Related papers: Abstract elementary classes stable in $\aleph_0$
This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the non-elementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of…
We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming…
We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a…
This paper continues the study of superstability in abstract elementary classes (AECs) satisfying the amalgamation property. In particular, we consider the definition of $\mu$-superstability which is based on the local character…
In this paper, we examine the locality condition for non-splitting and determine the level of uniqueness of limit models that can be recovered in some stable, but not superstable, abstract elementary classes. In particular we prove (note…
We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality $\lambda$ and a superstable-like forking notion for models of cardinality…
There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that \emph{stability} is enough to…
We study the class of acts with embeddings as an abstract elementary class. We show that the class is always stable and show that superstability in the class is characterized algebraically via weakly noetherian monoids. The study of these…
Given a cover $\mathbb{U}$ of a family of smooth complex algebraic varieties, we associate with it a class $\mathcal{U},$ containing $\mathbb{U}$, of structures locally definable in an o-minimal expansion of the reals. We prove that the…
In this paper we address a problem posed by Shelah in 1999 to find a suitable notion for superstability for abstract elementary classes in which limit models of cardinality $\mu$ are saturated. Theorem 1. Suppose that $\mathcal{K}$ is a…
For an abstract elementary class $\mathbf{K}$ and a cardinal $\lambda \geq LS(\mathbf{K})$, we prove under mild cardinal arithmetic assumptions, categoricity in two succesive cardinals, almost stability for $\lambda^+$-minimal types and…
The assumption that an AEC is tame is a powerful assumption permitting development of stability theory for AECs with the amalgamation property. Lately several upward categoricity theorems were discovered where tameness replaces strong…
Tame abstract elementary classes are a broad nonelementary framework for model theory that encompasses several examples of interest. In recent years, progress toward developing a classification theory for them have been made. Abstract…
In the framework of graphs, we study abstract elementary classes (aecs). In this work we analyze several properties of Forb(G) and versions of Forb-Con(G) in the context of aecs and we present some examples of classes of graphs which…
Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called…
We show how to build primes models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: $\mathbf{Theorem.}$ Let $K$ be an almost fully good AEC that is categorical in $\text{LS}…
We prove the uniqueness of high cofinality limit models in stable abstract elementary classes (AECs) with amalgamation, assuming the existence of a rather weak independence relation. $\textbf{Theorem.}$ Suppose $\mathbf{K}$ is a…
We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper we explore stability results in this context. We…
$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $\lambda > \text{LS} (K)$. If $K$ is categorical in $\lambda$, then the model of cardinality $\lambda$ is Galois-saturated. This…
The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah's Categoricity Conjecture in this…