Weakly o-minimal structures and real closed fields

Authors

Dugald Macpherson, David Marker, Charles Steinhorn

Type: Article

Publication Date: 2000-04-13

Citations: 155

DOI: https://doi.org/10.1090/s0002-9947-00-02633-7

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Abstract

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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Transactions of the American Mathematical Society
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Additive reducts of real closed fields and strongly bounded structures

2022-01-01

Hind Abu Saleh , Ya’acov Peterzil

Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, … Given a real closed field $R$, we identify exactly four proper reducts of $R$ which expand the underlying (unordered) $R$-vector space structure. Towards this theorem we introduce a new notion, of strongly bounded reducts of linearly ordered structures: A reduct $\mathcal M$ of a linearly ordered structure $\langle R;<,\cdots\rangle $ is called \emph{strongly bounded} if every $\mathcal M$-definable subset of $R$ is either bounded or co-bounded in $R$. We investigate strongly bounded additive reducts of o-minimal structures and as a corollary prove the above theorem on additive reducts of real closed fields.

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Zilber's conjecture for some o-minimal structures over the reals

1993-06-01

Ya’acov Peterzil

On VC-minimal fields and dp-smallness

2013-01-01

Vincent Guingona

In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this … In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.

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On VC-minimal fields and dp-smallness

2013-07-30

Vincent Guingona

The Non-Axiomatizability of O-Minimality

2012-01-01

Alex Rennet

Fix a language L extending the language of real closed fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent … Fix a language L extending the language of real closed fields by at least one new predicate or function symbol. Call an L-structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L-sentences \Lambda, there is a real closed field R satisfying \Lambda, which is not pseudo-o-minimal. In particular, there are locally o-minimal, definably complete real closed fields which are not pseudo-o-minimal. This answers negatively a question raised by Schoutens, and shows that the theory consisting of those L-sentences true in all o-minimal L-structures, called the theory of o-minimality (for L), is not recursively axiomatizable.

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The Non-Axiomatizability of O-Minimality

2012-03-13

Alex Rennet

Selection problems in O-minimal structures

2017-01-01

Saronsad Sokantika

In this dissertation, we improve the Definable Michael's Selection Theorem in o-minimal expansions of real closed fields. Then applications of this theorem are established; for instance, we prove the following … In this dissertation, we improve the Definable Michael's Selection Theorem in o-minimal expansions of real closed fields. Then applications of this theorem are established; for instance, we prove the following statement: Let be an o-minimal expansion of and T be a definable set-valued map where n = 1 or m=1. If T has a continuous selection, then T has a definable continuous selection. Moreover, we prove the statement: Let be an o-minimal expansion of a real closed field and be a closed subset of Rn. If T: E --> Rm is a definable continuous set-valued map and T is bounded for each in the boundary of E, then T has a definable continuous extension.

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Strongly minimal expansions of (ℂ, +) definable in o-minimal fields

2008-01-04

Assaf Hasson , Piotr Kowalski

We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly … We characterize those functions f:ℂ → ℂ definable in o-minimal expansions of the reals for which the structure (ℂ,+, f) is strongly minimal: such functions must be complex constructible, possibly after conjugating by a real matrix. In particular we prove a special case of the Zilber Dichotomy: an algebraically closed field is definable in certain strongly minimal structures which are definable in an o-minimal field.

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2005-01-26

Stefano Leonesi , Carlo Toffalori

Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for … Discrete weakly o-minimal structures, although not so stimulating as their dense counterparts, do exhibit a certain wealth of examples and pathologies. For instance they lack prime models and monotonicity for definable functions, and are not preserved by elementary equivalence. First we exhibit these features. Then we consider a countable theory of weakly o-minimal structures with infinite definable discrete (convex) subsets and we study the Boolean algebra of definable sets of its countable models. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

THE NON-AXIOMATIZABILITY OF O-MINIMALITY

2014-03-01

Alex Rennet

Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L -structure R pseudo-o-minimal if it is (elementarily equivalent … Abstract Fix a language extending the language of ordered fields by at least one new predicate or function symbol. Call an L -structure R pseudo-o-minimal if it is (elementarily equivalent to) an ultraproduct of o-minimal structures. We show that for any recursive list of L -sentences , there is a real closed field satisfying which is not pseudo-o-minimal. This shows that the theory T o−min consisting of those -sentences true in all o-minimal -structures, also called the theory of o-minimality (for L) , is not recursively axiomatizable. And, in particular, there are locally o-minimal, definably complete expansions of real closed fields which are not pseudo-o-minimal.

