December 2003 Categoricity and U-rank in excellent classes
Olivier Lessmann
J. Symbolic Logic 68(4): 1317-1336 (December 2003). DOI: 10.2178/jsl/1067620189


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


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Olivier Lessmann. "Categoricity and U-rank in excellent classes." J. Symbolic Logic 68 (4) 1317 - 1336, December 2003.


Published: December 2003
First available in Project Euclid: 31 October 2003

zbMATH: 1055.03025
MathSciNet: MR2017357
Digital Object Identifier: 10.2178/jsl/1067620189

Rights: Copyright © 2003 Association for Symbolic Logic


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Vol.68 • No. 4 • December 2003
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