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Minimal but not strongly minimal structures with arbitrary finite dimensions

Published online by Cambridge University Press:  12 March 2014

Koichiro Ikeda
Koichiro Ikeda*
Affiliation:
Department of Mathematics, Toyota National College of Technology, 2-1 Eiseicho, Toyota 471-8525., Japan, E-mail: [email protected]
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Abstract

An infinite structure is said to be minimal if each of its definable subset is finite or cofinite. Modifying Hrushovski's method we construct minimal, non strongly minimal structures with arbitrary finite dimensions. This answers negatively to a problem posed by B. I Zilber.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 66 , Issue 1 , March 2001 , pp. 117 - 126
Copyright
Copyright © Association for Symbolic Logic 2001

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References

REFERENCES

[1]Baldwin, J. T., New families of stable structures, Notes on Hrushovski's construction intended for lectures in Beijing.Google Scholar
[2]Baldwin, J. T., An almost strongly minimal non-Desarguesian projective plane, Transactions of the American Mathematical Society, vol. 342 (1996), pp. 695–711.CrossRefGoogle Scholar
[3]Baldwin, J. T. and Itai, M., K-generic projective planes have Morley rank two or infinity, Mathematical Logic Quarterly, vol. 40 (1994), pp. 143–152.CrossRefGoogle Scholar
[4]Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 79–96.Google Scholar
[5]Baldwin, J. T. and Shi, N., Stable generic structures, Annals of Pure and Applied Logic, vol. 79 (1996), pp. 1–35.CrossRefGoogle Scholar
[6]Baudish, A., A new uncountably categorical group, Transactions of the American Mathematical Society, vol. 348 (1996), no. 10, pp. 3889–3940.CrossRefGoogle Scholar
[7]Belegradek, O. V., On minimal structures, this Journal, vol. 63 (1998), pp. 421–426.Google Scholar
[8]Goode, J., Hrushovski's geometries, Proceedings of the Seventh Easter Conference on Model Theory (Wolter, H. and Dahn, B., editors), 1989, pp. 106–118.Google Scholar
[9]Hrushovski, E., A stable ℵ0-categorical pseudoplane, preprint, 1988.Google Scholar
[10]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 46 (1990), pp. 235–264.Google Scholar
[11]Pillay, A., An introduction to stability theory, Clarendon Press, Oxford.Google Scholar
[12]Wagner, F. O., Relational structures and dimensions, Automorphisms of first-order structures (Kaye, R.et al., editors), Clarendon Press, Oxford, 1994, pp. 153–180.CrossRefGoogle Scholar
[13]Wagner, F. O., Small filds, this Journal, vol. 63 (1998), pp. 995–1002.Google Scholar