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Maximum antichains in the product of chains (1984)

Griggs, Jerrold R.

Springer

Order 1 (1984), S. 21-28

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ISSN: 1572-9273

Keywords: 06A10 ; 05A05 ; Partially ordered sets ; Sperner's Theorem ; LYM property ; product of chains

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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 $$M = k_1 - \sum\nolimits_{i = 2}^n {(k_i - 1)} $$ . 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= ... =k n =2. We also determine the number and location of the maximum ranks of P.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00396270

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