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It is well known that whenever a class of structures K₁ is interpretable in a class of structures K₂, then the hereditary undecidability of (a fragment of) the theory of K₁ implies the hereditary undecidability of (a suitable fragment of) the theory of K₂. In the present paper, we construct a 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 ₂-theory of the second class. Next, we construct a 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 ₂-theory of the second class. The corresponding results are, in a sense, optimal, since the ₂-theories of the classes under consideration are decidable. Keywords: undecidability, elementary theories, prefix fragments
Vladimir E. Karpov (Tue,) studied this question.