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Non-locally modular regular types in classifiable theories

Elisabeth Bouscaren, Bradd Hart, Ehud Hrushovski, Michael C. Laskowski

Abstract

We introduce the notion of strong $p$-semi-regularity and show that if $p$ is a regular type which is not locally modular then any $p$-semi-regular type is strongly $p$-semi-regular. Moreover, for any such $p$-semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that $T$ is countable, classifiable and $M$ is any model. If $p\in S(M)$ is regular but not locally modular and $b$ is any realization of $p$ then every model $N$ containing $M$ that is dominated by $b$ over $M$ is both constructible and minimal over $Mb$.

Non-locally modular regular types in classifiable theories

Abstract

We introduce the notion of strong -semi-regularity and show that if is a regular type which is not locally modular then any -semi-regular type is strongly -semi-regular. Moreover, for any such -semi-regular type, "domination implies isolation" which allows us to prove the following: Suppose that is countable, classifiable and is any model. If is regular but not locally modular and is any realization of then every model containing that is dominated by over is both constructible and minimal over .

Paper Structure

This paper contains 18 sections, 52 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. A brief historical background
  3. Strongly $p$-semiregular types
  4. $p$-triples
  5. Proof of Theorem \ref{['strong']}
  6. Applications of Theorem \ref{['strong']}
  7. Constructible, minimal models over realizations of non-locally modular types
  8. Witnesses to non-modularity
  9. A definable witness to non-modularity
  10. Depth-zero like types and minimality
  11. The main theorem
  12. When domination implies isolation
  13. Appendix
  14. Domination and isolation in stable theories
  15. $p$-simplicity and locally modular regular types
  16. ...and 3 more sections

Key Result

Corollary 2.3

If $T$ is any countable, complete theory and its spectrum of uncountable models is any of 4,5,7, or 10 (in the notation of Theorem 6.1 of HHL) then an infinite group is interpretable in models of $T$. In particular, if $I(T,\aleph_{\alpha})=\min\{2^{\aleph_\alpha},\beth_2\}$ (i.e., the spectrum of $

Theorems & Definitions (76)

  • Definition 2.1
  • Definition 2.2
  • Corollary 2.3
  • Example 2.4
  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • Definition 3.5
  • Definition 3.6
  • ...and 66 more