Length of a partially ordered set
From Encyclopedia of Mathematics - Reading time: 1 min
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[edit]
| [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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