I was always thinking, in an informal way, that the powerset lattices ${\cal P}(X)$ (where $X$ is an infinite set) are the "widest" bounded distributive lattices with respect to their height. (In ${\cal P}(\omega)$ every chain is at most countably infinite, but there are uncountable collections of infinite subsets of $\omega$ such that the pairwise intersection of the subsets is finite.)

Since I am not sure whether there are always chains of maximum cardinality in bounded distributive lattices, I want to put the thought I started this post with, in a loose, but (hopefully) precise manner.

**Question.** If $(L,\leq)$ is a bounded, distributive lattice, is it true that for every antichain $A\subseteq L$ there is a chain $C\subseteq L$ such that $2^{|C|} \geq |A|$?

**Note.** An antichain of a lattice $L$ is meant to be a set $A\subseteq L$ such that for $a\neq b\in A$ we have both $a\not\leq b$ and $b\not\leq a$.

**Note 2.** My remark above about countable chains is false, mind-bogglingly so as I think, see Andreas Blass' comment.