Alienation of the quadratic, exponential and d’Alembert functional equations

Let \((S,+,0)\) be a commutative monoid, \(\sigma :S\rightarrow S\) be an endomorphism with \(\sigma ^2=id\) and let K be a field of characteristic different from 2. We study the solutions \(f,g,h:S\rightarrow K\) of the Pexider type functional equation

$$\begin{aligned} f(x+y)+f(x+\sigma y)+g(x+y)=2f(x)+2f(y)+g(x)g(y) \end{aligned}$$

resulting from summing up the well known quadratic and exponential functional equations side by side. We show that under some additional assumptions the above equation forces f and g to split back into the system of two equations

$$\begin{aligned} \left\{ \begin{array}{ll}f(x+y)+f(x+\sigma y)=2f(x)+2f(y)\\ g(x+y)=g(x)g(y)\end{array}\right. \end{aligned}$$

for all \(x,y\in S\) (alienation phenomenon). We also consider an analogous problem for the quadratic and d’Alembert functional equations as well as for the quadratic, exponential and d’Alembert functional equations.