Linear maps preserving products equal to primitive idempotents of an incidence algebra

TL;DR

This paper characterizes bijective linear maps on incidence algebras that preserve products equal to primitive idempotents, showing such maps are either automorphisms or their negatives.

Contribution

It fully characterizes when such product-preserving maps exist on incidence algebras and proves they are either automorphisms or their negatives.

Findings

Such maps exist only under specific conditions.

They are either automorphisms or negatives of automorphisms.

The characterization applies to incidence algebras of finite connected posets.

Abstract

Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $\varphi:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $\varphi(a_1)\varphi(a_2)=b$. In this paper we study bijective linear maps $\varphi:I(X,F)\to I(X,F)$ preserving products equal to primitive idempotents of $I(X,F)$, where $I(X,F)$ is the incidence algebra of a finite connected poset $X$ over a field $F$. We fully characterize the situation, when such a map $\varphi$ exists, and whenever it does, $\varphi$ is either an automorphism of $I(X,F)$ or the negative of an automorphism of $I(X,F)$.