Canonical forms for operation tables of finiate connected quandles

TL;DR

This paper introduces natural orderings for finite connected quandles, classifies automorphisms based on these orderings, and explores the structure of their operation tables, providing insights into their algebraic properties.

Contribution

It defines natural orderings for finite connected quandles and classifies automorphisms when the operation table exhibits certain cyclic properties.

Findings

Operation tables coincide when permutation *q is a cycle of length n-1.

Every row and column of the operation table contains all elements of Q.

Automorphisms correspond to natural orderings under specific conditions.

Abstract

We introduce a notion of natural orderings of elements of finite connected quandles of order $n$. When the elements of such a quandle $Q$ are already ordered naturally, any automophism on $Q$ is a natural ordering. Although there are many natural orderings, the operation tables for such orderings coincide when the permutation $*q$ is a cycle of length $n-1$. This leads to the classification of automorphisms on such a quandle. Moreover, it is also shown that every row and column of the operation table of such a quandle contains all the elements of $Q$, which is due to K. Oshiro. We also consider the general case of finite connected quandles.