A monotonicity theorem for dp-minimal densely ordered groups

Authors

John Goodrick

Type: Article

Publication Date: 2010-01-25

Citations: 35

DOI: https://doi.org/10.2178/jsl/1264433917

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Abstract

Abstract Dp-minimality is a common generalization of weak minimality and weak o-minimality. If T is a weakly o-minimal theory then it is dp-minimal (Fact 2.2), but there are dp-minimal densely ordered groups that are not weakly o-minimal. We introduce the even more general notion of inp-minimality and prove that in an inp-minimal densely ordered group, every definable unary function is a union of finitely many continuous locally monotonic functions (Theorem 3.2).

Locations

Journal of Symbolic Logic
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Weakly o-minimal types

2024-04-12

Slavko Moconja , Predrag Tanović

We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal with respect to $<$, a relatively … We introduce and study weak o-minimality in the context of complete types in an arbitrary first-order theory. A type $p\in S(A)$ is weakly o-minimal with respect to $<$, a relatively $A$-definable linear order on $p(\mathfrak C)$, if every relatively definable subset has finitely many convex components; we prove that in that case the latter holds for all orders. Notably, we prove: (i) a monotonicity theorem for relatively definable functions on the locus of a weakly o-minimal type; (ii) weakly o-minimal types are dp-minimal, and the weak and forking non-orthogonality are equivalence relations on weakly o-minimal types. For a weakly o-minimal pair $\mathbf p=(p,<)$, we introduce the notions of the left- and right-$\mathbf p$-genericity of $a\models p$ over $B$; the latter is denoted by $B\triangleleft^{\mathbf p}a$. We prove that $\triangleleft^{\mathbf p}$ behaves particularly well on realizations of $p$: the $\triangleleft^{\mathbf p}$-incomparability and forking-dependence of $x$ and $y$ over the domain of $p$ are the same equivalence relation and the quotient order is dense linear. We show that this naturally generalizes to the set of realizations of weakly o-minimal types from a fixed $\not\perp^w$-class.

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On the function depth in an o-stable ordered group of a finite convexity rank

2025-05-20

Viktor Verbovskiy

We investigate the monotonicity properties of unary functions definable in ordered groups whose elementary theories are o-stable and have finite convexity rank. The notion of o-stability, combining o-minimality and stability, … We investigate the monotonicity properties of unary functions definable in ordered groups whose elementary theories are o-stable and have finite convexity rank. The notion of o-stability, combining o-minimality and stability, ensures tameness of types around cuts. Prior work established piecewise or local monotonicity of definable functions in weakly o-minimal structures, with key contributions by Pillay, Steinhorn, Wencel, and others. We build on these results by focusing on local monotonicity, $n$-tidiness, and the depth of definable functions. In particular, we show that any such function is piecewise $n$-tidy for some finite $n$, extending the theory of monotonicity beyond weakly o-minimal structures to a broader o-stable context.

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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Dp-minimality: basic facts and examples

2009-01-01

Alfred Dolich , John Goodrick , David Lippel

We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. The rest of the paper is dedicated … We study the notion of dp-minimality, beginning by providing several essential facts, establishing several equivalent definitions, and comparing dp-minimality to other minimality notions. 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.

Definable Sets in Ordered Structures. III

1988-10-01

Anand Pillay , Charles Steinhorn

We show that any $o$-minimal structure has a strongly $o$-minimal theory. We show that any $o$-minimal structure has a strongly $o$-minimal theory.

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Lattice Ordered O-Minimal Structures

1998-10-01

Carlo Toffalori

We propose a notion of $o$-minimality for partially ordered structures. Then we study $o$-minimal partially ordered structures $(A, \leq, \ldots)$ such that $(A,\leq)$ is a Boolean algebra. We prove that … We propose a notion of $o$-minimality for partially ordered structures. Then we study $o$-minimal partially ordered structures $(A, \leq, \ldots)$ such that $(A,\leq)$ is a Boolean algebra. We prove that they admit prime models over arbitrary subsets and we characterize $\omega$-categoricity in their setting. Finally, we classify $o$-minimal Boolean algebras as well as $o$-minimal measure spaces.

