CD-independent subsets in meet-distributive lattices

TL;DR

This paper investigates the size and structure of CD-independent subsets in finite meet-distributive lattices, providing bounds, recursive descriptions, and applications to counting islands on a rectangular board.

Contribution

It extends previous results from distributive to meet-distributive lattices, offering new bounds, characterizations, and a novel application in combinatorics.

Findings

Maximum size of CD-independent subsets is bounded by atoms plus lattice length.

Recursive description of maximal CD-independent subsets under certain conditions.

New method to count islands on a rectangular grid.

Abstract

A subset $X$ of a finite lattice $L$ is CD-independent if the meet of any two incomparable elements of $X$ equals 0. In 2009, Cz\'edli, Hartmann and Schmidt proved that any two maximal CD-independent subsets of a finite distributive lattice have the same number of elements. In this paper, we prove that if $L$ is a finite meet-distributive lattice, then the size of every CD-independent subset of $L$ is at most the number of atoms of $L$ plus the length of $L$. If, in addition, there is no three-element antichain of meet-irreducible elements, then we give a recursive description of maximal CD-independent subsets. Finally, to give an application of CD-independent subsets, we give a new approach to count islands on a rectangular board.