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Forking and independence in o-minimal theories

Authors

Alfred Dolich
Type: Article
Publication Date: 2004-03-01
Citations: 43
DOI: https://doi.org/10.2178/jsl/1080938838

Abstract

In the following we try to answer a simple question, “what does forking look like in an o-minimal theory”, or more generally, “what kinds of notions of independence with what kinds of properties are admissible in an o-minimal theory?” The motivation of these question begin with the study of simple theories and generalizations of simple theories. In [3] Kim and Pillay prove that the class of simple theories may be described exactly as those theories bearing a notion of independence satisfying various axioms. Thus it is natural to ask, if we weaken the assumptions as to which axioms must hold, what kind of theories do we get? Another source of motivation, also stemming from the study of simple theories, comes from the work of Shelah in [8] and [7]. Here Shelah addresses a “classification” type problem for class of models of a theory, showing that a theory will have the appropriate “structure” type property if one can construct a partially ordered set, satisfying various properties, of models of the theory. Using this criterion Shelah shows that the class of simple theories has this “structure” property, yet also that several non-simple examples do as well (though it should be pointed out that o-minimal theories can not be among these since any theory with the strict order property will have the corresponding “non-structure” property [8]). Thus one is lead to ask, what are the non-simple theories meeting this criterion, and one is once again led to study the types of independence relation a theory might bear. Finally, Shelah in [6] provides some possible definitions of what axioms for a notion of independence one should possibly look for in order to hope that theories bearing such a notion of independence should be amenable closer analysis. In studying all of the above mentioned situations it readily becomes clear that dividing and forking play a central role in all of them, even though we are no longer dealing with the simple case where we know that dividing and forking are very well behaved. All of these considerations lead one to look for classes of non-simple theories of which something is known where one can construct interesting notions of independence and consequently also say something about the nature of forking and dividing in these contexts. Given this one is naturally lead to one of the most well behaved classes of non-simple theories, namely the o-minimal theories.

Locations

  • Journal of Symbolic Logic
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A GEOMETRIC INTRODUCTION TO FORKING AND THORN-FORKING

2009-06-01
Hans H. Adler
A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of … A ternary relation [Formula: see text] between subsets of the big model of a complete first-order theory T is called an independence relation if it satisfies a certain set of axioms. The primary example is forking in a simple theory, but o-minimal theories are also known to have an interesting independence relation. Our approach in this paper is to treat independence relations as mathematical objects worth studying. The main application is a better understanding of thorn-forking, which turns out to be closely related to modular pairs in the lattice of algebraically closed sets.

Properties and Consequences of th-Independence

2002-01-01
Alf Onshuus
We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of … We develop a new notion of independence suggested by Scanlon (th-independence). We prove that in a large class of theories (which includes all simple theories) this notion has many of the properties needed for an adequate geometric structure in these models. We also prove that this definition agrees with the usual independence notions in stable, supersimple and o-minimal theories. Finally, many of the proofs and results we get for th-independence when restricted to simple theories seem to show there is some connection between th-independence and the stable forking conjecture. In particular, we prove that in any simple theory where this conjecture holds, our definition agrees with the classic definition.

Properties and consequences of Thorn-independence

2006-03-01
Alf Onshuus
Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of … Abstract We develop a new notion of independence (ϸ-independence, read “thorn”-independence) that arises from a family of ranks suggested by Scanlon (ϸ-ranks). We prove that in a large class of theories (including simple theories and o-minimal theories) this notion has many of the properties needed for an adequate geometric structure. We prove that ϸ-independence agrees with the usual independence notions in stable, supersimple and o-minimal theories. Furthermore, we give some evidence that the equivalence between forking and ϸ-forking in simple theories might be closely related to one of the main open conjectures in simplicity theory, the stable forking conjecture. In particular, we prove that in any simple theory where the stable forking conjecture holds, ϸ-independence and forking independence agree.

Vaught's conjecture for o-minimal theories

1988-03-01
Laura L. Mayer
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is … The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

Vaught's conjecture for o-minimal theories

1988-03-01
Laura L. Mayer
The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is … The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is o-minimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2 ω countable models. This proviso (viz. that T has fewer than 2 ω countable models) is in the tradition of classification theory: given a cardinal α , if T has the maximum possible number of models of size α , i.e. 2 α , then no structure theorem is expected (cf. [Sh1]). O-minimality is introduced in §1. §1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In §2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In §3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called “components” in Pillay and Steinhorn's analysis of ω -categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2 ω countable models.

