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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
DOI: https://doi.org/10.1090/S0002-9947-1985-0773049-3
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.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0773049-3
Article copyright: © Copyright 1985 American Mathematical Society

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