Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way we establish an AZ{identity (see [9]) and its consequences, the LYM{inequality and the Sperner{property, for the Boolean interval lattice. Further, the Bollobas{inequality for the Boolean interval lattice turns out to be just the LYM{ inequality for the Boolean lattice. We also present an Intersection Theorem for this lattice. Perhaps more surprising is that by our approach the conjecture of Peter Erdos, Seress, Sz ekely [1] and F uredi concerning an Erdos{Ko{Rado{type intersection property for the poset of Boolean chains could also be established. Actually we give two seemingly elegant proofs.