Comparing the isomorphism types of equivalence structures and preorders

Abstract

A general theme of computable structure theory is to investigate when structures have copies of a given complexity Γ. We discuss such problem for the case of equivalence structures and preorders. We show that there is a Π 0 1 equivalence structure with no Σ 0 1 copy, and in fact that the isomorphism types realized by the Π 0 1 equivalence structures coincide with those realized by the ∆ 0 2 equivalence structures. We also construct a Σ 0 1 preorder with no Π 0 1 copy.2010 Mathematics Subject Classification. 03D45.