# Commuting maps on certain incidence algebras

## TL;DR

This paper characterizes when commuting maps on incidence algebras over certain rings are proper, focusing on cases where the underlying pre-ordered set has finitely many connected components.

## Contribution

It provides a necessary and sufficient condition for all commuting maps on these incidence algebras to be proper, extending understanding of their algebraic structure.

## Key findings

- Characterization of proper commuting maps on incidence algebras
- Condition applies when the pre-ordered set has finitely many connected components
- Advances the theory of algebraic mappings in incidence algebras

## Abstract

Let $\mathcal{R}$ be a $2$-torsion free commutative ring with unity, $X$ a locally finite pre-ordered set and $I(X,\mathcal{R})$ the incidence algebra of $X$ over $\mathcal{R}$. If $X$ consists of a finite number of connected components, in this paper we give a sufficient and necessary condition for each commuting map on $I(X,\mathcal{R})$ being proper.