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Tameness Results for Expansions of the Real Field by Groups

2013-01-01

Michael Tychonievich

Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we … Expanding on the ideas of o-minimality, we study three kinds of expansions of the real field and discuss certain tameness properties that they possess or lack. In Chapter 1, we introduce some basic logical concepts and theorems of o-minimality. In Chapter 2, we prove that the ring of integers is definable in the expansion of the real field by an infinite convex subset of a finite-rank additive subgroup of the reals. We give a few applications of this result. The main theorem of Chapter 3 is a structure theorem for expansions of the real field by families of restricted complex power functions. We apply it to classify expansions of the real field by families of locally closed trajectories of linear vector fields. Chapter 4 deals with polynomially bounded o-minimal structures over the real field expanded by multiplicative subgroups of the reals. The main result is that any nonempty, bounded, definable d-dimensional submanifold has finite d-dimensional Hausdorff measure if and only if the dimension of its frontier is less than d.

Ultraproducts of O-Minimal Structures

2012-01-01

Alexander David Rennet

There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures.In the first part we investigate (for a fixed first-order language L ) what we call … There are three main parts to this thesis, all centred around ultraproducts of o-minimal structures.In the first part we investigate (for a fixed first-order language L ) what we call the L-theory of o-minimality . It is the theory consisting of those L -sentences true in all o-minimal L -structures. We find that when L expands the language of real closed fields by at least one new function or relation symbol, the L -theory of o-minimality is not recursively axiomatizable. In particular, for any recursive list ofaxioms A which is consistent with the L -theory of o-minimality, we find that there are locally o-minimal, definably complete structures satisfying A which are not elementarily equivalent to an ultraproduct of o-minimal structures. We call the latter sort of structures pseudo-o-minimal.In the second part we investigate uniform finiteness and cell decomposition in the pseudo-o-minimal setting. To do this, we introduce the notion of a pseudo-o-minimal structure tallying a discrete definable set. Investigating this notion, we answer some questions of uniqueness and existence. Finally, we show that under certain assumptions about the discrete definable sets that a given pseudo-o-minimal structure can tally, we have a version of uniform finiteness, at least in the planar case. This is the first step towards a cell decomposition theorem in this setting.In the final section, we look into two classes of examples of ultraproducts of o-minimal structures. For the first class, we note the o-minimality of a certain subset of these structures, and show the non-o-minimality of another. In particular, we derive the o-minimality of a new structure related to the real field with the exponential function. The second class is relatively intractable, but we discuss its relation to an important open problem in o-minimality.

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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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O-minimal residue fields of o-minimal fields

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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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Jana Maříková

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Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner

Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets … Abstract A structure ( M , <, …) is called quasi-o-minimal if in any structure elementarily equivalent to it the definable subsets are exactly the Boolean combinations of 0-definable subsets and intervals. We give a series of natural examples of quasi-o-minimal structures which are not o-minimal; one of them is the ordered group of integers. We develop a technique to investigate quasi-o-minimality and use it to study quasi-o-minimal ordered groups (possibly with extra structure). Main results: any quasi-o-minimal ordered group is abelian; any quasi-o-minimal ordered ring is a real closed field, or has zero multiplication; every quasi-o-minimal divisible ordered group is o-minimal; every quasi-o-minimal archimedian densely ordered group is divisible. We show that a counterpart of quasi-o-minimality in stability theory is the notion of theory of U -rank 1.

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Definable Sets in Ordered Structures. III

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Definable linear orders definably embed into lexicographic orders in o-minimal structures

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DISTRIBUTIONS OF COUNTABLE MODELS OF QUITE O-MINIMAL EHRENFEUCHT THEORIES

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follows the Code of Conduct of the Committee on Publication Ethics (COPE), and follows the COPE Flowcharts for Resolving Cases of Suspected Misconduct (http://publicationethics. follows the Code of Conduct of the Committee on Publication Ethics (COPE), and follows the COPE Flowcharts for Resolving Cases of Suspected Misconduct (http://publicationethics.

