The translation equation on certain \(n\)-groups (Q1317959)

Let \(\Gamma\) be an arbitrary set and \((G, \circ)\) be a group. Define an \(n\)-group operation in \(G\) as follows: \([x_ 1, x_ 2 \dots x_ n] = x_ 1 \cdot x_ 2 \cdot \cdots \cdot x_ n\). Let \(n \geq 3\) be a natural number and \(\varphi\) be a map from \(\Gamma \times G\) to \(\Gamma\). The author considers the translation equation \[ \varphi \Bigl( \varphi \biggl( \varphi \bigl( \dots \varphi (\alpha,x_ 1), x_ 2 \bigr), \dots,x_{n-1}\biggr), x_ n \Bigr) = \varphi (\alpha, x_ 1 \cdot x_ 2 \cdot \cdots \cdot x_ n), \] where \(\alpha \in \Gamma\) and \(x_ 1, x_ 2, \dots, x_ n \in G\), and shows, that in order to characterize all solutions of the above said translation equation it is necessary and sufficient to describe all solutions of the translation equation \(F(F(\alpha,x), y) = F(\alpha,x\cdot y)\) where \(F:\Gamma \times G \to \Gamma\) is a map fulfilling certain conditions.