Each semipolynomial on a group is a polynomial

Given a semigroup S and a uniquely divisible group H , for every h ∈ S we define the right difference operator Δ h on functions f : S ⟶ H as follows: Δ h f ( g ) = f ( g h ) − f ( g ) . Each of the following two conditions on a function f can be considered as a characterization of polynomial mappings on S : 1) Δ h 1 ⋯ Δ h m f = 0 for some m ∈ N and every h 1 , … , h m ∈ S . 2) there is m ∈ N such that Δ h m f = 0 for every h ∈ S ; We prove that if S possesses the property g S = S g for every g ∈ S , then two classes of functions, defined by the above conditions respectively, coincide.