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Categoricity and U-rank in excellent classes. (English) Zbl 1055.03025

Summary: Let \(\mathcal K\) be the class of atomic models of a countable first-order theory. We prove that if \(\mathcal K\) is excellent and categorical in some uncountable cardinal, then each model is prime and minimal over the basis of a definable pregeometry given by a quasiminimal set. This implies that \(\mathcal K\) is categorical in all uncountable cardinals. We also introduce a U-rank to measure the complexity of complete types over models. We prove that the U-rank has the usual additivity properties, that quasiminimal types have U-rank 1, and that the U-rank of any type is finite in the uncountably categorical, excellent case. However, in contrast to the first-order case, the supremum of the U-rank over all types may be \(\omega\) (and is not achieved). We illustrate the theory with the example of free groups, and Zilber’s pseudo-analytic structures.

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
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References:

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