On and around the balanced Cauchy equation

Let G be a unital semigroup, \((K, +)\) an Abelian group. We extend to this case results of several authors on functions \(f: G\rightarrow K\) satisfying the equations

$$\begin{aligned} f(x_1x_2\ldots x_n) = \sum _{i=1}^n F_i(x_1, \ldots ,\widehat{x_i}, \ldots ,x_n) \end{aligned}$$

and

$$\begin{aligned} \sum _{k=0}^n(-1)^{k}\sum _{|S|=n-k}f \left( \prod _Sx \right) = 0. \end{aligned}$$

We also study a more general class of equations:

$$\begin{aligned} f(x_1x_2\ldots x_n) = \sum _{i=1}^n \sum _{j=1}^{J_i}p_{ij}(x_i)(F_{ij}(x_1,\ldots ,\widehat{x_i}, \ldots , x_n)), \end{aligned}$$

where all \(p_{ij}\) are polynomial maps from G to the group of all endomorphisms of K.