In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ-levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ-level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ${\mathrm{\Sigma }}_{\alpha }^{-1}$- and ${\mathrm{\Delta }}_{\alpha }^{-1}$-bound for every infinite computable ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ${\mathrm{\Sigma }}_{\alpha }^{-1}$- or ${\mathrm{\Delta }}_{\alpha }^{-1}$-bound of a c.e. set A is equivalent to the fact that $A^{\prime }\in {\mathrm{\Delta }}_{\alpha }^{-1}$. Finally, we consider the generalized truth-table reducibilities ${⩽}_{gtt(\alpha )}$ and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, $A\in {\mathrm{\Delta }}_{\alpha }^{-1}$ iff $A{⩽}_{gtt(\alpha )}{⌀}^{\prime }$.

中文翻译：

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低 ce 集的分类和 Ershov 层次结构

在本文中，我们证明了关于低 ce 集图灵跳跃的几个结果。我们证明，只有 Ershov 层次结构的 Δ 级才能正确包含 ce 集的图灵跳跃，并且存在具有正常符号的任意大的可计算序数，使得相应的 Δ 级适合某些 ce 集的图灵跳跃。接下来，我们将跳跃可追踪性的概念推广到跳跃可追踪性：${\mathrm{\Sigma }}_{\alpha }^{-1}$- 和${\mathrm{\Delta }}_{\alpha }^{-1}$-对于每个无限可计算序数 α 都有限制。众所周知，跳跃可追溯性和超低性在 ce 集上一致，并且我们证明对于每个无限可计算序数 α，跳跃可追溯性与${\mathrm{\Sigma }}_{\alpha }^{-1}$- 或者${\mathrm{\Delta }}_{\alpha }^{-1}$ce 集合A的 -bound等价于以下事实$A^{\prime }\epsilon {\mathrm{\Delta }}_{\alpha }^{-1}$。最后，我们考虑广义真值表可约性${⩽}_{Gtt（\alpha ）}$并证明对于每个（不一定是 ce 集的图灵跳跃）集合A和每个极限可计算序数 α，$A\epsilon {\mathrm{\Delta }}_{\alpha }^{-1}$当且仅当$A{⩽}_{Gtt（\alpha ）}{⌀}^{\prime }$。