Difference between revisions of "Length of a partially ordered set"
From Encyclopedia of Mathematics
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| − | 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 | + | 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 [[#References|[1]]] states that in a finite partially ordered set the length is equal to the minimal number of [[anti-chain]]s (sets of mutually incomparable elements) that cover the set. | ||
====References==== | ====References==== | ||
| − | + | <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 {{ZBL|0038.02003}}</TD></TR> | ||
| + | <TR><TD valign="top">[2]</TD> <TD valign="top"> George Grätzer, ''General Lattice Theory'', Springer (2003) {{ISBN|3764369965}} {{ZBL|1152.06300}}</TD></TR> | ||
| + | </table> | ||
Latest revision as of 07:27, 14 November 2023
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
| [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:
Length of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Length_of_a_partially_ordered_set&oldid=35423
Length of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Length_of_a_partially_ordered_set&oldid=35423
This article was adapted from an original article by T.S. Fofanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article