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Stability over a predicate and prime closure

Alexander Usvyatsov

Abstract

We prove that in a theory $T$ stable over a predicate $P$, for any $λ> |T|$, there is a $λ$-prime model over any complete set A with a $λ$-saturated $P$-part.

Stability over a predicate and prime closure

Abstract

We prove that in a theory stable over a predicate , for any , there is a -prime model over any complete set A with a -saturated -part.
Paper Structure (6 sections, 9 theorems, 4 equations)

This paper contains 6 sections, 9 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The setting
  3. Completeness, stability, and rank
  4. Completeness and relevant types
  5. Stability and rank
  6. Stability and primary models

Key Result

Corollary 1.1

Let $T$ be fully stable over $P$, $A$ be a complete set (e.g., $A \models T$) with $P^A$$\lambda$-saturated for some $\lambda>|T|$. Assume that the class of models is non-empty. Then $\mathcal{K}$ has a prime member over $P$: that is, there exists $N_0 \in \mathcal{K}$, which is elementarily embeddable into any $N \in \mathcal{K}$ over $P^A$.

Theorems & Definitions (19)

  • Corollary 1.1
  • Definition 3.1
  • Lemma 3.5
  • Definition 3.6
  • Remark 3.7
  • Remark 3.8
  • Lemma 3.10
  • Proposition 3.11
  • Definition 3.13
  • Theorem 3.16
  • ...and 9 more