B₃ — the Boolean lattice on three atoms

$B_3$ is the powerset $\mathcal{P}(\{a,b,c\})$ ordered by inclusion. With eight elements arranged on the vertices of a cube, it is the smallest Boolean lattice with three atoms and the prototypical distributive lattice. Every Boolean lattice is isomorphic to $\mathcal{P}(X)$ for some set $X$; $B_3$ is the case $|X| = 3$.

Where it appears

• As the lattice of subspaces of any 3-dimensional vector space when one restricts to coordinate subspaces.
• As the lattice of divisors of any squarefree integer with three prime factors — e.g. divisors of $30$.
• As the truth-value structure of three independent yes/no propositions; the basis of elementary propositional reasoning.

In Formal Concept Analysis, $B_n$ is the concept lattice of the “identity context” with $n$ objects and $n$ attributes — every combination of attributes picks out a distinct extent.