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Remarks on unimodularity

  • Charlotte Kestner (a1) and Anand Pillay (a1)

Abstract

We clarify and correct some statements and results in the literature concerning unimodularity in the sense of Hrushovski [7], and measurability in the sense of Macpherson and Steinhorn [8], pointing out in particular that the two notions coincide for strongly minimal structures and that another property from [7] is strictly weaker, as well as “completing” Elwes' proof [5] that measurability implies 1-basedness for stable theories.

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References

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[1]Buechler, S., The geometry of weakly minimal types, this Journal, vol. 50 (1985), pp. 10441053.
[2]Buechler, S., Locally modular theories of finite rank, Annals of Pure and Applied Logic, vol. 30 (1986), pp. 8395.
[3]Buechler, S., On nontrivial types of U-rank 1, this Journal, vol. 52 (1987), pp. 548–541.
[4]Chatzidakis, Z., van den Dries, L., and Macintyre, A., Definable sets over finite fields, Journal für die Reine und Angewandte Mathematik, vol. 427 (1992), pp. 107135.
[5]Elwes, R., Asymptotic classes of finite structures, this Journal, vol. 72 (2007), pp. 418438.
[6]Elwes, R. and Macpherson, D., A survey of asymptotic classes and measurable structures. Model theory with applications to algebra and analysis. Vol. 2 (Chatzidakis, Z., Macpherson, D., Pillay, A., and Wilkie, A., editors), London Mathematical Society Lecture Notes Series, no. 350, Cambridge University Press, 2008.
[7]Hrushovski, E., Unimodular minimal structures, Journal of the London Mathematical Society, vol. 46 (1992), pp. 385396.
[8]Macpherson, D. and Steinhorn, C., One-dimensional asymptotic classes of finite structures. Transactions of the American Mathematical Society, vol. 360 (2008), pp. 411448.
[9]Pillay, A., Geometric stability theory, Oxford University Press, 1996.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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