# Width of a partially ordered set

From Encyclopedia of Mathematics

2010 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 |

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Width of a partially ordered set.

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