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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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Monotone theories

2018-11-13

Slavko Moconja , Predrag Tanović

We study binary reducts of linearly ordered structures obtained by naming all monotone relations between $(\mathfrak{C},<)$ and $(\mathfrak{C},\lhd)$ for $\lhd\in \mathcal{F}$, where $\mathcal{F}$ is a set of definable linear orders. … We study binary reducts of linearly ordered structures obtained by naming all monotone relations between $(\mathfrak{C},<)$ and $(\mathfrak{C},\lhd)$ for $\lhd\in \mathcal{F}$, where $\mathcal{F}$ is a set of definable linear orders. We give a sufficient condition for the elimination of quantifiers in the reduct and show that it is satisfied by $\{ \}$. We introduce the weakly monotone reduct, show that it eliminates quantifiers and that it has all the original binary structure if and only if the underlying theory is weakly quasi-o-minimal. This offers a description of binary definable sets in such theories and, as a corollary, we obtain that the theory of the full binary reduct of a weakly quasi-o-minimal theory eliminates quantifiers. We also prove that weak quasi-o-minimality of a theory does not depend on the choice of a definable linear order.

Maximality and minimality under limitwise monotonic reducibility

2014-10-01

M. Kh. Faĭzrahmanov , I. Sh. Kalimullin , D. Kh. Zainetdinov

Topological properties of sets definable in weakly o-minimal structures

2010-07-09

Roman Wencel

Abstract The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and … Abstract The paper is aimed at studying the topological dimension for sets definable in weakly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the weakly o-minimal context. We prove that the topological dimension of a set definable in a weakly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [2] for sets and functions definable in models of weakly o-minimal theories. We pay special attention to large subsets of Cartesian products of definable sets, showing that if X, Y and S are non-empty definable sets and S is a large subset of X × Y. then for a large set of tuples , where k = dim( Y ), the union of fibers is large in Y. Finally, given a weakly o-minimal structure , we find various conditions equivalent to the fact that the topological dimension in enjoys the addition property.

Uniform local weak o-minimality and $*$-local weak o-minimality

2024-05-09

Masato Fujita

We propose the notions of uniform local weak o-minimality and $*$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the … We propose the notions of uniform local weak o-minimality and $*$-local weak o-minimality. Local monotonicity theorems hold in definably complete locally o-minimal structures and uniformly locally o-minimal structures of the second kind. In this paper, we demonstrate new local monotonicity theorems for uniformly locally weakly o-minimal structures of the second kind and for locally o-minimal structures under the assumption called the univariate $*$-continuity property. We also prove that several formulas for dimension of definable sets which hold in definably complete locally o-minimal structures also hold in $*$-locally weakly o-minimal structures possessing the univariate $*$-continuity property.

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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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A definition of order convergence in a weakly ordered set

1962-01-01

D. C. Kent

A correction to an example in "Dp-minimality: basic facts and examples"

2015-05-24

Alfred Dolich , John Goodrick

This note contains a new, corrected proof of the dp-minimality of an example previously published by the same authors in Dp-minimality: basic facts and examples. The example is of an … This note contains a new, corrected proof of the dp-minimality of an example previously published by the same authors in Dp-minimality: basic facts and examples. The example is of an ordered divisible abelian group which is dp-minimal and contains a unary predicate defining an open subset with infinitely many connected components.

Uniformly locally o-minimal structures and locally o-minimal structures admitting local definable cell decomposition

2019-11-13

Masato Fujita

Types, transversals and definable compactness in o-minimal structures.

2021-11-06

Pablo Andújar Guerrero

Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded … Through careful analysis of types inspired by [AGTW21] we characterize a notion of definable compactness for definable topologies in general o-minimal structures, generalizing results from [PP07] about closed and bounded definable sets in o-minimal expansions of ordered groups. Along the way we prove a parameter version for o-minimal theories of the connection between dividing and definable types known in the more general dp-minimal context [SS14], through an elementary proof that avoids the use of existing forking and VC literature. In particular we show that, if an $A$-definable family of sets has the $(p,q)$-property, for some $p\geq q$ with $q$ large enough, then the family admits a partition into finitely many subfamilies, each of which extends to an $A$-definable type.

Types, transversals and definable compactness in o-minimal structures

2021-01-01

Pablo Andújar Guerrero

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Maximal elements of weakly continuous relations

1990-04-01

Donald E. Campbell , Mark Walker

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."

