The paper introduces the partial order polytope of a digraph D, defined as the convex hull of the incidence vectors of all transitive acyclic arc sets of D. For this polytope it proves some classes of inequalities to be facet-defining and show that there is a polynomial separation algorithm for each of these classes. The results imply a polynomial separation algorithm for a class of valid inequalities of the clique partioning polytope that includes the two-chorded odd cycle inequalities. The polyhedral results concerning the partial order polytope are of interest since a cutting plane based algorithm to solve the maximum weighted transitive acyclic subdigraph problem can be used to solve the maximum weighted acyclic subdigraph problem, the maximum weighted linear ordering problem and a flexible manufacturing problem. For the acyclic subdigraph polytope the paper shows that the separation of simple t-reinforced k-fence-inequalities is 𝒩𝒫-complete.
