Geodesic
Groupoids with the identity defining the commutative Moufang loops
Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 6, pp. 33-39.

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It is known that in the class of loops the identity $xy\cdot xz=x^2\cdot yz$ defines the commutative Moufang loops. In the article, we consider groupoids with this identity and establish properties of some classes of such groupoids. 
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G. B. Belyavskaya; A. Kh. Tabarov. Groupoids with the identity defining the commutative Moufang loops. Fundamentalʹnaâ i prikladnaâ matematika, Tome 14 (2008) no. 6, pp. 33-39. http://geodesic.mathdoc.fr/item/FPM_2008_14_6_a1/

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