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Differential equations of geometric odule. (English. Russian original) Zbl 0890.53023
Dokl. Math. 52, No. 2, 268-270 (1995); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 344, No. 6, 745-748 (1995).
By definition, a geometric odule is a left loop $$M(\cdot)$$ such that (1) power associativity law holds in $$M$$ and (2) the operator $$x\to x^t$$ commutes with the operators $$L^{-1}_{a\cdot b}\circ L_a\circ L_b$$, where $$L_a(x)=a\cdot x$$. Differential equations of a geometric odule are found and it is proved that any geometric odule is a geodesic one for some affine connection.
##### MSC:
 53B05 Linear and affine connections 20N05 Loops, quasigroups
##### Keywords:
analytic loop; geometric odule; affine connection