Minimal Equivalence Relations in Hyperarithmetical and Analytical Hierarchies

TL;DR

This paper investigates the structure of minimal equivalence relations within the hyperarithmetical and analytical hierarchies, revealing the existence of infinitely many incomparable minimal relations in various complexity classes.

Contribution

It establishes the existence of infinitely many pairwise incomparable minimal equivalence relations in classes like al^0_\u03b1, al^1_n, expanding understanding of the degree structure under computable reducibility.

Findings

Existence of infinitely many incomparable minimal relations in al^0_\u03b1

Existence of infinitely many incomparable minimal relations in al^1_n

Minimal relations are properly within their classes

Abstract

A standard tool for classifying the complexity of equivalence relations on $\omega$ is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let $\Gamma$ be one of the following classes: $\Sigma^0_{\alpha}$, $\Pi^0_{\alpha}$, $\Sigma^1_n$, or $\Pi^1_n$, where $\alpha \geq 2$ is a computable ordinal and $n$ is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in $\Gamma$.