THORN-FORKING AS LOCAL FORKING

2009-06-01
Hans H. Adler
We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as … We introduce the notion of a preindependence relation between subsets of the big model of a complete first-order theory, an abstraction of the properties which numerous concrete notions such as forking, dividing, thorn-forking, thorn-dividing, splitting or finite satisfiability share in all complete theories. We examine the relation between four additional axioms (extension, local character, full existence and symmetry) that one expects of a good notion of independence. We show that thorn-forking can be described in terms of local forking if we localize the number k in Kim's notion of "dividing with respect to k" (using Ben-Yaacov's "k-inconsistency witnesses") rather than the forking formulas. It follows that every theory with an M-symmetric lattice of algebraically closed sets (in T eq ) is rosy, with a simple lattice theoretical interpretation of thorn-forking.

NSOP-LIKE INDEPENDENCE IN AECATS

2022-12-12
Mark Kamsma
Abstract The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them … Abstract The classes stable, simple, and NSOP $_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP $_1$ theories it must come from Kim-dividing. We generalise this work to the framework of Abstract Elementary Categories (AECats) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple, and NSOP $_1$ -like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking, and long Kim-dividing.
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Non-forking independence in stable theories

2024-10-13
Amador Martin-Pizarro
We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically … We observe that a simple condition suffices to describes non-forking independence over models in a stable theory. Under mild assumptions, this description can be extended to non-forking independence over algebraically closed subsets, without having to use the full strength of the work of the seminal work of Kim and Pillay. The results in this note (which are surely well-known among most model theorists) essentially use that types over models in a stable theory are stationary.
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Explanation of Independence

2005-01-01
Hans H. Adler
An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is … An axiomatic treatment of `independence relations' (notions of independence) for complete first-order theories is presented, the principal examples being forking (due to Shelah) and thorn-forking (due to Onshuus). Thorn-forking is characterised in terms of modular pairs in the lattice of algebraically closed sets. Wherever possible, forking and thorn-forking are treated in a uniform way. They are dual in the sense that forking is the finest (most restrictive) and thorn-forking the coarsest independence relation worth examining. We finish by defining the kernel of a sequence of indiscernibles and studying its relation to canonical bases.

The strong continuity in weakly o-minimal structures (Model theoretic aspects of the notion of independence and dimension)

2014-04-01
Hiroshi Tanaka

The strong continuity in weakly o-minimal structures (Model theoretic aspects of the notion of independence and dimension)

2014-04-01
広志 田中

An introduction to forking

1979-09-01
Daniel Lascar , Bruno Poizat
The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is … The notion of forking has been introduced by Shelah, and a full treatment of it will appear in his book on stability [S1]. The principal aim of this paper is to show that it is an easy and natural notion. Consider some well-known examples of ℵ 0 -stable theories: vector spaces over Q , algebraically closed fields, differentially closed fields of characteristic 0; in each of these cases, we have a natural notion of independence: linear, algebraic and differential independence respectively. Forking gives a generalization of these notions. More precisely, if are subsets of some model and c a point of this model, the fact that the type of c over does not fork over means that there are no more relations of dependence between c and than there already existed between c and . In the case of the vector spaces, this means that c is in the space generated by only if it is already in the space generated by . In the case of differentially closed fields, this means that the minimal differential equations of c with coefficient respectively in and have the same order. Of course, these notions of dependence are essential for the study of the above mentioned structures. Forking is no less important for stable theories. A glance at Shelah's book will convince the reader that this is the case. What we have to do is the following. Assuming T stable and given and p a type on , we want to distinguish among the extensions of p to some of them that we shall call the nonforking extensions of p .

