Autoparatopisms of Quasigroups and Latin Squares

TL;DR

This paper investigates autoparatopisms of Latin squares, characterizes their properties, determines the set for orders up to 17, and analyzes their asymptotic proportion among all paratopisms.

Contribution

It introduces new properties of autoparatopisms, explicitly determines the set for small orders, and studies their asymptotic behavior as the order grows.

Findings

Determined autoparatopism sets for Latin squares of order up to 17.

Established general properties of autoparatopisms.

Analyzed the asymptotic proportion of autoparatopisms among all paratopisms.

Abstract

Paratopism is a well known action of the wreath product $\mathcal{S}_n\wr\mathcal{S}_3$ on Latin squares of order $n$. A paratopism that maps a Latin square to itself is an autoparatopism of that Latin square. Let $\mathrm{Par}(n)$ denote the set of paratopisms that are an autoparatopism of at least one Latin square of order $n$. We prove a number of general properties of autoparatopisms. Applying these results, we determine $\mathrm{Par}(n)$ for $n\le17$. We also study the proportion of all paratopisms that are in $\mathrm{Par}(n)$ as $n\rightarrow\infty$.