Abstract We continue the study of the connection between the “geometric” properties of SU-rank 1 structures and the properties of “generic” pairs of such structures, started in [8]. In particular, we show that the SU-rank of the (complete) theory of generic pairs of models of an SU-rank 1 theory T can only take values 1 (iff T is trivial), 2 (iff T is linear) or ω, generalizing the corresponding results for a strongly minimal T in [3]. We also use pairs to derive the implication from pseudolinearity to linearity for ω-categorical SU-rank 1 structures, established in [7], from the conjecture that an ω-categorical supersimple theory has finite SU-rank, and find a condition on generic pairs, equivalent to pseudolinearity in the general case.

An elementary pair of models of T (T -pair) is a structure (M, P ) in the language LP obtained by augmenting L with a new unary predicate P , where M |= T and P (M ) ¹ M . The notion of a “generic T -pair” for a supersimple SU-rank 1 theory T was introduced in [8]. If T is a supersimple first order L-theory of SU-rank 1, with quantifier elimination (for convenience), then a T -pair (M, P ) is called generic, if for any finite dimensional A ⊂ M and a non-algebraic type p ∈ SL1 (A), p is realized both in P (M ) and in M \aclL (AP (M )). One motivation was to find the “right” kind of a pair construction in the simple unstable 1

context, which would preserve simplicity, similarly to Poizat’s “beautiful pairs” preserving stability. Generic pairs turned out to provide such a construction in the SU-rank 1 case. Namely, in [8], we have shown that any two generic T -pairs are elementarily equivalent, any saturated model of the (complete) theory TP of the generic T -pairs is again a generic T -pair, and TP is supersimple. This construction was later generalized to arbitrary simple theories (with certain “weak” non-fcp property) in [2]. Another motivation was to use generic pairs to study geometric properties of SU-rank 1 structures, by analogy with [3], where pairs were used to show that pseudomodular strongly minimal structures are locally modular (a proof using the group configuration technique rather than pairs was later found by Hrushovski [5]). Since the algebraic closure induces a (in general, non-homogeneous) pregeometry on any SU-rank 1 structure, the usual notions of modularity and local modularity make sense. Recall that a pregeometry (X, cl) is k-pseudomodular (0 < k < ω), if for any a, b ∈ X and C ⊂ X, if a ∈ cl(bC), then there is a subset C 0 of cl(C) of size ≤ k such that a ∈ cl(bC 0 ). So, 1-pseudomodularity coincides with modularity. By a plane curve in an SU-rank 1 structure (or type) we mean an SU-rank 1 strong type of a 2-tuple. An SU-rank 1 theory T (or type) is kpseudolinear (0 < k < ω), if the canonical base of any plane curve has SU-rank ≤ k, and linear, if it is 1 − pseudolinear. In the strongly minimal case, pseudolinearity, pseudomodularity, linearity and local modularity are all equivalent. In the SU-rank 1 case, local modularity is strictly weaker than linearity, linearity is equivalent to 1-basedness, and does not imply pseudomodularity (see example in [8]). Pseudomodularity clearly implies pseudolinearity, and in [7] it is shown that under additional assumption of ω-categoricity, pseudolinearity implies linearity. For a strongly minimal T , generic T -pairs are simply the T -pairs (M, P ) where dim(P (M )) and dim(M/P (M )) are infinite, and are essentially the “beau-

