A new matching property for posets and existence of disjoint chains (Q1881683)

\textit{E. Lehman} and \textit{D. Ron} [J. Comb. Theory, Ser. A 94, 399--404 (2001; Zbl 0989.06002)] proved that the Boolean lattice satisfies the following: given distinct \(k\)-sets \(A_1,A_2,\dots,A_m\), and distinct \(\ell\)-sets, \(B_1,B_2,\dots,B_m\), with \(A_i\subseteq B_i\) for all \(i\), there exist \(m\) disjoint saturated (skipless) chains from the \(A_i\) to the \(B_j\). The authors of this paper used the ideas of Lehman and Ron's proof to extend the result to a large class of posets. These they call shadow-matching posets: graded posets in which whenever \(x\) covers \(y\) there is an injective matching from the elements covered by \(y\) to the elements covered by \(x\) and an injective matching from the covers of \(x\) to the covers of \(y\). The set of shadow-matching posets is closed under taking intervals and direct products. The poset of intervals of a shadow-matching poset is shadow-matching. Any lattice containing no interval with exactly two elements is shadow-matching. Consequently, the following posets are shadow-matching, and hence satisfy the Lehman-Ron property: geometric lattices (e.g., subspace lattices, partition lattices, and intersection lattices of central hyperplane arrangements), the divisor lattice, face lattices of convex polytopes, and the lattice of noncrossing partitions. Finally, there is a comparison of the shadow-matching property with other poset matching properties.