Weakly o-minimal structures and real closed fields

Macpherson, Dugald; Marker, David; Steinhorn, Charles

doi:10.1090/s0002-9947-00-02633-7

Cited by 141 publications

(119 citation statements)

References 22 publications

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“…Let us further mention that as an

L_{V}

‐structure, R is weakly o‐minimal, i.e., any definable set

X \subseteq R

is a finite union of convex definable sets. The paper includes a detailed study of definable sets in such weakly o‐minimal fields and it is shown there that in fact any weakly o‐minimal field must be real closed.…”

Section: Tangent Cones In Real Closed (Valued) Fieldsmentioning

confidence: 99%

Non‐archimedean stratifications of tangent cones

Ramírez

2017

Mathematical Logic Qtrly

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We study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.

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“…Let us further mention that as an

L_{V}

‐structure, R is weakly o‐minimal, i.e., any definable set

X \subseteq R

Section: Tangent Cones In Real Closed (Valued) Fieldsmentioning

confidence: 99%

Non‐archimedean stratifications of tangent cones

Ramírez

2017

Mathematical Logic Qtrly

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“…(It follows that H(M, H) is weakly o-minimal-that is, every unary definable set is a finite union of convex definable sets-but we give a much more detailed statement at the end of this section. See, e.g., Macpherson, Marker and Steinhorn [24] for basic information on weak o-minimality.) .…”

Section: Formentioning

confidence: 99%

Expansions of o-minimal structures by dense independent sets

Dolich

Miller

Steinhorn

2016

Annals of Pure and Applied Logic

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Let M be an o-minimal expansion of a densely ordered group and H be a pairwise disjoint collection of dense subsets of M such that H is definably independent in M. We study the structure (M, (H) H∈H ). Positive results include that every open set definable in (M, (H) H∈H ) is definable in M, the structure induced in (M, (H) H∈H ) on any H 0 ∈ H is as simple as possible (in a sense that is made precise), and the theory of (M, (H) H∈H ) eliminates imaginaries and is strongly dependent and axiomatized over the theory of M in the most obvious way. Negative results include that (M, (H) H∈H ) does not have definable Skolem functions and is neither atomic nor satisfies the exchange property. We also characterize (model-theoretic) algebraic closure and thorn forking in such structures. Throughout, we compare and contrast our results with the theory of dense pairs of o-minimal structures.

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“…Therefore,

f ∣_{V}

is a definable function from open definable set V to M . So, by [, Theorem 4.8] there is an open box

B \subseteq V

such that

f ∣_{B}

is continuous.

□

…”

Section: Prime Models In Weakly O‐minimal Theoriesmentioning

confidence: 99%

A note on prime models in weakly o‐minimal structures

Tari

2017

Mathematical Logic Qtrly

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Let M = (M, <, . . .) be a weakly o-minimal structure with the strong cell decomposition property. In this note, we show that the canonical o-minimal extension M of M is the unique prime model of the full first order theory of M over any set A ⊆ M. We also show that if two weakly o-minimal structures with the strong cell decomposition property are isomorphic then, their canonical o-minimal extensions are isomorphic too. Finally, we show the uniqueness of the prime models in a complete weakly o-minimal theory with prime models.

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Weakly o-minimal structures and real closed fields

Cited by 141 publications

References 22 publications

Non‐archimedean stratifications of tangent cones

Non‐archimedean stratifications of tangent cones

Expansions of o-minimal structures by dense independent sets

A note on prime models in weakly o‐minimal structures

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