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arXiv · 2206.01788

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

Jorge J. Garcés·Mykola Khrypchenko
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Abstract

Let $A$, $B$ be algebras and $a\in A$, $b\in B$ a fixed pair of elements. We say that a map $φ:A\to B$ preserves products equal to $a$ and $b$ if for all $a_1,a_2\in A$ the equality $a_1a_2=a$ implies $φ(a_1)φ(a_2)=b$. In this paper we study bijective linear maps $φ:I(X,F)\to I(X,F)$ 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 $φ$ exists, and whenever it does, $φ$ is either an automorphism of $I(X,F)$ or the negative of an automorphism of $I(X,F)$.

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Jorge J. Garcés, Mykola Khrypchenko. 2022-09-09. Linear maps preserving products equal to primitive idempotents of an incidence algebra. https://arxiv.org/abs/2206.01788

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