DP-MINIMALITY: INVARIANT TYPES AND DP-RANK

2014-12-01

Pierre Simon

Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version … Abstract This paper has two parts. In the first one, we prove that an invariant dp-minimal type is either finitely satisfiable or definable. We also prove that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In the second part, we study dp-rank in dp-minimal theories and show that it enjoys many nice properties. It is continuous, definable in families and it can be characterised geometrically with no mention of indiscernible sequences. In particular, if the structure expands a divisible ordered abelian group, then dp-rank coincides with the dimension coming from the order.

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On uniform definability of types over finite sets

2012-04-04

Vincent Guingona

Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” … Abstract In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called “uniform definability of types over finite sets” (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

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Piecewise Monotonicity for Unary Functions in o-Stable Groups

2021-03-01

Viktor Verbovskiy , A. B. Dauletiyarova

Theories without the tree property of the second kind

2013-10-28

Artem Chernikov

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The canonical topology on dp-minimal fields

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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Linear orders in NIP structures

2021-11-17

Pierre Simon

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On Definability of Types in Dependent Theories

2011-01-01

Vincent Guingona

Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types … Title of dissertation: ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES Vincent Guingona, Doctor of Philosophy, 2011 Dissertation directed by: Professor Michael Chris Laskowski Department of Mathematics Using definability of types for stable formulas, one develops the powerful tools of stability theory, such as canonical bases, a nice forking calculus, and stable embeddability. When one passes to the class of dependent formulas, this notion of definability of types is lost. However, as this dissertation shows, we can recover suitable alternatives to definability of types for some dependent theories. Using these alternatives, we can recover some of the power of stability theory. One alternative is uniform definability of types over finite sets (UDTFS). We show that all formulas in dp-minimal theories have UDTFS, as well as formulas with VC-density < 2. We also show that certain Henselian valued fields have UDTFS. Another alternative is isolated extensions. We show that dependent formulas are characterized by the existence of isolated extensions, and show how this gives a weak stable embeddability result. We also explore the idea of UDTFS rank and show how it relates to VC-density. Finally, we use the machinery developed in this dissertation to show that VCminimal theories satisfy the Kueker Conjecture. ON DEFINABILITY OF TYPES IN DEPENDENT THEORIES

On VC-minimal fields and dp-smallness

2014-02-24

Vincent Guingona

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DISCRETE SETS DEFINABLE IN STRONG EXPANSIONS OF ORDERED ABELIAN GROUPS

2024-12-13

Alfred Dolich , John Goodrick

Abstract We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the … Abstract We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with a particular emphasis on the set $D'$ comprised of differences between successive elements. In particular, if the burden of the structure is at most n , then the result of applying the operation $D \mapsto D'\ n$ times must be a finite set (Theorem 1.1). In the case when the structure is densely ordered and has burden $2$ , we show that any definable unary discrete set must be definable in some elementary extension of the structure $\langle \mathbb{R}; <, +, \mathbb{Z} \rangle $ (Theorem 1.3).

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On dp-minimal ordered structures

2011-05-19

Pierre Simon

Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and … Abstract We show basic facts about dp-minimal ordered structures. The main results are: dp-minimal groups are abelian-by-finite-exponent, in a divisible ordered dp-minimal group, any infinite set has nonempty interior, and any theory of pure tree is dp-minimal.

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Externally definable sets and dependent pairs

2012-05-12

Artem Chernikov , Pierre Simon

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Topological properties of definable sets in ordered Abelian groups of burden 2

2023-05-01

Alfred Dolich , John Goodrick

Abstract We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the … Abstract We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp‐rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle‐third set (Theorem 2.9). If it has burden 2 and both an infinite discrete set D and a dense‐codense set X are definable, then translates of X must witness the Independence Property (Theorem 2.26). In the last section, an explicit example of an ordered Abelian group of burden 2 is given in which both an infinite discrete set and a dense‐codense set are definable.

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On dp-minimality, strong dependence and weight

2011-07-06

Alf Onshuus , Alexander Usvyatsov

Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight. Abstract We study dp-minimal and strongly dependent theories and investigate connections between these notions and weight.

Dp-Minimality: Basic Facts and Examples

2011-07-01

Alfred Dolich , John Goodrick , David Lippel

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Dp-minimality: basic facts and examples

2009-01-01

Alfred Dolich , John Goodrick , David Lippel

Strong theories of ordered abelian groups

2015-11-26

Alfred Dolich , John Goodrick

We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets … We consider strong expansions of the theory of ordered abelian groups. We show that the assumption of strength has a multitude of desirable consequences for the structure of definable sets in such theories, in particular as relates to definable infinite discrete sets. We also provide a range of examples of strong expansions of ordered abelian groups which demonstrate the great variety of such theories.

