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A free pseudospace

Published online by Cambridge University Press:  12 March 2014

Andreas Baudisch  and
Anand Pillay
Andreas Baudisch
Affiliation:
Humboldt Universität Zu Berlin, Institut für Mathematik, 10 099 Berlin, Germany, E-mail: baudisch@mathematik.hu-berlin.de
Anand Pillay
Affiliation:
University of Illinoisat Urbana-Champaign, Altgeld Hall, 1409 W Green St., Urbana, Illinois 61801, USA, E-mail: pillay@math.uiuc.edu
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Extract

In this paper we construct a non-CM -trivial stable theory in which no infinite field is interpretable. In fact our theory will also be trivial and ω-stable, but of infinite Morley rank. A long term aim would be to find a non CM-trivial theory which has finite Morley rank (or is even strongly minimal) and does not interpret a field. The construction in this paper is direct, and is a “3-dimensional” version of the free pseudoplane. In a sense we are cheating: the original point of the notion of CM-triviality was to describe the geometry of a strongly minimal set, or even of a regular type. In our example, non-CM-triviality will come from the behaviour of three orthogonal regular types.

A stable theory is said to be CM-trivial if whenever A ⊆ B and acl(Ac) ∩ acl(B) = acl(A) in Teq, then Cb(stp(c/A)) ⊆ Cb(stp(c/B)). ( An infinite stable field will not be CM-trivial.) The notion is due to Hrushovski [3], where he gave several equivalent definitions, as well as showing that his new strongly minimal sets constructed “ab ovo” were CM-trivial. The notion was studied further in [6] where it was shown that CM-trivial groups of finite Morley rank are nilpotent-by-finite. These results were generalized in various ways to the superstable case in [8].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 443 - 460
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Baudisch, A., A new uncountably categorical group, Transactions of the American Mathematical Society, vol. 348 (1996), pp. 3889–3940.CrossRefGoogle Scholar
[2]Baudisch, A., Mekler's construction preserves CM-triviality, preprint, 1997.Google Scholar
[3]Hrushovski, E., A new strongly minimal set, Annals of Pure and Applied Logic, vol. 62 (1983), pp. 147–166.CrossRefGoogle Scholar
[4]Hrushovski, E. and Srour, G., On stable non-equational theories, manuscript, 1989.Google Scholar
[5]Pillay, A., CM-triviality and the geometry of forking, to appear in this Journal.Google Scholar
[6]Pillay, A., The geometry of forking and groups of finite Morley rank, this Journal, vol. 60 (1995), pp. 1251–1259.Google Scholar
[7]Pillay, A., Geometric stability theory, Oxford University Press, 1996.CrossRefGoogle Scholar
[8]Wagner, F., CM-triviality and stable groups, to appear in this Journal.Google Scholar