Reduced finitary incidence algebras and their automorphisms

Let [Formula: see text] denote the incidence algebra of a locally finite poset [Formula: see text] over a field [Formula: see text] and [Formula: see text] some equivalence relation on the set of generators of [Formula: see text]. Then [Formula: see text] is the subset of [Formula: see text] of all the elements that are constant on the equivalence classes of [Formula: see text]. If [Formula: see text] satisfies certain conditions, then [Formula: see text] is a subalgebra of [Formula: see text] called a reduced incidence algebra. We extend this notion to finitary incidence algebras [Formula: see text] for any poset [Formula: see text]. We investigate reduced finitary incidence algebras [Formula: see text] and determine their automorphisms in some special cases.

On incidence algebras and directed graphs

The incidence algebraI(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 graphGwithout any cycle, multiple edges, and loops is represented by a partially ordered setVG. So in this paper, we define an incidence algebraI(G,ℤ)for(G,≤)overℤ, the ring of integers, byI(G,ℤ)={fi,fi*:V×V→ℤ}wherefi(u,v)denotes the number of directed paths of lengthifromutovandfi*(u,v)=−fi(u,v). WhenGis finite of ordern,I(G,ℤ)is isomorphic to a subring ofMn(ℤ). Principal idealsIvof(V,≤)induce the subdigraphs〈Iv〉which are the principal idealsℐvof(Gv,≤). They generate the idealsI(ℐv,ℤ)ofI(G,ℤ). These results are extended to the incidence algebra of the digraph representing a locally finite weak poset both bounded and unbounded.