论文标题
插值融合II：保存结果
Interpolative fusions II: Preservation results
论文作者
论文摘要
我们研究插值融合，一种结合理论的方法$ t_1 $和$ t_2 $以“通用”方式以“通用”方式上的方式，以$ t_ \ cap $，以获取理论$ t_ \ cup^*$。当每个$ t_i $都是模型完成时，$ t_ \ cup^*$是联合$ t_1 \ cup t_2 $的模型伴侣。我们的目标是证明保存结果，即找到足够的条件，在这些条件下，$ t_ \ cup^*$继承了$ t_1 $和$ t_2 $的模型理论属性。 我们首先证明了消除量化器，模型完整性和相关特性的保存结果。然后，我们应用这些工具以表明在温和的假设下，包括$ t_ \ cap $的稳定性，属性$ \ mathrm {nsop} _1 $被保留。我们还表明，在代数关闭的更强假设下，简单性保留在$ t_1 $和$ t_2 $中。这概括了许多先前的结果；例如，$ \ mathrm {acfa} $和随机$ n $ - hypergraph的简单性都是非明显的推论。我们还解决了稳定性的保存，$ \ mathrm {nip} $和$ \ aleph_0 $ -cateporicity，我们描述了这些结果表明这些结果很敏锐。
We study interpolative fusion, a method of combining theories $T_1$ and $T_2$ in distinct languages in a "generic" way over a common reduct $T_\cap$, to obtain a theory $T_\cup^*$. When each $T_i$ is model-complete, $T_\cup^*$ is the model companion of the union $T_1\cup T_2$. Our goal is to prove preservation results, i.e., to find sufficient conditions under which model-theoretic properties of $T_1$ and $T_2$ are inherited by $T_\cup^*$. We first prove preservation results for quantifier elimination, model-completeness, and related properties. We then apply these tools to show that, under mild hypotheses, including stability of $T_\cap$, the property $\mathrm{NSOP}_1$ is preserved. We also show that simplicity is preserved under stronger hypotheses on algebraic closure in $T_1$ and $T_2$. This generalizes many previous results; for example, simplicity of $\mathrm{ACFA}$ and the random $n$-hypergraph are both non-obvious corollaries. We also address preservation of stability, $\mathrm{NIP}$, and $\aleph_0$-categoricity, and we describe examples which witness that these results are sharp.
