Potent preservers of incidence algebras

Abstract

Let $X$ be a finite connected poset, $F$ a field and $I(X,F)$ the incidence algebra of $X$ over $F$. We describe the bijective linear idempotent preservers $\varphi:I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $\varphi$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $\varphi$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F=\mathbb{Z}_2$, then $\varphi$ is the composition of a bijective shift map and a Lie automorphism of $I(X,F)$. Under certain restrictions on the characteristic of $F$ we also obtain descriptions of the bijective linear maps which preserve tripotents and, more generally, $k$-potents of $I(X,F)$ for $k\ge 3$.