Relating Two Hopf Algebras Built from An Operad

Starting from an operad, one can build a family of posets. From this family of posets, one can define an incidence Hopf algebra. By another construction, one can also build a group directly from the operad. We then consider its Hopf algebra of functions. We prove that there exists a surjective morphism from the latter Hopf algebra to the former one. This is illustrated by the case of an operad built on rooted trees, the NAP operad, where the incidence Hopf algebra is identified with the Connes–Kreimer Hopf algebra of rooted trees.