Decomposition theorem on matchable distributive lattices

TL;DR

This paper characterizes matchable distributive lattices (MDLs) derived from perfect matchings of plane bipartite graphs, providing a decomposition theorem and identifying specific types of MDLs.

Contribution

It establishes a decomposition theorem for MDLs and identifies two specific classes of MDLs based on product of chains and tree orientations.

Findings

A plane bipartite graph's perfect matching lattice is irreducible if the graph is elementary.

A finite distributive lattice is an MDL iff all factors in any product decomposition are MDLs.

Two classes of MDLs are identified: product of chains and posets from tree orientations.

Abstract

A distributive lattice structure ${\mathbf M}(G)$ 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 ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.