Maximum antichains in the product of chains
JR Griggs - Order, 1984 - Springer
JR Griggs
Order, 1984•SpringerLet P be the poset k 1×...× kn, which is a product of chains, where n≥ 1 and k 1≥...≥ kn≥
2. Let M= k_1-∑ i= 2^ n (k_i-1). 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 M≤ 1. This is a generalization of a
classical result, Sperner's Theorem, which is the case k 1=...= kn= 2. We also determine the
number and location of the maximum ranks of P.
2. Let M= k_1-∑ i= 2^ n (k_i-1). 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 M≤ 1. This is a generalization of a
classical result, Sperner's Theorem, which is the case k 1=...= kn= 2. We also determine the
number and location of the maximum ranks of P.
Abstract
Let P be the poset k 1 × ... × k n , which is a product of chains, where n≥1 and k 1≥ ... ≥k n ≥2. 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 M≤1. This is a generalization of a classical result, Sperner's Theorem, which is the case k 1= ... =k n =2. We also determine the number and location of the maximum ranks of P.
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