Positive Preorders

Abstract

We consider positive preorders, i.e., computably enumerable equivalences, endowed with the structure of a partial order between equivalence classes. On positive preorders, a computable reducibility relation and the corresponding notion of degree of a positive preorder are introduced in the natural way. It is proved that the degree of any positive preorder contains either exactly one computable isomorphism class or an infinite set of computable isomorphism classes.