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Categoricity and U-rank in excellent classes

Published online by Cambridge University Press:  12 March 2014

Olivier Lessmann
Affiliation:
Mathematical Institute, Oxford University, Oxford, OX1 3LB, England, E-mail: lessmann@maths.ox.ac.uk
Corresponding
E-mail address:

Abstract

Let be the class of atomic models of a countable first order theory. We prove that if 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 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 ω (and is not achieved). We illustrate the theory with the example of free groups, and Zilber's pseudo analytic structures.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2003

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