Results 1 
4 of
4
Independence in finitary abstract elementary classes
 Ann. Pure Appl. Logic
, 2006
"... In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order nonforking over complete types. Our approach generalizes the context of ℵ0stable ho ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order nonforking over complete types. Our approach generalizes the context of ℵ0stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over ∅, LöwenheimSkolem number being ω, LS(K)tameness and a property we call finite character. We also start the studies of these classes from the ℵ0stable case. Stability in ℵ0 and LS(K)tameness can be replaced by categoricity above the Hanf number. Finite character is the main novelty of this paper. Almost all examples of AEC’s have this property and it allows us to use weak types, as we call them, in place of Galois types. 1
Generalized amalgamation in simple theories and characterization of dependence in nonelementary classes
, 2004
"... We examine the properties of dependence relations in certain nonelementary classes and firstorder simple theories. There are two major parts. The goal of the first part is to identify the properties of dependence relations in certain nonelementary classes that, firstly, characterize the modelthe ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We examine the properties of dependence relations in certain nonelementary classes and firstorder simple theories. There are two major parts. The goal of the first part is to identify the properties of dependence relations in certain nonelementary classes that, firstly, characterize the modeltheoretic properties of those classes; and secondly, allow to uniquely describe an abstract dependence itself in a very concrete way. I investigate totally transcendental atomic models and finite diagrams, stable finite diagrams, and a subclass of simple homogeneous models from this point of view. The second part deals with simple firstorder theories. The main topic of this part is investigation of generalized amalgamation properties for simple theories. Namely, we are trying to answer the question of when does a simple theory have the property of ndimensional amalgamation, where 2dimensional amalgamation is the Independence theorem for simple theories. We develop the notion of nsimplicity and strong nsimplicity for 1 ≤ n ≤ ω, where both “1simple ” and “strongly 1simple ” is the same as “simple. ” We present examples of simple unstable theories in each subclass and prove a characteristic property of nsimplicity in terms of ndividing, a strengthening of the dependence relation called dividing in simple theories. We prove 3dimensional amalgamation property for 2simple theories, and, under an additional assumption, a strong (n + 1)dimensional amalgamation property for strongly nsimple theories. Stable theories are strongly ωsimple, and the idea behind developing extra simplicity conditions is to show that, for instance, ωsimple theories are almost as nice as stable theories. The third part of the thesis contains an application of ωsimplicity to construct a Morley sequence without the construction of a long independent sequence. ii
A Topology for Galois Types in AECs
, 906
"... elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the modeltheoretic properties of classes and their constituent models and the topologica ..."
Abstract
 Add to MetaCart
elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the modeltheoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. 1