The structure of idempotent translatable quasigroups

TL;DR

This paper characterizes the structure of idempotent k-translatable quasigroups, providing algebraic conditions, exploring their connections with geometry and combinatorics, and analyzing their parastrophes and quadratic properties.

Contribution

It offers a complete characterization of idempotent k-translatable quasigroups via modular arithmetic and explores their structural properties and relationships with other algebraic and geometric concepts.

Findings

A necessary and sufficient condition for a groupoid to be an idempotent k-translatable quasigroup.

Description of the structure of various idempotent, k-translatable quasigroups and their connections to affine geometry.

Conditions under which k-translatable quasigroups are quadratic.

Abstract

We prove the main result that a groupoid of order n is an idempotent k-translatable quasigroup if and only if its multiplication is given by x.y = (ax+by)(mod n), where a+b = 1(mod n), a+bk = 0(mod n) and (k,n)= 1. We describe the structure of various types of idempotent, k-translatable quasigroups, some of which are connected with affine geometry and combinatorial algebra, and their parastrophes. We prove that such parastrophes are also idempotent, translatable quasigroups and determine when they are of the same type as the original quasigroup. In addition, we find several different necessary and sufficient conditions making a k-translatable quasigroup quadratical.