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

Homomorphism Complexes and Maximal Chains in Graded Posets