Classifying equivalence relations in the Ershov hierarchy

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 $$ . This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the $$ 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 $$ on the $$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.