TOPICS
Search

Boolean Lattice

The Boolean lattice B_n is the partially ordered set consisting of all subsets of an n-set, ordered by inclusion. Equivalently, it is the power set of an n-set regarded as a lattice with meet and join given by intersection and union, respectively.

The elements of rank k in B_n are the k-element subsets, so the kth rank has (n; k) elements. The Hasse diagram of B_n is the n-hypercube graph.

Meyer (2026) proved that B_6 is not a sphere order.

See also

Antichain, Boolean Algebra, Daykin-Frankl Conjecture, Hypercube Graph, Lattice, Order-Convex Set, Partially Ordered Set, Power Set, Sphere Order, Sperner's Theorem

Explore with Wolfram|Alpha

References

Balogh, J. and Krueger, R. A. "A Sharp Threshold for a Random Version of Sperner's Theorem." Random Structures Alg. 66, e21259, 2025. https://doi.org/10.1002/rsa.21259.Kierstead, H. A. and Trotter, W. T. "Explicit Matchings in the Middle Levels of the Boolean Lattice." Order 5, 163-171, 1988.Meyer, D. A. "The Boolean Lattice B_6 Is Not a Sphere Order." 1 Oct 2026. https://arxiv.org/abs/2610.02329.

Cite this as:

Weisstein, Eric W. "Boolean Lattice." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/BooleanLattice.html

Subject classifications