# Can a length n distributive lattice be embedded into Bn?

Let $\mathcal{L}$ be a finite distributive lattice, then it is known that it can be embedded into a finite boolean lattice (see theorem 8.5. p91 in this note).

Let $n$ be the length of $\mathcal{L}$ and let $\mathcal{B}_n$ be the boolean lattice of rank $n$.

Question: Can $\mathcal{L}$ be embedded into $\mathcal{B}_n$?

A distributive lattice $L$ of length $n$ is isomorphic to the set of order ideals of an $n$-element poset $P$, ordered by inclusion. This is the Fundamental Theorem of Finite Distributive Lattices, first proved by Garrett Birkhoff. Thus $L$ imbeds into the boolean algebra of all subsets of $P$.
• @DavidEppstein: if I'm not mistaken, this page does not precise that a length $n$ distributive lattice admits exactly $n$ join-irreducible elements. Anyway, I've found the more complete statement in the book Lattice theory of Garett Birkhoff, corollary 2 p59. – Sebastien Palcoux Nov 30 '15 at 4:59