Abstract. We investigate the functional equation(1) (1) was considered with constants a = 3, b = 2 and a = 9 and b = 4, respectively.1. Let (M, +) be an Abelian group in which the unique division by 2 and 3 is performable. Let (S, +) be an Abelian semigroup. Suppose that S contains the identity element 0 and for each λ ≥ 0 and s ∈ S, an element λ · s in S is defined. It is assumed that the multiplication [0, ∞)×S (λ, s) −→ λ·s ∈ S satisfies the following axioms: