Paires de structures O-minimales

Authors

Yerzhan Baisalov, Bruno Poizat

Type: Article

Publication Date: 1998-06-01

Citations: 68

DOI: https://doi.org/10.2307/2586850

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Abstract

Resumo Ni montras propecon de el j eteco de la kvantoro (∃y ∈ M ) pri la (sufi c e) belaj paroj de modeloj de una O -plimalpova teorio. G i havas korolaron ke, se ni aldonas malkavajn unarajn predikatojn a la lingvo de kelka O -plimalpova strukturo, ni ricevas malforte O -plimalpovan strukturon. Tui c i rezultato estis en speciala kaso pruvita de [5], kaj la g ia g eneralize c o estis anoncita en [1].

Locations

Journal of Symbolic Logic
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On elementary pairs ofo-minimal structures

1995-01-01

Adam H. Lewenberg

Some Remarks on Minimal Pairs

1982-01-01

Daniel L. Everett

Some remark on locally o-minimal structures (モデル理論における独立概念と次元の研究)

2018-08-01

久智前園

Regular projections in o-minimal structures

2022-10-13

Nguyễn Thị Thanh Nhàn

Levelled O-minimal structures

1997-01-01

David R. Marker , Chris Miller

Embedded o-minimal structures

2012-02-20

Assaf Hasson , Alf Onshuus

(Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of … (Bull. London Math. Soc. 42 (2010) 64–74) There is a serious mistake in the proof of Theorem 1 in the above mentioned paper. Consequently, we must withdraw the claim of having proved that theorem.

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MINIMAL STRUCTURE AND RELATED

2013-01-01

Alexandroff Spaces

A theorem of the complement and some new o-minimal structures

1999-12-01

A. J. Wilkie

A Whitney type theorem in o-minimal structures

2005-01-01

Tomohiro Kawakami , Wakayama Univiversity

Model theory of quasi-minimal structures

2002-01-01

Kentaro Wakai

On the bounded condition of an o-minimal structure

2007-01-01

優影山

Weakly quasi-o-minimal models

2010-10-01

K. Zh. Kudaîbergenov

Linear O-minimal structures

1993-02-01

James Loveys , Ya’acov Peterzil

TAME TOPOLOGY AND O-MINIMAL STRUCTURES (London Mathematical Society Lecture Note Series 248)

2000-05-01

A. J. Wilkie

By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998). By Lou van den Dries: 180 pp., £24.95 (US$39.95, LMS Members' price £18.70), isbn 0 521 59838 9 (Cambridge University Press 1998).

Model theory of quasi-minimal structures〔和文〕 (幾何的モデル理論の研究研究集会報告集)

2002-09-01

健太郎若井

Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 pp.

2000-06-01

Alessandro Berarducci

Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 … Lou van den Dries. Tame topology and o-minimal structures. London Mathematical Society lecture note series, no. 248. Cambridge University Press, Cambridge, New York, and Oakleigh, Victoria, 1998, x + 180 pp. - Volume 6 Issue 2

Minimal m-functions

1981-01-01

Bogusław Hajduk

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Łojasiewicz inequalities in o-minimal structures

2015-11-25

Ta Lê Loi

Locally o-minimal structures

2012-07-01

Tomohiro Kawakami , Kota Takeuchi , Hiroshi Tanaka +1 more

In this paper we study (strongly) locally o-minimal structures.We first give a characterization of the strong local o-minimality.We also investigate locally o-minimal expansions of (R, +, <). In this paper we study (strongly) locally o-minimal structures.We first give a characterization of the strong local o-minimality.We also investigate locally o-minimal expansions of (R, +, <).

