  The Boolean lattice $2^{[n]}$ is the power set of $[n]$ ordered by inclusion.
A chain $c_{0}\subset...\subset c_{k}$ in $2^{[n]}$ is rank-symmetric, if
$|c_{i}|+|c_{k-i}|=n$ for $i=0,...,k$; and it is symmetric, if
$|c_{i}|=(n-k)/2+i$. We show that there exist a bijection $$p: [n]^{(\geq
n/2)}\rightarrow [n]^{(\leq n/2)}$$ and a partial ordering $<$ on $[n]^{(\geq
n/2)}$ satisfying the following properties:
  (i) $\subset$ is an extension of $<$ on $[n]^{(\geq n/2)}$;
  (ii) if $C\subset [n]^{(\geq n/2)}$ is a chain with respect to $<$, then
$p(C)\cup C$ is a rank-symmetric chain in $2^{[n]}$, where $p(C)=\{p(x): x\in
C\}$;
  (iii) the poset $([n]^{(\geq n/2)},<)$ has the so called normalized matching
property.
  We show two applications of this result.
  A conjecture of F\"{u}redi asks if $2^{[n]}$ can be partitioned into
$\binom{n}{\lfloor n/2\rfloor}$ chains such that the size of any two chains
differ by at most 1. We prove an asymptotic version of this conjecture with the
additional condition that every chain in the partition is rank-symmetric:
$2^{[n]}$ can be partitioned into $\binom{n}{\lfloor n/2\rfloor}$
rank-symmetric chains, each of size $\Theta(\sqrt{n})$.
