Characterizing rosy theories

Authors

Clifton Ealy, Alf Onshuus

Type: Article

Publication Date: 2007-09-01

Citations: 50

DOI: https://doi.org/10.2178/jsl/1191333848

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Abstract

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.

Locations

Journal of Symbolic Logic
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2017-06-01

Gabriel Conant

Abstract We use axioms of abstract ternary relations to define the notion of a free amalgamation theory . These form a subclass of first-order theories, without the strict order property, … Abstract We use axioms of abstract ternary relations to define the notion of a free amalgamation theory . These form a subclass of first-order theories, without the strict order property, encompassing many prominent examples of countable structures in relational languages, in which the class of algebraically closed substructures is closed under free amalgamation. We show that any free amalgamation theory has elimination of hyperimaginaries and weak elimination of imaginaries. With this result, we use several families of well-known homogeneous structures to give new examples of rosy theories. We then prove that, for free amalgamation theories, simplicity coincides with NTP 2 and, assuming modularity, with NSOP 3 as well. We also show that any simple free amalgamation theory is 1-based. Finally, we prove a combinatorial characterization of simplicity for Fraïssé limits with free amalgamation, which provides new context for the fact that the generic K n -free graphs are SOP 3 , while the higher arity generic $K_n^r$ -free r -hypergraphs are simple.

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Expansiones no rosaceas del cuerpo de los reales

2009-01-01

Walther Muete

Estudiamos las estructuras (R,+, ·, 0, 1, S), donde 2Z es el subgrupo multiplicativo de los n´umeros reales positivos que consiste en las potencias enteras de 2 y S es … Estudiamos las estructuras (R,+, ·, 0, 1, S), donde 2Z es el subgrupo multiplicativo de los n´umeros reales positivos que consiste en las potencias enteras de 2 y S es el conjunto {(et cos(t), etsen(t)) | t 2 R}, y demostraremos que no son rosaceas. En general demostraremos que cualquier expansion del cuerpo de los numeros (R,+, ·, 0, 1,

Fields interpretable in superrosy groups with NIP (the non-solvable case)

2010-01-25

Krzysztof Krupiński

Abstract Let G be a group definable in a monster model of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U b … Abstract Let G be a group definable in a monster model of a rosy theory satisfying NIP. Assume that G has hereditarily finitely satisfiable generics and 1 < U b ( G ) < ∞. We prove that if G acts definably on a definable set of U р -rank 1, then, under some general assumption about this action, there is an infinite field interpretable in . We conclude that if G is not solvable-by-finite and it acts faithfully and definably on a definable set of U р -rank 1, then there is an infinite field interpretable in . As an immediate consequence, we get that if G has a definable subgroup H such that U р ( G ) = U р ( H ) + 1 and G /⋂ g ∈ G H g is not solvable-by-finite, then an infinite field interpretable in also exists.

Thorn-forking in continuous logic

2012-01-20

Clifton Ealy , Isaac Goldbring

Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example … Abstract We study thorn forking and rosiness in the context of continuous logic. We prove that the Urysohn sphere is rosy (with respect to finitary imaginaries), providing the first example of an essentially continuous unstable theory with a nice notion of independence. In the process, we show that a real rosy theory which has weak elimination of finitary imaginaries is rosy with respect to finitary imaginaries, a fact which is new even for discrete first-order real rosy theories.

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References (8)

Dense embeddings I: A theorem of Robinson in a general setting

1975-01-01

Angus Macintyre

Dimension of definable sets, algebraic boundedness and Henselian fields

1989-12-01

Lou van den Dries

Stability in geometric theories

2004-12-15

Jerry Gagelman

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].

Groups definable in local fields and pseudo-finite fields

1994-02-01

Ehud Hrushovski , Anand Pillay

Perfect pseudo-algebraically closed fields are algebraically bounded

2003-11-21

Zoé Chatzidakis , Ehud Hrushovski

Properties and consequences of Thorn-independence

2006-03-01

Alf Onshuus

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.