2

tiful pairs” as defined by Poizat in [6] (a T -pair is beautiful, if P (M ) is |T |+ saturated and any L-type over P (M ) and a finite tuple in M is realized in M ). Theory TP in this case is ω-stable. In [3] it was shown that TP has U-rank 1 iff T is trivial, U-rank 2 iff T is non-trivial locally modular, and U-rank ω otherwise. In [8], we showed that TP has SU-rank 1 iff T is trivial, SU-rank 2 iff T is linear (1-based). Note that linearity (1-basedness) is strictly weaker than local modularity in the SU-rank 1 context. In general, TP was shown to have SU-rank ≤ ω. In the present paper, we “fill the gap” and show that the only possible values of the SU-rank of TP are 1,2 and ω, just as in the strongly minimal case, with local modularity replaced by linearity. Then we show that pseudolinearity of T is equivalent to boundedness of a certain rank associated with the generic pair. In the case when T is ω-categorical, this easily gives us the equivalence of pseudolinearity of T and ω-categoricity of TP . In [7], it is shown, using the group configuration in simple theories, that a pseudolinear SU-rank 1 type in an ω-categorical simple theory is linear. This implies that the same is true for a pseudolinear ω-categorical SU-rank 1 theory. In this paper we show how to derive this last fact from the conjecture “there is no supersimple ω-categorical theory of infinite SU-rank”, using a pairs argument. We also give a characterization of pseudolinearity of T in terms of the properties of SU-rank in TP , namely, T is k-pseudolinear iff TP has the following property: whenever a forks with b over ∅ SU (a/b) ≤ k. As mentioned above, in a joint work with Ben-Yaacov and Pillay [2], we have generalized both the generic pairs of SU-rank 1 structures and Poizat’s beautiful pairs of stable structures to the class of simple theories. The resulting notion of “lovely pair” provided simple pair expansions of simple theories satisfying a weak version of the non-fcp property. In the SU-rank 1 case, lovely pairs are simply the sufficiently saturated generic pairs, and the theory theory of lovely pairs coincides with TP . However we will still use the term “generic” in this paper, to be consistent with [8].

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As in [8], tpL and tpLP refer to types in T and TP respectively (same for | , dim, stp, acl, SU and Cb). A set A ⊂ (M, P ) is called P -independent, if ^ L | P (A) P (M ). In [8], we showed that an LP -type of a P -independent set in a A^ generic pair is determined by its quantifier free LP -type. For any pair of sets A ⊂ B in the universal domain of T we can “embed” them in a generic T -pair (M, P ) in a “P -independent” way, i.e. we can find B 0 ⊂ M such that tpL (B 0 , P (B 0 )) = tpL (B, A) and B 0 is P -independent. The following characterization of forking in TP is given in [8] (see also [2] for a more general characterizations): Let A ⊂ B, a some element. Then tpLP (a/B) forks over A iff one of the following two conditions hold: (i) a ∈ aclL (BP (M ))\aclL (AP (M )) (ii) a ∈ aclL (AP (M )) and there is no ¯b = b1 . . . bn ∈ P (M ) such that a ∈ aclLP (A¯b), and bi 6∈ aclLP (Bb1 . . . bi−1 ). We also have the following bound on SU-rank in TP : If a ∈ aclL (Ab1 . . . bn ), where ¯b ∈ P (M ), then SULP (a/A) ≤ n.

1

Rank of TP in the non-linear case

In this section we prove Theorem 1.1 If T is non-linear, then TP has SU-rank ω (moreover, there is a 1-type of SU-rank ω in TP ). The proof of the Theorem 1.1 will follow the ideas from [3] (where it was one of the key steps in proving that pseudomodularity implies local modularity), with some modifications due to the lack of stationarity. Assume that T is noneq ¯ linear, witnessed by a plane curve stpL (b0 b1 /¯ a) (i.e. CbL (¯b/¯ a) 6⊂ aclL (b)). Note

that by elimination of hyperimaginaries in supersimple theories [4], stp = Lstp, and thus we can work with the strong types. We may assume that a ¯ is an L

| ∅ P (M ), b0 ∈ P (M ), and b1 ∈ M \P (M ) L-independent tuple in M \P (M ), a ¯^

4

(as in Lemma 3.2, [3]). Our goal is to prove that TP has infinite SU-rank, by showing that tpLP (b1 /∅) has infinite SU-rank. In [3] Buechler showed that for a non-locally modular strongly minimal set, the LP - type p of an element of M \P (M ) over ∅ is bad. A type q(¯ x, A) is called bad, if there is a pairwise A-independent indiscernible sequence in q which is not A-independent. A type is called good otherwise. Then it is shown that a regular type of finite U-rank in a superstable theory is good. Since p is regular and bad, it has an infinite U-rank, and hence so does TP . Here, in order to show that tpLP (b1 /∅) has infinite SU-rank, we need to show that the corresponding type in TP is not only bad itself, but so is any of its non-forking extensions (this came for free in s.m. case, since goodness is preserved under parallelism in the stable case, and TP was ω-stable). Lemma 1.2 Any non-forking extension of tpLP (b1 /∅) is bad. L

| ∅ P A. We need to show that tpLP (b1 /A) is bad. Proof: Assume A ⊂ M , b1 ^ L L | b P A. Then, by transitivity, a | ∅ P A. Let (¯b, c¯, . . .) We may assume that a ¯ b0 ^ ¯¯b^ 1

