On incidence algebras and directed graphs - Ancykutty Joseph (International Journal of Mathematics and Mathematical Sciences)
The incidence algebra I(X,ℝ) of a locally finite poset (X,≤) has been defined and studied by Spiegel and O'Donnell (1997). A poset (V,≤) has a directed graph (Gv,≤) representing it. Conversely, any directed graph G without any cycle, multiple edges, and loops is represented by a partially ordered set VG. So in this paper, we define an incidence algebra I(G,ℤ) for (G,≤) over ℤ, the ring of integers, by I(G,ℤ)={fi,fi*:V×V→ℤ} where fi(u,v) denotes the number of directed paths of length i from u to v and fi*(u,v)=−fi(u,v). When G is finite of order n, I(G,ℤ) is isomorphic to a subring of Mn(ℤ). Principal ideals Iv of (V,≤) induce the subdigraphs 〈Iv〉 which are the principal ideals ℐv of (Gv,≤). They generate the ideals I(ℐv,ℤ) of I(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.