Figure 2: We use (b) and (c) to find a G -sourcing set for the relation of (a). relation on X$_{2}$ = { 1 , 2 , 3 , 4 , 5 } determined by (b) in Figure 2 then ρ$_{2}$ is a preorder on X$_{2}$ and there is a compression map θ : X$_{2}$ → X$_{1}$ given by θ (1) = 1 , θ (2) = 2 , θ (3) = 2 , θ (4) = 3 , and θ (5) = 4 . By equating paired elements we may relate ( X$_{2}$, ρ$_{2}$ ) to a partial order. This gives ρ$_{3}$ , the minimally-connected partial order on X$_{3}$ = { 1 , 2 , 3 , 4 } determined by (c) in Figure 2. By Theorem 3.8 a G -grading set for ρ$_{3}$ is σ$_{3}$ = { (1 , 2) , (3 , 4) } . Following the proof of Theorem 5.1 a G -grading set for ρ$_{2}$ is σ$_{2}$ = β ∪ γ with β = { (1 , 2) , (3 , 4) } and γ = { (4 , 5) } . The proof of Theorem 4.3 shows a G -grading set of ρ$_{1}$ is σ$_{1}$ = { (1 , 2) , (2 , 3) , (3 , 4) } . Thus for any associative ring with unity R and any group G the G -gradings of S = I ( X, ρ, R ) are uniquely determined by functions from σ$_{1}$ to G . Suppose σ is the Hasse diagram for a partial order ( X, ≤ ) and F is a field. The incidence algebra A = I ( X, ≤ , F ) is said to have the free-extension property (see [4, Definition 2]) if there exists γ ⊆ σ such that for any group G and any function φ : γ → G , there is a unique grading of I ( X, ≤ , F ) such that e$_{xy}$ ∈ A$_{φ}$$_{(}$$_{x,y}$$_{)}$ for all x, y ∈ X such that ( x, y ) ∈ γ . Theorem 3.4 shows a good group grading of a generalized incidence ring induced by a homomorphism must be a finite grading. This kind of result was already known for incidence algebras over partial orders (see [5, Theorem 3.3]). The free-extension property may fail for partial orders on an infinite set even if it contains a cross-cut 18