NSOP$_1$-like independence in AECats

2021-01-01
Mark Kamsma
The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is … The classes stable, simple and NSOP$_1$ in the stability hierarchy for first-order theories can be characterised by the existence of a certain independence relation. For each of them there is a canonicity theorem: there can be at most one nice independence relation. Independence in stable and simple first-order theories must come from forking and dividing (which then coincide), and for NSOP$_1$ theories it must come from Kim-dividing. We generalise this work to the framework of AECats (Abstract Elementary Categories) with the amalgamation property. These are a certain kind of accessible category generalising the category of (subsets of) models of some theory. We prove canonicity theorems for stable, simple and NSOP$_1$-like independence relations. The stable and simple cases have been done before in slightly different setups, but we provide them here as well so that we can recover part of the original stability hierarchy. We also provide abstract definitions for each of these independence relations as what we call isi-dividing, isi-forking and long Kim-dividing.
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Types in locally o-minimal structures (Model theoretic aspects of the notion of independence and dimension)

2020-09-01
Hisatomo Maesono

Three surprising instances of dividing

2023-01-01
Gabriel Conant , Alex Kruckman
We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no … We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type $p$ over a set $B$ does not divide over $C\subseteq B$, then no extension of $p$ to a complete type over $\text{acl}(B)$ divides over $C$. Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an NSOP$_1$ theory with a complete type that forks, but does not divide, over a model (answering a question of d'Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$-free incidence structures.
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Independence Structures in Set Theory

1996-06-20
Michiel van Lambalgen
Abstract The axioms for ‘independent choices’ presented in [Lambalgen, 1992) are strengthened here, so that they can be seen as introducing a new type of indiscernibles in set theory. The … Abstract The axioms for ‘independent choices’ presented in [Lambalgen, 1992) are strengthened here, so that they can be seen as introducing a new type of indiscernibles in set theory. The resulting system allows for the construction of natural inner models. The article is organised as follows. Section 1 introduces the axioms, some preliminary lemmas are proved and the relation with the axiom of choice is investigated. Section O gives a philosophical motivation for the axioms; the reader who is not interested in such matters can skip this part. In section 2 we compare the structure introduced by the axioms, here called an independence structure, with two constructions from model theory, indiscernibles and minimal sets. Section 3 contains the construction of inner models, while section 4 presents some concluding philosophical remarks.
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Vaught's conjecture for quite o-minimal theories

2016-09-14
B. Sh. Kulpeshov , S. V. Sudoplatov

A New Kim’s Lemma

2024-08-12
Alex Kruckman , Nicholas Ramsey
Kim's Lemma is a key ingredient in the theory of forking independence in simple theories.It asserts that if a formula divides, then it divides along every Morley sequence in type … Kim's Lemma is a key ingredient in the theory of forking independence in simple theories.It asserts that if a formula divides, then it divides along every Morley sequence in type of the parameters.Variants of Kim's Lemma have formed the core of the theories of independence in two orthogonal generalizations of simplicity -namely, the classes of NTP 2 and NSOP 1 theories.We introduce a new variant of Kim's Lemma that simultaneously generalizes the NTP 2 and NSOP 1 variants.We explore examples and nonexamples in which this lemma holds, discuss implications with syntactic properties of theories, and ask several questions. 1. Introduction 825 2. Preliminaries 828 3. A diversity of Kim's lemmas 838 4. Examples 844 5.
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THREE SURPRISING INSTANCES OF DIVIDING

2024-03-21
Gabriel Conant , Alex Kruckman
Abstract We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$ , … Abstract We give three counterexamples to the folklore claim that in an arbitrary theory, if a complete type p over a set B does not divide over $C\subseteq B$ , then no extension of p to a complete type over $\operatorname {acl}(B)$ divides over C . Two of our examples are also the first known theories where all sets are extension bases for nonforking, but forking and dividing differ for complete types (answering a question of Adler). One example is an $\mathrm {NSOP}_1$ theory with a complete type that forks, but does not divide, over a model (answering a question of d’Elbée). Moreover, dividing independence fails to imply M-independence in this example (which refutes another folklore claim). In addition to these counterexamples, we summarize various related properties of dividing that are still true. We also address consequences for previous literature, including an earlier unpublished result about forking and dividing in free amalgamation theories, and some claims about dividing in the theory of generic $K_{m,n}$ -free incidence structures.
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Stationarily ordered types and the number of countable models

2019-11-28
Slavko Moconja , Predrag Tanović
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On Pillay's conjecture in the general case

2017-03-17
Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more

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.

