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Formalized Mathematics

(a computer assisted approach)

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Dilworth's Decomposition Theorem for Posets

Piotr Rudnicki
  • University of Alberta, Edmonton, Canada
Published Online: 2010-07-08 | DOI: https://doi.org/10.2478/v10037-009-0028-4

Dilworth's Decomposition Theorem for Posets

The following theorem is due to Dilworth [8]: Let P be a partially ordered set. If the maximal number of elements in an independent subset (anti-chain) of P is k, then P is the union of k chains (cliques).

In this article we formalize an elegant proof of the above theorem for finite posets by Perles [13]. The result is then used in proving the case of infinite posets following the original proof of Dilworth [8].

A dual of Dilworth's theorem also holds: a poset with maximum clique m is a union of m independent sets. The proof of this dual fact is considerably easier; we follow the proof by Mirsky [11]. Mirsky states also a corollary that a poset of r x s + 1 elements possesses a clique of size r + 1 or an independent set of size s + 1, or both. This corollary is then used to prove the result of Erdős and Szekeres [9].

Instead of using posets, we drop reflexivity and state the facts about anti-symmetric and transitive relations.

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  • [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

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Published Online: 2010-07-08

Published in Print: 2009-01-01


Citation Information: Formalized Mathematics. Volume 17, Issue 4, Pages 223–232, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-009-0028-4, July 2010

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[1]
Piotr Rudnicki and Lorna Stewart
Formalized Mathematics, 2011, Volume 19, Number 1

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