Differential equations of geometric odule (Q1363296)

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.