  This inquiry is based on both the construction of generalized incidence rings
due to Gene Abrams and the construction of good group gradings of incidence
algebras due to Molli Jones. We provide conditions for a generalized incidence
ring to be graded isomorphic to a subring of an incidence ring over a preorder.
We also extend Jones's construction to good group gradings for incidence
algebras over preorders with crosscuts of length one or two.
