Algebras with parastrophically uncancellable quasigroup equations

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 obtained.