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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Coordinatization in superstable theories. I. Stationary types
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by Steven Buechler PDF
Trans. Amer. Math. Soc. 288 (1985), 101-114 Request permission

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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Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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