On the homotopy type of definable groups in an o-minimal structure

2009-01-01
Alessandro Berarducci , Marcello Mamino
Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result … Given a definably compact group G in a saturated o-minimal structure, there is a canonical homomorphism from G to a compact real Lie group F(G). We establish a similar result for the (o-mininimal) universal cover of a definably compact group. We also show that F(G) determines the definable homotopy type of G. A crucial step is to show that the fundamental group of an open subset of F(G) is isomorphic to the definable fundamental group of its preimage in G. Our results depend on the study of the o-minimal fundamental groupoid of G.

Connected components of definable groups and o-minimality I

2011-01-29
Annalisa Conversano , Anand Pillay
We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is … We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.

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.

Definable quotients of locally definable groups

2012-03-23
Pantelis E. Eleftheriou , Ya’acov Peterzil
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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).

Generic sets in definably compact groups

2007-01-01
Ya’acov Peterzil , Anand Pillay
A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably … A subset $X$ of a group $G$ is called <em>left generic</em> if finitely many left translates of $X$ cover $G$. Our main result is that if $G$ is a definably compact group in an o-minimal structure and a definable $X\subseteq G$ is not right generic then i
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Definable quotients of locally definable groups

2011-03-24
Pantelis E. Eleftheriou , Ya’acov Peterzil
We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to … We study locally definable abelian groups $\CU$ in various settings and examine conditions under which the quotient of $\CU$ by a discrete subgroup might be definable. This turns out to be related to the existence of the type-definable subgroup $\CU^{00}$ and to the divisibility of $\CU$.

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.

2009 North American Annual Meeting of the Association for Symbolic Logic

2009-10-14
Alasdair Urquhart
An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content. An abstract is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Equivariant homotopy of definable groups

2009-05-07
Alessandro Berarducci , Marcello Mamino

A note on generic subsets of definable groups

2011-01-01
Mário J. Edmundo , Giuseppina Terzo
We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably … We generalize the theory of generic subsets of definably compact definable groups to arbitrary o-minimal structures. This theory is a crucial part of the solution to Pillay's conjecture connecting definably compact definable groups with Lie groups.
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Definable groups in models of Presburger Arithmetic

2020-03-09
Alf Onshuus , Mariana Vicaría
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Forking and Dividing in NTP<sub>2</sub> theories

2012-01-20
Artem Chernikov , Itay Kaplan
Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in … Abstract We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP 2 .
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Forking in VC-minimal theories

2012-11-02
Sarah Cotter , Sergei Starchenko
Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable … Abstract We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M , generalizing a result of Dolich on o-minimal theories in [4].
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Definable versions of theorems by Kirszbraun and Helly

2010-10-26
Matthias Aschenbrenner , Andreas Fischer
Kirszbraun's theorem states that every Lipschitz map S → ℝn, where S⊆ ℝm, has an extension to a Lipschitz map ℝm → ℝn with the same Lipschitz constant. Its proof … Kirszbraun's theorem states that every Lipschitz map S → ℝn, where S⊆ ℝm, has an extension to a Lipschitz map ℝm → ℝn with the same Lipschitz constant. Its proof relies on Helly's theorem: every family of compact subsets of ℝn, having the property that each of its subfamilies consisting of at most (n + 1) sets share a common point, has a non-empty intersection. We prove versions of these theorems valid for definable maps and sets in arbitrary definably complete expansions of ordered fields.
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Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

2009-06-16
Alessandro Berarducci
Abstract By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free … Abstract By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.
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On generically stable types in dependent theories

2009-01-04
Alexander Usvyatsov
Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories. Abstract We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent theories.
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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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Independence in Model Theory

2016-01-01
Åsa Hirvonen

The domination monoid in o-minimal theories

2020-08-04
Rosario Mennuni
We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by … We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with arXiv:1702.06504, this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.

The domination monoid in o-minimal theories

2021-12-09
Rosario Mennuni
We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by … We study the monoid of global invariant types modulo domination-equivalence in the context of o-minimal theories. We reduce its computation to the problem of proving that it is generated by classes of 1-types. We show this to hold in Real Closed Fields, where generators of this monoid correspond to invariant convex subrings of the monster model. Combined with [C. Ealy, D. Haskell and J. Maríková, Residue field domination in real closed valued fields, Notre Dame J. Formal Logic 60(3) (2019) 333–351], this allows us to compute the domination monoid in the weakly o-minimal theory of Real Closed Valued Fields.
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Connected components of definable groups and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>o</mml:mi></mml:math>-minimality I

2012-06-29
Annalisa Conversano , Anand Pillay
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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.

