随机化和保存定理的简单模型

Karim Khanaki, Massoud Pourmahdian
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引用次数: 0

摘要

本文的主要目的是对这些定理(见 [5] 和 [4])提出新的、更统一的模型理论/组合证明:具有 $NIP$/stability 的完整一阶理论 $T$ 的随机化 $T^{R}$ 是具有 $NIP$/stability 的(完整)一阶连续理论。这两个定理的证明方法都基于对 $T^{R}$ 的一种特殊模型,即简单模型和某些不可辨别阵列的大量使用。最后,我们将注意力转向随机化中的 $NSOP$。我们证明,根据 [11] 给出的 $NSOP$ 定义,当且仅当 $NIP$ 和 $NSOP$ 时,$T^R$ 是稳定的。
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Simple Models of Randomization and Preservation Theorems
The main purpose of this paper is to present new and more uniform model-theoretic/combinatorial proofs of the theorems (in [5] and [4]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$/stability is a (complete) first-order continuous theory with $NIP$/stability. The proof method for both theorems is based on the significant use of a particular type of models of $T^{R}$, namely simple models, and certain indiscernible arrays. Using simple models of $T^R$ gives the advantage of re-proving these theorems in a simpler and quantitative manner. We finally turn our attention to $NSOP$ in randomization. We show that based on the definition of $NSOP$ given [11], $T^R$ is stable if and only if it is $NIP$ and $NSOP$.
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