Dilworth's Decomposition Theorem for Posets

Formalized Mathematics (2009)

• Volume: 17, Issue: 4, page 223-232
• ISSN: 1426-2630

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Abstract

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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.

How to cite

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Piotr Rudnicki. "Dilworth's Decomposition Theorem for Posets." Formalized Mathematics 17.4 (2009): 223-232. <http://eudml.org/doc/266978>.

@article{PiotrRudnicki2009,
abstract = {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.},
author = {Piotr Rudnicki},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {223-232},
title = {Dilworth's Decomposition Theorem for Posets},
url = {http://eudml.org/doc/266978},
volume = {17},
year = {2009},
}

TY - JOUR
AU - Piotr Rudnicki
TI - Dilworth's Decomposition Theorem for Posets
JO - Formalized Mathematics
PY - 2009
VL - 17
IS - 4
SP - 223
EP - 232
AB - 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.
LA - eng
UR - http://eudml.org/doc/266978
ER -

References

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1. [1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.
2. [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
3. [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
4. [4] Grzegorz Bancerek. Bounds in posets and relational substructures. Formalized Mathematics, 6(1):81-91, 1997.
5. [5] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Formalized Mathematics, 6(1):93-107, 1997.
6. [6] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
7. [7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
8. [8] R. P. Dilworth. A Decomposition Theorem for Partially Ordered Sets. Annals of Mathematics, 51(1):161-166, 1950.[Crossref] Zbl0038.02003
9. [9] P. Erdős and G. Szekeres. A combinatorial problem in geometry. Compositio Mathematica, 2:463-470, 1935. Zbl0012.27010
10. [10] Adam Grabowski. Auxiliary and approximating relations. Formalized Mathematics, 6(2):179-188, 1997.
11. [11] L. Mirsky. A Dual of Dilworth's Decomposition Theorem. The American Mathematical Monthly, 78(8). Zbl0263.06002
12. [12] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147-152, 1990.
13. [13] M. A. Perles. A Proof of Dilworth's Decomposition Theorem for Partially Ordered Sets. Israel Journal of Mathematics, 1:105-107, 1963. Zbl0115.01702
14. [14] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441-444, 1990.
15. [15] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
16. [16] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387-393, 1990.
17. [17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

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