Non-matchable distributive lattices

Based on an acyclic orientation of the Z-transformation graph, a finite distributive lattice (FDL for short) M(G) is established on the set of all 1-factors of a plane (weakly) elementary bipartite graph G. For an FDL L, if there exists a plane bipartite graph G such that L is isomorphic to M(G), then L is called a matchable FDL. A natural question arises: Is every FDL a matchable FDL? In this paper we give a negative answer to this question. Further, we obtain a series of non-matchable FDLs by characterizing sub-structures of matchable FDLs with cut-elements.