Boolean dimension of a Boolean lattice

TL;DR

This work investigates the Boolean dimension of natural posets, focusing on the Boolean lattice $\mathcal{B}_{n}$. It develops a product-structure approach and a concrete Boolean realizer for $\mathcal{B}_{6}$, showing $\operatorname{bdim}(\mathcal{B}_{n}) \le \left\lceil \frac{5}{6} n \right\rceil$ for all $n$ and establishing $\operatorname{bdim}(\mathcal{M}_{n})=\operatorname{dim}(\mathcal{M}_{n})=n$ for multisets. The main technical contribution is a constructive reduction for products of posets and a signature-based lower bound that yields exactness for $\mathcal{M}_{n}$ and tight bounds for $\mathcal{B}_{n}$ in many cases, complemented by SAT-based verification for the $\mathcal{B}_{6}$ case. The paper also raises open questions about the true growth rate of $\operatorname{bdim}(\mathcal{B}_{n})$ and related relaxed notions, highlighting directions for future research in poset dimension theory.

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 $n$.

Figures (2)

• Figure 1: For every positive integer $n \geqslant 2$, the standard example of order $n$, denoted by $S_n$, is isomorphic to the subposet of $\mathcal{B}_{n}$ induced by the singletons and co-singletons. On the right, we show a poset diagram of $S_4$ and on the left, we show $S_4$ as a subposet of $\mathcal{B}_{4}$.
• Figure 2: Consider the poset $\mathcal{M}_{3, 2} = \mathcal{B}_{3}$. $L_1, L_2, L_3$ are some linear orders on elements of the poset (in the figure the least element is on the bottom). The singletons $\mathcal{S} = \{\{1\},\{2\},\{3\}\}$ are highlighted with purple color. We fix $A = \{2,3\}$ (in blue) and $B = \{1,2,3\}$ (in red). We have $s_1(A) = 2$, since $A$ is greater than $\{2\}$ and $\{3\}$ and less than $\{1\}$. Similarly, one can check that $s_2(A) = 1$ and $s_3(A) = 3$. It follows that the signature of $A$ is equal to $(2,1,3)$. The signature of $B$ turns out to be the same. We claim that this prevents $L_1,L_2,L_3$ from being a Boolean realizer of $\mathcal{B}_{3}$ regardless of $\phi : \{0,1\}^{3} \rightarrow \{0,1\}$. Indeed, $1 \in B \backslash A$, however, the relations between $\{1\}$ and $A$ are the same as the relations between $\{1\}$ and $B$ in $L_1,L_2,L_3$. This is a contradiction since $\{1\} \leqslant B$ and $\{1\} \not\leqslant A$.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Proposition 3
• Lemma 4
• proof
• Theorem 5
• Lemma 6
• proof
• Lemma 7
• proof