  We exhibit several posets arising from commutative algebra, order theory,
tropical convexity as potential face posets of tropical polyhedra, and we
clarify their inclusion relations. We focus on monomial tropical polyhedra, and
deduce how their geometry reflects properties of monomial ideals. Their
vertex-facet lattice is homotopy equivalent to a sphere and encodes the Betti
numbers of an associated monomial ideal.
