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Strongly 2-dimensional theories

Published online by Cambridge University Press:  12 March 2014

Akito Tsuboi
Akito Tsuboi*
Affiliation:
Institute of Mathematics, University of Tsukuba, Ibaraki, 305., Japan
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Extract

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved:

Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent:

(i) T is almost strongly minimal;

(ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal.

In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general.

Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 53 , Issue 3 , September 1988 , pp. 931 - 936
Copyright
Copyright © Association for Symbolic Logic 1988

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References

REFERENCES

[1]Baldwin, J. T. and Lachlan, A. H., On strongly minimal sets, this Journal, vol. 36 (1971), pp. 487–493.Google Scholar
[2]Hrushovski, E. U., Countable stable unidimensional theories are superstable (to appear).Google Scholar
[3]Hrushovski, E. U., Contributions to stable model theory, Ph. D. thesis, University of California, Berkeley, California, 1986.Google Scholar
[4]Kikyo, H., A note on strongly ω-dimensional theories (to appear).Google Scholar
[5]Pillay, A., An introduction to stability theory, Oxford University Press, London, 1983.Google Scholar
[6]Poizat, B., Paires de structures stables, this Journal, vol. 48 (1983), pp. 239–249.Google Scholar
[7]Shelah, S., Classification theory and the number of non-isomorphic models, North-Holland, Amsterdam, 1978.Google Scholar
[8]Tsuboi, A., Categoricity and non-orthogonality of types, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 33 (1987), pp. 335–338.CrossRefGoogle Scholar