L | A¯Pa ¯b and c¯ |= stpLP (¯b/A¯ be a Morley sequence in stpLP (¯b/A¯ a). Then c¯^ a). LP LP LP L | ∅ A implies A^ | a¯ ¯b, and together with c¯^ | A¯a ¯b this implies c¯^ | a¯ P ¯b. Now, a ¯¯b^ L | a¯ P ¯b Note that SULP (¯ c/¯ a) = 1, since c1 ∈ aclL (c0 a ¯), with c0 ∈ P (M ). Thus c¯^ L | ¯b. Also, clearly c¯ |= stpL (¯b/¯ implies c¯^ a). Then, since CbL (¯b/¯ a) 6⊂ acleq (¯b), a ¯

L

L

L | ¯b. Otherwise, we would have SUL (¯ | ¯b a ¯, and thus CbL (¯b/¯ a) = c¯^ c/¯b) = 1, so c¯^ CbL (¯ c/¯ a) ⊂ acleq (¯b), a contradiction. Now, exactly as in the claim in the proof of L L | ∅ P b1 . Namely, assume c1 6 ^ | ∅ P b1 . Then there is d¯ ⊂ P (M ), Lemma 3.2 [3], c1^ L ¯ Let d¯ = d¯0 d¯00 , d¯0^ | ∅ c¯¯b¯ such that c1 ∈ aclL (b1 d). a, and d¯00 ∈ aclL (¯ c¯b¯ ad¯0 ). Since,

b1 ∈ aclL (b0 a ¯), c1 ∈ aclL (c0 a ¯), we have d¯00 ∈ aclL (c0 b0 a ¯d¯0 ). But by the choice L ¯ c0 , b0 ∈ P (M ), we have d¯00 ∈ aclL (c0 b0 d¯0 ). So, | ∅ P (M ). Since, d, of a ¯, a ¯^ L | ∅ c¯¯b, we have c1 ∈ aclL (c0¯b). Contradiction with c1 ∈ aclL (c0¯bd¯0 ). Since d¯0 ^ L | ¯b. c¯^ ∅

L | A¯b. Now, we claim that c¯^ L L | A¯b. By c¯^ | A¯Pa ¯b, we have c¯ 6∈ aclL (A¯ Assume c¯6 ^ a¯b), and thus SUL (¯ c/A¯b) =

5

1 = SUL (¯ c/A¯ a¯b). So, stpL (¯ c/¯ a) and stpL (¯ c/A¯b) have a common non-forking eq extension stpL (¯ c/A¯ a¯b), and therefore CbL (¯b/¯ a) ⊂ aclL (A¯b). But CbL (¯b/¯ a) ⊂ L eq eq ¯ | ∅ P A. Thus CbL (¯b/¯ aclL (¯ a), and thus we have CbL (¯b/¯ a)¯b^ (b). Since a) ⊂ aclL P eq c¯ |= stpL (¯b/¯ a), c1 ∈ aclL (c0 CbL (¯b/¯ a)). So, c1 ∈ aclLP (c0¯b) ⊂ aclL (P (M )b1 ). L

| ∅ P b1 . Contradiction with c1^ L

L

L

| P (M ) | P (M ) A, which together with a | ∅ P A implies a ¯^ ¯^ Note also that a ¯^ L | A P (M ). implies a ¯^ L

L

| A¯b and a | A P (M ), and as in the claim in the proof of Now, we have c¯^ ¯^ L | ∅ to ^ | A , and acl(−) to acl(− ∪ A), we get c1^ | AP b1 . Lemma 3.2 [3], changing ^ Note that I = (b1 , c1 , . . .) is an A-indiscernible sequence in tpLP (b1 /A), and I ⊂ aclL (¯ aP (M )). Therefore dimL (I/P (M )) is finite. If I was an LP independent sequence over A, dimL (I/P (M )) would be infinite, a contradiction. Thus I witnesses the badness of tpLP (b1 /A). 2 Now we are ready to prove the theorem.

