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Difference between revisions of "Length of a partially ordered set"

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The greatest possible length of a [[Chain|chain]] in this set. There exist infinite partially ordered sets of finite length.
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{{TEX|done}}{{MSC|06A}}
  
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The greatest possible length of a [[chain]] (totally ordered subset) in a [[partially ordered set]] (the length of a finite chain is one less than the number of elements). There exist infinite partially ordered sets of finite length.  A partially ordered set of length zero is a [[trivial order]].
  
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Dilworth's theorem [[#References|[1]]] states that in a finite partially ordered set the length is equal to the minimal number of [[anti-chain]]s (sets of mutually incomparable elements) that cover the set.
  
====Comments====
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====References====
See also [[Partially ordered set|Partially ordered set]].
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> R.P. Dilworth, "A decomposition theorem for partially ordered sets"  ''Ann. of Math.'' , '''51'''  (1950)  pp. 161–166  {{ZBL|0038.02003}}</TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> George Grätzer, ''General Lattice Theory'', Springer (2003) {{ISBN|3764369965}} {{ZBL|1152.06300}}</TD></TR>
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</table>

Latest revision as of 07:27, 14 November 2023

2020 Mathematics Subject Classification: Primary: 06A [MSN][ZBL]

The greatest possible length of a chain (totally ordered subset) in a partially ordered set (the length of a finite chain is one less than the number of elements). There exist infinite partially ordered sets of finite length. A partially ordered set of length zero is a trivial order.

Dilworth's theorem [1] states that in a finite partially ordered set the length is equal to the minimal number of anti-chains (sets of mutually incomparable elements) that cover the set.

References

[1] R.P. Dilworth, "A decomposition theorem for partially ordered sets" Ann. of Math. , 51 (1950) pp. 161–166 Zbl 0038.02003
[2] George Grätzer, General Lattice Theory, Springer (2003) ISBN 3764369965 Zbl 1152.06300
How to Cite This Entry:
Length of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Length_of_a_partially_ordered_set&oldid=12809
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article