Enumerating finite racks, quandles and kei

TL;DR

This paper establishes exponential bounds on the number of isomorphism classes of finite racks, quandles, and kei, by linking their structure to permutation groups and extending previous theoretical results.

Contribution

It provides new upper and lower bounds on the count of finite racks, quandles, and kei, using the relationship between racks and their operator groups.

Findings

Derived bounds of the form 2^{cn^2} for the number of isomorphism classes

Extended the relationship between racks and their operator groups beyond previous work

Applied permutation group theory to classify and count algebraic structures

Abstract

A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all $y\in X$, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all $x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arbitrary rack is related to its operator group (the permutation group on $X$ generated by the maps $x\mapsto x\rack y$ for $y\in Y$), and then applying some of the theory of permutation groups. The relationship between a rack and its operator group extends results of Joyce and of Ryder; this relationship might be of independent interest.