ON LINEARLY ORDERED STRUCTURES OF FINITE RANK

2009-12-01
Alf Onshuus , Charles Steinhorn
O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable … O-minimal structures have long been thought to occupy the base of a hierarchy of ordered structures, in analogy with the role that strongly minimal structures play with respect to stable theories. This is the first in an anticipated series of papers whose aim is the development of model theory for ordered structures of rank greater than one. A class of ordered structures to which a notion of finite rank can be assigned, the decomposable structures, is introduced here. These include all ordered structures definable (as subsets of n-tuples of the universe) in o-minimal structures. The principal result in this paper, Theorem 5.1, asserts roughly that a decomposable structure [Formula: see text] can be partitioned into finitely many definable subsets such that on each set the restriction of &lt; is a "twisted lexicographic" order. As a consequence (Corollary 5.1), for all n and linear orders ≺ definable on a subset X ⊆ M n in an o-minimal structure [Formula: see text], there is a definable partition of X such that the restriction of ≺ to each set in the partition is "lexicographic".

On the homotopy type of definable groups in an o-minimal structure

2011-02-24
Alessandro Berarducci , Marcello Mamino
We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 … We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1 → G00 → G → G/G00 → 1, where G00 is the ‘infinitesimal subgroup’ of G and G/G00 is a compact real Lie group. We show that given a connected open subset U of G/G00, there is a canonical isomorphism between the fundamental group of U and the o-minimal fundamental group of its preimage under the projection p: G→ G/G00. We apply this result to show that there is a natural exact sequence 1 → G 00 → G ~ → G / G 00 ~ → 1 , where G ~ is the (o-minimal) universal cover of G, and G / G 00 ~ is the universal cover of the real Lie group G/G00. We also prove that, up to isomorphism, each finite covering H → G/G00, with H a connected Lie group, is of the form H/H00→ G/G00 for some definable group extension H→G. Finally we prove that the (Lie-)isomorphism type of G/G00 determines the definable homotopy type of G. In the semisimple case a stronger result holds: G/G00 determines G up to definable isomorphism. Our results depend on the study of the o-minimal fundamental groupoid of G and the homotopy properties of the projection G→ G/G00.
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On the o-minimal Hilbert's fifth problem

2015-01-01
Mário J. Edmundo , Marcello Mamino , Luca Prelli +2 more
Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal … Let ${\mathbb M}$ be an arbitrary o-minimal structure. Let $G$ be a definably compact definably connected abelian definable group of dimension $n$. Here we compute the new the intrinsic o-minimal fundamental group of $G;$ for each $k&gt;0$, the $k$-torsion subgroups of $G;$ the o-minimal cohomology algebra over ${\mathbb Q}$ of $G.$ As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

On generically stable types in dependent theories

2007-01-01
Alexander Usvyatsov
We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories. We develop the theory of generically stable types, independence relation based on nonforking and stable weight in the context of dependent (NIP) theories.

polynomially bounded structures

2008-01-01
Marcus Tressl

Definable groups in models of Presburger Arithmetic and G^{00}

2016-12-29
Alf Onshuus , Mariana Vicaría
This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem … This paper is devoted to understand groups definable in Presburger arithmetic. We prove the following theorems: Theorem 1. Every group definable in a model of Presburger Arithmetic is abelian-by-finite. Theorem 2. Every bounded group definable in a model (Z,+,<) of Presburger Arithmetic is definably isomorphic to (Z, +)^{n} mod out by a lattice.

Invariant types in dp-minimal theories

2012-10-16
Pierre Simon
We try to analyse general invariant types in dp-minimal theories in terms of finitely satisfiable and definable ones. We prove in particular that an invariant dp-minimal type is either finitely … We try to analyse general invariant types in dp-minimal theories in terms of finitely satisfiable and definable ones. We prove in particular that an invariant dp-minimal type is either finitely satisfiable or definable and that a definable version of the (p,q)-theorem holds in dp-minimal theories of small or medium directionality. In an appendix with Sergei Starchenko, we prove that in dp-minimal theories with Skolem functions, every non-forking formula extends to a definable type.

