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 $\phi :I(X,F)\to I(X,F)$. Namely, we prove that, whenever $\mathrm{char}(F)\ne 2$, $\phi$ is either an automorphism or an anti-automorphism of $I(X,F)$. If $\mathrm{char}(F)=2$ and $|F|>2$, then $\phi$ is a (in general, non-proper) Lie automorphism of $I(X,F)$. Finally, if $F={\mathbb{Z}}_2$, then $\phi$ 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 $k\ge 3$.

Final version published in Linear Algebra and its Applications

References in corpus (1)

• The singular linear preservers of non-singular matrices

Cited by in corpus (4)

• Non-associative algebraic structures: classification and structure
• Jordan embeddings and linear rank preservers of structural matrix algebras
• An extension of Petek-Šemrl preserver theorems for Jordan embeddings of structural matrix algebras
• Multiplicative and Jordan multiplicative maps on structural matrix algebras