Incomparability and Intersection Properties of Boolean Interval Lattices and Chain Posets

In a canonical way, we establish an AZ-identity (see [2]) and its consequences, the LYM-inequality and the Sperner property, for the Boolean interval lattice. Furthermore, 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. s more surprising is that by our approach the conjecture of P. L. Erdöset al.[7] and Z. Füredi concerning an Erdös–Ko–Rado-type intersection property for the poset of Boolean chains could also be established. In fact, we give two seemingly elegant proofs.