Weakly o-minimal algebraic structures

2005-01-01

Hiroshi Tanaka

Open core and small groups in dense pairs of topological structures

2020-07-16

Elías Baro , Amador Martín-Pizarro

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Distality in valued fields and related structures

2022-02-16

Matthias Aschenbrenner , Artem Chernikov , Allen Gehret +1 more

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style … We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an Ax-Kochen-Eršov-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

On NIP and invariant measures

2007-01-01

Ehud Hrushovski , Anand Pillay

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if … We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of $NIP$ (not the independence property), continuing aspects of math.LO/0607442. Among key results are: (i) if $p = tp(b/A)$ does not fork over $A$ then the Lascar strong type of $b$ over $A$ coincides with the compact strong type of $b$ over $A$ and any global nonforking extension of $p$ is Borel definable over $bdd(A)$ (ii) analogous statements for Keisler measures and definable groups, including the fact that $G^{000} = G^{00}$ for $G$ definably amenable, (iii) definitions, characterizations and properties of "generically stable" types and groups (iv) uniqueness of translation invariant Keisler measures on groups with finitely satisfiable generics (vi) A proof of the compact domination conjecture for definably compact commutative groups in $o$-minimal expansions of real closed fields.

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On the strong cell decomposition property for weakly o-minimal structures

2013-10-31

Roman Wencel

We consider a class of weakly o-minimal structures admitting an o-minimal style cell decomposition, for which one can construct certain canonical o-minimal extension. The paper contains several fundamental facts concerning … We consider a class of weakly o-minimal structures admitting an o-minimal style cell decomposition, for which one can construct certain canonical o-minimal extension. The paper contains several fundamental facts concerning the structures in question. Among other things, it is proved that the strong cell decomposition property is preserved under elementary equivalences. We also investigate fiberwise properties (of definable sets and definable functions), definable equivalence relations, and conditions implying elimination of imaginaries.

O-minimal fields with standard part map

2009-01-01

Jana Maříková

Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion … Let R be an o-minimal field and V a proper convex subring with residue field k and standard part (residue) map st: V \to k. Let k_{ind} be the expansion of k by the standard parts of the definable relations in R. We investigate the definable sets in k_{ind} and conditions on (R,V) which imply o-minimality of k_{ind}. We also show that if R is omega-saturated and V is the convex hull of the rationals in R, then the sets definable in k_{ind} are exactly the standard parts of the sets definable in (R,V).

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The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

2007-01-01

Jana Maříková

Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n … Let R be a sufficiently saturated o-minimal expansion of a real closed field, let O be the convex hull of the rationals in R, and let st: O^n \to \mathbb{R}^n be the standard part map. For X \subseteq R^n define st(X):=st(X \cap O^n). We let \mathbb{R}_{\ind} be the structure with underlying set \mathbb{R} and expanded by all sets of the form st(X), where X \subseteq R^{n} is definable in R and n=1,2,.... We show that the subsets of \mathbb{R}^n that are definable in \mathbb{R}_{\ind} are exactly the finite unions of sets of the form st(X) \setminus st(Y), where X,Y \subseteq R^n are definable in R. A consequence of the proof is a partial answer to a question by Hrushovski, Peterzil and Pillay about the existence of measures with certain invariance properties on the lattice of bounded definable sets in R^n.

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Pillay's conjecture and its solution—a survey

2010-06-07

Ya’acov Peterzil

Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the … Introduction. These notes were originally written for a tutorial I gave in a Modnet Summer meeting which took place in Oxford 2006. I later gave a similar tutorial in the Wroclaw Logic colloquium 2007. The goal was to survey recent work in model theory of o-minimal structures, centered around the solution to a beautiful conjecture of Pillay on definable groups in o-minimal structures. The conjecture (which is now a theorem in most interesting cases) suggested a connection between arbitrary definable groups in o-minimal structures and compact real Lie groups.

Groups, measures, and the NIP

2007-02-02

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce the new notion of "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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Groups, measures, and the NIP

2006-01-01

Ehud Hrushovski , Ya’acov Peterzil , Anand Pillay

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating … We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author's conjectures relating definably compact groups $G$ in saturated $o$-minimal structures to compact Lie groups. We also prove some other structural results about such $G$, for example the existence of a left invariant finitely additive probability measure on definable subsets of $G$. We finally introduce a new notion "compact domination" (domination of a definable set by a compact space) and raise some new conjectures in the $o$-minimal case.

