Invariantly universal analytic quasi-orders

TL;DR

This paper introduces the concept of invariant universality for analytic quasi-orders and equivalence relations, providing a general theorem and demonstrating its widespread occurrence across various mathematical contexts.

Contribution

It defines invariant universality for pairs (S, E), proves a sufficient condition for it, and shows its prevalence among many complete analytic quasi-orders in mathematics.

Findings

Invariant universality occurs for many complete analytic quasi-orders.

A general theorem provides a sufficient condition for invariant universality.

The phenomenon is widespread across different mathematical areas.

Abstract

We introduce the notion of an invariantly universal pair (S,E) where S is an analytic quasi-order and E \subseteq S is an analytic equivalence relation. This means that for any analytic quasi-order R there is a Borel set B invariant under E such that R is Borel bireducible with the restriction of S to B. We prove a general result giving a sufficient condition for invariant universality, and we demonstrate several applications of this theorem by showing that the phenomenon of invariant universality is widespread. In fact it occurs for a great number of complete analytic quasi-orders, arising in different areas of mathematics, when they are paired with natural equivalence relations.