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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Coordinatization in superstable theories. I. Stationary types

Author: Steven Buechler
Journal: Trans. Amer. Math. Soc. 288 (1985), 101-114
MSC: Primary 03C45
MathSciNet review: 773049
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Abstract: Suppose $ T$ is superstable and $ P$ is a complete type over some finite set with $ U(p) = \alpha + 1$ for some $ \alpha $. We show how to associate with $ p$ an incidence geometry which measures the complexity of the family of extensions of $ p$ of rank $ \alpha $. When $ p$ is stationary we give a characterization of the possible incidence geometries. As an application we prove

Theorem. Suppose $ M$ is superstable and has only one $ 1$-type $ p \in S(\emptyset )$. Further suppose $ p$ is stationary with $ U(p) = \alpha + 1$ for some $ \alpha $. Then one of the following holds:

(i) There is an equivalence relation $ E \subset {M^2}$ with infinitely many infinite classes definable over $ \emptyset $.

(ii) $ M$ is the algebraic closure of a set of Morley rank $ 1$. In particular, $ M$ is $ {\aleph _0}$-stable of finite rank.

References [Enhancements On Off] (What's this?)

  • [B] S. Buechler, A note on normalization preprint, 1983.
  • [CHL] G. Cherlin, L. Harrington and A. H. Lachlan, $ {\aleph _0}$-categorical, $ {\aleph _0}$-stable structures, preprint; Ann. Pure Appl. Logic (to appear). MR 779159 (86g:03054)
  • [D] P. Dembowski. Finite geometries, Springer-Verlag, Berlin and New York, 1968. MR 0233275 (38:1597)
  • [HH] V. Harnik and L. Harrington, Fundamentals of forking, Ann. Pure Appl. Logic 126 (1984), 245-286. MR 747686 (86c:03032)
  • [La] A. H. Lachlan, Two conjectures on the stability of $ {\aleph _0}$-categorical theories, Fund. Math. 81 (1974), 133-145. MR 0337572 (49:2341)
  • [L] D. Lascar, Ranks and definability in superstable theories, Israel J. Math. 23 (1976), 53-87. MR 0409169 (53:12931)
  • [LP] D. Lascar and B. Poizat, An introduction to forking, J. Symbolic Logic 44 (1979), 330-350. MR 540665 (80k:03030)
  • [M] M. Makkai, A survey of basic stability theory, with particular emphasis on orthogonality and regular types, Israel J. Math. (to appear). MR 788268 (86h:03055)
  • [P] A. Pillay, An introduction to stability theory, Oxford Univ. Press, New York and London, 1983. MR 719195 (85i:03104)
  • [Sh] S. Shelah, Classification theory, North-Holland, Amsterdam, 1978. MR 513226 (81a:03030)
  • [Z] B. I. Zil'ber, Totally categorical structures and combinatorial geometries, Soviet Math. Dokl. 24 (1981), 149-151.

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Article copyright: © Copyright 1985 American Mathematical Society

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