Group configurations and germs in simple theories

Ben-Yaacov, Itay

doi:10.2178/jsl/1190150301

Cited by 5 publications

(9 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hence t ∈ dcl(t, uu ) ; as t | uu by Lemma 2.6.1, we see that Func(π) is a set of generic functions on p (i.e. π is a well-defined generic action on p in the terminology of [Ben02]). We know that π is closed under inverse and generic composition; in a stable theory, the reduction of a well-defined generic action is well-defined (i.e.…”

Section: Almost Orthogonalitymentioning

confidence: 92%

“…Getting a poly-chunk. We shall assume familiarity with the theory of generic actions, as developed in [Ben02]. We recall a few of the notions that are used below:…”

Section: Almost Orthogonalitymentioning

confidence: 99%

“…In the second part of the paper we shall give a construction corresponding to the third and weakest condition, and therefore to the largest group (or in fact, polygroup). In fact we shall construct a generic poly-chunk multi-acting on p in the sense of [Ben02]. This allows us to invoke the machinery of [BTW,Ben03,TW01] to obtain a coreless almost A-hyperdefinable polygroup, or, over some additional parameters, an almost hyperdefinable group acting transitively on an almost hyperdefinable set X, whose generic elements are interbounded (over independent parameters) with realizations of p. If the theory was stable to start with, the group obtained will be the original binding group.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On almost orthogonality in simple theories

Ben-Yaacov¹,

Wagner²

2004

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Almost Orthogonalitymentioning

confidence: 92%

“…Getting a poly-chunk. We shall assume familiarity with the theory of generic actions, as developed in [Ben02]. We recall a few of the notions that are used below:…”

Section: Almost Orthogonalitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On almost orthogonality in simple theories

Ben-Yaacov¹,

Wagner²

2004

J. symb. log.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Moreover, many unrelated results which were proved in the context of first order theories can be seen, upon careful reading, to use only thickness assumption, and therefore hold in any thick simple cat. One example for this are the series of articles on group constructions in simple theories [1,15,3,9]. Another is the theory of lovely pairs discussed below.…”

Section: Tarski-vaught Propertymentioning

confidence: 99%

Compactness and Independence in Non First Order Frameworks

Ben-Yaacov

2005

Bull. symb. log.

Self Cite

View full text Add to dashboard Cite

Abstract. This communication deals with positive model theory, a non first order model theoretic setting which preserves compactness at the cost of giving up negation. Positive model theory deals transparently with hyperimaginaries, and accommodates various analytic structures which defy direct first order treatment. We describe the development of simplicity theory in this setting, and an application to the lovely pairs of models of simple theories without the weak non finite cover property. BackgroundEvery first order theory admitting quantifier elimination (and up to a change of language, we may always assume that it does) is the model companion of a universal theory (in fact, its model completion). The converse is known to be false, namely, not every universal theory even has a first order model companion: it has one if and only if the class of its existentially closed (e.c.) models is elementary. Nevertheless, it was observed by more than one person that much of first order model theory can be repeated in the class of e.c. models of a universal theory T , whether this class is elementary or not; how much of classical model theory can be repeated may depend on additional conditions one imposes on T . This can be viewed as studying the class of models of the model companion of T , even though such may not exist as a first order theory.A universal theory has a model completion if and only if it first has a model companion, and second, the model companion eliminates quantifiers. The second property can be re-stated in a way which does not depend on the first: a universal theory having this property is called a Robinson theory. Robinson theories were defined (with a little more generality) and studied by Hrushovski in [Hru97]. From a model-theoretic point of view, the class of e.c. models of a Robinson theory behaves very much like the class of models of a first order theory with quantifier elimination, and thus the framework of Robinson theories is just a small step outside the scope of first order theories. Simplicity and stability theory extend to Robinson theories in an obvious and straightforward manner. Robinson theories were used, among other things, to provide an example of a simple theory where the Lascar strong type differs from that of Shelah (the existence of a first order theory with this property is still open).

show abstract

“…[1, Theorem 3.9] Let be a generic multi-chunk, and P = Germ( ). Then composition • = induces a hyperdefinable multifunction * : P × P → P, which is defined up to a bounded non-zero number of possible values, satisfying 1.…”

mentioning

confidence: 99%

The Group Configuration in Simple Theories and its Applications

Ben-Yaacov¹,

Tomašić²,

Wagner³

2002

Bull. symb. log.

Self Cite

View full text Add to dashboard Cite

In recent work, the authors have established the group configuration theorem for simple theories, as well as some of its main applications from geometric stability theory, such as the binding group theorem, or, in the ω-categorical case, the characterization of the forking geometry of a finitely based non-trivial locally modular regular type as projective geometry over a finite field and the equivalence of pseudolinearity and local modularity.The proof necessitated an extension of the model-theoretic framework to include almost hyperimaginaries, and the study of polygroups.

show abstract

Group configurations and germs in simple theories

Cited by 5 publications

References 12 publications

On almost orthogonality in simple theories

On almost orthogonality in simple theories

Compactness and Independence in Non First Order Frameworks

The Group Configuration in Simple Theories and its Applications

Contact Info

Product

Resources

About