Special classes of positive preorders

We study positive preorders relative to computable reducibility. An approach is suggested to lift well-known notions from the theory of ceers to positive preorders. It is shown that each class of positive preoders of a special type (precomplete, $e$-complete, weakly precomplete, effectively finite precomplete, and effectively inseparable ones) contains infinitely many incomparable elements and has a universal object. We construct a pair of incomparable dark positive preorders that possess an infimum. It is shown that for every non-universal positive preorder $P$, there are infinitely many pairwise incomparable minimal weakly precomplete positive preorders that are incomparable with $P$.