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 ) → 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 .