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

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''Dilworth number'', ''Sperner number''
 
''Dilworth number'', ''Sperner number''
  
The greatest possible length of an [[anti-chain]] (set of mutually incomparable elements) in a [[partially ordered set]]. A partially ordered set of width 1 is a chain ([[totally ordered set]]).   
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The greatest possible size of an [[anti-chain]] (set of mutually incomparable elements) in a [[partially ordered set]]. A partially ordered set of width 1 is a chain ([[totally ordered set]]).   
  
 
Dilworth's theorem [[#References|[1]]] states that in a finite partially ordered set the width is equal to the minimal number of chains that cover the set.
 
Dilworth's theorem [[#References|[1]]] states that in a finite partially ordered set the width is equal to the minimal number of chains that cover the set.

Revision as of 06:19, 27 September 2016

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

Dilworth number, Sperner number

The greatest possible size of an anti-chain (set of mutually incomparable elements) in a partially ordered set. A partially ordered set of width 1 is a chain (totally ordered set).

Dilworth's theorem [1] states that in a finite partially ordered set the width is equal to the minimal number of chains that cover the set.


See also Sperner property.


References

[1] R.P. Dilworth, "A decomposition theorem for partially ordered sets" Ann. of Math. , 51 (1950) pp. 161–166
[2] George Grätzer, General Lattice Theory, Springer (2003) ISBN 3764369965
How to Cite This Entry:
Width of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Width_of_a_partially_ordered_set&oldid=39318