A new matching property for posets and existence of disjoint chains

Let P be a graded poset. Assume that x₁,...,x_m are elements of rank k and y₁,...,y_m are elements of rank l for some k<l. Further suppose x_i≤y_i, for 1≤i≤m. Lehman and Ron (J. Combin. Theory Ser. A 94 (2001) 399) proved that, if P is the subset lattice, then there exist m disjoint skipless chains in P that begin with the x's and end at the y's. One complication is that it may not be possible to have the chains respect the original matching and hence, in the constructed set of chains, x_i and y_i may not be in the same chain. In this paper, by introducing a new matching property for posets, called shadow-matching, we show that the same property holds for a much larger class of posets including the divisor lattice, the subspace lattice, the lattice of partitions of a finite set, the intersection poset of a central hyperplane arrangement, the face lattice of a convex polytope, the lattice of noncrossing partitions, and any geometric lattice.