On incidence algebras and directed graphs

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 ( G v , ≤ ) 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 , ℤ ) for ( G , ≤ ) over ℤ , the ring of integers, by I ( G , ℤ ) = { f i , f i * : V × V → 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 , ℤ ) is isomorphic to a subring of M n ( ℤ ) . Principal ideals I v of ( V , ≤ ) induce the subdigraphs 〈 I v 〉 which are the principal ideals ℐ v of ( G v , ≤ ) . They generate the ideals I ( ℐ v , 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.