Potent preservers of incidence algebras

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