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Commutativity preservers of incidence algebras (2207.10713v1)
Published 21 Jul 2022 in math.RA
Abstract: Let $I(X,K)$ be the incidence algebra of a finite connected poset $X$ over a field $K$ and $D(X,K)$ its subalgebra consisting of diagonal elements. We describe the bijective linear maps $\varphi:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $\varphi(D(X,K))=D(X,K)$. We prove that such a map $\varphi$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(\theta,\sigma,c,\kappa)$ of simpler maps $\theta$, $\sigma$, $c$ and a sequence $\kappa$ of elements of $K$.