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Matthias Aschenbrenner , Alf Dolich , Deirdre Haskell +2 more

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LOCALLY O-MINIMAL STRUCTURES WITH TAME TOPOLOGICAL PROPERTIES

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D. Yu. Emel’yanov , B. Sh. Kulpeshov , S. V. Sudoplatov

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Roman Wencel

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Generalized Euler characteristic in power-bounded T-convex valued fields

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Yimu Yin

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Masato Fujita

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

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Strongly NIP almost real closed fields

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Matthias Aschenbrenner , Lou van den Dries

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Viktor Verbovskiy , A. B. Dauletiyarova

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A. Alibek , B. S. Baizhanov , B. Sh. Kulpeshov +1 more

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Ayhan Günaydın , Philipp Hieronymi

Abstract We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative … Abstract We prove that certain pairs of ordered structures are dependent. Among these structures are dense and tame pairs of o-minimal structures and further the real field with a multiplicative subgroup with the Mann property, regardless of whether it is dense or discrete.

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Stefano Leonesi , Carlo Toffalori

Abstract We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic … Abstract We consider the sets definable in the countable models of a weakly o‐minimal theory T of totally ordered structures. We investigate under which conditions their Boolean algebras are isomorphic (hence T is p‐ ω ‐categorical), in other words when each of these definable sets admits, if infinite, an infinite coinfinite definable subset. We show that this is true if and only if T has no infinite definable discrete (convex) subset. We examine the same problem among arbitrary theories of mere linear orders. Finally we prove that, within expansions of Boolean lattices, every weakly o‐minimal theory is p‐ ω ‐categorical. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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Abstract We introduce and study some local versions of o‐minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood … Abstract We introduce and study some local versions of o‐minimality, requiring that every definable set decomposes as the union of finitely many isolated points and intervals in a suitable neighbourhood of every point. Motivating examples are the expansions of the ordered reals by sine, cosine and other periodic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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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, +, <).

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Almost o-minimal structures and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:mi mathvariant="fraktur">X</mml:mi></mml:math>-structures

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2018-05-23

Will Johnson

We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] … We construct a nontrivial definable type V field topology on any dp-minimal field [Formula: see text] that is not strongly minimal, and prove that definable subsets of [Formula: see text] have small boundary. Using this topology and its properties, we show that in any dp-minimal field [Formula: see text], dp-rank of definable sets varies definably in families, dp-rank of complete types is characterized in terms of algebraic closure, and [Formula: see text] is finite for all [Formula: see text]. Additionally, by combining the existence of the topology with results of Jahnke, Simon and Walsberg [Dp-minimal valued fields, J. Symbolic Logic 82(1) (2017) 151–165], it follows that dp-minimal fields that are neither algebraically closed nor real closed admit nontrivial definable Henselian valuations. These results are a key stepping stone toward the classification of dp-minimal fields in [Fun with fields, Ph.D. thesis, University of California, Berkeley (2016)].

The independence property in generalized dense pairs of structures

2011-05-19

Alexander Berenstein , Alf Dolich , Alf Onshuus

Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply … Abstract We provide a general theorem implying that for a (strongly) dependent theory T the theory of sufficiently well-behaved pairs of models of T is again (strongly) dependent. We apply the theorem to the case of lovely pairs of thorn-rank one theories as well as to a setting of dense pairs of first-order topological theories.

