The middle translations of finite involutory latin quandles

TL;DR

This paper explores the structure of finite involutory Latin quandles, focusing on middle translations and spins, revealing their algebraic properties and construction methods from cyclic groups.

Contribution

It introduces a novel approach to constructing involutory Latin quandles from cyclic groups and investigates the properties of spins within these structures.

Findings

Left involutory Latin quandle of odd order n can be constructed from a cyclic group of order n.

The set of all right spins forms a cyclic group of odd order n.

Middle translations are key to understanding the structure of these quandles.

Abstract

This paper studies the left (right) middle translations on finite involutory latin quandles and their representations. It also shows that a left involutory latin quandle of odd order n can be constructed from a cyclic group of odd order by the application of left middle translations. Furthermore, the concept of spins of involutory latin quandle is investigated, and it is shown that if Q is a left involutory latin quandle of odd order n, then the set of all right spins (r-spins) is a cyclic group of odd order n under composition of mapping.