Valued difference fields and NTP2

Chernikov, Artem; Hils, Martin

doi:10.1007/s11856-014-1094-z

Cited by 23 publications

(37 citation statements)

References 22 publications

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“…As an application of the above result, we show in § that the constructed model

M ⊧ {sans-serifLA}_{2}

endowed with a natural structure of a (discretely) ordered module has the independence property. This answers negatively the question of Chernikov and Hils [, Question 5.9.1] whether all ordered modules have the NIP.…”

Section: Introductionmentioning

confidence: 79%

“…A structure

scriptA

has the NIP (negation of the independence property; cf. for an extensive introduction to NIP structures and theories) if there is no formula

φ (\bar{x}, \bar{y})

such that for every

n \in ω

, there are

\bar{a_{i}} \in A^{ℓ (\bar{x})}

, with

i < n

, and

\bar{b_{J}} \in A^{ℓ (\bar{y})}

, with

J \subseteq n

, such that

φ (\bar{a_{i}}, \bar{b_{J}}) \Leftrightarrow i \in J .

Chernikov and Hils [, Question 5.9.1] asked whether all ordered modules have the NIP. We answer their question negatively:…”

Section: A Non‐nip Discretely Ordered Modulementioning

confidence: 99%

See 1 more Smart Citation

A wild model of linear arithmetic and discretely ordered modules

Glivický

Pudlák

2017

Mathematical Logic Qtrly

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ABSTRACT. Linear arithmetics are extensions of Presburger arithmetic (Pr) by one or more unary functions, each intended as multiplication by a fixed element (scalar), and containing the full induction schemes for their respective languages.In this paper we construct a model M of the 2-linear arithmetic LA 2 (linear arithmetic with two scalars) in which an infinitely long initial segment of "Peano multiplication" on M is ∅-definable. This shows, in particular, that LA 2 is not model complete in contrast to theories LA 1 and LA 0 = Pr that are known to satisfy quantifier elimination up to disjunctions of primitive positive formulas.As an application, we show that M, as a discretely ordered module over the discretely ordered ring generated by the two scalars, is not NIP, answering negatively a question of Chernikov and Hils.

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“…As an application of the above result, we show in § that the constructed model

M ⊧ {sans-serifLA}_{2}

Section: Introductionmentioning

confidence: 79%

“…A structure

scriptA

has the NIP (negation of the independence property; cf. for an extensive introduction to NIP structures and theories) if there is no formula

φ (\bar{x}, \bar{y})

such that for every

n \in ω

, there are

\bar{a_{i}} \in A^{ℓ (\bar{x})}

, with

i < n

, and

\bar{b_{J}} \in A^{ℓ (\bar{y})}

, with

J \subseteq n

, such that

φ (\bar{a_{i}}, \bar{b_{J}}) \Leftrightarrow i \in J .

Chernikov and Hils [, Question 5.9.1] asked whether all ordered modules have the NIP. We answer their question negatively:…”

Section: A Non‐nip Discretely Ordered Modulementioning

confidence: 99%

A wild model of linear arithmetic and discretely ordered modules

Glivický

Pudlák

2017

Mathematical Logic Qtrly

View full text Add to dashboard Cite

show abstract

“…In this paper, we shall follow the definition given in . (The same notion is called strongly indiscernible array in .) Definition We may view the Cartesian product

ω \times ω

as a model in the language

{scriptL}_{ar} : = {<_{1}, <_{2}}

where < 1 and < 2 are binary relation symbols interpreted in

ω \times ω

as follows:

(a, b) <_{1} (c, d) \Leftrightarrow a < c

(a, b) <_{2} (c, d) \Leftrightarrow (a = c) \land (b < d)

Given a set of tuples

A : = {{\overset{a}{true}}_{μ} ∣ μ \in ω \times ω}

where all

{\bar{a}}_{μ}

have the same arity, …”

Section: Tp2 Burden and Indiscernible Arraysmentioning

confidence: 99%

“…Several variations of the notion indiscernible array exist in the literature [2,7,8,15]. In this paper, we shall follow the definition given in [15].…”

Section: Remark 32mentioning

confidence: 99%