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 \(\varphi :I(X,K)\rightarrow I(X,K)\) 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.