Uncountably categorical local tame abstract elementary classes with disjoint amalgamation

We prove Baldwin-Lachlan theorem for local (LS(K)-)tame abstract elementary classes K with disjoint amalgamation property and with LS(K)=ω. Partially supported by the Academy of Finland, grant 40734.     

Quasiminimal abstract elementary classes

We propose the notion of a quasiminimal abstract elementary class (AEC). This is an AEC satisfying four semantic conditions: countable Löwenheim–Skolem–Tarski number, existence of a prime model, closure under intersections, and uniqueness of the generic orbital type over every countable model. We exhibit a correspondence between Zilber’s quasiminimal pregeometry classes and quasiminimal AECs: any quasiminimal pregeometry class induces a quasiminimal AEC (this was known), and for any quasiminimal AEC there is a natural functorial expansion that induces a quasiminimal pregeometry class. We show in particular that the exchange axiom is redundant in Zilber’s definition of a quasiminimal pregeometry class.     

Category-theoretic aspects of abstract elementary classes

We highlight connections between accessible categories and abstract elementary classes (AECs), and provide a dictionary for translating properties and results between the two contexts. We also illustrate a few applications of purely category-theoretic methods to the study of AECs, with model-theoretically novel results. In particular, the category-theoretic approach yields two surprising consequences: a structure theorem for categorical AECs, and a partial stability spectrum for weakly tame AECs.     

Shelah's eventual categoricity conjecture in tame abstract elementary classes with primes

A new case of Shelah's eventual categoricity conjecture is established:

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A topology for galois types in abstract elementary classes

We present a way of topologizing sets of Galois types over structures in abstract 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 model‐theoretic 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. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Building prime models in fully good abstract elementary classes

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well‐behaved independence relation: Let be an almost fully good AEC that is categorical in and has the ‐existence property for domination triples. For any , the class of Galois saturated models of of size λ has prime models over every set of the form . This generalizes an argument of Shelah, who proved the result when λ is a successor cardinal.     

On the uniqueness property of forking in abstract elementary classes

In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent.     

Limit models in metric abstract elementary classes: the categorical case

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Lattice varieties with non-elementary amalgamation classes

In this paper it is shown that under certain general conditions, if the amalgamation class of a lattice variety contains a member which does not have a 2-congruence, then the amalgamation class is not closed under ultrapowers or direct products. In particular, the amalgamation class is not first order axiomatizable. Received October 1, 1997; accepted in final form Feburary 26 1999.     

Uniqueness of limit models in classes with amalgamation

We prove the following main theorem: Let be an abstract elementary class satisfying the joint embedding and the amalgamation properties with no maximal models of cardinality μ. Let μ be a cardinal above the the Löwenheim‐Skolem number of the class. If is μ‐Galois‐stable, has no μ‐Vaughtian Pairs, does not have long splitting chains, and satisfies locality of splitting, then any two ‐limits over M, for , are isomorphic over M.     

A presentation theorem for continuous logic and metric abstract elementary classes

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Around independence and domination in metric abstract elementary classes: assuming uniqueness of limit models

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Genericity and amalgamation of classes of Banach spaces

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Symmetry classes of tensors associated with certain groups

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Computable axiomatizability of elementary classes

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