On incidence algebras and directed graphs

Abstract

The incidence algebra I (X,□) 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,□) for (G,≤) over □, the ring of integers, by I (G,□) ≡ {fi,fi*:V×V→□} 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,□) is isomorphic to a subring of Mn (□). Principal ideals Iv of (V,≤) induce the subdigraphs □ Iv□ which are the principal ideals □v of (Gv,≤). They generate the ideals I (□v,□) of I (G,□). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded. Copyright © 2002 Hindawi Publishing Corporation. All rights reserved.