Classifying equivalence relations in the Ershov hierarchy

TL;DR

This paper extends the classification of computably enumerable equivalence relations to the $ ext{Delta}^0_2$ level within the Ershov hierarchy, analyzing algebraic properties and the existence of infima and suprema in the degree-structure.

Contribution

It introduces a new study of $c$-degrees at the $ ext{Delta}^0_2$ level using the Ershov hierarchy, revealing algebraic properties and the (non)existence of infima and suprema.

Findings

Algebraic properties of the degree-structure are established.

Results on the (non)existence of infima and suprema of $c$-degrees.

Extension of classification to the $ ext{Delta}^0_2$ case.

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.