Uniform Martin’s conjecture, locally,Proceedings of the American Mathematical Society

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Uniform Martin’s conjecture, locally
Proceedings of the American Mathematical Society ( IF 1 ) Pub Date : 2020-09-17 , DOI: 10.1090/proc/15159
Vittorio Bard

Abstract:We show that part I of the uniform Martin's conjecture follows from a local phenomenon, namely that every non-constant uniformly Turing invariant $f:[x]_{\equiv _T}\to [y]_{\equiv _T}$ satisfies $x\le _T y$ . Besides improving our knowledge about part I of the uniform Martin's conjecture (which turns out to be equivalent to Turing determinacy), the discovery of such local phenomenon also leads to new results that did not look strictly related to Martin's conjecture before. In particular, we get that computable reducibility $\le _c$ on equivalence relations on $\mathbb{N}$ has a very complicated structure, as $\le _T$ is Borel reducible to it. We conclude by raising the question: Is part II of the uniform Martin's conjecture implied by local phenomena, too? and briefly indicating possible directions.

中文翻译：

统一马丁猜想，局部

摘要：我们证明了统一马丁猜想的第一部分来自一个局部现象，即每个非恒定一致图灵不变式都满足。除了提高我们对统一的马丁猜想的第一部分的知识（事实证明这等效于图灵确定性）之外，这种局部现象的发现还导致新的结果，这些结果看起来似乎与马丁的猜想没有严格的关系。尤其是，我们得到等价关系上的可计算约简具有非常复杂的结构，就像Borel可约简一样。最后，我们提出一个问题：局部现象也隐含着统一的马丁猜想的第二部分？并简要指出可能的方向。 $f：[x] _ {\ equiv _T} \ to [y] _ {\ equiv _T}$ $x \ le _T y$ $\ le _c$ $\ mathbb {N}$ $\ le _T$

更新日期：2020-10-19

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