  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.
