Comparing the isomorphism types of equivalence structures and preorders

A general theme of computable structure theory is to investigate when structures have copies of a given complexity $\mathrm{\Gamma }$. We discuss such problem for the case of equivalence structures and preorders. We show that there is a ${\mathrm{\Pi }}_1^0$ equivalence structure with no ${\mathrm{\Sigma }}_1^0$ copy, and in fact that the isomorphism types realized by the ${\mathrm{\Pi }}_1^0$ equivalence structures coincide with those realized by the ${\mathrm{\Delta }}_2^0$ equivalence structures. We also construct a ${\mathrm{\Sigma }}_1^0$ preorder with no ${\mathrm{\Pi }}_1^0$ copy.