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A THEORY OF PAIRS FOR NON-VALUATIONAL STRUCTURES

2019-01-25

ELITZUR BAR-YEHUDA , Assaf Hasson , Ya’acov Peterzil

Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ … Given a weakly o-minimal structure $\mathcal M$ and its o-minimal completion $\bar {\mathcal M}$, we first associate to $\bar {\mathcal M}$ a canonical language and then prove that $Th(\mathcal M)$ determines $Th(\bar {\mathcal M})$. We then investigate the theory of the pair $\mathcal M^P=(\bar {\mathcal M};M)$ in the spirit of the theory of dense pairs of o-minimal structures, and prove, among other results, that it is near model complete, and every $\mathcal M^P$-definable open subset of $\bar M^n$ is already definable in $\bar {\mathcal M}$. We give an example of a weakly o-minimal structure which interprets $\bar {\mathcal M}$ and show that it is not elementarily equivalent to any reduct of an o-minimal trace.

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On definable matchings in o-minimal bipartite graphs

2024-01-01

Jana Maříková

We consider bipartite graphs definable in o-minimal structures, in which the edge relation $G$ is a finite union of graphs of certain measure-preserving maps. We establish the existence of definable … We consider bipartite graphs definable in o-minimal structures, in which the edge relation $G$ is a finite union of graphs of certain measure-preserving maps. We establish the existence of definable matchings with few short augmenting paths.

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Generic pairs of SU-rank 1 structures

2002-10-10

Evgueni Vassiliev

The elementary theory of Dedekind cuts in polynomially bounded structures

2005-02-05

Marcus Tressl

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O-minimal residue fields of o-minimal fields

2011-02-09

Jana Maříková

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Unexpected imaginaries in valued fields with analytic structure

2013-05-15

Deirdre Haskell , Ehud Hrushovski , Dugald Macpherson

Abstract We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries … Abstract We give an example of an imaginary defined in certain valued fields with analytic structure which cannot be coded in the ‘geometric’ sorts which suffice to code all imaginaries in the corresponding algebraic setting.

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Definable choice for a class of weakly o-minimal theories

2016-05-25

M. Laskowski , Christopher Shaw

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Embedded o-minimal structures

2012-02-20

Assaf Hasson , Alf Onshuus

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Hausdorff measure on o-minimal structures

2012-04-04

Antongiulio Fornasiero , Elisa Vasquez Rifo

Abstract We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy–Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every … Abstract We introduce the Hausdorff measure for definable sets in an o-minimal structure, and prove the Cauchy–Crofton and co-area formulae for the o-minimal Hausdorff measure. We also prove that every definable set can be partitioned into “basic rectifiable sets”, and that the Whitney arc property holds for basic rectifiable sets.

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Model completeness of o-minimal structures expanded by Dedekind cuts

2005-03-01

Marcus Tressl

§1. Introduction . Let M be a totally ordered set. A (Dedekind) cut p of M is a couple ( p L , p R ) of subsets p L … §1. Introduction . Let M be a totally ordered set. A (Dedekind) cut p of M is a couple ( p L , p R ) of subsets p L , p R of M such that p L ⋃ p R = M and p L < p R , i.e., a < b for all a ∈ p L , b ∈ p R . In this article we are looking for model completeness results of o-minimal structures M expanded by a set p L for a cut p of M . This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M . Then we are looking for a natural, definable expansion of the L ( D )-structure ( M , p L ) which is model complete. The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z + for the cut p with p R = { a ∈ M ∣ a > Z } and Z − for the cut q with q L = { a ∈ M ∣ a < Z }.

Weakly o-minimal structures and real closed fields

2000-04-13

Dugald Macpherson , David Marker , Charles Steinhorn

A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures … A linearly ordered structure is weakly o-minimal if all of its definable sets in one variable are the union of finitely many convex sets in the structure. Weakly o-minimal structures were introduced by Dickmann, and they arise in several contexts. We here prove several fundamental results about weakly o-minimal structures. Foremost among these, we show that every weakly o-minimal ordered field is real closed. We also develop a substantial theory of definable sets in weakly o-minimal structures, patterned, as much as possible, after that for o-minimal structures.

