  It is well known that whenever a class of structures $\mathcal{K}_1$ is
interpretable in a class of structures $\mathcal{K}_2$, then the hereditary
undecidability of (a fragment of) the theory of $\mathcal{K}_1$ implies the
hereditary undecidability of (a suitable fragment of) the theory of
$\mathcal{K}_2$. In the present paper, we construct a $\Sigma_1$-interpretation
of the class of all finite bipartite graphs in the class of all pairs of
equivalence relations on the same finite domain; from this we obtain the
hereditary undecidability of the $\Sigma_2$-theory of the second class. Next,
we construct a $\Sigma_1$-interpretation of the class of all pairs of
equivalence relations on the same finite domain in the class of all pairs
consisting of a linear ordering and an equivalence relation on the same finite
domain; this gives us the hereditary undecidability of the $\Sigma_2$-theory of
the second class. The corresponding results are, in a sense, optimal, since the
$\Pi_2$-theories of the classes under consideration are decidable.
  Keywords: undecidability, elementary theories, prefix fragments
