(see [8]). If G is a cancellative monoid then Theorem 4.3 shows compression maps provide a correspondence between good G -gradings. Stable relations are used to describe a class of generalized incidence rings which are isomorphic to subrings of incidence rings over preorders by Theorem 4.5. The isomorphism, which is described in Lemma 4.2, is an isomorphism of G -graded rings. Good group gradings are considered in section 5. The main result of this section is Theorem 5.1, which extends [4, Theorem 4] to good group gradings for incidence algebras over preorders with crosscuts of length one or two. The conclusion of our theorem is modified to account for finite gradings. We show our result for preorders is related to generalized incidence rings in Corollary 5.2 and Example 5.3. We finish with example 5.4, which describes a partial order with a minimal element whose incidence algebra does not have the free-extension property. Unfortunately this is a counterexample to [4, Theorem 4]. 2 Generalized Incidence Rings By definition a relation ρ on a set X is a subset of X × X . We adopt the usual convention of writing xρy if x, y ∈ X satisfy ( x, y ) ∈ ρ . The notation xρy is often shorter and more convenient, but the notation ( x, y ) ∈ ρ will be used when it is helpful to describe the relation as a set of ordered pairs. The directed graphs shown in Figure 1 represent reflexive relations. We omit loops in all diagrams so that the arrows match up with elements of the off-diagonal subset of relations. The construction of generalized incidence rings does not require the relation to 3