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 $P$.

  • $\begingroup$ en.wikipedia.org/wiki/Birkhoff%27s_representation_theorem $\endgroup$ – David Eppstein Nov 29 '15 at 17:31
  • $\begingroup$ @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. $\endgroup$ – Sebastien Palcoux Nov 30 '15 at 4:59
  • $\begingroup$ Another reference is Section 3.4 of my book Enumerative Combinatorics, vol. 1, second edition. $\endgroup$ – Richard Stanley Nov 30 '15 at 13:16

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