Ranked poset
Encyclopedia
In mathematics
, a ranked partially ordered set
- or poset - may be either:
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. Stanley
defines a graded poset of length n as one in which all maximal chains have length n.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a ranked partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
- or poset - may be either:
- a graded posetGraded posetIn mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to N compatible with the ordering such that whenever y covers x, then...
, or - a poset that has the property that for every element x, all maximal chainTotal orderIn set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
s 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 more strict in that it excludes posets with infinite chains and also requires all maximal elements to have the same rank. Richard P. Stanley
Richard P. Stanley
Richard Peter Stanley is the Norman Levinson Professor of Applied Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota...
defines a graded poset of length n as one in which all maximal chains have length n.