International Journal of Mathematics and Mathematical Sciences
Volume 31 (2002), Issue 5, Pages 301-305
On incidence algebras and directed graphs
Department of Mathematics, St. Dominic's College, Kanjirapally 686512, Kerala, India
Received 20 June 2001
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.
The incidence algebra of a locally finite poset has been defined and studied by Spiegel and O'Donnell (1997). A poset has a directed graph representing it. Conversely, any directed graph without any cycle, multiple edges, and loops is represented by a partially ordered set . So in this paper, we define an incidence
algebra for over , the ring of integers, by where denotes the number of directed paths of length from to and . When is finite of order , is isomorphic to a subring of . Principal ideals of induce the subdigraphs which are the principal ideals of . They generate the ideals of . These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.