Weak Rudin-Keisler reductions on projective ideals

TL;DR

This paper introduces a modified Rudin-Keisler order on ideals, constructs complete ideals in projective classes, and provides a new proof of Hjorth's theorem, extending it under Projective Determinacy.

Contribution

It presents a new approach to the Rudin-Keisler order, constructs complete ideals in various projective classes, and generalizes Hjorth's theorem to higher classes under PD.

Findings

Existence of complete ideals in various projective classes.

A simplified proof of Hjorth's theorem on $oldsymbol{ ext{Pi}}^1_1$ equivalence relations.

Extension of Hjorth's theorem to $oldsymbol{ ext{Pi}}^{1}_{2n+1}$ classes under PD.

Abstract

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 $\mathbf \Pi^1_1$ equivalence relation. Our proof of Hjorth's theorem enables us (under PD) to generalize his result to the classes of $\mathbf \Pi^1_{2n+1}$ equivalence relations.