Acyclicity and reduction

TL;DR

This paper explores how acyclicity assumptions enable broad reductions of Borel relations, providing finite bases and conditions for reducing complex relations like G 0 within Borel classes.

Contribution

It introduces new conditions under which Borel relations can be reduced globally, including finite bases and criteria for reducing G 0, advancing understanding of Borel reducibility.

Findings

Existence of a finite =< c-antichain basis for certain Borel relations.

Acyclic symmetrization allows for broader reductions.

Conditions for < c-reducing G 0 are established.

Abstract

The literature provides dichotomies involving homomorphisms (like the G 0 dichotomy) or reductions (like the characterization of sets potentially in a Wadge class of Borel sets, which holds on a subset of a product). However, part of the motivation behind the latter result was to get reductions on the whole product, like in the classical notion of Borel reducibility considered in the study of analytic equivalence relations. This is not possible in general. We show that, under some acyclicity (and also topological) assumptions, this is widely possible. In particular, we prove that, for any non-self dual Borel class {\Gamma}, there is a concrete finite =< c-antichain basis for the class of Borel relations, whose closure has acyclic symmetrization, and which are not potentially in {\Gamma}. Along similar lines, we provide a sufficient condition for =< c-reducing G 0. We also prove a similar result giving a minimum set instead of an antichain if we allow rectangular reductions.