A characterization of generalized exponential polynomials in terms of decomposable functions

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\text{\hspace{0.17em}⁣}:G\to \text{cc}$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1+\dots +x_n)$ is decomposable; that is, if $f(x_1+\dots +x_n)=\text{sumik}{u}_i\text{cd}{v}_i$, where the function $u_i$ only depends on the variables belonging to a set $\emp \ne E_i \subsetneq { x_1 \stb x_n }$,and$v_i$ only depends on the variables belonging to $\{x_1\text{stb}{x}_n\}\text{se}{E}_i$ $(i=1\text{stb}k)$.