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 $fcolon Gto cc$ is a generalized exponential polynomial if and only if there is an $nge 2$ such that $f(x_1 +ldots +x_n )$ is decomposable; that is, if $f(x_1 +ldots +x_n )=sumik u_i cd v_i$, where the function $u_i$ only depends on the variables belonging to a set $emp e E_i subsetneq { x_1 stb x_n }$, and $v_i$ only depends on the variables belonging to ${ x_1 stb x_n } se E_i$ $(i=1stb k)$.