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On expansions of weakly o-minimal non-valuational structures by convex predicates

2008-12-19

Roman Wencel

We prove that if ${\cal M}=(M,\leq,+,\ldots)$ is a weakly o-minimal non-valuational structure expanding an ordered group $(M,\leq,+)$, then its expansion by a family of “non-valuational” unary predicates remains non-valuational. The … We prove that if ${\cal M}=(M,\leq,+,\ldots)$ is a weakly o-minimal non-valuational structure expanding an ordered group $(M,\leq,+)$, then its expansion by a family of “non-valuational” unary predicates remains non-valuational. The paper is b

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Heirs of box types in polynomially bounded structures

2009-10-05

Marcus Tressl

Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of … Abstract A box type is an n -type of an o-minimal structure which is uniquely determined by the projections to the coordinate axes. We characterize heirs of box types of a polynomially bounded o-minimal structure M . From this, we deduce various structure theorems for subsets of M k , definable in the expansion of M by all convex subsets of the line. We show that after naming constants, is model complete provided M is model complete.

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Compact domination for groups definable in linear o-minimal structures

2009-07-15

Pantelis E. Eleftheriou

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Value groups and residue fields of models of real exponentiation

2019-01-01

Lothar Sebastian Krapp

Let F be an archimedean field, G a divisible ordered abelian group and h a group exponential on G. A triple (F,G,h) is realised in a non-archimedean exponential field (K,exp) … Let F be an archimedean field, G a divisible ordered abelian group and h a group exponential on G. A triple (F,G,h) is realised in a non-archimedean exponential field (K,exp) if the residue field of K under the natural valuation is F and the induced exponential group of (K,exp) is (G,h). We give a full characterisation of all triples (F,G,h) which can be realised in a model of real exponentiation in the following two cases: i) G is countable. ii) G is of cardinality kappa and kappa-saturated for an uncountable regular cardinal kappa with kappa^(<kappa) = kappa.

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Ellis enveloping semigroups in real closed fields

2024-03-13

Elías Baro , Daniel Palacín

Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the … Abstract We introduce the Boolean algebra of d -semialgebraic (more generally, d -definable) sets and prove that its Stone space is naturally isomorphic to the Ellis enveloping semigroup of the Stone space of the Boolean algebra of semialgebraic (definable) sets. For definably connected o-minimal groups, we prove that this family agrees with the one of externally definable sets in the one-dimensional case. Nonetheless, we prove that in general these two families differ, even in the semialgebraic case over the real algebraic numbers. On the other hand, in the semialgebraic case we characterise real semialgebraic functions representing Boolean combinations of d -semialgebraic sets.

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Externally definable sets and dependent pairs

2012-05-12

Artem Chernikov , Pierre Simon

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Weakly one-based geometric theories

2012-04-04

Alexander Berenstein , Evgueni Vassiliev

Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find … Abstract We study the class of weakly locally modular geometric theories introduced in [4], a common generalization of the classes of linear SU-rank 1 and linear o-minimal theories. We find new conditions equivalent to weak local modularity: “weak one-basedness”, absence of type definable “almost quasidesigns”, and “generic linearity”. Among other things, we show that weak one-basedness is closed under reducts. We also show that the lovely pair expansion of a non-trivial weakly one-based ω -categorical geometric theory interprets an infinite vector space over a finite field.

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Dp-Minimality: Basic Facts and Examples

2011-07-01

Alfred Dolich , John Goodrick , David Lippel

We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of … We study the notion of dp-minimality, beginning by providing several essential facts about dp-minimality, establishing several equivalent definitions for dp-minimality, and comparing dp-minimality to other minimality notions. The majority of the rest of the paper is dedicated to examples. We establish via a simple proof that any weakly o-minimal theory is dp-minimal and then give an example of a weakly o-minimal group not obtained by adding traces of externally definable sets. Next we give an example of a divisible ordered Abelian group which is dp-minimal and not weakly o-minimal. Finally we establish that the field of p-adic numbers is dp-minimal.

