Existentially closed ordered difference fields and rings

Point, Françoise

doi:10.1002/malq.200910007

Mathematical Logic Qtrly

2010

DOI: 10.1002/malq.200910007

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Existentially closed ordered difference fields and rings

Françoise Point

Abstract: Key words Existentially closed difference field, ordered field, valued field, lattice-ordered ring. MSC (2010) 03C60, 12H10, 12J10, 12J15, 13J25We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields. Existentially closed real-closed difference fieldsIn the first part of this paper we will consider on one hand totally ordered difference fields (a difference field is a field with a distinguished automorphis… Show more

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Cited by 3 publications

(10 citation statements)

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“…This corrigendum concerns [, § ] on ordered difference existentially closed valued fields where we overlooked the problem of immediate extensions.…”

mentioning

confidence: 99%

“…We correct [, Lemma 2.9]. In the course of its proof we used a strong form of σ‐Hensel Lemma, stronger than the one stated in [, Definition 2.8].…”

mentioning

confidence: 99%

“…We correct [, Lemma 2.9]. In the course of its proof we used a strong form of σ‐Hensel Lemma, stronger than the one stated in [, Definition 2.8]. This strong form does not hold in our context, as can be shown by considering an example analogous to the ones used by Kaplansky (cf.…”

mentioning

confidence: 99%

“…Note that later in [, § ], we only used the statement of the σ‐Hensel Lemma (and not the stronger form appearing in the proof). However, a further mistake was pointed out to us by Durhan in the proof of Proposition 2.16, in part (iii) where we considered immediate extensions and so Corollary 2.19 doesn't hold, and further one cannot in general embed a valued ordered difference field

scriptK

in one of the form:

(k ((G)), τ)

(even using factor sets), where k is a real‐closed field (containing the residue field of K ), G is a totally ordered abelian group (containing the value group of K ) which is in addition a divisible

Z [T]

‐module, and the automorphism τ defined by

τ (\sum_{i} a_{i} \cdot x^{g}) : = \sum_{i} a_{i} \cdot x^{g \cdot T}

.…”

mentioning

confidence: 99%

“…We shall be using the notations and definitions of and we now restate [, Lemma 2.9] replacing complete by maximally complete and give a corrected proof. The former proof used an induction hypothesis stronger than the one we needed and which does not necessarily hold at limit stages.…”

mentioning

confidence: 99%

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Corrigendum to F. Point, Existentially closed ordered difference fields and rings

Point

2015

Mathematical Logic Qtrly

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“…This corrigendum concerns [, § ] on ordered difference existentially closed valued fields where we overlooked the problem of immediate extensions.…”

mentioning

confidence: 99%

“…We correct [, Lemma 2.9]. In the course of its proof we used a strong form of σ‐Hensel Lemma, stronger than the one stated in [, Definition 2.8].…”

mentioning

confidence: 99%

mentioning

confidence: 99%

scriptK

in one of the form:

(k ((G)), τ)

Z [T]

‐module, and the automorphism τ defined by

τ (\sum_{i} a_{i} \cdot x^{g}) : = \sum_{i} a_{i} \cdot x^{g \cdot T}

.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Corrigendum to F. Point, Existentially closed ordered difference fields and rings

Point

2015

Mathematical Logic Qtrly

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ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

Pillay¹,

Yao

2023

J. symb. log.

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The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$ , with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$ . We call such groups definable f-generic groups. So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$ , and the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context. In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$ , so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$ . This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case.

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Multiplicative valued difference fields

Pal¹

2012

J. symb. log.

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The theory of valued difference fields (K, σ, v) depends on how the valuation v interacts with the automorphism σ. Two special cases have already been worked out -the isometric case, where v(σ(x)) = v(x) for all x ∈ K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon [2]; and the contractive case, where v(σ(x)) > nv(x) for all n ∈ N and x ∈ K × with v(x) > 0, has been worked out by Salih Azgin [4]. In this paper we deal with a more general version, called the multiplicative case, where v(σ(x)) = ρ · v(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.If there is a non-zero L ∈ Z[ρ] such that ∀x > 0(L · x = 0), we say ρ is algebraic (over the integers); otherwise we say ρ is transcendental. If ρ is algebraic, there is a minimal (degree) polynomial that it satisfies.Note that the kernel of Φ need not be trivial. For example, if ρ · x = 2x for all x, then ρ − 2 ∈ Ker(Φ). In particular, Ker(Φ) is non-trivial iff ρ is algebraic. We then form the following ring:Definition 2.3. Let M ODDAG be the L ρ,< -theory of non-trivial multiplicative ordered divisible difference abelian groups. This theory is axiomatized by the above axioms along with ∃x(x = 0) and the following additional infinite list of axioms:i.e., all non-zero linear difference operators are surjective. Thus, M ODDAG is an ∀∃-theory. Similarly as above, we denote by M ODDAG ρ the theory M ODDAG where we fix the order type of ρ.We would now like to show that M ODDAG is the model companion of M ODAG. By abuse of terminology, we would refer to any model of M ODAG (respectively M ODDAG) also as M ODAG (respectively M ODDAG).Remark. It might already be clear from the definitions above that for a given ρ, M ODDAG ρ is basically the theory of non-trivial ordered vector spaces over the ordered field Q(ρ) and then quantifier elimination actually follows from well-known results. However, here we are doing things a little differently. Instead of proving the result for a particular ρ, we are proving it uniformly across all ρ using Axiom OM. And even though in the completion the type of ρ is determined and the theory actually reduces to the above well-known theory, nevertheless it makes sense to write down some of the trivial details just to make sure that nothing fishy happens.Lemma 2.4. M ODAG and M ODDAG are co-theories.Proof. We will actually prove something stronger: for a fixed ρ, M ODAG ρ and M ODDAG ρ are co-theories. Any model of M ODDAG ρ is trivially a model of M ODAG ρ . So all we need to show is that we can embed any model G of M ODAG ρ into a model of M ODDAG ρ .If G is trivial, we can embed it into Q with any given ρ. So without loss of generality we may assume, G is non-trivial.Let Z[ρ] + := {L ∈ Z[ρ] : L > 0}. Define an equivalence relation ∼ on G × Z[ρ] + as follows:(g, L) ∼ (g ′ , L ′ ) ⇐⇒ L ′ · g = L · g ′ .We have thus shown that for a fixed ρ, M ODAG ρ and M ODDAG ρ are cotheories. In fact, since (M ODDAG ρ ) ∀ = ...

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Existentially closed ordered difference fields and rings

Cited by 3 publications

References 20 publications

Corrigendum to F. Point, Existentially closed ordered difference fields and rings

Corrigendum to F. Point, Existentially closed ordered difference fields and rings

ON GROUPS WITH DEFINABLE F-GENERICS DEFINABLE IN P-ADICALLY CLOSED FIELDS

Multiplicative valued difference fields

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