Ancykutty Joseph

Copyright © 2002 Ancykutty Joseph. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.