Geodesic
Algebras with parastrophically uncancellable quasigroup equations
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2016), pp. 41-63

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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. 
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Amir Ehsani; Aleksandar Krapež; Yuri Movsisyan. Algebras with parastrophically uncancellable quasigroup equations. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2016), pp. 41-63. http://geodesic.mathdoc.fr/item/BASM_2016_1_a3/