A class of non-matchable distributive lattices

TL;DR

This paper characterizes non-matchable distributive lattices by exploring their relation to perfect matchings in plane bipartite graphs, introducing meet-irreducible cells, and extending existing results to identify new classes of such lattices.

Contribution

It introduces meet-irreducible cells in plane bipartite graphs and extends the classification of non-matchable distributive lattices with new examples.

Findings

Characterization of meet-irreducible cells in bipartite graphs

Extension of non-matchable lattice classification

Identification of new non-matchable distributive lattices

Abstract

The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$-transformation directed graph of the graph. A finite distributive lattice is matchable if its Hasse diagram is isomorphic to a $Z$-transformation directed graph of a plane weakly elementary bipartite graph, otherwise non-matchable. We introduce the meet-irreducible cell with respect to a perfect matching of a plane (weakly) elementary bipartite graph and give its equivalent characterizations. Using these, we extend a result on non-matchable distributive lattices, and obtain a class of new non-matchable distributive lattices.