Abstract

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice.

中文翻译：

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关于图灵度族的可计算编号

摘要

在这项工作中，我们研究了由 Arslanov 引入并首先研究的图灵度的可计算族及其编号。我们证明存在有限的图灵 ce 度族，无论有没有可计算的主编号，并且图灵度族的每个可计算的主编号对于该族的任何元素都是完整的。我们还表明，每个可计算的图灵度系列都具有关于其每个元素的完整可计算编号，即使它没有主编号。从 Mal'tsev 和 Ershov 的结果可以看出，完全编号具有良好的编程工具和计算属性，例如 Kleene 递归定理、Rice 定理、Visser 的 ADN 定理等。因此，每个可计算的图灵度族都具有具有这些属性的可计算编号。最后，我们证明每个这样的非空非单族的罗杰斯半格是无限的并且不是格。