Une généralisation d'un résultat de J. Aczél et M. Hosszú sur l'équation de translation. (A generalization of a result by J. Aczél and M. Hosszú on the translation equation) (Q1120081)

Let \(\Gamma\) be an arbitrary set and G a group. The author offers necessary and sufficient conditions for the general solution F: \(\Gamma\) \(\times G\to \Gamma\) of the translation equation \((T)\quad F(F(\alpha,x),y)=F(\alpha,xy)\) to be of the form \(F(\alpha,x)=f^{- 1}(f(\alpha)\ell (x)),\) where f: \(\Gamma\) \(\to G_ 1\) is a bijection, \(\ell: G\to G_ 1\) a homomorphism, \(G_ 1\) being isomorphic to G. If, in particular, \(G={\mathbb{R}}^ m\) and \(\Gamma\) is a proper interval, then \[ F(\alpha,x_ 1,...,x_ m)=f^{-1}(f(\alpha)+c_ 1x_ 1+...+c_ mx_ m) \] is the general solution of (T) if, and only if, F is transitive (\(\forall \alpha,\beta \in \Gamma\), \(\exists (x_ 1,...,x_ m)\in {\mathbb{R}}^ m:\) \(F(\alpha,x_ 1,...,x_ m)=\beta)\) and there exists an \(\alpha_ 0\) such that \(F(\alpha_ 0,.)\) is continuous.