The translation equation on certainn-groups

Summary

In this note the characterization of all solutions of the equation

$$\Phi (\Phi (\Phi ( \ldots \Phi (\alpha , x_1 ), x_2 ), \ldots ), x_{n - 1} ), x_n ) = \Phi (\alpha , x_1 \cdot x_2 \cdot \ldots \cdot x_n ), wheren \geqslant 3, \Phi :\Gamma \times G \to \Gamma $$

and (G, ⋅) is a group, is given. This equation represents the translation equation on particularn-groups which are obtained from a binary group (G, ⋅) by definition of ann-group operation as follows: [x ₁...x ₂]≔x ₁⋅x ₂⋅...⋅x _n . We will show that, in order to characterize all solutions of the above mentioned equation, it is necessary and sufficient to describe all solutions of the translation equationF(F(α, x), y) = F(α, x ⋅ y) fulfilling certain conditions.