Tapani Hyttinen

Independence in finitary abstract elementary classes

Abstract 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 non-forking over complete types. Our approach generalizes the context of ℵ 0 -stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over 0 , Lowenheim–Skolem number being ω , LS ( K ) -tameness and a property we call finite character. We also start the studies of these classes from the ℵ 0 -stable case. Stability in ℵ 0 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.

Strong splitting in stable homogeneous models

Abstract In this paper we study elementary submodels of a stable homogeneous structure. We improve the independence relation defined in Hyttinen (Fund. Math. 156 (1998) 167–182). We apply this to prove a structure theorem. We also show that dop and sdop are essentially equivalent, where the negation of dop is the property we use in our structure theorem and sdop implies nonstructure, see Hyttinen (1998).

Constructing strongly equivalent nonisomorphic models for unstable theories

If T is an unstable theory of cardinality <λ or countable stable theory with OTOP or countable superstable theory with DOP, λω λω1 in the superstable with DOP case) is regular and λ<λ=λ, then we construct for T strongly equivalent nonisomorphic models of cardinality λ. This can be viewed as a strong nonstructure theorem for such theories. We also consider the case when T is unsuperstable and develop further a result of Shelah about the existence of L∞,λ-equivalent nonisomorphic models for such T. In addition, we show that a natural analogue of Scotts isomorphism theorem fails for models of power κ, if κω is regular, assuming κ<κ=κ.

Quasiminimal structures and excellence

We show that the excellence axiom in the definition of Zilbers quasiminimal excellent classes is redundant, in that it follows from the other axioms. This substantially simplifies a number of categoricity proofs.

Categoricity in homogeneous complete metric spaces

We introduce a new approach to the model theory of metric structures by defining the notion of a metric abstract elementary class (MAEC) closely resembling the notion of an abstract elementary class. Further we define the framework of a homogeneous MAEC were we additionally assume the existence of arbitrarily large models, joint embedding, amalgamation, homogeneity and a property which we call the perturbation property. We also assume that the Löwenheim-Skolem number, which in this setting refers to the density character of the set instead of the cardinality, is

Interpreting groups and fields in some nonelementary classes

{\aleph_0}

Quantum Team Logic and Bell’s Inequalities

. In these settings we prove an analogue of Morley’s categoricity transfer theorem. We also give concrete examples of homogeneous MAECs.

Constructing strongly equivalent nonisomorphic models for unsuperstable theories, part A

Abstract. We discuss the relationships between the notions of