Maximum antichains in the product of chains

Let P be the poset k₁ × ... × k_n, which is a product of chains, where n1 and k₁ ... k_n2. Let . P is known to have the Sperner property, which means that its maximum ranks are maximum antichains. Here we prove that its maximum ranks are its only maximum antichains if and only if either n=1 or M1. This is a generalization of a classical result, Sperner's Theorem, which is the case k₁= ... =k_n=2. We also determine the number and location of the maximum ranks of P.Research supported in part by the National Science Foundation 10/25/83.