Difference between revisions of "Width of a partially 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]]). | 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. | + | [[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==== | ||
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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</TD></TR> | + | <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> | + | <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=39318
Width of a partially ordered set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Width_of_a_partially_ordered_set&oldid=39318