Generic pairs of SU-rank 1 structures

Vassiliev, Evgueni

doi:10.1016/s0168-0072(02)00060-x

Cited by 25 publications

(68 citation statements)

References 21 publications

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“…Then clearly P cannot be modular over W , and similarly for ¬P . (We noticed later that a similar example is described in [21]. )…”

Section: Acl(a B ) ∩ Acl(f B) ⊆ Acl(a B F ) ∩ Acl(f B) ⊆ Acl(f B)supporting

confidence: 59%

“…As in [21], one can easily construct a counterexample by adding a generic predicate P (x) to a vector space over a finite field. In Remark 3.7, we will deal with this example again.…”

Section: Independent Over B (= Bp) By the Remark Above Acl(ap) = Acmentioning

confidence: 99%

“…From [21], it follows that when we consider D as the universe of a modelM , D is 1-based iff DM 1 is modular. (He also proved the equivalence of 1 and 2 in Theorem 3.6 using a generic pair argument.…”

Section: Acl(a B ) ∩ Acl(f B) ⊆ Acl(a B F ) ∩ Acl(f B) ⊆ Acl(f B)mentioning

confidence: 99%

“…Furthermore, in the case of a supersimple rank-1 theory, since any model is the disjoint union of acl(∅) and the solution sets of rank-1 Lascar strong types (over ∅), we can restrict our attention to the solution set of a Lascar strong type. In [21], using his notion of "generic pair", Vassiliev investigated the rank-1 theory. Here we study rank-1 types using a straightforward argument so that we can provide, for example, a direct proof of the fact that linearity implies 1-basedness.…”

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confidence: 99%

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The geometry of 1-based minimal types

Piro

Kim

2003

Trans. Amer. Math. Soc.

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Abstract. In this paper, we study the geometry of a (nontrivial) 1-based SU rank-1 complete type. We show that if the (localized, resp.) geometry of the type is modular, then the (localized, resp.) geometry is projective over a division ring. However, unlike the stable case, we construct a locally modular type that is not affine. For the general 1-based case, we prove that even if the geometry of the type itself is not projective over a division ring, it is when we consider a 2-fold or 3-fold of the geometry altogether. In particular, it follows that in any ω-categorical, nontrivial, 1-based theory, a vector space over a finite field is interpretable.Geometric stability theory is one of the most important themes in model theory. Originally developing as a pure subject, it has turned out to be the major technical bridge connecting pure model theory and its applications to algebraic geometry and number theory. It mainly focuses on rank-1 types/structures or regular types where a canonical combinatorial geometry can be assigned. In other words, it is primarily concerned with geometric aspects of Shelah's stability theory [19], the study of stable structures. Arguably, the first major achievements of geometric stability theory are Zilber's results from the early 1980s (the translated version is [23]) on a strongly minimal ω-categorical structure. He showed that the geometry assigned to the structure is locally modular, hence, if nontrivial, must either be affine or projective over a finite field. A different proof was discovered independently by Cherlin, Harrington and Lachlan [14]. Refined notions such as 1-basedness, regular types and p-weight have also been introduced. Pillay, in his book [17], makes a complete exposition of the subject. Hrushovski has now shifted the direction of his research towards applications in algebraic geometry and number theory, which has deepened and broadened the subject. It is well known that, using geometric stability theory and in particular Zilber's principle on "Zariski structures", he solved the Mordell-Lang conjecture [12] and other problems in number theory.From the mid 1990s, after the initial papers [15], [16] of Kim and Pillay, simplicity theory, introduced by Shelah [18], has developed rapidly and extensively. Simplicity

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“…Then clearly P cannot be modular over W , and similarly for ¬P . (We noticed later that a similar example is described in [21]. )…”

Section: Acl(a B ) ∩ Acl(f B) ⊆ Acl(a B F ) ∩ Acl(f B) ⊆ Acl(f B)supporting

confidence: 59%

“…As in [21], one can easily construct a counterexample by adding a generic predicate P (x) to a vector space over a finite field. In Remark 3.7, we will deal with this example again.…”

Section: Independent Over B (= Bp) By the Remark Above Acl(ap) = Acmentioning

confidence: 99%

Section: Acl(a B ) ∩ Acl(f B) ⊆ Acl(a B F ) ∩ Acl(f B) ⊆ Acl(f B)mentioning

confidence: 99%

mentioning

confidence: 99%

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The geometry of 1-based minimal types

Piro

Kim

2003

Trans. Amer. Math. Soc.

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“…This situation has been studied extensively, typically in the case where A is an elementary submodel of M [20], [3], [24] or in the context of stable theories, when the induced structure on the predicate is stable [7], [2].…”

Section: Introductionmentioning

confidence: 99%

The independence property in generalized dense pairs of structures

Berenstein

Dolich²,

Onshuus

2011

J. symb. log.

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On pseudolinearity and generic pairs

Vassiliev¹

2010

Mathematical Logic Qtrly

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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 (if and only if T is trivial), 2 (if and only if 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.

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Generic pairs of SU-rank 1 structures

Cited by 25 publications

References 21 publications

The geometry of 1-based minimal types

The geometry of 1-based minimal types

The independence property in generalized dense pairs of structures

On pseudolinearity and generic pairs

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