2. A subset γ of ρ is a G -essential set for ρ if for all homomorphisms Φ$_{1}$ , Φ$_{2}$ : ρ → G such that Φ$_{2}$ = Φ$_{1}$ there exists c ∈ γ such that Φ$_{2}$ ( c ) = Φ$_{1}$ ( c ) . 3. A G -extendible and G -essential subset is called a G -grading set for ρ . Remark 3.7 σ is a G -grading set for ρ if and only if for every function φ : σ → G there is a unique homomorphism Φ : ρ → G such that Φ |$_{σ}$ = φ . If ρ is balanced and ρ contains a G -grading set then all good G -gradings of a generalized incidence ring S = I ( X, ρ, R ) such that e$_{xx}$ ∈ S$_{1}$ for all x ∈ X are uniquely determined by grading admissible functions from the G -grading set to G . We finish this section with a result on partial orders. Recall ( X, ≤ ) is minimally connected if ( X, ≤ ) is a connected, locally finite partial-order and [ x, y ] is either empty or a chain for all x, y ∈ X . The Hasse diagram of X is the directed graph H with vertex set X and arrow set { ( a, b ) : a, b ∈ X and b covers a } . Theorem 3.8 Assume G is a semigroup, R is an associative ring with unity, and ( X, ≤ ) is a minimally-connected partial order. Then the arrow set of the Hasse diagram of ( X, ≤ ) is a G -grading set for ≤ . Proof. Let φ : σ → G be given, where σ is the set of arrows in the Hasse diagram for ( X, ≤ ). We use φ to define a function Φ : ≤→ G . Let x, y ∈ X such that x ≤ y be given. Since ( X, ≤ ) is a minimally connected there is a unique chain { x$_{1}$, . . . , x$_{m}$ } in σ such that x = x$_{1}$ ≤ x$_{2}$ ≤ . . . ≤ x$_{m}$ = y and ( x$_{i}$, x$_{i}$$_{+1}$ ) ∈ σ for each i < m . We set Φ ( x, y ) = φ ( x$_{1}$, x$_{2}$ ) · · · φ ( x$_{m}$$_{−}$$_{1}$, x$_{m}$ ). A straightforward check proves Φ is a homomorphism such that Φ |$_{σ}$ = φ . Thus σ is a G -extendible set for ≤ . 10