S = ⊕ a ∈ $_{G}$S$_{a}$ , as a group under the addition of S , such that S$_{b}$S$_{c}$ ⊆ S$_{bc}$ for all b,c ∈ G . The subgroups S$_{a}$ , a ∈ G , are called the homogeneous components, the elements of ∪$_{a}$$_{∈}$$_{G}$ S$_{a}$ are called the homogeneous elements, and every element is a sum of finitely many homogeneous elements. We let ∂s be the unique element of G such that s ∈ S$_{∂s}$ for any nonzero homogeneous s ∈ S . The support of S is the set Supp$_{G}$ S = { a ∈ G : S$_{a}$ = 0 } . The grading is called finite if Supp$_{G}$ S is a finite set. An important type of matrix algebra grading is a good grading (for example, see references [2], [3], and [4]). This definition extends easily to incidence algebras (see [4], [6], and [7]). In section 2 we state the definition of balanced relation introduced by Abrams (see [1]) and go over the construction of generalized incidence rings. Good semigroup gradings of generalized incidence rings are defined in section 3. Theorems 3.2 and 3.4 are fundamental for our constructions since they categorize good gradings of generalized incidence rings in terms of homomorphisms from the relations to the semigroup. Theorem 3.8 shows how to construct good semigroup gradings of incidence algebras over minimally connected partial orders. Suppose S = ⊕ a ∈ $_{G}$S$_{a}$ and T = ⊕ a ∈ $_{G}$T$_{a}$ are G -graded rings. A homomorphism of G -graded rings is a ring homomorphism h : S → T such that h ( S$_{a}$ ) ⊆ T$_{a}$ for all a ∈ Supp$_{G}$ S . An isomorphism which is a homomorphism of G -graded rings is called an isomorphism of G -graded rings. In the case of matrix algebras there are gradings which are not good gradings but are isomorphic to good gradings (see [3, Example 1.3]). Isomorphic gradings for good group gradings of incidence algebras over partial orders have been studied by Miller and Spiegel (see [6]). In section 4 we state the definitions of compression maps and stable relations 2