DP-MINIMAL VALUED FIELDS

2017-03-01

Franziska Jahnke , Pierre Simon , Erik Walsberg

Abstract We show that dp-minimal valued fields are henselian and give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure. Abstract We show that dp-minimal valued fields are henselian and give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.

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On theories without the tree property of the second kind

2012-10-08

Artem Chernikov

We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is sub-multiplicative in arbitrary … We initiate a systematic study of the class of theories without the tree property of the second kind - NTP2. Most importantly, we show: the burden is sub-multiplicative in arbitrary theories (in particular, if a theory has TP2 then there is a formula with a single variable witnessing this); NTP2 is equivalent to the generalized Kim's lemma and to the boundedness of ist-weight; the dp-rank of a type in an arbitrary theory is witnessed by mutually indiscernible sequences of realizations of the type, after adding some parameters - so the dp-rank of a 1-type in any theory is always witnessed by sequences of singletons; in NTP2 theories, simple types are co-simple, characterized by the co-independence theorem, and forking between the realizations of a simple type and arbitrary elements satisfies full symmetry; a Henselian valued field of characteristic (0,0) is NTP2 (strong, of finite burden) if and only if the residue field is NTP2 (the residue field and the value group are strong, of finite burden respectively), so in particular any ultraproduct of p-adics is NTP2; adding a generic predicate to a geometric NTP2 theory preserves NTP2.

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Externally definable sets and dependent pairs II

2012-02-13

Artem Chernikov , Pierre Simon

We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; … We continue investigating the structure of externally definable sets in NIP theories and preservation of NIP after expanding by new predicates. Most importantly: types over finite sets are uniformly definable; over a model, a family of non-forking instances of a formula (with parameters ranging over a type-definable set) can be covered with finitely many invariant types; we give some criteria for the boundedness of an expansion by a new predicate in a distal theory; naming an arbitrary small indiscernible sequence preserves NIP, while naming a large one doesn't; there are models of NIP theories over which all 1-types are definable, but not all n-types.

Naming an indiscernible sequence in NIP theories

2009-01-01

Artem Chernikov , Pierre Simon

In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a … In this short note we show that if we add predicate for a dense complete indiscernible sequence in a dependent theory then the result is still dependent. This answers a question of Baldwin and Benedikt and implies that every unstable dependent theory has a dependent expansion interpreting linear order.

Dp-minimal valued fields

2015-07-14

Franziska Jahnke , Pierre Simon , Erik Walsberg

We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-trivial definable … We show that dp-minimal valued fields are henselian and that a dp-minimal field admitting a definable type V topology is either real closed, algebraically closed or admits a non-trivial definable henselian valuation. We give classifications of dp-minimal ordered abelian groups and dp-minimal ordered fields without additional structure.

Tame Topology over dp-Minimal Structures

2019-01-01

Pierre Simon , Erik Walsberg

In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is … In this article, we develop tame topology over dp-minimal structures equipped with definable uniformities satisfying certain assumptions. Our assumptions are enough to ensure that definable sets are tame: there is a good notion of dimension on definable sets, definable functions are almost everywhere continuous, and definable sets are finite unions of graphs of definable continuous "multivalued functions." This generalizes known statements about weakly o-minimal, C-minimal, and P-minimal theories.

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DP AND OTHER MINIMALITIES

2025-04-11

Pierre Simon , Erik Walsberg

A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups … A first order expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We prove analogous results for algebraic closures of finite fields, $p$-adic fields, ordered abelian groups with only finitely many convex subgroups (in articular archimedean ordered abelian groups), and abelian groups equipped with archimedean cyclic group orders. The latter allows us to describe unary definable sets in dp-minimal expansions of $(\mathbb{Z},+,C)$, where $C$ is a cyclic group order. Along the way we describe unary definable sets in dp-minimal expansions of ordered abelian groups. In the last section we give a canonical correspondence between dp-minimal expansions of $(\mathbb{Q},+,<)$ and o-minimal expansions $\mathcal{R}$ of $(\mathbb{R},+,<)$ such that $(\mathcal{R},\mathbb{Q})$ is a dense pair.

