Local automorphisms of finitary incidence algebras.

Let $R$ be a commutative, indecomposable ring with identity and let $(P,\le )$ be a locally finite partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le )$ over $R$. In this case, the finitary incidence algebra exactly coincides with the incidence space. We will give an explicit criterion for when local automorphisms of $FI(P)$ are actually $R$-algebra automorphisms. We will also show that cases which do not meet that criterion may have nonsurjective local automorphisms- in particular, those maps are not $R$-algebra automorphisms, showing a strict inclusion of the collection of $R$-algebra automorphisms inside the collection of local 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. Finally, we will explore the automorphisms and local automorphisms of the $EndR(V),$ where $V$ is a free $R$-module and $R$ is no longer necessarily indecomposable.