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 \(\varphi :I(X,K)\rightarrow I(X,K)\) $\phi :I(X,K)\to I(X,K)$ that strongly preserve the commutativity and satisfy \(\varphi (D(X,K))=D(X,K)\) $\phi (D(X,K))=D(X,K)$ . We prove that such a map \(\varphi \) $\phi$ is a composition of a commutativity preserver of shift type and a commutativity preserver associated to a quadruple \((\theta ,\sigma ,c,\kappa )\) $(\theta ,\sigma ,c,\kappa )$ of simpler maps \(\theta \) $\theta$ , \(\sigma \) $\sigma$ , c and a sequence \(\kappa \) $\kappa$ of elements of K.