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A D-polyhedron is a polyhedron $P$ such that if $x,y$ are in $P$ then so aretheir componentwise max and min. In other words, the point set of aD-polyhedron forms a distributive lattice with the dominance order. We providea full characterization of the bounding hyperplanes of D-polyhedra.  Aside from being a nice combination of geometric and order theoreticconcepts, D-polyhedra are a unifying generalization of several distributivelattices which arise from graphs. In fact every D-polyhedron corresponds to adirected graph with arc-parameters, such that every point in the polyhedroncorresponds to a vertex potential on the graph. Alternatively, an edge-baseddescription of the point set can be given. The objects in this model are dualto generalized flows, i.e., dual to flows with gains and losses.  These models can be specialized to yield some cases of distributive latticesthat have been studied previously. Particular specializations are: lattices offlows of planar digraphs (Khuller, Naor and Klein), of $\alpha$-orientations ofplanar graphs (Felsner), of c-orientations (Propp) and of $\Delta$-bonds ofdigraphs (Felsner and Knauer). As an additional application we exhibit adistributive lattice structure on generalized flow of breakeven planardigraphs.

Distributive Lattices, Polyhedra, and Generalized Flow