Linear orders in NIP structures

2018-07-20

Pierre Simon

We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interprets an infinite … We show that every unstable NIP theory admits a V-definable linear quasi-order, over a finite set of parameters. In particular, if the theory is omega-categorical, then it interprets an infinite linear order. This partially answers a longstanding open question.

On dp-minimal fields

2015-07-09

Will Johnson

We classify dp-minimal pure fields up to elementary equivalence. Most are equivalent to Hahn series fields $K((t^\Gamma))$ where $\Gamma$ satisfies some divisibility conditions and $K$ is $\mathbb{F}_p^{alg}$ or a local … We classify dp-minimal pure fields up to elementary equivalence. Most are equivalent to Hahn series fields $K((t^\Gamma))$ where $\Gamma$ satisfies some divisibility conditions and $K$ is $\mathbb{F}_p^{alg}$ or a local field of characteristic zero. We show that dp-small fields (including VC-minimal fields) are algebraically closed or real closed.

Tame Topology over Definable Uniform Structures

2022-02-01

Alfred Dolich , John Goodrick

A visceral structure on M is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite … A visceral structure on M is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite definable subset X⊆M has nonempty interior. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable function f:M→M has a finite set of discontinuities; any definable function f:Mn→Mm is continuous on a nonnempty open set; and assuming definable finite choice, we obtain a cell decomposition result for definable sets. Under an additional topological assumption ("no space-filling functions"), we prove that the natural notion of topological dimension is invariant under definable bijections. These results generalize some of the theorems proved by Simon and Walsberg, who assumed dp-minimality in addition to viscerality. In the final section, we construct new examples of visceral structures.

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On VC-minimal fields and dp-smallness

2013-07-30

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.

The reconstruction of cycle-free partial orders from their automorphism groups

2014-04-01

Robert Barham

Is a cycle-free partial order recognisable from its abstract automorphism group? This thesis resolves that question for two disjoint families: those cycle-free partial orders which share an automorphism group with … Is a cycle-free partial order recognisable from its abstract automorphism group? This thesis resolves that question for two disjoint families: those cycle-free partial orders which share an automorphism group with a tree; and those which satisfy certain transitivity conditions, before giving a method for combining the two. Chapter 1, the introduction, as well as introducing some notation and defining the cyclefree partial orders (CFPOs), gives a list of the results that this thesis calls upon. Chapter 2 gives a structure theorem for ℵ0-categorical trees, which is of particular interest here as their reconstruction problem is completely solved, and for the ℵ0- categorical CFPOs, which when combined with the results in Chapter 3, gives a complete reconstruction result for ℵ0-categorical CFPOs. Chapter 3 asks which CFPOs have an automorphism group isomorphic to one of a tree. It gives conditions on the CFPO and the automorphism group that allow the invocation of the work done by Rubin on the reconstruction of trees. In a brief epilogue the results are also used to show that many of the model theoretic properties of the trees are also properties of the CFPOs. The second family is addressed in Chapter 4 using a method used by Shelah and Truss on the symmetric groups of cardinals, which uses the alternating group on five elements. In Chapter 5 I give a method of attaching structures of the first kind to structures of the second, which admits a second order definition in the abstract automorphism group of the automorphism groups of the components. The last chapter is a discussion of how the work done here can be made more complete. I have included an appendix, which lists the formulas used in Chapters 4 and 5, which the reader can tear out and keep at hand to save flicking between pages.

Externally definable quotients and NIP expansions of the real ordered additive group

2019-10-23

Erik Walsberg

Let $\mathscr{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathscr{R}$ is generically locally o-minimal. It follows that … Let $\mathscr{R}$ be an $\mathrm{NIP}$ expansion of $(\mathbb{R},<,+)$ by closed subsets of $\mathbb{R}^n$ and continuous functions $f : \mathbb{R}^m \to \mathbb{R}^n$. Then $\mathscr{R}$ is generically locally o-minimal. It follows that if $X \subseteq \mathbb{R}^n$ is definable in $\mathscr{R}$ then the $C^k$-points of $X$ are dense in $X$ for any $k \geq 0$. This follows from a more general theorem on $\mathrm{NIP}$ expansions of locally compact groups, which itself follows from a result on quotients of definable sets by equivalence relations which are externally definable and $\bigwedge$-definable. We also show that $\mathscr{R}$ is strongly dependent if and only if $\mathscr{R}$ is either o-minimal or interdefinable with $(\mathbb{R},<,+, \mathcal{B}, \alpha\mathbb{Z})$ for some $\alpha > 0$ and collection $\mathcal{B}$ of bounded subsets of $\mathbb{R}^n$ such that $(\mathbb{R},<,+,\mathcal{B})$ is o-minimal.