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The Non-Axiomatizability of O-Minimality

2012-03-13

Alex Rennet

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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Remarks on Tarski's problem concerning (R, +, *, exp)

1984-01-01

Lou van den Dries

Lectures on Formally Real Fields

1984-01-01

Alexander Prestel

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Real closed rings II. model theory

1983-12-01

Gregory Cherlin , Max Dickmann

Cell decompositions of C-minimal structures

1994-03-01

Deirdre Haskell , Dugald Macpherson

Definable sets in ordered structures

1984-01-01

Anand Pillay , Charles Steinhorn

On introduit la notion de theorie O-minimale des structures ordonnees, une telle theorie etant telle que les sous-ensembles definissables de ses modeles soient particulierement simples On introduit la notion de theorie O-minimale des structures ordonnees, une telle theorie etant telle que les sous-ensembles definissables de ses modeles soient particulierement simples

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Elimination of quantifiers for ordered valuation rings

1987-03-01

Max Dickmann

Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for … Cherlin and Dickmann [2] proved that the theory RCVR of real closed (valuation) rings admits quantifier-elimination (q.e.) in the language ℒ = {+, −, ·, 0, 1, <, ∣} for ordered rings augmented by the divisibility relation “∣”. The purpose of this paper is to prove a form of converse of this result: Theorem. Let T be a theory of ordered commutative domains (which are not fields), formulated in the language ℒ. In addition we assume that : (1) The symbol “∣” is interpreted as the honest divisibility relation : (2) The following divisibility property holds in T : If T admits q.e. in ℒ, then T = RCVR. We do not know at present whether the restriction imposed by condition (2) can be weakened. The divisibility property (DP) has been considered in the context of ordered valued fields; see [4] for example. It also appears in [2], and has been further studied in Becker [1] from the point of view of model theory. Ordered domains in which (DP) holds are called in [1] convexly ordered valuation rings , for reasons which the proposition below makes clear. The following summarizes the basic properties of these rings: Proposition I [2, Lemma 4]. (1) Let A be a linearly ordered commutative domain. The following are equivalent : (a) A is a convexly ordered valuation ring . (b) Every ideal (or, equivalently, principal ideal) is convex in A . (c) A is a valuation ring convex in its field of fractions quot( A ). (d) A is a valuation ring and its maximal ideal M A is convex (in A or, equivalently, in quot (A)) . (e) A is a valuation ring and its maximal ideal is bounded by ± 1.

T-convexity and tame extensions

1995-03-01

Lou van den Dries , Adam H. Lewenberg

Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that … Abstract Let T be a complete o-minimal extension of the theory of real closed fields. We characterize the convex hulls of elementary substructures of models of T and show that the residue field of such a convex hull has a natural expansion to a model of T . We give a quantifier elimination relative to T for the theory of pairs (ℛ, V ) where ℛ ⊨ T and V ≠ ℛ is the convex hull of an elementary substructure of ℛ. We deduce that the theory of such pairs is complete and weakly o-minimal. We also give a quantifier elimination relative to T for the theory of pairs with ℛ a model of T and a proper elementary substructure that is Dedekind complete in ℛ. We deduce that the theory of such “tame” pairs is complete.

On variants of o-minimality

1996-06-01

Dugald Macpherson , Charles Steinhorn

Vapnik-Chervonenkis Classes of Definable Sets

1992-04-01

M. Laskowski

We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the … We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the independence property. Via this connection we obtain several new examples of Vapnik-Chervonenkis classes, including sets of positivity of finitely subanalytic functions.

Elimination of quantifiers in algebraic structures

1983-01-01

Angus Macintyre , K.F. McKenna , Lou van den Dries

The universality of formal power series fields

1939-01-01

Saunders MacLane

In a recent paper,f André Gleyzal has constructed ordered fields consisting of certain "transfinite real numbers" and has established the interesting result that any ordered field can be considered as … In a recent paper,f André Gleyzal has constructed ordered fields consisting of certain "transfinite real numbers" and has established the interesting result that any ordered field can be considered as a subfield of one of these transfinite fields.These fields prove to be identical with fields of formal power series in which the exponents are allowed to range over a suitable ordered abelian group.Such fields were first introduced by Hahn,$ while they have been analyzed in terms of generalized valuations by Krull.§ Gleyzal applied his construction of transfinite real numbers not only to the case when the coefficient field consisted of real numbers, but also to suitable fields of characteristic p.He conjectured that this construction should yield a "universal" field of characteristic p.We show here that KrmTs technique can be used to establish Gleyzal's conjecture.

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Paires de structures O-minimales

1998-06-01

Yerzhan Baisalov , Bruno Poizat

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 … 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].

