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

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(Start article: 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.
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[[Dilworth theorem|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.
  
  
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====References====
 
====References====
 
<table>
 
<table>
<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</TD></TR>
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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>
<TR><TD valign="top">[2]</TD> <TD valign="top">  George Grätzer, ''General Lattice Theory'', Springer (2003) ISBN 3764369965</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>
 
</table>
 
</table>

Latest revision as of 08:46, 26 November 2023

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 Zbl 0038.02003
[2] George Grätzer, General Lattice Theory, Springer (2003) ISBN 3764369965 Zbl 1152.06300
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=35425