S = ⊕ a ∈ $_{G}$S$_{a}$ then we may write h as the sum of homogeneous elements. Thus there is a positive integer p and distinct b$_{1}$, . . . , b$_{p}$ ∈ G such that h = h$_{1}$ + · · · + h$_{p}$ where ∂h$_{i}$ = b$_{i}$ for each i = 1 , . . . , p . We will show Im Φ ⊆ { b$_{1}$, . . . , b$_{p}$ } . For all x, y ∈ X if xρy we have h ( x, y ) = 1 hence h$_{1}$ ( x, y ) + · · · + h$_{p}$ ( x, y ) = 1. Then h$_{i}$ ( x, y ) = 0 for some i ≤ p so, by construction, we have Φ ( x, y ) = b$_{i}$ . Since x, y ∈ X with xρy were arbitrarily chosen we can conclude Im Φ ⊆ { b$_{1}$, . . . , b$_{p}$ } . This gives the desired result, Im Φ is finite. Definition 3.3 We say a grading is induced by a homomorphism Φ if it can be constructed in the setting of Theorem 3.2 with Im Φ finite. Theorem 3.4 Let S = I ( X, ρ, R ) be a generalized incidence ring and let G be a monoid. Suppose S = ⊕ a ∈ $_{G}$S$_{a}$ is a G -graded ring. 1. If the grading is good then there is a homomorphism Φ : ρ → G given by Φ ( x, y ) = ∂e$_{x,y}$ for all x, y ∈ X with xρy . 2. If the grading is good and e$_{xx}$ ∈ S$_{1}$ for all x ∈ X then the grading is induced by Φ . Moreover, the grading is finite. Proof. Part 1 can be proved directly from equation 4. To prove part 2 we let a ∈ Supp$_{G}$ S be arbitrarily chosen. We will show S$_{a}$ is given by equation 5 so the grading is induced by Φ. Then the grading is finite since Supp$_{G}$ S = Im Φ. Suppose f ∈ I ( X, ρ, R ) and for all x, y ∈ X with xρy if f ( x, y ) = 0 then Φ ( x, y ) = a . We must prove such an f is contained in S$_{a}$ . Since S is graded there is a positive integer m and nonzero homogeneous f$_{1}$, . . . , f$_{m}$ ∈ S such that 8