  We apply the homomorphism complex construction to partially ordered sets,
introducing a new topological construction based on the set of maximal chains
in a graded poset. Our primary objects of study are distributive lattices, with
special emphasis on finite products of chains. For the special case of a
Boolean algebra, we observe that the corresponding homomorphism complex is
isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this
work can be interpreted as a generalization of the study of these complexes. We
provide a detailed investigation when our poset is a product of chains, in
which case we find an optimal discrete Morse matching and prove that the
corresponding complex is torsion-free.
