Sperner theory in a difference of Boolean lattices Original Research Article

Consider any sets x⊆y⊆{1,…,n}. Remove the interval [x,y]={z⊆y | x⊆z} from the Boolean lattice of all subsets of {1,…,n}. We show that the resulting poset, ordered by inclusion, has a nested chain decomposition and has the normalized matching property. We also classify the largest antichains in this poset. This generalizes results of Griggs, who resolved these questions in the special case x=∅.