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 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.
Highlights
For any notation a for a non-zero computable ordinal, we prove several algebraic properties of the degree structure induced by c on the
Computable reducibility is a longstanding notion that allows classifying equivalence relations on natural numbers according to their complexity
We follow and extend the work of Andrews and Sorbi [3], that provides a very extensive analysis of the degree structure induced by computable reducibility on ceers
Summary
Computable reducibility is a longstanding notion that allows classifying equivalence relations on natural numbers according to their complexity. 1 1 equivalence relations are computably reducible to the isomorphism relations on several classes of computable structures (e.g., graphs, trees, torsion Abelian groups, fields of characteristic 0 or p, linear orderings). The goal of the present paper is to contribute to this vast (yet somehow unsystematic) research program by making use of computable reducibility to initiate a throughout classification of the complexity of. In this endeavour, we follow and extend the work of Andrews and Sorbi [3], that provides a very extensive analysis of the degree structure induced by computable reducibility on ceers. It is easy to see that i , s Rb j , t ⇔ Li,s ∼= L j,t , the relation Rb can be treated as (one of the possible formalizations of) the relation of isomorphism on the class of finite linear orderings
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