Commutativity preservers of incidence algebras

TL;DR

This paper characterizes bijective linear maps on incidence algebras of finite posets that strongly preserve commutativity and fix diagonal subalgebras, showing they decompose into simpler preserver maps.

Contribution

It provides a complete description of commutativity-preserving bijections on incidence algebras, including their decomposition into shift-type and quadruple-associated maps.

Findings

Characterization of bijective linear maps preserving commutativity

Decomposition of such maps into shift and quadruple-based components

Maps fix the diagonal subalgebra D(X,K)

Abstract

Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $\varphi:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $\varphi(D(X,K))=D(X,K)$. We prove that such a map $\varphi$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(\theta,\sigma,c,\kappa)$ of simpler maps $\theta$, $\sigma$, $c$ and a sequence $\kappa$ of elements of $K$.