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 $\Gamma$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a $\Pi^0_1$ equivalence structure with no $\Sigma^0_1$ copy, and in fact that the isomorphism types realized by the $\Pi^0_1$ equivalence structures coincide with those realized by the $\Delta^0_2$ equivalence structures. We also construct a $\Sigma^0_1$ preorder with no $\Pi^0_1$ copy.