View source for Length of a partially ordered set

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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 order 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 incompable elements) that cover the set.

====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">[2]</TD> <TD valign="top">  George Grätzer, ''General Lattice Theory'', Springer (2003) ISBN 3764369965</TD></TR>
</table>