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The quotient of a Boolean algebra by a cyclic group is proven to have asymmetric chain decomposition. This generalizes earlier work of Griggs, Killianand Savage on the case of prime order, giving an explicit construction for anyorder, prime or composite. The combinatorial map specifying how to proceeddownward 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.

Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals