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Classifying equivalence relations in the Ershov hierarchy
Archive For Mathematical Logic ( IF 0.3 ) Pub Date : 2020-02-13 , DOI: 10.1007/s00153-020-00710-1
Nikolay Bazhenov , Manat Mustafa , Luca San Mauro , Andrea Sorbi , Mars Yamaleev

Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility \(\leqslant _c\). This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the \(\Delta ^0_2\) case. In doing so, we rely on the Ershov hierarchy. For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by \(\leqslant _c\) on the \(\Sigma ^{-1}_{a}\smallsetminus \Pi ^{-1}_a\) equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.



中文翻译:

在Ershov层次结构中对等价关系进行分类

可计算的等价关系(子代)在文献中受到了很多关注。可计算的归约度\(\ leqslant _c \)提供了对cees分类的标准工具。这产生了丰富的学位结构。在本文中,我们将c度的研究提升到\(\ Delta ^ 0_2 \)情况。在此过程中,我们依靠Ershov层次结构。对于非零可计算序数的任何符号a,我们证明了由\(\ Sigma ^ {-1} _ {a} \ smallsetminus \ Pi ^上的\(\ leqslant _c \)诱导的度数结构的若干代数性质。{-1} _a \)等价关系。我们工作的一个特别重点是c的infima和suprema的不存在度。

更新日期:2020-02-13
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