This is a sequel to [1]. Here we give careful attention to the difficulties of calculating Morley and U-rank of the infinite rank ω-stable theories constructed by variants of Hrushovski's methods. Sample result: For every k < ω, there is an ω-stable expansion of any algebraically closed field which has Morley rank ω × k. We include a corrected proof of the lemma in [1] establishing that the generic model is ω-saturated in the rank 2 case.
We continue the work of Shelah and Casanovas on the cardinality of families of pairwise inconsistent types in simple theories. We prove that, in a simple theory, there are at most pairwise inconsistent types of size μ over a set of size λ. This bound improves the previous bounds and clarifies the role of κ(T). We also compute exactly the maximal cardinality of such families for countable, simple theories. The main tool is the fact that, in simple theories, the collection of nonforking extensions...
Dans une belle paire de modèles d’une théorie stable ayant élimination des imaginaires sans la propriété de recouvrement fini, tout groupe définissable se projette, à isogénie près, sur les points -rationnels d’un groupe définissable dans le réduit à paramètres dans . Le noyau de cette projection est un groupe définissable dans le réduit.Un groupe interprétable dans une paire de corps algébriquement clos où est une extension propre de est, à isogénie près, l’extension des points -rationnels...
We characterize the stable theories T for which the saturated models of T admit decompositions. In particular, we show that countable, shallow, stable theories with NDOP have this property.
The theorem of Ax says that any regular selfmapping of a complex algebraic variety is either surjective or non-injective; this property is called surjunctivity and investigated in the present paper in the category of proregular mappings of proalgebraic spaces. We show that such maps are surjunctive if they commute with sufficiently large automorphism groups. Of particular interest is the case of proalgebraic varieties over infinite graphs. The paper intends to bring out relations between model theory,...
We define an abstract setting suitable for investigating perturbations of metric structures generalizing the notion of a metric abstract elementary class. We show how perturbation of Hilbert spaces with an automorphism and atomic Nakano spaces with bounded exponent fit into this framework, where the perturbations are built into the definition of the class being investigated. Further, assuming homogeneity and some other properties true in the example classes, we develop a notion of independence for...
Originally, m-independence, ℳ -rank, m-stability and m-normality were defined only for small stable theories. Here we extend the definitions to an arbitrary small countable complete theory. Then we investigate these notions in the new, broader context. As a consequence we show that any superstable theory with countable models is m-normal. In particular, any *-algebraic group interpretable in such a theory is abelian-by-finite.
We give a model-theoretic interpretation of a result by Campana and Fujiki on the algebraicity of certain spaces of cycles on compact complex spaces. The model-theoretic interpretation is in the language of canonical bases, and says that if b,c are tuples in an elementary extension 𝓐* of the structure 𝓐 of compact complex manifolds, and b is the canonical base of tp(c/b), then tp(b/c) is internal to the sort (ℙ¹)*. The Zilber dichotomy in 𝓐* follows immediately (a type of U-rank 1 is locally...
By results of [9] there are models and for which the Ehrenfeucht-Fraïssé game of length ω₁, , is non-determined, but it is consistent relative to the consistency of a measurable cardinal that no such models have cardinality ≤ ℵ₂. We now improve the work of [9] in two ways. Firstly, we prove that the consistency strength of the statement “CH and is determined for all models and of cardinality ℵ₂” is that of a weakly compact cardinal. On the other hand, we show that if , T is a countable complete...