Properties and consequences of Thorn-independence

Authors

Alf Onshuus

Type: Article

Publication Date: 2006-03-01

Citations: 61

DOI: https://doi.org/10.2178/jsl/1140641160

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Abstract

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.

Locations

Journal of Symbolic Logic
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Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to … Abstract We discuss the role of weakly normal formulas in the theory of thorn forking, as part of a commentary on the paper [5]. We also give a counterexample to Corollary 4.2 from that paper, and in the process discuss “theories with selectors.”

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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 < 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".

Expansions of o-minimal structures by dense independent sets

2016-04-29

Alfred Dolich , Chris Miller , Charles Steinhorn

NIP ω$\omega$‐categorical structures: The rank 1 case

2022-10-03

Pierre Simon

$\mathbb{Q}$ACFA

2015-08-25

Alice Medvedev

We show that many nice properties of a theory $T$ follow from the corresponding properties of its reducts to finite subsignatures. If $\{ T_i \}_{i \in I}$ is a directed … We show that many nice properties of a theory $T$ follow from the corresponding properties of its reducts to finite subsignatures. If $\{ T_i \}_{i \in I}$ is a directed family of conservative expansions of first-order theories and each $T_i$ is stable (respectively, simple, rosy, dependent, submodel complete, model complete, companionable), then so is the union $T := \cup_i T_i$. In most cases, (thorn)-forking in $T$ is equivalent to (thorn)-forking of algebraic closures in some $T_i$. This applies to fields with an action by $(\mathbb{Q}, +)$, whose reducts to finite subsignatures are interdefinable with the theory of fields with one automorphism. We show that the model companion $\mathbb{Q}$ACFA of this theory is strictly simple and has the same level of quantifier elimination and the same algebraic characterization of algebraic closure and forking independence as ACFA. The lattice of the fixed fields of the named automorphisms breaks supersimplicity in $\mathbb{Q}$ACFA, but away from these we find many (weakly) minimal formulas.

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Independence in randomizations

2018-12-02

Uri Andrews , Isaac Goldbring , H. Jerome Keisler

The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: … The randomization of a complete first-order theory [Formula: see text] is the complete continuous theory [Formula: see text] with two sorts, a sort for random elements of models of [Formula: see text] and a sort for events in an underlying atomless probability space. We study independence relations and related ternary relations on the randomization of [Formula: see text]. We show that if [Formula: see text] has the exchange property and [Formula: see text], then [Formula: see text] has a strict independence relation in the home sort, and hence is real rosy. In particular, if [Formula: see text] is o-minimal, then [Formula: see text] is real rosy.

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Stable embeddedness and NIP

2010-01-04

Anand Pillay

We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has finite rank, P has NIP in … We give sufficient conditions for a predicate P in a complete theory T to be stably embedded: P with its induced 0-definable structure has finite rank, P has NIP in T and P is 1-stably embedded. This generalizes recent work by Hasson and Onshuus in the case where P is o-minimal in T.

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Expansions which introduce no new open sets

2010-02-19

Gareth Boxall , Philipp Hieronymi

We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This … We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in practice are often easy to check. We give a further characterisation in the special case of an expansion by a generic predicate.

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Independence relations in randomizations

2014-01-01

Uri Andrews , Isaac Goldbring , H. Jerome Keisler

The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort … The randomization of a complete first order theory $T$ is the complete continuous theory $T^R$ with two sorts, a sort for random elements of models of $T$, and a sort for events in an underlying probability space. We study various notions of independence in models of $T^R$.

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NIP omega-categorical structures: the rank 1 case

2018-07-18

Pierre Simon

We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures … We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures.

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References (4)

Coordinatisation and canonical bases in simple theories

2000-03-01

Bradd Hart , Byunghan Kim , Anand Pillay

In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical … In this paper we discuss several generalization of theorems from stability theory to simple theories. Cherlin and Hrushovski, in [2] develop a substitute for canonical bases in finite rank, ω-categorical supersimple theories. Motivated by methods there, we prove the existence of canonical bases (in a suitable sense) for types in any simple theory. This is done in Section 2. In general these canonical bases will (as far as we know) exist only as “hyperimaginaries”, namely objects of the form a / E where a is a possibly infinite tuple and E a type-definable equivalence relation. (In the supersimple, ω-categorical case, these reduce to ordinary imaginaries.) So in Section 1 we develop the general theory of hyperimaginaries and show how first order model theory (including the theory of forking) generalises to hyperimaginaries. We go on, in Section 3 to show the existence and ubiquity of regular types in supersimple theories, ω-categorical simple structures and modularity is discussed in Section 4. It is also shown here how the general machinery of simplicity simplifies some of the general theory of smoothly approximable (or Lie-coordinatizable) structures from [2]. Throughout this paper we will work in a large, saturated model M of a complete theory T . All types, sets and sequences will have size smaller than the size of M . We will assume that the reader is familiar with the basics of forking in simple theories as laid out in [4] and [6]. For basic stability-theoretic results concerning regular types, orthogonality etc., see [1] or [9].

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.

Supersimple theories

2000-09-20

Steven Buechler , Anand Pillay , Frank Olaf Wagner

We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined … We prove elimination of hyperimaginaries in supersimple theories. This means that if an equivalence relation on the set of realisations of a complete type (in a supersimple theory) is defined by a possibly infinite conjunction of first order formulas, then it is the intersection of definable equivalence relations.