A distributive lattice structure M(G) has been established on the
set of perfect matchings of a plane bipartite graph G. We call a lattice {\em
matchable distributive lattice} (simply MDL) if it is isomorphic to such a
distributive lattice. It is natural to ask which lattices are MDLs. We show
that if a plane bipartite graph G is elementary, then M(G) is
irreducible. Based on this result, a decomposition theorem on MDLs is obtained:
a finite distributive lattice L is an MDL if and only if each factor
in any cartesian product decomposition of L is an MDL. Two types of
MDLs are presented: J(m×n) and J(T), where
m×n denotes the cartesian product between m-element
chain and n-element chain, and T is a poset implied by any
orientation of a tree.Comment: 19 pages, 7 figure