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Canonical bases in excellent classes

Published online by Cambridge University Press:  12 March 2014

Tapani Hyttinen  and
Olivier Lessmann
Tapani Hyttinen
Affiliation:
Department of Mathematics and Statistics, P.O. BOX 4, 00014, University of Helsinki, Finland, E-mail: tapani.hyttinen@helsinki.fi
Olivier Lessmann
Affiliation:
Collège Rousseau, Case Postale 216, 1211 Geneva 28, Switzerland, E-mail: lessmann@gmail.com
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Abstract

We show that any (atomic) excellent class can be expanded with hyperimaginaries to form an (atomic) excellent class which has canonical bases. When is, in addition, of finite U-rank, then is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class gives rise to an excellent class, which is simple if is of finite U-rank.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 1 , March 2008 , pp. 165 - 180
Copyright
Copyright © Association for Symbolic Logic 2008

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References

REFERENCES

[BY]Yaacov, Itay Ben, Discouraging results with ultra-imaginaries, preprint.Google Scholar
[HKP]Hart, Bradd, Kim, Byunghan, and Pillay, Anand, Coordinatisation and canonical bases in simple theories, this Journal, vol. 65 (2000), no. 1, pp. 293–309.Google Scholar
[Hy]Hyttinen, Tapani, Finiteness of U-rank implies simplicity in homogeneous structures, Mathematical Logic Quaterly, vol. 49 (2003), no. 6, pp. 576–578.CrossRefGoogle Scholar
[HyLe]Hyttinen, Tapani and Lessmann, Olivier, Simplicity and uncountable categoricity in excellent classes, Annals of Pure and Applied Logic, vol. 139 (2006), no. 1-3, pp. 110–137.CrossRefGoogle Scholar
[HLS]Hyttinen, Tapani, Lessmann, Olivier, and Shelah, Saharon, Interpreting groups and fields in some nonelementary classes, Journal of Mathematical Logic, vol. 5 (2005), no. 1, pp. 1–47.CrossRefGoogle Scholar
[Le1]Lessmann, Olivier, Categoricity and U-rank in excellent classes, this Journal, vol. 68 (2003), no. 4, pp. 1317–1336.Google Scholar
[Le2]Lessmann, Olivier, An introduction to excellent classes, Logic and its applications, Contemporary Mathematics, vol. 380, AMS, Providence, RI, 231–259 2005.CrossRefGoogle Scholar
[Sh48]Shelah, Saharon, Categoricity in ℵ1 of sentences in Lω1ω (Q), Israel Journal of Mathematics, vol. 20 (1975), pp. 127–148.CrossRefGoogle Scholar
[Sh87a]Shelah, Saharon, Classification theory for nonelementary classes. I. The number of uncountable models of ψ ∈ Lω1ω. 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 ∈ Lω1ω. Part B, Israel Journal of Mathematics, vol. 46 (1983),pp. 241–273.CrossRefGoogle Scholar
[Zi]Zilber, Boris, Pseudo-exponentiation on algebraically closed fields of characteristic zero, Annals of Pure and Applied Logic, vol. 132 (2005), no. 1, pp. 67–95.CrossRefGoogle Scholar