Graphings of arithmetical equivalence relations

TL;DR

This paper explores when arithmetical equivalence relations can be represented as connectedness relations of simpler graphs, providing examples and methods for such graphings, including for complex relations like Friedman-Stanley jumps.

Contribution

It introduces new techniques for representing complex arithmetical equivalence relations as graph connectedness relations, including the first arithmetical construction for Friedman-Stanley jumps.

Findings

The $ ext{Σ}^0_3$ relation of computable isomorphism is $ ext{Π}^0_2$-graphable.

Several examples of equivalence relations are shown to be realizable as graph connectedness.

A method for constructing graphings of Friedman-Stanley jumps is developed.

Abstract

This paper studies when an arithmetical equivalence relation $E$ can be realized as the connectedness relation of a graph $G$ which is simpler to define than $E$. Several examples of such equivalence relations are established. In particular, it is proved that the $\Sigma^0_3$ relation of computable isomorphism of structures on $\N$ in a computable first-order language is $\Pi^0_2$-graphable, i.e., is the connectedness relation of a $\Pi^0_2$ graph. Graphings of Friedman-Stanley jumps are studied, including an arithmetical construction of a graphing of the Friedman-Stanley jump of $E$ from a graphing of $E$.