Proof of the Theorem 1.1: It suffices to show that p = tpLP (b1 /∅) has infinite SU-rank.

Assume that SULP (p) is finite.

Note that p is regular,

since any forking extension of p over some set A is realized by an element of aclL (AP (M )), while any non-forking extension of p over A is realized by L

| AP a, by characterization of forking in some b 6∈ aclL (AP (M )), and hence b ^ TP . Now, since p is regular and of finite rank, there is an SU-rank 1 (hence regular) type q in TP eq , non-orthogonal to p. Over some A ⊂ M , there are non-forking (hence regular) extensions p0 and q 0 of p and q respectively, and L

| AP b. By Lemma 1.2, p0 is bad. realizations a |= p0 and b |= q 0 such that a6 ^ Take (a0 , a1 , . . .) an A-indiscernible sequence in p0 witnessing its badness, and L

| AP bi . By the Erd¨os-Rado Theorem, we for each i, choose bi |= q 0 such that ai 6 ^ may assume that (a0 b0 , a1 b1 , . . .) is A-indiscernible (in TP ). As in Proposition 2.2 [3], we conclude that q 0 is bad. Indeed, by regularity, LP -forking on p0 ∪ q 0 is

6

L

| AP bi implies that any LP - dependence over A that holds for transitive, and ai 6 ^ ai ’s, holds for bi ’s as well. So, the A-indiscernible (in TP ) sequence (bi |i ∈ ω) witnesses the badness of q 0 . But q 0 has SU-rank 1, and hence is good (since any pairwise independent indiscernible sequence is infinite, and any infinite sequence in an SU-rank 1 type is independent). Contradiction. 2

2

Pseudolinearity and ω-categoricity

Definition. For a ∈ M , A ⊂ M , define RP (a/A) = minimal n such that L

| P (M ) b). a ∈ aclL (Ac1 . . . cn ) for some c¯ ∈ P (M ) (∞ if no such c¯ exist, i.e. a^ By properties of SU-rank in TP , we have the following Lemma 2.1 For any a and A SULP (a/A) ≤ R(a/A). 2 L

| P (M ) b (i.e. RP (a/b) < ∞). Lemma 2.2 Assume that a, b ∈ M \P (M ) and a6 ^ L | A b. Then Take any small (finite) A ⊂ P (M ) such that a6 ^ SUL (CbL (ab/A)) ≤ RP (a/b) ≤ 2 SUL (CbL (ab/A)). Proof: Let C = CbL (ab/A). The first inequality is trivial. To show the second, take a Morley sequence I = (ai bi |i ∈ ω) in stpL (ab/A). We may assume that ai ∈ P (M ). Since A ⊂ P (M ), we also have bi ∈ P (M ). We know: C ⊂ eq eq aclL (I). Take the minimal n such that C ⊂ aclL (a0 b0 . . . an−1 bn−1 ). Clearly,

SUL (a0 b0 . . . an−1 bn−1 /C) = n. We claim that SUL (a0 b0 . . . an−1 bn−1 ) = 2n. Assume it is < 2n. Note that SUL (a0 . . . an−1 ) = n. Take a minimal i such that bi+1 ∈ aclL (¯ ab0 . . . bi ). Then SUL (ai+1 bi+1 /a0 b0 . . . ai bi ai+2 bi+2 . . . an−1 bn−1 ) = 1 = SUL (ai+1 bi+1 /a0 b0 . . . ai bi ai+2 bi+2 . . . an−1 bn−1 A).

