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Abstract
Given a multiplicative group G, an algebra A over the ring R is G-graded if A = [special characters omitted] Ag, where each Ag is an R-subspace of A and, for all g, h ∈ G, Ag Ah ⊆ Agh. Group gradings of the incidence algebra I(X, F), where X is a locally-finite partially-ordered set and F is a field are studied. In order to obtain a better understanding of the gradings of incidence algebras, we relate our techniques to those used in the study of gradings of full matrix algebras. We begin with several observations about the structure of any group grading of an incidence algebra, but then restrict our attention to two special classes of gradings: good gradings and elementary gradings. These classes of gradings are equivalent in the case of matrix algebras, but there exist incidence algebras that have good gradings that are not elementary gradings. We have provided an algorithmic method for finding all good gradings of an incidence algebra when X has a cross-cut of length one or two, and we have discussed the complications that arise in this method when X has a cross-cut of length greater than two. It is shown that the number of good gradings of an incidence algebra when X has a cross-cut of length one or two depends only on the size of G and not on its structure, but this is no longer true when X has a cross-cut of length greater than two.
