ON LINEARLY ORDERED STRUCTURES OF FINITE RANK

Authors

Alf Onshuus, Charles Steinhorn

Type: Article

Publication Date: 2009-12-01

Citations: 8

DOI: https://doi.org/10.1142/s0219061309000896

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Abstract

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 < 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".

Locations

Journal of Mathematical Logic
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Decomposable ordered groups

2014-01-01

Eliana Barriga , Alf Onshuus , Charles Steinhorn

Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a … Decomposable ordered structures were introduced in \cite{OnSt} to develop a general framework to study `finite-dimensional' totally ordered structures. This paper continues this work to include decomposable structures on which a ordered group operation is defined on the structure. The main result at this level of generality asserts that any such group is supersolvable, and that topologically it is homeomorphic to the product of o-minimal groups. Then, working in an o-minimal ordered field $\mathcal R$ satisfying some additional assumptions, in Sections 3-7 definable ordered groups of dimension 2 and 3 are completely analyzed modulo definable group isomorphism. Lastly, this analysis is refined to provide a full description of these groups with respect to definable ordered group isomorphism.

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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Quasi-o-minimal structures

2000-09-01

Oleg Belegradek , Ya’acov Peterzil , Frank Olaf Wagner

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

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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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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On the Boolean algebras of definable sets in weakly o‐minimal theories

2004-04-07

Stefano Leonesi , Carlo Toffalori

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

First order topological structures and theories

1987-09-01

Anand Pillay

In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing … In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o -minimal structures [PS] in a general topological context. Note, however, that the p -adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.

First order topological structures and theories

1987-09-01

Anand Pillay

Definable sets in ordered structures

1984-01-01

Anand Pillay , Charles Steinhorn

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

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ALMOST 1-TRANSITIVITY IN LINEARLY ORDERED STRUCTURES

2023-04-01

B. Sh. Kulpeshov , S. V. Sudoplatov

The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally deeply studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization … The present paper concerns the notion of weak o-minimality introduced by M. Dickmann and originally deeply studied by D. Macpherson, D. Marker, and C. Steinhorn. Weak o-minimality is a generalization of the notion of o-minimality introduced by A. Pillay and C. Steinhorn in series of joint papers. As is known, the ordered field of real numbers is an example of an o-minimal structure. We continue studying model-theoretic properties of o-minimal and weakly o-minimal structures. In particular, we introduce the notion of almost 1-transitivity in linearly ordered structures and study tits properties. Almost 1-transitive o-minimal and weakly o-minimal linear orderings have been described. It has been established that an almost 1-transitive weakly o-minimal linear ordering is isomorphic to a finite number of concatenations of almost 1-transitive o-minimal linear orderings. Properties of expansions of families of almost 1-transitive linearly ordered theories are studied. Rank values for families of almost 1-transitive o-minimal and weakly o-minimal linear orderings have been found. A criterion for preserving both the almost 1-transitivity and weak o-minimality has been found at expanding an almost 1-transitive weak o-minimal theory by an arbitrary unary predicate. Dense ordering of an almost 1-transitive weakly o-minimal theory that is almost omega-categorical has been established.

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Definable Sets in Ordered Structures. I

1986-06-01

Anand Pillay , Charles Steinhorn

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

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The Borel Complexity of Isomorphism for Complete Theories of Linear Orders With Unary Predicates

2015-01-01

Richard Rast

We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has … We show that if A is a linear order then Th(A) is either $\aleph_0$-categorical or Borel complete (in the sense of Friedman and Stanley). We generalize this; if A has countably many unary predicates attached, then Th(A) is $\aleph_0$-categorical, has finitely many countable models (at least three), is Borel equivalent to equality on the reals, is Borel equivalent to "countable sets of reals," or is Borel complete. Furthermore, each of these cases corresponds in a natural way to a count of models of all sizes, up to back-and-forth equivalence. All these cases are possible and we compute precise model-theoretic conditions indicated which case occurs. This complements work on o-minimal theories where analogous results were shown. A large portion of the machinery under the proof is based on work by Matatyahu Rubin in "Theories of Linear Order," where it was shown that such a theory satisfies Vaught's Conjecture and cannot have precisely $\aleph_0$ countable models.

