Related papers: Uniqueness of Limit Models in Classes with Amalgam…

In S. 1 we deal with amalgamation bases, e.g., we define when an a.e.c. $k$ has $(\lambda,\kappa)$-amalgamation which means "many" M in $K^k_\lambda$ are amalgamation bases. We then consider what happens for the class of lf groups. In S. 2…

We study amalgamation properties in a family of abstract elementary classes that we call coloring classes. The family includes the examples previously studied in previous work of Baldwin, Kolesnikov, and Shelah. We establish that the…

Malliaris and Shelah famously proved that Keisler's order $\trianglelefteq$ has infinitely many classes. In more detail, for each $2 \leq k < n < \omega$, let $T_{n, k}$ be the theory of the random $k$-ary $n$-clique free hypergraph.…

For a fixed natural number $n \geq 1$, the Hart-Shelah example is an abstract elementary class (AEC) with amalgamation that is categorical exactly in the infinite cardinals less than or equal to $\aleph_n$. We investigate recently-isolated…

Let K be an Abstract Elementary Class. Under the asusmptions that K has a nicely behaved forking-like notion, regular types and existence of some prime models we establish a decomposition theorem for such classes. The decomposition implies…

We study abstract elementary classes (AECs) that, in $\aleph_0$, have amalgamation, joint embedding, no maximal models and are stable (in terms of the number of orbital types). Assuming a locality property for types, we prove that such…

Two automorphisms of a simple stable AF algebra with a finite dimensional lattice of lower semicontinuous traces are shown to be outer conjugate if they act in the same way on the K-group and the extremal traces are scaled by numbers which…

The idea of this paper is to explore the existence of canonical countably saturated models for different classes of structures. It is well-known that, under CH, there exists a unique countably saturated linear order of cardinality…

We provide here the first steps toward Classification Theory of Abstract Elementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some lambda greater than its Lowenheim-Skolem…

The best-known version of Shelah's celebrated singular cardinal compactness theorem states that if the cardinality of an abelian group is singular, and all its subgroups of lesser cardinality are free, then the group itself is free. The…

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for $\mu$-forking, a certain independence notion arising from splitting. This had been a longstanding technical…

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 introduce the notion of a `pure` Abstract Elementary Class to block trivial counterexamples. We study classes of models of bipartite graphs and show: Main Theorem (cf. Theorem 3.5.2 and Corollary 3.5.6): If $(\lambda_i : i \le…

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…

The disjoint amalgamation property (DAP), which asserts that all spans of a class of models can be amalgamated with minimal intersection, is an important property in the context of abstract elementary classes, with connections to both…

We show that $\beth_{(2^{\operatorname{LS}({\bf K})})^+}$ is the lower bound to the Hanf numbers for the length of the order property and for stability in stable abstract elementary classes (AECs). Our examples satisfy the joint embedding…

We try to build, provably in ZFC, for a first order T a model in which any isomorphism between two Boolean algebras is definable. The problem, compared to [Sh:384], is with pseudo-finite Boolean algebras. A side benefit is that we do not…

Keisler proved that if $\theta$ is a strong limit cardinal and $\lambda$ is a singular cardinal, then the transfer relation $\theta\longrightarrow\lambda$ holds. In a previous paper, we studied initial elementary submodels of the…

We prove that if two homomorphisms from O_{\infty} to a purely infinite simple C*-algebra have the same class in KK-theory, and if either both are unital or both are nonunital, then they are approximately unitarily equivalent. It follows…

We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^{\omega}/ \mathcal{U} \simeq \mathbf{N}^\omega / \mathcal{U}$ for some ultrafilter…