论文标题
$ω$ - 分类稳定结构的分类
Classification of $ω$-categorical monadically stable structures
论文作者
论文摘要
一阶结构$ \ mathfrak {a} $被称为单声稳定，如果Fimary Predicates的每一个$ \ Mathfrak {a} $的每个扩展都稳定。在本文中，我们将$ \ MATHCAL的类$ \ Mathcal {M} $分类从$ω$ - 分类的单模稳定结构根据其自动形态组。反过来，我们证明了$ \ Mathcal {M} $是最小的结构类，其中包含单元素纯集，在同构中关闭，并在有限的分离工会，无限副本和有限的索引索引一阶减少下关闭。使用我们的分类，我们表明$ \ Mathcal {M} $中的每个结构都是具有有限界均匀结构的一阶互互定。我们还证明，$ \ Mathcal {m} $中的每个结构都有有限的可降低到可互定性的，从而证实了Thomas对类$ \ Mathcal {M} $的猜想。
A first-order structure $\mathfrak{A}$ is called monadically stable iff every expansion of $\mathfrak{A}$ by unary predicates is stable. In this article we give a classification of the class $\mathcal{M}$ of $ω$-categorical monadically stable structures in terms of their automorphism groups. We prove in turn that $\mathcal{M}$ is smallest class of structures which contains the one-element pure set, closed under isomorphisms, and closed under taking finitely disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal{M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal{M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathcal{M}$.
