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arXiv · 1810.03559

Classifying equivalence relations in the Ershov hierarchy

Nikolay Bazhenov·Manat Mustafa·Luca San Mauro·Andrea Sorbi·Mars Yamaleev
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Abstract

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 $\leq_c$. 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 $\leq_c$ 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.

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Nikolay Bazhenov, Manat Mustafa, Luca San Mauro, Andrea Sorbi, Mars Yamaleev. 2018-10-08. Classifying equivalence relations in the Ershov hierarchy. https://doi.org/10.1007/s00153-020-00710-1

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