Theorem 3.2 Assume G is a semigroup, I ( X, ρ, R ) is a generalized incidence ring, and Φ : ρ → G is a homomorphism. Let S$_{a}$ be given by equation 5 for each a ∈ G . Then I ( X, ρ, R ) = ⊕ a ∈ $_{G}$S$_{a}$ is a G -graded ring if and only if Im Φ is finite. S$_{a}$ = { f ∈ I ( X, ρ, R ) : f ( r ) = 0 implies Φ ( r ) = a for all r ∈ ρ } (5) Proof. It is easy to see S$_{a}$ is an R -submodule for all a ∈ G and S$_{a}$ ∩ S$_{b}$ = { 0 } if b ∈ G and b = a . We show S$_{a}$S$_{b}$ ⊆ S$_{ab}$ for all a, b ∈ G . Suppose f ∈ S$_{a}$ , g ∈ S$_{b}$ and ( fg ) ( x, y ) = 0 for some x, y ∈ X with xρy . By equation 1 there exists z ∈ [ x, y ] such that xρz , zρy , and f ( x, z ) g ( z, y ) = 0. Thus f ( x, z ) = 0 and g ( z, y ) = 0 which implies Φ ( x, z ) = a and Φ ( z, y ) = b . Moreover, ab = Φ ( x, y ) since ( x, z, y ) is a transitive triple and Φ is a homomorphism. This proves fg ∈ S$_{ab}$ as desired. To complete the proof we show Im Φ is finite if and only if I ( X, ρ, R ) = ⊕ a ∈ $_{G}$S$_{a}$ . First assume Im Φ is a finite subset of G . Then there is a positive integer m and a$_{1}$, . . . , a$_{m}$ ∈ G such that Im Φ = { a$_{1}$, . . . , a$_{m}$ } . We must prove an arbitrarily chosen f ∈ S is a sum of finitely many homogeneous elements. For each i = 1 , . . . , m let f$_{i}$ ∈ I ( X, ρ, R ) be the the function satisfying equation 6 for all ( x, y ) ∈ X . By construction f$_{i}$ ∈ S$_{a}$$_{i}$ for all i ≤ m . It is easy to prove f is the sum of finitely many homogeneous elements, f$_{1}$, . . . , f$_{m}$ , as desired. To prove the other direction assume S = ⊕ a ∈ $_{G}$S$_{a}$ . We choose h ∈ I ( X, ρ, R ) so that for all x, y ∈ X we have h ( x, y ) = 1 if xρy and otherwise h ( x, y ) = 0. If 7 f$_{i}$ ( x, y ) =      f ( x, y ) if xρy in X and Φ ( x, y ) = a$_{i}$ 0 otherwise (6)