Width of a partially ordered set

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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.


[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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