Tame topology over definable uniform structures: viscerality and dp-minimality

2015-05-24

Alfred Dolich , John Goodrick

A visceral structure on a model is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any … A visceral structure on a model is given by a definable base for a uniform topology on its universe M in which all basic open sets are infinite and any infinite definable subset X of M has non-empty interior. Assuming only viscerality, we show that the definable sets in M satisfy some desirable topological tameness conditions. For example, any definable unary function has a finite set of discontinuities; any definable function from some Cartesian power of M into M is continuous on an open set; and assuming definable finite choice, we obtain a cell decomposition result for definable sets. Under an additional topological assumption (no space-filling functions), we prove that the natural notion of topological dimension is invariant under definable bijections. These results generalize some of the theorems proved by Simon and Walsberg, who assumed dp-minimality in addition to viscerality. In the final two sections, we construct new examples of visceral structures a subclass of which are dp-minimal yet not weakly o-minimal.

On uniform definability of types over finite sets

2010-05-26

Vincent Guingona

In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). … In this paper, using definability of types over indiscernible sequences as a template, we study a property of formulas and theories called uniform definability of types over finite sets (UDTFS). We explore UDTFS and show how it relates to well-known properties in model theory. We recall that stable theories and weakly o-minimal theories have UDTFS and UDTFS implies dependence. We then show that all dp-minimal theories have UDTFS.

The Reconstruction of Cycle-free Partial Orders from their Abstract Automorphism Groups III : Decorated CFPOs

2015-02-09

Robert Barham

In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the … In this triple of papers, we examine when two cycle-free partial orders can share an abstract automorphism group. This question was posed by M. Rubin in his memoir concerning the reconstruction of trees. In this first paper, we give a variety of conditions that guarantee when a CFPO shares an automorphism group with a tree. Some of these conditions are conditions on the abstract automorphism group, while some are one the CFPO itself. Some of the lemmas used have corollaries concerning the model theoretic properties of a CFPO.

Sur les théories sans la propriété de l'arbre du second type

2012-10-08

Artem Chernikov

Cette these en theorie des modeles pure presente la premiere etude systematique de la classe des theories NTP2 introduites par Shelah, avec un accent particuliere sur le cas NIP. Dans … Cette these en theorie des modeles pure presente la premiere etude systematique de la classe des theories NTP2 introduites par Shelah, avec un accent particuliere sur le cas NIP. Dans les premier et deuxieme chapitres, nous developpons la theorie de la bifurcation sur des bases d'extension (par exemple, nous prouvons l'existence de suites de Morley universelles, l'egalite de la bifurcation avec la division, un theoreme d'independance et d'egalite du type Lascar avec le type compact). Ceci rend possible de considerer les resultats de Kim et Pillay sur des theories simples comme un cas particulier, tout en fournissant une contrepartie manquante pour le cas des theories NIP. Cela repond a des questions de Adler, Hrushovski et Pillay. Dans le troisieme chapitre, nous developpons les rudiments de la theorie du fardeau (une generalisation du calcul du poids), en particulier, nous montrons qu'il est sous-multiplicatif, repondant a une question de Shelah. Nous etudions ensuite les types simples et NIP en theories NTP2: nous montrons que les types simples sont co-simples, caracterises par le theoreme de coindependance, et que la bifurcation entre les realisations d'un type simple et des elements arbitraires satisfait la symetrie complete; nous montrons qu'un type est NIP si et seulement si toutes ses extensions ont un nombre borne d'extensions globales non-bifurquantes. Nous prouvons aussi une preservation de type d'Ax-Kochen pour NTP2, montrant que, par exemple, tout ultraproduit de p-adics est NTP2. Nous continuons a etudier le cas particulier des theories NIP. Dans le chapitre 4, nous introduisons les definitions honnetes et les utilisons pour donner une nouvelle preuve du theoreme de l'expansion de Shelah et un critere general pour la dependance d'une paire elementaire. Comme une application, nous montrons que le fait de nommer une petite suite indiscernable preserve NIP. Dans le chapitre 5, nous combinons les definitions honnetes avec des resultats combinatoires plus profonds de la theorie de Vapnik- Chervonenkis pour deduire que, dans theories NIP, des types sur ensembles finis sont uniformement definissables. Cela confirme une conjecture de Laskowski pour les theories NIP. Par ailleurs, nous donnons une nouvelle condition suffisante pour une theorie d'une paire d'eliminer les quantificateurs en des quantificateurs sur le predicat et quelques exemples concernant la definissabilite de 1-types vs la definissabilite de n-types sur les modeles. Le dernier chapitre concernes la classification des taux de croissance du nombre des extensions non-bifurquantes. Nous avancons vers la conjecture qu'il existe un nombre fini de possibilites differentes et developpons une technique generale pour la construction de theories avec un nombre prescrit d'extensions non- bifurquantes que nous appelons la circularisation. En particulier, nous repondons par la negative a une question d'Adler en donnant un exemple d'une theorie qui a IP ou le nombre des extensions non- bifurquantes de chaque type est bornee. Par ailleurs, nous resolvons une question de Keisler sur le nombre de coupures de Dedekind dans les ordres lineaires: il est compatible avec ZFC que κ < (ded κ)ω

