Geodesic
When is it hard to show that a quasigroup is a loop?
Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 241-247

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We contrast the simple proof that a quasigroup which satisfies the Moufang identity $(x\cdot yz)x = xy\cdot zx$ is necessarily a loop (Moufang loop) with the remarkably involved prof that a quasigroup which satisfies the Moufang identity $(xy\cdot z)y=x(y\cdot zy)$ is likewise necessarily a Moufang loop and attempt to explain why the proofs are so different in complexity.
Classification : 20N05
Keywords: Moufang quasigroups; Moufang loops; identities of Bol-Moufang type 
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Keedwell, A. D. When is it hard to show that a quasigroup is a loop?. Commentationes Mathematicae Universitatis Carolinae, Tome 49 (2008) no. 2, pp. 241-247. http://geodesic.mathdoc.fr/item/CMUC_2008__49_2_a5/