Dilworth's Decomposition Theorem for Posets

• Autorzy: Piotr Rudnicki
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2009
• Tom: 17
• Numer: 4
• Strony: 223-232

Daty

wydano

2009-01-01

online

2010-07-08

Twórcy

autor

Piotr Rudnicki

• University of Alberta, Edmonton, Canada

Bibliografia

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Identyfikatory

DOI

10.2478/v10037-009-0028-4