Hereditary undecidability of fragments of some elementary theories

Abstract

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