Fields with a dense-codense linearly independent multiplicative subgroup

Berenstein, Alexander; Vassiliev, Evgueni

doi:10.1007/s00153-019-00683-w

Cited by 7 publications

(18 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…) is an H -structure. We call such a pair (R, P(R)) a G-structure [4]. Similarly, we can obtain a G-structure expansion of an algebraically closed field K by taking a multiplicative group generated by the set H (K ) in an H -structure (K , H (K )).…”

Section: Properties Of Definable Sets and Types In Dense/codense Expansionsmentioning

confidence: 99%

“…Note that such expansions are not necessarily dense/co-dense. It is shown in [4] that G-structures satisfy the Mann property.…”

Section: Properties Of Definable Sets and Types In Dense/codense Expansionsmentioning

confidence: 99%

“…Let (R, G) be a G-structure expanding a real closed field (see [4]), the natural models of this theory come by starting with an H -structure (R, H ) and then considering the multiplicative group G generated by H (R). In this section we will study some basic properties of subgroups of G. In order to do so, we will need some basic definitions (originally coming from [10]) and a characterization of types from [4].…”

Section: Basic Properties Of G-structuresmentioning

confidence: 99%

See 2 more Smart Citations

Definable groups in dense pairs of geometric structures

Berenstein¹,

Vassiliev

2021

Arch. Math. Logic

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Properties Of Definable Sets and Types In Dense/codense Expansionsmentioning

confidence: 99%

“…Note that such expansions are not necessarily dense/co-dense. It is shown in [4] that G-structures satisfy the Mann property.…”

Section: Properties Of Definable Sets and Types In Dense/codense Expansionsmentioning

confidence: 99%

Section: Basic Properties Of G-structuresmentioning

confidence: 99%

See 1 more Smart Citation

Definable groups in dense pairs of geometric structures

Berenstein¹,

Vassiliev

2021

Arch. Math. Logic

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then, Mann's result was generalized in [5]. For the model-theoretic approaches, the reader might consult [1,4,8]. In [9], the second author and Sertbaş proved that a number field K is either Q or an imaginary quadratic fied if and only if Equation (1.1) has only finitely many non-degenerate solutions with coordinates in O K for all a 1 , ..., a k ∈ O K .…”

Section: Introductionmentioning

confidence: 99%

Number fields and divisible groups via model theory

Çelik¹,

Göral²

2021

Turk J Math

View full text Add to dashboard Cite

“…Let G be the disjoint union of K \(P +πP ) and P 2 , equipped with the group structure induced from K via the identity map on the first part and via f on the second. Then G is not strongly large, since dim G = 1 and dim cl(G) = 2, but it is definably isomorphic to the L-definable group K. Theorem 1.1 puts a constraint on the existence of new definable groups, which has already been the theme of previous research, such as in [4] and [5]. Let N = M, P denote one of the aforementioned pairs from [17].…”

Section: Introductionmentioning

confidence: 99%

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

Eleftheriou¹

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let N be an expansion of an o-minimal structure M that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of M by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil's group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely their dimension equals the dimension of their topological closure, (3) if N expands M by a dense independent set, then every definable group is o-minimal.

show abstract

Fields with a dense-codense linearly independent multiplicative subgroup

Cited by 7 publications

References 14 publications

Definable groups in dense pairs of geometric structures

Definable groups in dense pairs of geometric structures

Number fields and divisible groups via model theory

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

Contact Info

Product

Resources

About