The M\"obius Function on Implication sublattices of a Boolean algebra

TL;DR

This paper investigates the structure of implication sublattices within finite Boolean algebras by computing the M"obius function on their poset, offering new insights into their combinatorial properties.

Contribution

It provides two distinct methods for calculating the M"obius function on the poset of implication sublattices of finite Boolean algebras, advancing understanding of their algebraic structure.

Findings

Explicit formulas for the M"obius function on the poset of implication sublattices.

Comparison of two different computational approaches.

Enhanced understanding of the combinatorial structure of implication sublattices.

Abstract

Let $B$ be a finite Boolean algebra. Let $\mathcal A$ be the partial order of all implication sublattices of $B$. We will compute the M\"obius function on $\mathcal A$ in two different ways.