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A Note on Weakly O-Minimal Structures and Definable Completeness

2007-04-01

Alfred Dolich

We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures. IntroductionIn this short … We consider the extent to which certain properties of definably complete structures may persist in structures which are not definably complete, particularly in the weakly o-minimal structures. IntroductionIn this short note we study weakly o-minimal theories and how they relate to general ordered theories which are not definably complete.First, we consider the degree to which topological properties of definable sets in weakly o-minimal structures mirror those in o-minimal structures.Second, we consider the degree to which weakly o-minimal theories may be characterized as the "best-behaved," densely ordered theories among those theories which are not definably complete.Here we are motivated by results characterizing o-minimal theories as those definably complete theories bearing certain desirable properties.For the problems we consider that our answers are negative.Recall the definition of weak o-minimality.Definition 1.1 A structure (M, <, . . . ) in a language L with a symbol < for a dense linear order is called weakly o-minimal if any definable X ⊆ M is a finite union of convex sets.A theory T is weakly o-minimal if all of its models are.(See, for example, [8] and the references therein.)Also recall the definition of definable completeness (for a discussion of this, see [15]).Definition 1.2 A structure (M, <, . . . ) in a language L with a symbol < for a dense linear order is said to be definably complete if, for any definable subset X ⊆ M, if X is bounded above then there is a supremum a ∈ M of X .Similarly, we demand

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A note on prime models in weakly o‐minimal structures

2017-03-31

Somayyeh Tari

Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the … Let be a weakly o‐minimal structure with the strong cell decomposition property. In this note, we show that the canonical o‐minimal extension of is the unique prime model of the full first order theory of over any set . 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.

Triangulation in o-minimal fields with standard part map

2009-01-01

Lou van den Dries , Jana Maříková

In answering questions from arXiv:0901.2337v1 we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, … In answering questions from arXiv:0901.2337v1 we prove a triangulation result that is of independent interest. In more detail, let R be an o-minimal field with a proper convex subring V, and let st: V \to k be the corresponding standard part map. Under a mild assumption on (R,V) we show that definable subsets X of V^n admit a triangulation that induces a triangulation of its standard part st(X).

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The structure on the real field generated by the standard part map on an o-minimal expansion of a real closed field

2009-06-01

Jana Maříková

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Beautiful pairs.

2021-12-01

Pablo Cubides Kovacsics , Martin Hils , Jinhe Ye

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called beautiful pairs, which comprises Poizat's belles paires of stable … We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called beautiful pairs, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with classical Fra\{i}ss\'{e} properties. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically, $p$-adically and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces such as Berkovich and Huber's analytifications. In particular, we recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields.

Distality in valued fields and related structures

2020-08-22

Matthias Aschenbrenner , Artem Chernikov , Allen Gehret +1 more

We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style … We investigate distality and existence of distal expansions in valued fields and related structures. In particular, we characterize distality in a large class of ordered abelian groups, provide an AKE-style characterization for henselian valued fields, and demonstrate that certain expansions of fields, e.g., the differential field of logarithmic-exponential transseries, are distal. As a new tool for analyzing valued fields we employ a relative quantifier elimination for pure short exact sequences of abelian groups.

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Triangulation in o-minimal fields with standard part map

2010-01-01

Lou van den Dries , Jana Maříková

In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let $R$ be an o-minimal field with … In answering questions of J. Maříková [Fund. Math. 209 (2010)] we prove a triangulation result that is of independent interest. In more detail, let $R$ be an o-minimal field with a proper convex subring $V$, and let $\mathop{\rm st}: V \to \boldsymbol k$

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A model theoretic application of Gelfond-Schneider theorem

2003-01-01

Roman Wencel

We prove that weakly o-minimal expansions of the ordered field of all real algebraic numbers are polynomially bounded. Apart of this we make a couple of observations concerning weakly o-minimal … We prove that weakly o-minimal expansions of the ordered field of all real algebraic numbers are polynomially bounded. Apart of this we make a couple of observations concerning weakly o-minimal expansions of ordered fields of finite transcendence degree over the rationals. We show for instance that if Schanuel’s conjecture is true and K R is a field of finite transcendence degree over the rationals, then weakly o-minimal expansions of (K, , +,·) are polynomially bounded.

