Structurable equivalence relations
Structurable equivalence relations
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可结构化的等价关系
DOI：
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发表时间：
2016
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通讯作者：
A. Kechris
中科院分区：
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作者：
Ruiyuan Chen;A. Kechris
对于可数关系结构的类$\mathcal{K}$，一个可数的博雷尔等价关系$E$被称为是$\mathcal{K}$ - 可构造的，如果存在一种博雷尔方式在每个$E$ - 等价类上放置一个$\mathcal{K}$中的结构。在本文中，我们研究了各种$\mathcal{K}$的$\mathcal{K}$ - 可构造等价关系类的整体结构。我们表明$\mathcal{K}$ - 可构造性与在可数博雷尔等价关系分类中常用的几种博雷尔同态和归约相互作用良好。我们考虑在包含关系下各种$\mathcal{K}$的$\mathcal{K}$ - 可构造等价关系类的偏序集，并表明它是一个分配格；这意味着在可数博雷尔等价关系之间的博雷尔可归约性预序包含一个大的子格。最后，我们考虑$\mathcal{K}$的各种模型论性质对$\mathcal{K}$ - 可构造性的影响。特别地，我们刻画了使得每个$\mathcal{K}$ - 可构造等价关系都是光滑的$\mathcal{K}$，回答了马克斯（Marks）的一个问题。
For a class $mathcal K$ of countable relational structures, a countable Borel equivalence relation $E$ is said to be $mathcal K$-structurable if there is a Borel way to put a structure in $mathcal K$ on each $E$-equivalence class. We study in this paper the global structure of the classes of $mathcal K$-structurable equivalence relations for various $mathcal K$. We show that $mathcal K$-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in the classification of countable Borel equivalence relations. We consider the poset of classes of $mathcal K$-structurable equivalence relations for various $mathcal K$, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on $mathcal K$-structurability of various model-theoretic properties of $mathcal K$. In particular, we characterize the $mathcal K$ such that every $mathcal K$-structurable equivalence relation is smooth, answering a question of Marks.