Local automorphisms of finitary incidence algebras

Let R be a commutative, indecomposable ring with identity and (P,≤) a partially ordered set. Let FI(P) denote the finitary incidence algebra of (P,≤) over R. We will show that, in most cases, local automorphisms of FI(P) are actually R-algebra automorphisms. In fact, the existence of local automorphisms which fail to be R-algebra automorphisms will depend on the chosen model of set theory and will require the existence of measurable cardinals. We will discuss local automorphisms of cartesian products as a special case in preparation of the general result.