The paper deals with the continuous solutions \(H: S\to [0,\infty)\) of \(H(x)=\int_{S}H(x+y)\sigma (dy)\) where S is a certain type of topological semigroup and \(\sigma\) is a measure on the Borel subsets of S. The results, too complicated to be given here, generalize earlier theorems where S was assumed to be a group. They rely on martingale arguments for their proofs and can be applied to characterize multivariate distributions.