Boolean dimension of a Boolean lattice

TL;DR

This paper proves that for large enough n, the Boolean dimension of the Boolean lattice formed by all subsets of an n-element set is always less than n, providing new bounds on this combinatorial property.

Contribution

The paper establishes a new upper bound on the Boolean dimension of Boolean lattices for all n ≥ 6, improving understanding of their structural complexity.

Findings

Boolean dimension of the lattice is less than n for n ≥ 6

Provides bounds that improve previous estimates

Enhances understanding of poset complexity in combinatorics

Abstract

For every integer $n$ with $n \geq 6$, we prove that the Boolean dimension of a poset consisting of all the subsets of $\{1,\dots,n\}$ equipped with the inclusion relation is strictly less than $n$.