Ranked poset

Inmathematics,aranked posetis apartially ordered setin which one of the following (non-equivalent) conditions hold: it is

• agraded poset,or
• a poset with the property that for every elementx,all maximalchainsamong those withxasgreatest elementhave the same finitelength,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 more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank.Richard P. Stanleydefines a graded poset of lengthnas one in which all maximal chains have lengthn.