Tame Topology and O-minimal Structures

1998-05-07

L. P. D. van den Dries

Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment … Following their introduction in the early 1980s o-minimal structures were found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry. These notes give a self-contained treatment of the theory of o-minimal structures from a geometric and topological viewpoint, assuming only rudimentary algebra and analysis. The book starts with an introduction and overview of the subject. Later chapters cover the monotonicity theorem, cell decomposition, and the Euler characteristic in the o-minimal setting and show how these notions are easier to handle than in ordinary topology. The remarkable combinatorial property of o-minimal structures, the Vapnik-Chervonenkis property, is also covered. This book should be of interest to model theorists, analytic geometers and topologists.

An introduction to stability theory

2018-04-06

Daniel Palacín

These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical … These lecture notes are based on the first section of Pillay's book \[3] and they cover fundamental notions of stability theory such as defi nable types, forking calculus and canonical bases, as well as stable groups and homogeneous spaces. The approach followed here is originally due to Hrushovski and Pillay \[2], who presented stability from a local point of view. Throughout the notes, some general knowledge of model theory is assumed. I recommend the book of Tent and Ziegler \[4] as an introduction to model theory. Furthermore, the texts of Casanovas \[1] and Wagner \[5] may also be useful to the reader to obtain a different approach to stability theory.

Definable Sets in Ordered Structures. I

1986-06-01

Anand Pillay , Charles Steinhorn

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly … This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures.The definition of this class and the corresponding class of theories, the strongly ©-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories.Theorems 2.1 and 2.3, respectively, provide characterizations of C-minimal ordered groups and rings.Several other simple results are collected in §3.The primary tool in the analysis of ¿¡-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2.This result states that any (parametrically) definable unary function in an (5-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals.The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all N0-categorical ¿¡¡-minimal structures (Theorem 6.1).

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Abelian groups with contractions. I

1994-01-01

Franz‐Viktor Kuhlmann

Definable Sets in Ordered Structures. II

1986-06-01

Julia F. Knight , Anand Pillay , Charles Steinhorn

It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved … It is proved that any O-minimal structure M (in which the underlying order is dense) is strongly O-minimal (namely, every N elementarily equivalent to M is O-minimal).It is simultaneously proved that if M is 0minimal, then every definable set of n-tuples of M has finitely many "definably connected components."0. Introduction.In this paper we study the structure of definable sets (of tuples) in an arbitrary O-minimal structure M (in which the underlying order is dense).Recall from [PSI, PS2] that the structure M is said to be O-minimal if M = (M, <, Ri)iei, where < is a total ordering on M and every definable (with parameters) subset of M is a finite union of points in M and intervals (a, b) where aE M or a = -co and b E M or b = +co.M is said to be strongly O-minimal if every N which is elementarily equivalent to M is O-minimal.We will always assume that the underlying order of M is a dense order with no first or least element.In this paper we also introduce the notion of a definable set X C Mn being definably connected, and we prove THEOREM 0.1.Let M be O-minimal.Then any definable X C Mn is a disjoint union of finitely many definably connected definable sets.THEOREM 0.2.If M is O-minimal, then M is strongly O-minimal.THEOREM 0.3.(a) Let M be O-minimal and let (p(xi,...,xn,yi,...,ym) be any formula of L (the language for M).Then there is K < uj such that for any b E Mm, the set <j>(x,b)M (= {5 E Mn:M t= 4>(a,b)}) has at most K definably connected components.(b) If M is a O-minimal expansion o/(R, <), then in (a) we can replace definably connected by connected.Theorems 0.1 and 0.2 are proved simultaneously by a rather complicated induction argument (outlined in §3 and undertaken in § §4 and 5).Theorem 0.3 follows from Theorems 0.1 and 0.2 by a compactness argument.Let us remark that if M is the field of real numbers, or more generally any real closed field, then by Tarski's quantifier elimination [T], M is (strongly) O-minimal and moreover the definable sets (of n-tuples) in M are precisely the semialgebraic

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Abelian Groups with Contractions II: Weak O-Minimality

1995-01-01

Franz‐Viktor Kuhlmann