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A definable continuous rank for nonmultidimensional superstable theories

Published online by Cambridge University Press:  12 March 2014

Ambar Chowdhury ,
James Loveys  and
Predrag Tanović
Ambar Chowdhury
Affiliation:
Department of Mathematics, Mcmaster University, E-mail: ambar@math.uconn.edu Department of Mathematics, University of Connecticut U-9, Storrs, CT 06269, USA, E-mail: ambar@math.uconn.edu
James Loveys
Affiliation:
Department of Mathematics, Mcgill University, Montreal, Quebec H3A 2K6, Canada, E-mail: loveys@triples.math.mcgill.ca
Predrag Tanović
Affiliation:
Department of Mathematics, Mcgill University, E-mail: tane@mi.sanu.ac.yu Matematički institut, Belgrade, Yugoslavia, E-mail: tane@mi.sanu.ac.yu
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Extract

Pillay studied nonmultidimensional superstable theories in [8], among other things defining a certain hierarchy of regular types in terms of which all other types may be analysed. Using this hierarchy, he showed that after naming a suitable ‘base’ of parameters, there are j-constructible (hence locally atomic) models over arbitrary sets (see Section 2 for definitions). It is asked at the end of [8] whether the parameter set can be removed. On a different note, it has been known for some time that in nonmultidimensional superstable theories, R∞-rank is definable for formulas having finite rank (see for example [9]). Definability of R∞-rank has had various applications in the literature, and so it is natural to ask whether the restriction to finite rank is necessary. In this paper we do not quite answer this question, but instead use Pillay's analysis to establish the existence of a ‘new’ continuous rank (the original idea for which is due to Tanović) which is defined on all complete types, reflects forking as does R∞-rank and satisfies certain definability properties.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 61 , Issue 3 , September 1996 , pp. 967 - 984
Copyright
Copyright © Association for Symbolic Logic 1996

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References

REFERENCES

[1]Chowdhury, A., A note on trivial nonmultidimensional superstable theories, to appear.Google Scholar
[2]Harnik, V. and Harrington, L., Fundamentals of forking, Annals of Pure and Applied Logic, vol. 26 (1984), pp. 245–286.CrossRefGoogle Scholar
[3]Hrushovski, E., Contributions to stable model theory, Ph.D. thesis, University of California, Berkeley, 1986.Google Scholar
[4]Hrushovski, E. and Shelah, S., A dichotomy theorem for regular types, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 157–169.CrossRefGoogle Scholar
[5]Makkai, M., A survey of basic stability theory, Israel Journal of Mathematics, vol. 49 (1984), no. 1–3, pp. 181–238.CrossRefGoogle Scholar
[6]Newelski, L., More on locally atomic models, Fundamenta Mathematicae, vol. 136 (1990), pp. 21–26.CrossRefGoogle Scholar
[7]Pillay, A., forthcoming book on geometric stability theory.Google Scholar
[8]Pillay, A., Some remarks on nonmultidimensional superstable theories, this Journal, vol. 59 (1994), pp. 151–165.Google Scholar
[9]Saffe, J., Categoricity and ranks, this Journal, vol. 49 (1984), pp. 1379–1382.Google Scholar
[10]Shelah, S., Classification theory, North Holland, Amsterdam and New York, 1990.Google Scholar
[11]Shelah, S. and Buechler, S., On the existence of regular types, Annals of Pure and Applied Logic, vol. 45 (1989), pp. 207–308.CrossRefGoogle Scholar