Coset-minimal groups

Belegradek, Oleg V.; Verbovskiy, Viktor; Wagner, Frank Olaf

doi:10.1016/s0168-0072(02)00084-2

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…Proof. Each of the examples has quasi-o-minimal theory: For (1) this was noted in [14, Section 1], and for ( 2) and (3) this is proved (based on a quantifier-elimination result from [99]) in [15,Theorem 15]. The corollary now follows from Theorem 6.4.…”

Section: Examples Of Vc D: Weakly O-minimal Theories and Variantsmentioning

confidence: 85%

Vapnik-Chervonenkis density in some theories without the independence property, I

Aschenbrenner

Dolich

Haskell

et al. 2015

Trans. Amer. Math. Soc.

107

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We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly quasi-o-minimal, and P -minimal theories.Our approach to Theorem 1.1, via definable types, was partly inspired by the use of Puiseux series in [11,81].Let ACVF denote the theory of (non-trivially) valued algebraically closed fields, in the ring language expanded by a predicate for the valuation divisibility. This has completions ACVF (0,0) (for residue characteristic 0), ACVF (0,p) (field characteristic 0, residue characteristic p), and ACVF (p,p) (field characteristic p). Because ACVF (0,0) is interpretable in RCVF, our methods give (non-optimal) density bounds for ACVF (0,0) (Corollary 6.3). However, they give no information on density in the theories ACVF (0,p) and ACVF (p,p) . The problems arise essentially because a definable set in 1-space in ACVF is a finite union of 'Swiss cheeses' but we have no way of choosing a particular Swiss cheese. This means that the definable types technique in our main tool (Theorem 5.7) breaks down. On the other hand, our methods do yield:Theorem 1.2. Suppose M = Q p is the field of p-adic numbers, construed as a firstorder structure in Macintyre's language L p . Then the VC density of every L p -formula ϕ(x; y) is at most 2|y| − 1.

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Section: Examples Of Vc D: Weakly O-minimal Theories and Variantsmentioning

confidence: 85%

Vapnik-Chervonenkis density in some theories without the independence property, I

Aschenbrenner

Dolich

Haskell

et al. 2015

Trans. Amer. Math. Soc.

107

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“…. , x n ) = 1 s ( i r i x i + b) with r i , s ∈ Z and b ∈ G. This result has been proven in the special case of groups of finite regular rank by Belegradek-Verbovskiy-Wagner [1] (using a version of quantifier elimination in this context from Weispfenning, [11]), but to our knowledge, it has yet not been written down in full generality before. Our interest in this result came from valued fields.…”

Section: Introductionmentioning

confidence: 83%

“…Lemma 2.8. For n ∈ N, a ∈ G, α ∈ A and β ∈ S n ∪ T n , we have the following equivalences, where for (1) =⇒, we additionally need a / ∈ nG, and for (3) =⇒, we additionally need a = 0.…”

Section: 2mentioning

confidence: 99%

“…Now φ(G) satisfies the prerequisites of Lemma 3.2: in that lemma, let H 0 +a 0 be the set defined by the positive literal of φ 0 (or H 0 = G, a 0 = 0 if there is no positive literal), let H i + a i be the sets excluded by the negative literals, and set G ′ := p r−1 G. (The a i are the representatives denoted by b * before.) To get our desired formula defining X ′ = φ 0 (G) + p r−1 G, it remains to verify that conditions (1) and (2)…”

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confidence: 99%

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Quantifier Elimination in Ordered Abelian Groups

Cluckers

Halupczok

2011

Confluentes Math.

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We give a new proof of quantifier elimination in the theory of all ordered abelian groups in a suitable language. More precisely, this is only "quantifier elimination relative to ordered sets" in the following sense. Each definable set in the group is a union of a family of quantifier free definable sets, where the parameter of the family runs over a set definable (with quantifiers) in a sort which carries the structure of an ordered set with some additional unary predicates.As a corollary, we find that all definable functions in ordered abelian groups are piecewise linear on finitely many definable pieces.

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“…A linearly ordered structure is weakly o-minimal if every definable subset is a finite union of convex sets [10]. A linearly ordered structure is quasio-minimal if every definable subset is a Boolean combination of intervals, points, and subsets definable over the empty set [12,13]. Proof.…”

Section: Examples Of O-stable Theoriesmentioning

confidence: 99%

O-Stable Theories

Baizhanov¹,

Verbovskii²

2011

Algebra Logic

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A well-developed technique created to study stable theories (M. Morley, S. Shelah) is applied in dealing with a class of theories with definable linear order. We introduce the notion of an o-stable theory, which generalizes the concepts of o-minimality, of weak o-minimality, and of quasi-o-minimality. It is proved that o-stable theories are dependent, but they do not exhaust the class of dependent theories with definable linear order, and that every linear order is o-superstable. DEFINITION OF AN O-STABLE THEORYA recent trend in model theory is searching for ways of applying methods of stable theories to nonstable theories. Of special interest among these are simple theories and theories without the independence property. (S. Shelah refers to the latter as dependent theories.) In the present paper, the apparatus created to study stable theories [1, 2] will be brought to bear on a class of theories with definable linear order. We introduce the notions of o-stable, ω-stable, and superstable theories (Defs. 1.2, 1.11). These generalize the concepts of o-minimality, of weak o-minimality, and of quasi-o-minimality (Lemma 4.1). It is proved that o-stable theories are dependent (Thm. 2.4), but they do not exhaust the class of dependent theories with definable linear order (Thms. 3.2 and 3.3), and that every linear order is o-superstable (Thm. 4.2). Definition 1.1. Let s be a partial 1-type over a set B and A be a set. Put S 1 s (A) def = {p ∈ S 1 (A) : p ∪ s is consistent}.

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Coset-minimal groups

Cited by 8 publications

References 10 publications

Vapnik-Chervonenkis density in some theories without the independence property, I

Vapnik-Chervonenkis density in some theories without the independence property, I

Quantifier Elimination in Ordered Abelian Groups

O-Stable Theories

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