7

eq eq Thus C = CbL (ab/A) ⊂ aclL (a0 b0 . . . ai bi ai+2 bi+2 . . . an−1 bn−1 ). By aclL (A)-

indiscernibility of I, this contradicts the minimality of n. So, SUL (C) = SUL (a0 b0 . . . an−1 bn−1 ) − SUL (a0 b0 . . . an−1 bn−1 /C) = 2n − n = n. eq But C ⊂ aclL (a0 b0 . . . an−1 bn−1 ), and a0 , b0 , . . . an−1 , bn−1 ∈ P (M ). So, L

| a b ...a bn−1 b, a6 ^ 0 0 n−1 and thus RP (a/b) ≤ 2n. 2 Proposition 2.3 The following are equivalent: (i) T is pseudolinear (ii) there is m such that RP (a/b), if finite, is ≤ m (iii) for any n there is mn , RP (a/b1 . . . bn ), if finite, is ≤ mn Proof: Since for any (non-trivial) plane curve stpL (ab/A) in T we may assume that A ⊂ P (M ) and a, b ∈ M \P (M ), (i → ii) and (ii → i) easily follow from Lemma 2.2. (iii → ii) is trivial, and (ii → iii) follows by induction from the fact that (M, aclL (¯ eP (M ))) is again a (saturated) generic T -pair (see [8], Lemma 5.6). We can actually take mn = mn . 2 Corollary 2.4 For an ω-categorical T , TP is ω-categorical iff T is pseudolinear. Proof: We need to show that TP is ω-categorical iff (iii) above holds. Left to right is trivial. To show right to left we first recall from [8] that in a generic T -pair (M, P ), | P (A) P (M ) (i.e. if A = aclL (A) and A ^

A is closed P -independent) then

qf tpLP (A) |= tpLP (A). Now, let A = {a1 , a2 , . . . , an } be a subset of M , and assume that ak+1 , . . . , an ∈ aclL (P (M ) ∪ {a1 , . . . , ak }) with a1 , . . . , ak aclL independent over P (M ). Then by (iii), we can find a set B ⊂ P (M ) of size at most (n − k)mk such that A ∪ B is P -independent. Also note that since TP is ω-categorical, aclL is uniformly locally finite. It follows that there is a 8

function f : ω → ω such that any n-tuple in M can be embedded in a closed P independent set of size ≤ f (n). Again by ω-categoricity of T , there are finitely many n-LP -types for every n, and thus TP is ω-categorical. 2 Using Theorem 1.1 and Corollary 2.4, we see that if T is ω-categorical pseudolinear and non-linear, then TP is ω-categorical and supersimple of SU-rank ω. Thus we get Corollary 2.5 The conjecture “any supersimple ω-categorical theory has finite SU-rank” implies that any ω-categorical pseudolinear SU-rank 1 theory is linear.2 While the conjecture above is still open, the statement “pseudolinearity implies linearity” was recently proved for SU-rank 1 Lascar strong types in an ω-categorical simple theory by Tomasic and Wagner [7], using the group configuration for simple theories [1]. Note that if an SU-rank 1 theory T is pseudolinear, then for any n ∈ ω there is a uniform (finite) bound on the SU-rank of canonical bases of n-types (easy to see by induction), and even types of the form tp(e/A) where e is an imaginary which is in acl of a real n-tuple. It easily follows that any SU-rank 1 type in T eq is pseudolinear. Now, if an SU-rank 1 theory T is pseudolinear and ω-categorical, then by the above argument and Tomasic-Wagner’s result, all SU-rank 1 types in T eq are linear (1-based), so by “coordinatization”, T is itself 1-based and hence linear. So one gets the following characterization (see also [2], Proposition 7.8). Proposition 2.6 For an ω-categorical SU-rank 1 theory T the following are equivalent: (i) T is linear; (ii) T is pseudolinear; (iii) TP is ω-categorical. 2

