Classifying equivalence relations in the Ershov hierarchy

Abstract

AbstractComputably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $$\leqslant _c$$ ⩽ c . This gives rise to a rich degree structure. In this paper, we lift the study of c-degrees to the $$\Delta ^0_2$$ Δ 2 0 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 $$\leqslant _c$$ ⩽ c on the $$\Sigma ^{-1}_{a}\smallsetminus \Pi ^{-1}_a$$ Σ a - 1 \ Π a - 1 equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of c-degrees.