Local automorphisms of finitary incidence algebras.


Let R be a commutative, indecomposable ring with identity and let (P,≤) be a locally finite partially ordered set. Let FI(P) denote the finitary incidence algebra of (P,≤) 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.



Local automorphism. Finitary incidence algebra.