ^ | A B} is closed in S(B) for any A ⊆ B. This properties are sufficient to show that if a type tp(c/B) is finitely satisfiable in A ⊆ B then c^ | A B. We do not assume the independence relation satisfies the Independence Theorem over a model nor the Independence Theorem for Lascar strong types. Simple theories and also o-minimal theories satisfy all the conditions, so the next proposition applies to these cases. In this context by a Morley sequence over A we understand an ordered sequence (ai : i ∈ I) which is indiscernible over A and independent over A. 6
This last condition means that every ai is independent of its predecessors {aj : j < i} over A or, equivalently, that every ai is independent of the remaining elements {aj : j 6= i} over A. Proposition 3.2 Let T be a theory with an independence relation ^ | with the properties of invariance, transitivity, symmetry, extension and such that {tp(c/B) : c ^ | A B} is closed in S(B) for any A ⊆ B. Then the following are equivalent. 1. a, b start an infinite Morley sequence. 2. For some infinite sequence I ^ | ab, I a b is indiscernible and I a a is indiscernible over b. 3. a, b are in nice position over some model M ^ | ab such that a ≡M b. 4. a, b are in coheir (heir) position over some model M ^ | a such that a ≡M b. Proof. As remarked above, our assumptions on the independence relation imply that if tp(c/B) is finitely satisfiable in A ⊆ B, then c ^ | A B. In particular coheirs are independent extensions. The rest is basically the same argument as in the proof of Proposition 3.1. In the proof of 3 from 2, since tp(b/N 0 ) = limU tp(ai /N 0 ) is finitely satisfiable in I, it follows that b ^ | I N 0. | N 0 and ab ^ | M . Thus we can We may assume that N 0 ^ | Ib a. Hence ab ^ construct the elementary chain (Mi : i < ω) with ab ^ | Mi for all i < ω. Then ab ^ | Mω . In the proof of 1 from 4 and also in the iteration done in the proof of 3 from 2 we use the fact that coheir extensions are independent extensions in order to construct an infinite Morley sequence over M starting with a, b. In the proof of 1, since a ^ | M , it is also a Morley sequence over the empty set. 2 Remarks 3.3 1. If T is simple one can also add in Proposition 3.2 the condition that a ^ | b and a, b are an indiscernible pair. To prove that a, b start an infinite Morley sequence under this hypothesis one needs the Independence Theorem. 7
2. As pointed out to us by Pillay, the proof of 3 from 1 in Proposition 3.2 for the particular case of T simple can also be done directly using the Independence Theorem and the results in [4]. In this article Lascar and Pillay show (Corollary 4) that for simple T any type p ∈ S(A) has a non-forking extension q over some model M ⊇ A such that every non-forking extension of q is a heir of q. If we start with p(x) = Lstp(a) = Lstp(b) and we assume that a ^ | b, it is easy to show, as indicated in the proof of Theorem 7 in [4], that we may assume that q = tp(a/M ) = tp(b/M ) and that ab ^ | M . This is the point where the Independence Theorem is used. Now it is clear that a, b are in a nice position over M .
References [1] B. Kim, Simple first order theories, Ph. D. Thesis, University of Notre Dame 1996. [2] B. Kim, Forking in simple unstable theories, Journal of the London Mathematical Society 57 (1998), 257-267. [3] B. Kim and A. Pillay, Simple theories, Annals of Pure and Applied Logic 88 (1997), 149-164. [4] D. Lascar and A. Pillay Forking and fundamental order in simple theories, The Journal of Symbolic Logic 64 (1999), 1155-1158. [5] D. Lascar and B. Poizat, An introduction to forking, The Journal of Symbolic Logic 44 (1979), 330-350. [6] Z. Shami, Definability in low simple theories, The Journal of Symbolic Logic 65 (2000), 1481-1490.
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