The structure of computably enumerable preorder relations

We study the structure Ceprs induced by degrees of computably enumerable preorder relations with respect to computable reducibility $\leq_c$. It is proved that the structure of computably enumerable equivalence relations is definable in Ceprs. This fact and results of Andrews, Schweber, and Sorbi imply that the theory of the structure Ceprs is computably isomorphic to first-order arithmetic. It is shown that a $\Sigma_1$-fragment of the theory is decidable, while its $\Pi_3$-fragment is hereditarily undecidable. It is stated that any two incomparable degrees in Ceprs do not have a least upper bound, and that among minimal degrees in Ceprs, exactly two are $c$-degrees of computably enumerable linear preorders.