A Sufficient Condition for a Quandle to be Latin

TL;DR

This paper establishes a sufficient condition based on cycle structures in the profile of a quandle, ensuring it is Latin, thereby linking algebraic properties to permutation cycle structures.

Contribution

It proves that if all right translations in a quandle have distinct cycle lengths, then the quandle is necessarily Latin, providing a new criterion for Latin quandles.

Findings

Cycle structures determine Latin property in quandles.

Distinct cycle lengths imply the quandle is Latin.

Provides a new algebraic characterization of Latin quandles.

Abstract

A quandle is an algebraic structure satisfying three axioms: idempotency, right-invertibility and right self-distributivity. In quandles, right translations are permutations. The profile of a quandle is the list of cycle structures, one per right translation in the quandle. In this note we prove that if, for each cycle structure in the profile of a quandle, no two cycle lengths are equal, then the quandle is latin -- this is the sufficient condition mentioned in the title.