论文标题
通用扩展和组配置定理
Generic expansions and the group configuration theorem
论文作者
论文摘要
我们在简单级别与$ \ m atrm {nsop} _ {1} $ - $ \ mathrm {sop} _ {3} $ gap之间展示了几何稳定理论与不稳定结构的分类之间的联系。特别是，我们介绍了与可定义的关系$ r $ $ t $相关的理论$ t $的通用扩展$ t^{r} $，这可以包括添加新的一单元谓词或新的等价关系。 When $T$ is weakly minimal and $R$ is a ternary fiber algebraic relation, we show that $T^{R}$ is a well-defined $\mathrm{NSOP}_{4}$ theory, and use one of the main results of geometric stability theory, the \textit{group configuration theorem} of Hrushovski, to give an exact correspondence between the $ r $的几何形状和$ t^{r} $的分类理论复杂性。也就是说，$ t^{r} $是$ \ mathrm {sop} _ {3} $，而$ \ mathrm {tp} _ {2} $恰好是$ r $在几何上等同于类型可定义的组集组操作时的几何等效;否则，$ t^{r} $要么很简单（在$ t^{r} $的谓词版本中，或$ \ mathrm {nsop} _ {1} $（在等价关系版本中）。这为我们提供了严格的$ \ mathrm {nsop} _ _ {1} $的新示例。
We exhibit a connection between geometric stability theory and the classification of unstable structures at the level of simplicity and the $\mathrm{NSOP}_{1}$-$\mathrm{SOP}_{3}$ gap. Particularly, we introduce generic expansions $T^{R}$ of a theory $T$ associated with a definable relation $R$ of $T$, which can consist of adding a new unary predicate or a new equivalence relation. When $T$ is weakly minimal and $R$ is a ternary fiber algebraic relation, we show that $T^{R}$ is a well-defined $\mathrm{NSOP}_{4}$ theory, and use one of the main results of geometric stability theory, the \textit{group configuration theorem} of Hrushovski, to give an exact correspondence between the geometry of $R$ and the classification-theoretic complexity of $T^{R}$. Namely, $T^{R}$ is $\mathrm{SOP}_{3}$, and $\mathrm{TP}_{2}$ exactly when $R$ is geometrically equivalent to the graph of a type-definable group operation; otherwise, $T^{R}$ is either simple (in the predicate version of $T^{R}$) or $\mathrm{NSOP}_{1}$ (in the equivalence relation version.) This gives us new examples of strictly $\mathrm{NSOP}_{1}$ theories.
