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Categoricity and U-rank in excellent classes

Published online by Cambridge University Press:  12 March 2014

Olivier Lessmann
Olivier Lessmann*
Affiliation:
Mathematical Institute, Oxford University, Oxford, OX1 3LB, England, E-mail: lessmann@maths.ox.ac.uk
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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
Information
The Journal of Symbolic Logic , Volume 68 , Issue 4 , December 2003 , pp. 1317 - 1336
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[BaLa] Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
[BCM] Baur, Walter, Cherlin, Gregory, and Macintyre, Angus, Totally categorical groups and rings, Journal of Algebra, vol. 57 (1979), pp. 407–440.CrossRefGoogle Scholar
[BuLe] Buechler, Steve and Lessmann, Olivier, Simple homogeneous models, Journal of the American Mathematical Society, vol. 16 (2003), no. 1, pp. 91–121.Google Scholar
[Gr1] Grossberg, Rami, A course on abstract elementary classes, Class notes from a course given at Carnegie Mellon University in 1997.Google Scholar
[Gr2] Grossberg, Rami, Classification theory for abstract elementary classes, Logic and algebra, Contemporary Mathematics, vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 165–204.Google Scholar
[GrHa] Grossberg, Rami and Hart, Bradd, The classification of excellent classes, this Journal, vol. 54 (1989), pp. 1359–1381.Google Scholar
[GrLe] Grossberg, Rami and Lessmann, Olivier, Shelah's stability spectrum and homogeneity spectrum in finite diagrams, Archive for Mathematical Logic, vol. 41 (2002), no. 1, pp. 1–31.Google Scholar
[Hy] Hyttinen, Tapani, Generalizing Morley's theorem, Mathematical Logic Quarterly, vol.44 (1998), pp. 176–184.Google Scholar
[HyLe] Hyttinen, Tapani and Lessmann, Olivier, A rank for the class of elementary submodels of a superstable homogeneous model, this Journal, vol. 67 (2002), no. 4, pp. 1469–1482.Google Scholar
[HyShl] Hyttinen, Tapani and Shelah, Saharon, Strong splitting in stable homogeneous models. Annuls of Pure and Applied Logic, vol. 103 (2000), pp. 201–228.Google Scholar
[HySh2] Hyttinen, Tapani, Main gap for locally saturated elementary submodels of a homogeneous structure, this Journal, vol. 66 (2001), no. 3, pp. 1286–1302.Google Scholar
[Ke] Keisler, H. Jerome, Model theory for infinitary logic, North-Holland Publishing Co., Amsterdam, 1971.Google Scholar
[Ko] Kolesnikov, Alexei, Dependence relations in nonelementary classes . Preprint.Google Scholar
[Le1] Lessmann, Olivier, Ranks and pregeometries in finite diagrams. Annals of Pure and Applied Logic, vol. 106 (2000), pp. 49–83.Google Scholar
[Le2] Lessmann, Olivier, Homogeneous model theory: existence and categoricity, Logic and algebra (Zhang, Yi, editor), Contemporary Mathematics, vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 149–164.Google Scholar
[Ma] Marcus, Leo, A minimal prime model with an infinite set of indiscernibles, Israel Journal of Mathematics, vol. 11 (1972), pp. 180–183.CrossRefGoogle Scholar
[Mo] Morley, Michael, Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.Google Scholar
[Sh600] Shelah, Saharon, Preprint.Google Scholar
[Sh576] Shelah, Saharon, Categoricity of an abstract elementary class in two successive cardinals, 115 pages, Preprint.Google Scholar
[Sh3] Shelah, Saharon, Finite diagrams stable in power, Annals of Mathematical Logic, vol. 2 (1970), pp. 69–118.Google Scholar
[Sh70] Shelah, Saharon, Solution to Łoš conjecture for uncountable languages, Notices of the American Mathematical Society, vol. 17 (1970), 968.Google Scholar
[Sh48] Shelah, Saharon, Categoricity in ℵ1 of sentences in L ω1ω (Q), Israel Journal of Mathematics, vol. 20 (1975), pp. 127–148.Google Scholar
[Sh54] Shelah, Saharon, The lazy model-theorist's guide to stability, Proceedings of a symposium in Louvain, March 1975 (Henrand, P., editor), vol. 18, Logique et Analyse, no. 71-72, 1975, pp. 241–308.Google Scholar
[Sh87a] Shelah, Saharon,Classification theory for nonelementary classes. I. The number of uncountable models of ψ ∈ L ω1ω , Part A, Israel Journal of Mathematics, vol. 46 (1983), pp. 212–240.CrossRefGoogle Scholar
[Sh87b] Shelah, Saharon, Classification theory for nonelementary classes, I. The number of uncountable models of ψ ∈ L ω1ω Part B, Classification theory (chicago, il, 1985),Lecture Notes in Mathematics, vol. 1292, Springer, Berlin, 1987, pp. 264–418, Israel Journal of Mathematics , vol. 46 (1983), pp. 241-273, Proceedings of the USA–Israel Conference on Classification Theory, Chicago (J. T. Baldwin, editor), December 1985.Google Scholar
[Zi2] Zilber, Boris, Analytic and pseudo-analytic structures , Preprint.Google Scholar
[Zi1] Zilber, Boris, Covers of the multiplicative group of an algebraically closed field of characteristic 0, Preprint.Google Scholar
[Zi3] Zilber, Boris, Covers of the multiplicative group under ℵ0-categoricity assumption . Preprint.Google Scholar