On the function depth in an o-stable ordered group of a finite convexity rank

2025-05-20

Viktor Verbovskiy

Strong theories, burden, and weight

2007-01-01

Hans H. Adler

We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a … We introduce the notion of the burden of a partial type in a complete first-order theory and call a theory strong if all types have almost finite burden. In a simple theory it is the supremum of the weights of all extensions of the type, and a simple theory is strong if and only if all types have finite weight. A theory without the independence property is strong if and only if it is strongly dependent. As a corollary, a stable theory is strongly dependent if and only if all types have finite weight. A strong theory does not have the tree property of the second kind.

Definable groups for dependent and 2-dependent theories

2007-01-01

Saharon Shelah

Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are … Let $T$ be a (first order complete) dependent theory, ${\mathfrak{C}}$ a $\barκ$-saturated model of $T$ and $G$ a definable subgroup which is abelian. Among subgroups of bounded index which are the union of $

Expansions of the real line by open sets: o-minimality and open cores

1999-01-01

Chris Miller , Patrick Speissegger

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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Generic structures and simple theories

1998-11-01

Zoé Chatzidakis , Anthony L. Pillay

Strongly dependent theories

2014-07-28

Saharon Shelah

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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.

Structures having o-minimal open core

2009-10-08

Alfred Dolich , Chris Miller , Charles Steinhorn

The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. … The open core of an expansion of a dense linear order is its reduct, in the sense of definability, generated by the collection of all of its open definable sets. In this paper, expansions of dense linear orders that have o-minimal open core are investigated, with emphasis on expansions of densely ordered groups. The first main result establishes conditions under which an expansion of a densely ordered group has an o-minimal open core. Specifically, the following is proved: <disp-quote> <italic>Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an expansion of a densely ordered group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper R comma greater-than comma asterisk right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:mo>></mml:mo> <mml:mo>,</mml:mo> <mml:mo>∗</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(R,>,*)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that is definably complete and satisfies the uniform finiteness property. Then the open core of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is o-minimal.</italic> </disp-quote> Two examples of classes of structures that are not o-minimal yet have o-minimal open core are discussed: dense pairs of o-minimal expansions of ordered groups, and expansions of o-minimal structures by generic predicates. In particular, such structures have open core interdefinable with the original o-minimal structure. These examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane, a phenomenon that can occur in dense pairs but not in expansions by generic predicates. The property of having no dense graphs is examined and related to uniform finiteness, definable completeness, and having o-minimal open core.

Introduction to theories without the independence property

2008-01-01

Hans H. Adler

We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories). We present an updated exposition of the classical theory of complete first order theories without the independence property (also called NIP theories or dependent theories).

On dp-minimality, strong dependence and weight

2011-07-06

Alf Onshuus , Alexander Usvyatsov

CLASSIFICATION THEORY FOR ELEMENTARY CLASSES WITH THE DEPENDENCE PROPERTY-A MODEST BEGINNING

2004-03-01

Saharon Shelah

Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the … Our thesis is that for the family of classes of the form EC(T),T a com- plete first order theory with the dependence property (which is just the negation of the independence property) there is a substantial theory which means: a substantial body of basic results for all such classes and some complimentary results for the first order theories with the independence property, as for the family of stable (and the family of simple) first order theories. We examine some properties.