  The quotient of a Boolean algebra by a cyclic group is proven to have a
symmetric chain decomposition. This generalizes earlier work of Griggs, Killian
and Savage on the case of prime order, giving an explicit construction for any
order, prime or composite. The combinatorial map specifying how to proceed
downward in a symmetric chain is shown to be a natural cyclic analogue of the
$\mathfrak{sl}_2$ lowering operator in the theory of crystal bases.
