Abstract—

This paper is devoted to the study of \(eT\)-reducibility, i.e., the most intuitively general algorithmic reducibility, which is both enumeration and decision reducibilities. The corresponding degree structure, that is, the upper semilattice \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) of \(eT\)-degrees is considered. It is shown that it is possible to correctly define a jump operation on \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) by using the \(T\)-jump or \(e\)-jump of sets. The local properties of \(eT\)-degrees, such as totality and computable enumerability, are considered. It is proven that all degrees between the smallest element and the first jump in \({{{\mathbf{D}}}_{{{\mathbf{eT}}}}}\) are computably enumerable; moreover, these degrees contain computably enumerable sets and only them. The existence of nontotal \(eT\)-degrees is established. Based on this, we obtain some results on the relations between degrees; in particular, the fact that every \(eT\)-degree is either completely contained in some \(T\)-degree or \(e\)-degree, or completely coincides with it, implies that nontotal \(e\)-degrees are contained in the \(T\)-degrees located above the second \(T\)-jump and coincide with the corresponding nontotal \(eT\)-degrees.

中文翻译：

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eT-集的可约性

摘要-

本文致力于\（eT \）-可约性的研究，即最直观的通用算法可约性，即枚举和决策可约性。相应的结构程度，也就是，上半格\（{{{\ mathbf {d}}} _ {{{\ mathbf {ET}}}}} \）的\（ET \） -degrees被考虑。显示可以通过使用\（T \）在\（{{{\ mathbf {D}}} __ {{{\ mathbf {eT}}}}}} \\上正确定义跳转操作-跳转或\（e \） -集的跳转。考虑\（eT \）-度的局部属性，例如总数和可计算的可枚举性。事实证明，最小元素与第一次跳跃之间的所有度数\（{{{\ mathbf {D}}} _ {{{\ mathbf {eT}}}}} \）可以计算得出；此外，这些学位仅包含可计算的可数集合。建立了非总\（eT \）-度的存在。在此基础上，我们得到了度之间关系的一些结果。特别是，每个\（eT \）-degree完全包含在某个\（T \）- degree或\（e \）- degree中，或者与之完全重合，这一事实意味着非总计\（e \） -degrees包含在\（T \）位于第二上面-degrees \（T \） -跳转并与相应的nontotal重合\（ET \） -degrees。