Hopf algebra structure of incidence algebras

The incidence algebra of a partially ordered set (poset) supports in a natural way also a coalgebra structure, so that it becomes a m-weak bialgebra even a m-weak Hopf algebra with M\"obius function as antipode. Here m-weak means that multiplication and comultiplication are not required to be coalgebra- or algebra-morphisms, respectively. A rich theory is obtained in computing modulo an equivalence relation on the set of intervals in the poset.