Geodesic
Characterization of linear and alinear quasigroups
Diskretnaya Matematika, Tome 4 (1992) no. 2, pp. 142-147
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A quasigroup $(Q,\,\cdot\,)$ is said to be linear [alinear] if, for all $x,y\in Q$, $xy=\phi x+c+\psi y$, where $(Q,+)$ is some group, $\phi$ and $\psi$ are its automorphisms[antiautomorphisms], $c\in Q$. We prove that (primitive) linear [alinear] quasigroups are characterized by one identity in four variables. 
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     title = {Characterization of linear and alinear quasigroups},
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G. B. Belyavskaya; A. Kh. Tabarov. Characterization of linear and alinear quasigroups. Diskretnaya Matematika, Tome 4 (1992) no. 2, pp. 142-147. http://geodesic.mathdoc.fr/item/DM_1992_4_2_a16/