On Characterization of Finite Geometric Distributive Lattices

TL;DR

This paper characterizes all finite geometric distributive lattices, providing a criterion that helps understand their structure, prove conjectures about their sublattices, and compute their Whitney numbers, with applications to coding theory.

Contribution

It introduces a unique criterion for a geometric lattice to be distributive, enabling new insights into lattice structure and related coding theory applications.

Findings

Characterization criterion for finite geometric distributive lattices

Proof of a conjecture on maximum size of distributive sublattices

Calculation of Whitney numbers for geometric distributive lattices

Abstract

A Lattice is a partially ordered set where both least upper bound and greatest lower bound of any pair of elements are unique and exist within the set. K\"{o}tter and Kschischang proved that codes in the linear lattice can be used for error and erasure-correction in random networks. Codes in the linear lattice have previously been shown to be special cases of codes in modular lattices. Two well known classifications of semimodular lattices are geometric and distributive lattices. Most of the frequently used coding spaces are examples of either or both. We have identified the unique criterion which makes a geometric lattice distributive, thus characterizing all finite geometric distributive lattices. Our characterization helps to prove a conjecture regarding the maximum size of a distributive sublattice of a finite geometric lattice and identify the maximal case. The Whitney numbers of the class of geometric distributive lattices are also calculated. We present a few other applications of this unique characterization to derive certain results regarding linearity and complements in the linear lattice.