The uniform Martin's conjecture for many-one degrees

TL;DR

This paper extends the uniform Martin's conjecture to many-one degrees, establishing a correspondence with Wadge degrees for functions invariant under degree equivalences, in a broad general setting.

Contribution

It generalizes the uniform Martin's conjecture to many-one degrees and broad classes of functions, connecting degree-invariant functions with Wadge degrees in a new way.

Findings

Functions from reals to reals that are uniformly degree-invariant correspond to Wadge degrees.

The proof applies to many-one degrees on $\\mathcal{Q}^\omega$ and Wadge degrees of functions from $\omega^\omega$ to a quasi-order.

The result broadens the understanding of degree-invariant functions in descriptive set theory.

Abstract

We study functions from reals to reals which are uniformly degree-invariant from Turing-equivalence to many-one equivalence, and compare them "on a cone." We prove that they are in one-to-one correspondence with the Wadge degrees, which can be viewed as a refinement of the uniform Martin's conjecture for uniformly invariant functions from Turing- to Turing-equivalence. Our proof works in the general case of many-one degrees on $\mathcal{Q}^\omega$ and Wadge degrees of functions $\omega^\omega\to\mathcal{Q}$ for any better quasi ordering $\mathcal{Q}$.