Reconstruction of homogeneous relational structures
Reconstruction of homogeneous relational structures
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同质关系结构的重构
DOI：
10.2178/jsl/1191333842
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发表时间：
2007
影响因子：
0.6
通讯作者：
D. Macpherson
中科院分区：
文献类型：
--
作者：
S. Barbina;D. Macpherson
本文给出了一类齐次传递ω-范畴结构从其自同构群重构的结果。所处理的结构是相关的。在证明中，它表明，他们的自同构群包含一个通用对（在一个稍微不标准的意义上，来自Baire范畴）。重构结果给出了ω-范畴结构的自同构群Aut（）的抽象群结构确定Aut（）上的拓扑的条件，从而由[1]确定了双可解释性;它们也可以给出抽象群Aut（）确定置换群Aut（），的条件。所以确定了双向可定义性。M. Rubin在[12]中，它与Aut（）中点稳定器的可定义性有关。如果这个条件成立，那么这个结构就被称为具有弱可解释性，而Aut（）则确定了双向可解释性，或者在某些情况下，确定了双向可定义性。一个更著名的重构方法是通过“小指数性质”：一个ω-范畴结构具有小指数性质，如果Aut（）的任何指数小于的子群是开的。这保证了Aut（）的抽象群结构决定了拓扑，所以如果是ω-范畴的且Aut（）≠ Aut（），那么和是可双向解释的。
This paper contains a result on the reconstruction of certain homogeneous transitive ω-categorical structures from their automorphism group. The structures treated are relational. In the proof it is shown that their automorphism group contains a generic pair (in a slightly non-standard sense, coming from Baire category). Reconstruction results give conditions under which the abstract group structure of the automorphism group Aut() of an ω-categorical structure determines the topology on Aut(), and hence determines up to bi-interpretability, by [1]; they can also give conditions under which the abstract group Aut() determines the permutation group ⟨Aut (), ⟩. so determines up to bi-definability. One such condition has been identified by M. Rubin in [12], and it is related to the definability, in Aut(), of point stabilisers. If the condition holds, the structure is said to have a weak ∀∃ interpretation, and Aut() determines up to bi-interpretability or, in some cases, up to bi-definability. A better-known approach to reconstruction is via the ‘small index property’: an ω-categorical stucture has the small index property if any subgroup of Aut() of index less than is open. This guarantees that the abstract group structure of Aut() determines the topology, so if is ω-categorical with Aut() ≅ Aut() then and are bi-interpretable.