Algebras with Parastrophically Uncancellable Quasigroup Equations

TL;DR

This paper classifies 48 specific quadratic functional equations involving quasigroup operations, providing linear representations for the operations satisfying these equations, which advances understanding of algebraic structures with complex identities.

Contribution

It introduces a classification of parastrophically uncancellable quadratic equations and derives linear representations for quasigroup operations satisfying these identities.

Findings

Linear representations for quasigroup operations are established.

Classification of 48 quadratic functional equations into two classes.

Results apply to binary algebras satisfying hyperidentities.

Abstract

We consider $48$ parastrophically uncancellable quadratic functional equations with four object variables and two quasigroup operations in two classes: balanced non--Belousov (consists of 16 equations) and non--balanced non--gemini (consists of 32 equations). A linear representation of a group (Abelian group) for a pair of quasigroup operations satisfying one of these parastrophically uncancellable quadratic equations is obtained. As a consequence of these results, a linear representation for every operation of a binary algebra satisfying one of these hyperidentities is presented.