Commutativity preservers of incidence algebras

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 $φ:I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy $φ(D(X,K))=D(X,K)$. We prove that such a map $φ$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple $(θ,σ,c,κ)$ of simpler maps $θ$, $σ$, $c$ and a sequence $κ$ of elements of $K$.

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