On incidence algebras and directed graphs (Q1607948)

Summary: The incidence algebra \(I (X,{\mathbb{R}})\) of a locally finite poset \((X,\leq)\) has been defined and studied by \textit{E. Spiegel} and \textit{C. J. O'Donnell} [Incidence algebras (Pure and Applied Mathematics 206, Marcel Dekker, New York) (1977; Zbl 0871.16001)]. A poset \((V,\leq)\) has a directed graph \((G_{v},\leq)\) representing it. Conversely, any directed graph \(G\) without any cycle, multiple edges, and loops is represented by a partially ordered set \(V_{G}\). So in this paper, we define an incidence algebra \(I (G,\mathbb{Z})\) for \((G,\leq)\) over \(\mathbb{Z} \), the ring of integers, by \(I (G,\mathbb{Z})= \{ f_{i}\mid f_i^*:V\times V\to \mathbb{Z} \}\) where \(f_{i}(u,v)\) denotes the number of directed paths of length \(i\) from \(u\) to \(v\) and \(f_i^*(u,v) = -f_{i}(u,v)\). When \(G\) is finite of order \(n\), \(I (G,\mathbb{Z})\) is isomorphic to a subring of \(M_{n}(\mathbb{Z})\). Principal ideals \(I_{v}\) of \((V,\leq)\) induce the subdigraphs \(\langle I_{v}\rangle\) which are the principal ideals \({\mathcal I}_{v}\) of \((G_{v},\leq)\). They generate the ideals \(I({\mathcal I}_{v},\mathbb{Z})\) of \(I(G,\mathbb{Z})\). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.