# Invariant universality for quandles and fields

## TL;DR

This paper proves that the embeddability relations for countable quandles and fields are maximally complex, meaning they encompass all analytic quasi-orders, highlighting their deep structural complexity in Borel reducibility theory.

## Contribution

It establishes the invariant universality of embeddability relations for countable quandles and fields, advancing understanding of their complexity in descriptive set theory.

## Key findings

- Embeddability relations are maximally complex for these algebraic structures.
- Countable quandles' embeddability relation is a complete analytic quasi-order.
- The results apply to fields of any characteristic other than 2.

## Abstract

We show that the embeddability relations for countable quandles and for countable fields of any given characteristic other than 2 are maximally complex in a strong sense: they are invariantly universal. This notion from the theory of Borel reducibility states that any analytic quasi-order on a standard Borel space essentially appears as the restriction of the embeddability relation to an isomorphism-invariant Borel set. As an intermediate step we show that the embeddability relation of countable quandles is a complete analytic quasi-order.