Ranked poset
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In mathematics, a partially ordered set (or "poset") may be called ranked as a synonym for "graded" (see graded poset), or alternatively if it has the property that, for every element x, every chain xm < xm−1 < ... < x0 = x has the same length m.
The latter definition differs from that of a graded poset in that not every maximal chain need have the same length. For instance, the disjoint union of two finite chains is ranked in the second definition but it is not graded unless the two chains have the same length.
Both meanings are in common use.
[edit] References
- Richard Stanley, Enumerative Combinatorics, vol.1, Cambridge Studies in Advanced Mathematics 49, CUP, ISBN 0-521-66351-2
