Index Sets for Classes of Positive Preorders

We study the complexity of index sets with respect to a universal computable numbering of the family of all positive preorders. Let ≤_c be computable reducibility on positive preorders. For an arbitrary positive preorder R such that the R-induced equivalence ∼R has infinitely many classes, the following results are obtained. The index set for preorders P with R ≤_c P is \( {\sum}_3^0-\mathrm{complete} \). A preorder R is said to be self-full if the range of any computable function realizing the reduction R ≤_c R intersects all ∼Rclasses. If L is a non-self-full positive linear preorder, then the index set of preorders P with P ≡_c L is \( {\sum}_3^0-\mathrm{complete} \). It is proved that the index set of self-full linear preorders is \( {\prod}_3^0-\mathrm{complete} \).