Characterizing O-Minimal Groups in Tame Expansions of O-Minimal Structures

Eleftheriou, Pantelis E.

doi:10.1017/s1474748019000392

Cited by 4 publications

(5 citation statements)

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“…In this section, we will show that in supersimple theories, all L H -definable groups in H-structures are definably isomorphic to L-definable groups. 4 We first introduce some basic notions and facts about H-structures developed in [2], as well as some results about groups in simple theories that we will use later.…”

Section: Groups In H-structuresmentioning

confidence: 99%

PSEUDOFINITE H-STRUCTURES AND GROUPS DEFINABLE IN SUPERSIMPLE H-STRUCTURES

Zou¹

2019

J. symb. log.

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In this paper we explore some properties of H-structures which are introduced in [2].We describe a construction of H-structures based on one-dimensional asymptotic classes which preserves pseudo-finiteness. That is, the H-structures we construct are ultraproducts of finite structures.We also prove that under the assumption that the base theory is supersimple of SU -rank one, there are no new definable groups in H-structures. This improves the corresponding result in [2]. 1 3. (Density/coheir property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, there is a ∈ H(M ) such that a |= q; (Extension property) IfA ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, then there is a ∈ M , a |= q and a ∈ acl L (A ∪ H(M )).Equivalently, we can replace density and extension properties with the following more general ones:• (Generalised density/coheir property) If A ⊆ M is finite dimensional and q ∈ S n (A) has dimension n, then there is a ∈ H(M ) n such that a |= q;• (Generalised extension property) If A ⊆ M is finite dimensional and q ∈ S n (A) is non-algebraic, then there is a ∈ M n , a |= q andA structure M is called an H-structure if it is an H-expansion of some model of a geometric theory.H-structures are closely related to lovely pairs, where, instead of an independent subset, a dense and co-dense elementary substructure is added. We recall the definition of lovely pairs in the special case that the base theory is geometric, see [1].Definition 2. Let T be a geometric theory in a language L and let L P be the expansion of L by a unary predicate P . An L P -structure (M, N ) is a lovely pair of models of T , if 1. M |= T ; 2. N is an L-elementary submodel of M ; 3. (Density/coheir property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, there is a ∈ N such that a |= q; 4. (Extension property) If A ⊆ M is finite dimensional and q ∈ S 1 (A) is nonalgebraic, then there is a ∈ M , a |= q and a ∈ acl L (A ∪ N ).Fact 3. [2], [1]. Properties of H-structures and lovely pairs. Let T be a complete geometric theory in a language L. • H-expansions of models of T exist and all of them are L H -elementary equivalent. Let T H be the corresponding theory. Similarly, lovely pairs of models of T exist, and all of them are L P -elementary equivalent.• If the geometry of T is nontrivial and T is strongly minimal/supersimple/superrosy of rank 1, then T H is ω-stable/supersimple/superrosy of rank ω.• Let (M, H(M )) be an H-structure. Then (M, acl L (H(M ))) is a lovely pair.Consider the theory of pseudofinite fields. It is supersimple of SU -rank one. By the fact above, H-expansions and lovely pairs of pseudofinite fields exist. However, the proof of existence uses general model theoretic techniques such as saturated models and union of chains. It is not clear whether it is possible to have H-expansions or lovely pairs of pseudofinite fields that are ultraproducts of finite structures.The answer turns out to be negative for lovely pairs.

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Section: Groups In H-structuresmentioning

confidence: 99%

PSEUDOFINITE H-STRUCTURES AND GROUPS DEFINABLE IN SUPERSIMPLE H-STRUCTURES

Zou¹

2019

J. symb. log.

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“…In the supersimple SU-rank 1 case, Zou [22] has shown that groups definable in H-structures are definably isomorphic to type-definable groups in the base language. Similarly, Eleftheriou [13] has shown that any large group definable in a dense-codense expansion of an o-minimal theory is definable in the original theory.…”

Section: Introductionmentioning

confidence: 89%

“…Example 4. 13 Pseudofinite fields (see [14]). Assume U is a non-principal ultrafilter over N and let F = U F i be the corresponding ultraproduct (in particular it is ℵ…”

Section: Proposition 411 Let G Be a Definably Amenable Group Definable In M | T Such That Dim(g) = K And Let μ Be A Left-invariant Measurmentioning

confidence: 99%

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Definable groups in dense pairs of geometric structures

Berenstein¹,

Vassiliev

2021

Arch. Math. Logic

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We study definable groups in dense/codense expansions of geometric theories with a new predicate P such as lovely pairs and expansions of fields by groups with the Mann property. We show that in such expansions, large (in the sense of dimension over the predicate) definable subgroups (the new language) of groups definable in the original language L are also L-definable, and definably amenable L-definable groups remain amenable in the expansion. We also show that if the underlying geometric theory is NIP, and G is a group definable in a model of T , then the (type-definable) connected component G 00 L P of G in the expansion agrees with the connected component G 00 in the original language. We prove similar preservation results for G ∩ P(M) n , the P-part of G in a lovely pair (M, P), and for subgroups of G in pairs (R, G) where R is a real closed field and G is a subgroup of R >0 or the unit circle S(R) with the Mann property. Keywords Unary predicate expansions • Definable groups • Amenable groups • Mann property • Connected component Mathematics Subject Classification 03C45 • 03C64A. Berenstein was partially supported by Colciencias' Project Teoría de modelos y dinámicas topológicas number 120471250707. E. Vassiliev was partially supported by the Ministry of Education and Science of the Republic of Kazakhstan (Grant AP05134992 Conservative extensions, countable ordered models, and closure operators). Both authors were also supported by the project Expansiones densas de teorías geométricas: grupos y medidas, Facultad de Ciencias, Universidad de los Andes. The authors thank Erik Walsberg for providing the proof for Proposition 4.11.

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“…By [5, Theorem 5.12] and Theorem 1.4 below, this strong structure theorem holds in many settings, among others when P is a dense

prefixdcl

‐independent set. In this last setting, the theorem has already been crucially used in applications, such as in the description of definable groups in

\overset{M}{true}

in [3], as well as in the point counting theorems in [4]. We expect that similar applications will soon be found in the rest of the settings of Theorem 1.4.…”

Section: Introductionmentioning

confidence: 93%