CEA-Operators and the Ershov Hierarchy. I

IF 0.4 3区数学 Q4 LOGIC

Algebra and Logic Pub Date : 2025-05-02 DOI:10.1007/s10469-025-09780-7

M. M. Arslanov, I. I. Batyrshin, M. M. Yamaleev

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引用次数: 0

Abstract

We consider the relationship between the CEA-hierarchy and the Ershov hierarchy in \({\Delta }_{2}^{0}\) Turing degrees. A degree c is called CEA(a) if c is computably enumerable in a, and a ≤ c. Soare and Stob [Stud. Logic Found. Math., 107, 299-324 (1982)] proved that for a noncomputable low c.e. degree a there exists a CEA(a) degree that is not c.e. Later, Arslanov, Lempp, and Shore [Ann. Pure Appl. Logic, 78, Nos. 1-3, 29-56 (1996)] formulated the problem of describing pairs of degrees a < e such that there exists a CEA(a) 2-c.e. degree d ≤ e which is not c.e. Since then the question has remained open as to whether a CEA(a) degree in the sense of Soare and Stob can be made 2-c.e. Here we answer this question in the negative, solving it in a stronger formulation: there exists a noncomputable low c.e. degree a such that any CEA(a) ω-c.e. degree is c.e. Also possible generalizations of the result obtained are discussed, as well as various issues associated with the problem mentioned.

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cea -算子和Ershov层次。我

我们考虑了在\({\Delta }_{2}^{0}\)图灵度中cea -层次和Ershov层次之间的关系。如果c在A中可计算枚举，且A≤c，则称度c为CEA(A)。逻辑发现。数学。[j]、[j]、[j]、[j]，证明了对于一个不可计算的低c.e.度a，存在一个不c.e.的CEA(a)度。纯苹果。逻辑，78,no . 1-3, 29-56(1996)]表述了描述度对的问题a ＆lt；因此存在CEA(a) 2- ce。度d≤e，不是c.e.。从那时起，关于Soare和Stob意义上的CEA(a)度是否可以构成2-c.e.的问题一直没有解决。在这里，我们以否定的形式回答这个问题，用一个更强的公式来解决它：存在一个不可计算的低c.e.度a，使得任何CEA(a) ω-c.e.。此外，还讨论了所得结果的可能概括，以及与所提到的问题相关的各种问题。

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来源期刊

Algebra and Logic 数学-数学

CiteScore

1.10

自引率

20.00%

发文量

审稿时长

>12 weeks

期刊介绍： This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions. Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences. All articles are peer-reviewed.