Hereditary undecidability of fragments of some elementary theories

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 ${\mathrm{\Sigma }}_1$-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 ${\mathrm{\Sigma }}_2$-theory of the second class. Next, we construct a ${\mathrm{\Sigma }}_1$-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 ${\mathrm{\Sigma }}_2$-theory of the second class. The corresponding results are, in a sense, optimal, since the ${\mathrm{\Pi }}_2$-theories of the classes under consideration are decidable. Keywords: undecidability, elementary theories, prefix fragments