# Local automorphisms of finitary incidence algebras

## TL;DR

This paper investigates local automorphisms of finitary incidence algebras over certain rings, showing that they are mostly automorphisms, with exceptions linked to set-theoretic assumptions like measurable cardinals.

## Contribution

It establishes conditions under which local automorphisms of finitary incidence algebras are actual automorphisms, connecting algebraic properties with set theory.

## Key findings

- Most local automorphisms are algebra automorphisms.
- Existence of non-automorphism local automorphisms depends on set theory.
- Results include special cases like automorphisms of Cartesian products.

## Abstract

Let $R$ be a commutative, indecomposable ring with identity and $(P,\le)$ a partially ordered set. Let $FI(P)$ denote the finitary incidence algebra of $(P,\le)$ 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.