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Now, if T is as above, by ω-categoricity of TP , the relation x ∈ aclL ({y1 . . . yn } ∪ P ) is definable in TP . Thus for any (M, P ) |= TP and a1 , . . . , an ∈ M , the structure (M, aclL (¯ aP )) is definable in (M, P ) (with parameters a ¯). Hence its LP -theory is also ω-categorical and supersimple. Note also that by [8], Lemma 5.6, aclL (¯ aP ) is an elementary substructure of M , so itself a model of T . For any b ∈ aclL (¯ aP ), the SU-rank of b over a ¯ in TP is ≤ 1 (since TP has SU-rank ≤ 2, and either b ∈ P , or b forks with a ¯ over ∅ in TP ). This implies that (aclL (¯ aP ), P, a ¯) is an SU-rank 1 structure, and hence so is its reduct (aclL (¯ aP ), P ). Note that taking a ¯ of dimension n over P , we get different ω-categorical SU-rank 1 pair expansions of T . Thus we get the following corollary. Corollary 2.7 If T is an ω-categorical linear SU-rank 1 theory, then for any n ∈ ω, there is an ω-categorical supersimple SU-rank 1 completion Tn of the theory of all T -pairs, such that for any (M, P ) |= Tn we have dim(M/P (M )) = n, and all Tn are distinct. 2 Whether pseudolinearity implies linearity is true in the non-ω-categorical SU-rank 1 case is still unknown (it is proved for strongly minimal structures in [3]). What we can easily show, for example, is that “strong pseudomodularity” implies linearity, namely Remark 2.8 Any SU-rank 1 structure with the property “there is m ∈ ω such that a ∈ acl(AB) implies a ∈ acl(Ab1 . . . bm ) where ¯b ∈ acl(B)” is linear. Proof: In this case TP has SU-rank ≤ m + 1, hence T is linear (and TP has SU-rank ≤ 2). 2

3

Pseudolinearity and SU-rank in TP

In this section we give a characterization of pseudolinearity of T in terms of SU-rank in TP . 10

eq Lemma 3.1 SULP (e) = SUL (e) for any e ∈ aclL (P (M )). eq ¯ Proof: Consider ¯b ∈ P (M ) such that e ∈ aclL (b). Take c¯ ∈ M realizing L | e ¯b. Write c¯ = c¯1 c¯2 with c¯1 aclL -independent over e stpL (¯b/e) such that c¯ ^ L | e ¯b, c¯1 is also aclL -independent over ¯b. By lemmas c¯2 ∈ aclL (e¯ c1 ). Since c¯^ 5.10,5.12 [8], we can find a ¯1 ∈ P (M ) realizing stpL (¯ c1 /¯b) Take a ¯2 such that

a ¯1 a ¯2 |= stpL (¯ c/¯b). Since a ¯1¯b ∈ P (M ), we also have a ¯2 ∈ P (M ). Note that a ¯2 ∈ aclL (e¯ a1 ). Since aclL = aclLP inside P (M ) and a ¯1 is aclL -independent over ¯b, it is also aclLP -independent over ¯b, and hence over e (meaning that no component of a ¯1 is in LP -algebraic closure of the other components and e). L

L

| ePa | ea Now, take a ¯0 |= stpLP (¯ a/e) such that a ¯^ ¯0 . We claim that a ¯^ ¯0 . Otherwise, since a2 ∈ aclL (e¯ a1 ), a1 is not aclL -independent over e¯ a0 . Since a ¯1 L

| ePa is aclLP -independent over e, this contradicts a ¯^ ¯0 . L L 0 eq eq 0 | ePa | ea Now, we have a ¯^ ¯0 , a ¯^ ¯ and e ∈ aclL (¯ a) ∩ aclL (¯ a ). So, SUL (¯ a) = SUL (¯ ae) = SUL (¯ a/e) + SUL (e), and thus SUL (e) = SUL (¯ a) − SUL (¯ a/e) = SUL (¯ a) − SUL (¯ a/e¯ a0 ) = SUL (¯ a) − SUL (¯ a/¯ a0 ). Exactly the same calculation for SULP shows that SULP (e) = SULP (¯ a) − SULP (¯ a/¯ a0 ). But L-forking coincides with LP -forking inside P (M ). So, SUL (¯ a) = SULP (¯ a) and SUL (¯ a/¯ a0 ) = SULP (¯ a/¯ a0 ). Thus SULP (e) = SUL (e). 2 By elimination of hyperimaginaries in supersimple theories [4], canonical base of a strong n-type is a set of imaginaries, and is contained in the algebraic closure (in T eq ) of a single imaginary contained in its definable closure. As in [8], we will identify canonical bases with such single imaginaries. L

| P (M ) b. Take any (small) B ⊂ Lemma 3.2 Assume a, b ∈ M \P (M ) and a6 ^ L | B b, and let C = CbL (ab/B). Then C does not depend on P (M ) such that a6 ^ 11

the choice of B and eq (i) C ∈ aclL (ab) P

(ii) SULP (a/b) = SUL (C) (and finite) Proof: (i) We will work in a large saturated elementary extension N eq of M eq . L

| B P (M ). Thus C is interalgebraic with CbL (ab/P (M )). Take any Clearly, ab^ automorphism f of (M eq , P ) fixing ab. Extend it to an automorphism f 0 of N eq . Clearly f 0 fixes stpL (ab/P (M )). Thus it fixes CbL (ab/P (M )), and there are finitely many possible images of C under such f . So, the Autab (M eq , P )eq orbit of C is finite, and thus C ∈ aclL (ab). P