One-basedness and groups of the form G/G 00

2011-07-15

Davide Penazzi

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Dependent first order theories, continued

2009-09-01

Saharon Shelah

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O-minimal residue fields of o-minimal fields

2010-01-11

Jana Maříková

Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures (R,V) such that k_ind, the corresponding residue field with structure induced … Let R be an o-minimal field with a proper convex subring V. We axiomatize the class of all structures (R,V) such that k_ind, the corresponding residue field with structure induced from R via the residue map, is o-minimal. More precisely, in previous work it was shown that certain first order conditions on (R,V) are sufficient for the o-minimality of k_ind. Here we prove that these conditions are also necessary.

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polynomially bounded structures

2008-01-01

Marcus Tressl

Paires de structures stables

1983-06-01

Bruno Poizat

Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec … Le paradigme de théorie stable est la théorie T d'un corps algébriquement clos; une autre théorie stable T ′ est celle de la structure formée d'un corps algébriquement clos, avec en outre un symbole relationnel unaire interprétant un de ses sous-corps propres algébriquement clos. C'est à l'éclaircissement des rapports de T et de T ′ qu'est consacré cet article. J'y considère une théorie complète T stable, et les structures formées d'un modèle N de T, avec en outre un symbole relationnel unaire ( x ) interprétant une restriction élémentaire M de N ; j'appelle ces structures paires de modèles de T . Et je dis que la paire ( N, M ) est belle si d'une part M est ∣ T ∣ + -saturé, et d'autre part pour tout n -uplet ā d'éléments de N , tout type, au sens de T , sur M ⋃ {α} est réalisé dans N . Le premier résultat (Théorème 4) est que deux belles paires sont élémentairement équivalentes. Plus précisément, si ( N 1 , M 1 ) et ( N 2 , M 2 ) sont deux belles paires, et si ā est dans la première, b¯ dans la seconde, le fait que le type de ā sur M 1 et celui de b¯ sur M 2 soient équivalents dans l'ordre fondamental au sens de T suffit (et est bien sûr nécessaire) pour que ā at b¯ aient même type (sur ⊘) au sens de la théorie T ′ des belles paires.

Definability of types, and pairs of O-minimal structures

1994-12-01

Anand Pillay

Abstract Let T be a complete O-minimal theory in a language L . We first give an elementary proof of the result (due to Marker and Steinhorn) that all types … Abstract Let T be a complete O-minimal theory in a language L . We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P . Let T * be the L *-theory of all pairs ( N, M ), where M is a Dedekind complete model of T and N is an ⅼ M ⅼ + -saturated elementary extension of N (and M is the interpretation of P ). Using the definability of types result, we show that T * is complete and we give a simple set of axioms for T *. We also show that for every L *-formula ϕ ( x ) there is an L -formula ψ ( x ) such that T * ⊢ (∀ x )( P ( x ) → ( ϕ ( x ) ↔ ψ ( x )). This yields the following result: Let M be a Dedekind complete model of T . Let ϕ ( x, y ) be an L -formula where l ( y ) – k . Let X = { X ⊂ M k : for some a in an elementary extension N of M, X = ϕ ( a, y ) N ∩ M k }. Then there is a formula ψ ( y, z ) of L such that X = { ψ ( y, b ) M : b in M }.

Definable sets in ordered structures. III

1988-01-01

Anand Pillay , Charles Steinhorn

We show that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure has a strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> … We show that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure has a strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="o"> <mml:semantics> <mml:mi>o</mml:mi> <mml:annotation encoding="application/x-tex">o</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal theory.

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Definable sets in ordered structures. II

1986-01-01

Julia F. Knight , Anand Pillay , Charles Steinhorn

It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> … It is proved that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal structure <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in which the underlying order is dense) is strongly <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal (namely, every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N"> <mml:semantics> <mml:mi>N</mml:mi> <mml:annotation encoding="application/x-tex">N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elementarily equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal). It is simultaneously proved that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0"> <mml:semantics> <mml:mn>0</mml:mn> <mml:annotation encoding="application/x-tex">0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-minimal, then every definable set of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-tuples of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has finitely many "definably connected components."

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