Wadge-like degrees of Borel bqo-valued functions

TL;DR

This paper generalizes Wadge theory to $Q$-valued functions with more complex reducibilities, establishing isomorphisms between degree structures and generalized homomorphism orders on $Q$-labeled forests.

Contribution

It extends existing Wadge theory by considering broader reducibilities and $Q$-valued functions, linking degree structures to $Q$-labeled forest homomorphisms.

Findings

Degree structures are isomorphic across certain Borel classes.

$ ext{Delta}^0_eta$-degrees correspond to homomorphism orders on $Q$-labeled forests.

Results unify and extend previous theories of Wadge degrees.

Abstract

We unite two well known generalisations of the Wadge theory. The first one considers more general reducing functions than the continuous functions in the classical case, and the second one extends Wadge reducibility from sets (i.e., $\{0,1\}$-valued functions) to $Q$-valued functions, for a better quasiorder $Q$. In this article, we consider more general reducibilities on the $Q$-valued functions and generalise some results of L. Motto Ros in the first direction and of T. Kihara and A. Montalb\'an in the second direction: Our main result states that the structure of the $\mathbf{\Delta}^0_\alpha$-degrees of $\mathbf{\Delta}^0_{\alpha+\gamma}$-measurable $Q$-valued functions is isomorphic to the $\mathbf{\Delta}^0_\beta$-degrees of $\mathbf{\Delta}^0_{\beta+\gamma}$-measurable $Q$-valued functions, and these are isomorphic to the generalized homomorphism order on the $\gamma$-th iterated $Q$-labeled forests.