(ii) First note that both SU-ranks are finite (by the properties of SU-rank in TP and supersimplicity of T ). Note also that since a, b 6∈ P (M ), SUL (a/B) = SUL (b/B) = 1. Note that SULP (Ca/b) = SULP (a/bC) + SULP (C/b) = 0 + L

| ∅ P C (follows from C being SULP (C/b) = SULP (C), since a ∈ aclL (bC) and b^ algebraic over P (M ), b ∈ M \P (M ) and the characterization of forking in TP ). On the other hand, SULP (Ca/b) = SULP (C/ab) + SULP (a/b) = SULP (a/b), by (i). By Lemma 3.1, SUL (C) = SULP (C), so SULP (a/b) = SUL (C). 2 Proposition 3.3 The following are equivalent for an SU-rank 1 theory T : (i) T is k-pseudolinear L

| ∅ P b then SULP (a/b) ≤ k. (ii) for any a, b ∈ (M, P ) |= TP , if a6 ^ Proof: (i → ii) Assume (i). If a, b ∈ M \P (M ), then (ii) follows from Lemma L

| ∅ P b). But if 3.2(ii). If one of a, b is in P (M ), so is the other one (otherwise a^ L | ∅ P b, then SULP (a/b) = 0. a, b ∈ P (M ) and a6 ^ (ii → i) For any (non-trivial) plane curve stpL (ab/B) in T we may assume that B ⊂ P (M ) and a, b ∈ M \P (M ). Then we apply Lemma 3.2(ii). 2 Corollary 3.4 The following are equivalent for an SU-rank 1 T : (i) T is pseudolinear L

| ∅ P b, (ii) there is k ∈ ω such that whenever a, b ∈ (M, P ) |= TP and a6 ^

12

SULP (a/b) ≤ k (iii) for any n ∈ ω there is kn ∈ ω such that whenever a, b1 , . . . , bn ∈ (M, P ) |= L | ∅ P ¯b, SULP (a/¯b) ≤ kn . TP and a6 ^

Proof: If suffices to show (i → iii). So, assume (i). If a ∈ P (M ), SULP (a/¯b) ≤ 1. If a ∈ M \P (M ), then (iii) follows from Proposition 2.3 and Lemma 2.1.

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References [1] Itay Ben-Yaacov, Ivan Tomasic, Frank O. Wagner, Constructing an almost hyperdefinable group, Journal of Mathematical Logic, volume 4 (2004), 181-212 [2] Itay Ben-Yaacov, Anand Pillay, Evgueni Vassiliev, Lovely pairs of models, Annals of Pure and Applied Logic, volume 122 (2003), 235-261. [3] Steven Buechler, Pseudoprojective strongly minimal sets are locally projective, Journal of Symbolic Logic, vol. 56, no. 4, Dec.1991. [4] Steven Buechler, Anand Pillay, Frank O. Wagner, Supersimple theories, J. AMS 14 (2001) 109124. [5] Ehud Hrushovski, Unimodular minimal structures, J. London Math. Soc. (2) 46 (1992), no. 3, 385–396. [6] Bruno Poizat, Paires de structures stables, Journal of Symbolic Logic, 48 (1983), 234-249. [7] Ivan Tomasic, Frank O. Wagner, Applications of the group configuration theorem in simple theories, Journal of Mathematical Logic, 3(2):239-255, 2003. [8] Evgueni Vassiliev, Generic pairs of SU-rank 1 structures, Annals of Pure and Applied Logic 120 (2003) 103-149.

Evgueni Vassiliev (new spelling: Yevgeniy Vasilyev) Sir Wilfred Grenfell College, Memorial University of Newfoundland Corner Brook, NL A2H 6P9, Canada E-mail: [email protected]

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