Addition theorems and representations of topological semigroups

Let a representation T of semigroup G on linear space X   be given. We call x∈Xx∈Xa finite vector   if its orbit T(G)T(G) is contained in a finite-dimensional subspace. In this paper some statements on finite vectors will be proved and applied to functional equationsf(x+y+z)=a1(x)b1(y,z)+a2(y)b2(x,z)+a3(z)b3(x,y)+∑k=1mαk(x)βk(y)γk(z) andf(x1+⋯+xn)=∑γ∏Δ∈γaΔγ(∑i∈Δxi), where γ   runs through all possible partitions of the set {1,…,n}=Δ1∪⋯∪Δr{1,…,n}=Δ1∪⋯∪Δr, r>1r>1, Δi≠∅Δi≠∅, and aΔγ-continuous functions on G.