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 ≤ 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 Σ 1 -fragment of the theory is decidable, while its Π 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.