Ranked poset

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In mathematics, a ranked partially ordered set (or poset) may be either:

• a graded poset, or
• a poset that has the property that for every element x, all maximal chains among those with x as greatest element have the same finite length, or
• a poset in which all maximal chains have the same finite length.

The second definition differs from the first, in that it requires all minimal elements to have the same rank; for posets with a least element however, the two requirements are equivalent. The third definition is even stricter in that it excludes posets with infinite chains, and also requires all maximal elements to have the same rank. Stanley defines a graded poset of length n as one in which all maximal chains have length n (Stanley 1995, p.99).

[edit] References

• Richard Stanley, Enumerative Combinatorics, vol.1, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, 1995, ISBN 0-521-66351-2