Abstract:A recent thread in computable structure theory has been the investigation of computable structures after relativizing, the key idea being that facts which are true for algebraic/structural reasons tend to relativize. On the other hand, there are pathological examples such as a structure which is computably categorical but not relatively computably categorical; but such behaviour must eventually stabilize, as for example a structure is either computably categorical relative to all degrees above $\mathbf {0}''$ or not computably categorical relative to all degrees above $\mathbf {0}''$. But what can happen in between? We show a surprising result: there is a structure which alternates between being computably categorical and not computably categorical relative to an infinite increasing sequence of c.e. degrees.

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中文翻译：

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相对化可计算的范畴

摘要：可计算结构理论中的一个最新主题是在相对化之后对可计算结构的研究，其关键思想是对于代数/结构​​原因为真的事实倾向于相对化。另一方面，也有病理例子，例如可计算分类但不是相对可计算分类的结构；但这种行为最终必须稳定，例如，一个结构要么相对于 $\mathbf {0}''$ 之上的所有度数都是可计算分类的，要么相对于 $\mathbf {0}''$ 之上的所有度数而言是不可计算分类的。但是这中间会发生什么呢？我们展示了一个令人惊讶的结果：相对于无限递增的 ce 度序列，存在一种在可计算分类和不可计算分类之间交替的结构。

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