Definable one dimensional structures in o-minimal theories

2010-10-31

Assaf Hasson , Alf Onshuus , Ya’acov Peterzil

On expansions of models of weakly o-minimal theories by binary predicates

2019-05-20

S. S. Baĭzhanov , B. Sh. Kulpeshov

Here questions of preservation of properties at expanding countably categorical weakly o-minimal structures non-being 1-indiscernible by an arbitrary binary predicate are studied.A criterion for preserving the countable categoricity of a … Here questions of preservation of properties at expanding countably categorical weakly o-minimal structures non-being 1-indiscernible by an arbitrary binary predicate are studied.A criterion for preserving the countable categoricity of a weakly o-minimal expansion of convexity rank 1 is obtained.

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Weakly o-minimal structures and some of their properties

1998-12-01

B. Sh. Kulpeshov

Abstract The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we … Abstract The main result of this paper is Theorem 3.1 which is a criterion for weak o-minimality of a linearly ordered structure in terms of realizations of 1-types. Here we also prove some other properties of weakly o-minimal structures. In particular, we characterize all weakly o-minimal linear orderings in the signature {<, =} . Moreover, we present a criterion for density of isolated types of a weakly o-minimal theory. Lastly, at the end of the paper we present some remarks on the Exchange Principle for algebraic closure in a weakly o-minimal structure.

On minimal ordered structures

2005-01-01

Predrag Tanović

We partially describe minimal, first-order structures which have a strong form of the strict order property. We partially describe minimal, first-order structures which have a strong form of the strict order property.

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Definability in classes of finite structures

2011-03-10

Dugald Macpherson , Charles Steinhorn

This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from … This paper provides an overview of recent work by the authors and others on two topics in the model theory of finite structures. The point of view here differs from that usually associated with the term 'finite model theory', as presented for example in [21] or [46], in which the emphasis and motivation come primarily from computer science. Instead, the inspiration for this work has its origins in contemporary (infinite) model theoretic themes such as dimension, independence, and various measures of the complexity of definable sets. Each of the topics deals with classes of finite structures for first-order logic that are isolated by conditions that are drawn from these model-theoretic considerations. Moreover, in both cases, connections exist to areas in infinite model theory such as stability and simplicity theory, and o-minimality. This survey is intended for both mathematical logicians and computer scientists whose work focuses on logical aspects of the subject.

Variations of Rigidity for Ordered Theories

2024-01-01

B. Sh. Kulpeshov , S. V. Sudoplatov

One of the important characteristics of structures is degrees of semantic and syntactic rigidity, as well as indices of rigidity, showing how much the given structure differs from semantically rigid … One of the important characteristics of structures is degrees of semantic and syntactic rigidity, as well as indices of rigidity, showing how much the given structure differs from semantically rigid structures, i.e., structures with one-element automorphism groups, as well as syntactically rigid structures, i.e., structures covered by definable closure of the empty set. Issues of describing the degrees and indices of rigidity represents interest both in a general context and in relation to ordering theories and their models. In the given paper, we study possibilities for semantic and syntactic rigidity for ordered theories, i.e., the rigidity with respect to automorphism group and with respect to definable closure. We describe values for indices and degrees of semantic and syntactic rigidity for well-ordered sets, for discrete, dense, and mixed orders and for countable models of $\aleph_0$-categorical weakly o-minimal theories. All possibilities for degrees of rigidity for countable linear orderings are described.

Unstable structures definable in o-minimal theories

2007-04-29

Assaf Hasson , Alf Onshuus

Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the … Let M be an o-minimal structure with elimination of imaginaries, N an unstable structure definable in M. Then there exists X, interpretable in N, such that X with all the structure induced from N is o-minimal. In particular X is linearly ordered. As part of the proof we show: Theorem 1: If the M-dimenson of N is 1 then any 1-N-type is either strongly stable or finite by o-minimal. Theorem 2: If N is N-minimal then it is 1-M-dimensional.

Unstable structures definable in o-minimal theories

2007-01-01

Assaf Hasson , Alf Onshuus

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Definable one-dimensional topologies in O-minimal structures

2019-05-31

Ya’acov Peterzil , Ayala Rosel

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Definable one dimensional topologies in o-minimal structures

2018-07-21

Ya’acov Peterzil , Ayala Rosel

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable … We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $\left(X,\tau\right)$ is definably homeomorphic to an affine definable space (namely, a definable subset of $M^{n}$ with the induced subspace topology). One of the main results says that it is sufficient for $X$ to be regular and decompose into finitely many definably connected components.

The Marker–Steinhorn Theorem via Definable Linear Orders

2019-09-06

Erik Walsberg

We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures. We give a short proof of the Marker–Steinhorn theorem for o-minimal expansions of ordered groups. The key tool is Ramakrishnan’s classification of definable linear orders in such structures.

