hold for all f ∈ I ( X, ρ, R ) and x, y, z, w ∈ X such that xρy and zρw . f ( x, y ) e$_{xy}$ = e$_{xx}$fe$_{yy}$ (3) e$_{xy}$e$_{zw}$ =      e$_{x,w}$ if y = z and xρw in X 0 otherwise (4) In any situation where we refer to a generalized incidence ring we mean an as- sociative ring with unity formed on the R -module of functions I ( X, ρ, R ) where R is a ring with unity and ρ is a locally finite balanced relation on X . The operation of R on I ( X, ρ, R ) does not play a significant role in our investigation. We reserve the term generalized incidence algebra for I ( X, ρ, R ) where R is a commutative ring with unity and ρ is a locally finite balanced relation on X . If, additionally, ρ is a partial order then I ( X, ρ, R ) is the usual incidence algebra over R (see [9]). 3 Good Gradings Definition 3.1 Assume G is a semigroup and ρ is a relation on X . 1. Set Trans ( X ) = { ( x, y, z ) : xρy , yρz , xρz , and x, y, z ∈ X } . A transitive triple in X is an ordered triple in Trans ( X ) . 2. We say Φ : ρ → G is a homomorphism if Φ ( x, y ) Φ ( y, z ) = Φ ( x, z ) holds for any x, y, z ∈ X such that ( x, y, z ) ∈ Trans ( X ) . 3. If I ( X, ρ, R ) is a G -graded generalized incidence ring then the grading is good if e$_{xy}$ is homogeneous for all x, y ∈ X such that xρy . 6