Weak Rudin-Keisler reductions on projective ideals

We consider a slightly modified form of the standard Rudin-Keisler order on ideals and demonstrate the existence of complete (with respect to this order) ideals in various projective classes. Using our methods, we obtain a simple proof of Hjorth's theorem on the existence of a complete $\mathbf Π^1_1$ equivalence relation. Our proof of Hjorth's theorem enables us (under PD) to generalize his result to the classes of $\mathbf Π^1_{2n+1}$ equivalence relations.

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