ON FORKING AND DEFINABILITY OF TYPES IN SOME DP-MINIMAL THEORIES

2014-12-01
Pierre Simon , Sergei Starchenko
We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types. We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst non-forking types.
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Types in o-minimal theories

2008-12-20
Janak Ramakrishnan
Types in o-minimal theories by Janak Daniel Ramakrishnan Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair We extend previous work on classifying o-minimal types, and … Types in o-minimal theories by Janak Daniel Ramakrishnan Doctor of Philosophy in Mathematics University of California, Berkeley Professor Thomas Scanlon, Chair We extend previous work on classifying o-minimal types, and develop several applications. Marker developed a dichotomy of o-minimal types into “cuts” and “noncuts,” with a further dichotomy of cuts being either “uniquely” or “non-uniquely realizable.” We use this classification to extend work by van den Dries and Miller on bounding growth rates of definable functions in Chapter 3, and work by Marker on constructing certain “small” extensions in Chapter 4. We further sub-classify “non-uniquely realizable cuts” into three categories in Chapter 2, and we give define the notion of a “decreasing” type in Chapter 5, which is a presentation of a type well-suited for our work. Using this definition, we achieve two results: in Chapter 5.2, we improve a characterization of definable types in o-minimal theories given by Marker and Steinhorn, and in Chapter 6 we answer a question of Speissegger’s about extending a continuous function to the boundary of its domain. As well, in Chapter 5.3, we show how every elementary extension can be presented as decreasing.
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Stable types in rosy theories

2010-10-04
Assaf Hasson , Alf Onshuus
Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to … Abstract We study the behaviour of stable types in rosy theories. The main technical result is that a non-þ-forking extension of an unstable type is unstable. We apply this to show that a rosy group with a þ-generic stable type is stable. In the context of super-rosy theories of finite rank we conclude that non-trivial stable types of U þ -rank 1 must arise from definable stable sets.

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

A survey on groups definable in o-minimal structures

2008-05-22
Margarita Otero
Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable … Groups definable in o-minimal structures have been studied for the last twenty years. The starting point of all the development is Pillay's theorem that a definable group is a definable group manifold (see Section 2). This implies that when the group has the order type of the reals, we have a real Lie group. The main lines of research in the subject so far have been the following:
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Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

2007-01-01
Alessandro Berarducci
By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an expansion of a compact Lie group by a torsion free normal … By recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an expansion of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

Forking in NTP_2 theories

2009-06-15
Artem Chernikov , Itay Kaplan
We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact … We prove that in theories without the tree property of the second kind (which include dependent and simple theories) forking and dividing over models are the same, and in fact over any extension base. As an application we show that dependence is equivalent to bounded non-forking assuming NTP_2.

Weak dividing, chain conditions, and simplicity

2004-02-01
Alfred Dolich

Toward classifying unstable theories

1996-08-01
Saharon Shelah

Omitting types in -minimal theories

1986-03-01
David Marker
Let L be a first order language containing a binary relation symbol &lt;. Definition. Suppose ℳ is an L -structure and &lt; is a total ordering of the domain of … Let L be a first order language containing a binary relation symbol &lt;. Definition. Suppose ℳ is an L -structure and &lt; is a total ordering of the domain of ℳ. ℳ is ordered minimal ( -minimal) if and only if any parametrically definable X ⊆ ℳ can be represented as a finite union of points and intervals with endpoints in ℳ. In any ordered structure every finite union of points and intervals is definable. Thus the -minimal structures are the ones with no unnecessary definable sets. If T is a complete L -theory we say that T is strongly ( - minimal if and only if every model of T is -minimal. The theory of real closed fields is the canonical example of a strongly -minimal theory. Strongly -minimal theories were introduced (in a less general guise which we discuss in §6) by van den Dries in [1]. Extending van den Dries' work, Pillay and Steinhorn (see [3], [4] and [2]) developed an extensive structure theory for definable sets in strongly -minimal theories, generalizing the results for real closed fields. They also established several striking analogies between strongly -minimal theories and ω -stable theories (most notably the existence and uniqueness of prime models). In this paper we will examine the construction of models of strongly -minimal theories emphasizing the problems involved in realizing and omitting types. Among other things we will prove that the Hanf number for omitting types for a strongly -minimal theory T is at most (2 ∣ T ∣ ) + , and characterize the strongly -minimal theories with models order isomorphic to ( R , &lt;).

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.