Non-affine latin quandles of order $2^k$

TL;DR

This paper characterizes the existence of non-affine latin quandles of order 2^k, showing they exist only for specific values of k, and provides a construction method using central extensions of affine quandles.

Contribution

It establishes the precise conditions for the existence of non-affine latin quandles of order 2^k and introduces a construction via central extensions.

Findings

Non-affine latin quandles of order 2^k exist only for k=6 or k≥8.

Construction method uses central extensions of affine quandles.

Provides a complete characterization of such quandles' orders.

Abstract

We prove that a non-affine latin quandle (also known as left distributive quasigroup) of order $2^k$ exists if and only if $k = 6$ or $k \geq 8$. The construction is expressed in terms of central extensions of affine quandles.