Definably extending partial orders in totally ordered structures

2014-05-01

Janak Ramakrishnan , Charles Steinhorn

We show, for various classes of totally ordered structures , including o‐minimal and weakly o‐minimal structures, that every definable partial order on a subset of extends definably in to a … We show, for various classes of totally ordered structures , including o‐minimal and weakly o‐minimal structures, that every definable partial order on a subset of extends definably in to a total order. This extends the result proved in for and o‐minimal.

Does weak quasi-o-minimality behave better than weak o-minimality?

2021-05-29

Slavko Moconja , Predrag Tanović

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

2012-10-10

Janak Ramakrishnan

We classify definable linear orders in o-minimal structures expanding groups. For example, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> … We classify definable linear orders in o-minimal structures expanding groups. For example, let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(P,\prec )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a linear order definable in the real field. Then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper P comma precedes right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mo>≺</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(P,\prec )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> embeds definably in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-struck upper R Superscript n plus 1 Baseline comma greater-than Subscript lex Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>lex</mml:mtext> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\mathbb {R}^{n+1},>_{\text {lex}})</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than Subscript lex Baseline"> <mml:semantics> <mml:msub> <mml:mo>></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext>lex</mml:mtext> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">>_{\text {lex}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the lexicographic order and <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> is the o-minimal dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This improves a result of Onshuus and Steinhorn in the o-minimal group context.

A Simple Proof of the Marker-Steinhorn Theorem for Expansions of Ordered Abelian Groups

2013-09-23

Erik Walsberg

We give a short and self-contained proof of the Marker-Steinhorn Theorem for o-minimal expansions of ordered groups, based on an analysis of linear orders definable in such structures. We give a short and self-contained proof of the Marker-Steinhorn Theorem for o-minimal expansions of ordered groups, based on an analysis of linear orders definable in such structures.

The Marker–Steinhorn Theorem

2025-01-01

Pablo Andújar Guerrero

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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Forking and independence in o-minimal theories

2004-03-01

Alfred Dolich

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 … 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.

An ordered structure of rank two related to Dulac's Problem

2008-01-01

Alf Dolich , Patrick Speissegger

For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a … For a vector field $\xi$ on $\mathbb{R}^2$ we construct, under certain assumptions on $\xi$, an ordered model-theoretic structure associated to the flow of $\xi$. We do this in such a way that the set of all limit cycles of $\xi$ is represente

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Characterizing rosy theories

2007-09-01

Clifton Ealy , Alf Onshuus

Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences … Abstract We examine several conditions, either the existence of a rank or a particular property of þ-forking that suggest the existence of a well-behaved independence relation, and determine the consequences of each of these conditions towards the rosiness of the theory. In particular we show that the existence of an ordinal valued equivalence relation rank is a (necessary and) sufficient condition for rosiness.

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.

Definable types in -minimal theories

1994-03-01

David Marker , Charles Steinhorn

Let L be a first order language. If M is an L -structure, let L M be the expansion of L obtained by adding constants for the elements of M … Let L be a first order language. If M is an L -structure, let L M be the expansion of L obtained by adding constants for the elements of M . Definition. A type is definable if and only if for any L -formula , there is an L M -formula so that for all iff M ⊨ d θ(¯). The formula d θ is called the definition of θ. Definable types play a central role in stability theory and have also proven useful in the study of models of arithmetic. We also remark that it is well known and easy to see that for M ≺ N , the property that every M -type realized in N is definable is equivalent to N being a conservative extension of M , where Definition. If M ≺ N , we say that N is a conservative extension of M if for any n and any L N -definable S ⊂ N n , S ∩ M n is L M -definable in M . Van den Dries [Dl] studied definable types over real closed fields and proved the following result. 0.1 (van den Dries), (i) Every type over ( R , +, -,0,1) is definable . (ii) Let F and K be real closed fields and F ⊂ K. Then, the following are equivalent : (a) Every element of K that is bounded in absolute value by an element of F is infinitely close (in the sense of F) to an element of F . (b) K is a conservative extension of F .

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.

Geometric Stability Theory

1996-09-12

Anand Pillay

Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine … Abstract This book gives an account of the fundamental results in geometric stability theory, a subject that has grown out of categoricity and classification theory. This approach studies the fine structure of models of stable theories, using the geometry of forking; this often achieves global results relevant to classification theory. Topics range from Zilber-Cherlin classification of infinite locally finite homogenous geometries, to regular types, their geometries, and their role in superstable theories. The structure and existence of definable groups is featured prominently, as is work by Hrushovski. The book is unique in the range and depth of material covered and will be invaluable to anyone interested in modern model theory.