A power set, partially ordered by inclusion, with rank defined as number of elements, forms a graded poset.
In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) P equipped with a rank functionρ from P to the set N of all natural numbers. ρ must satisfy the following two properties:
The rank function is compatible with the ordering, meaning that for all x and y in the order, if x < y then ρ(x) < ρ(y), and
The rank is consistent with the covering relation of the ordering, meaning that for all x and y, if y covers x then ρ(y) = ρ(x) + 1.
The value of the rank function for an element of the poset is called its rank. Sometimes a graded poset is called a ranked poset but that phrase has other meanings; see Ranked poset. A rank or rank level of a graded poset is the subset of all the elements of the poset that have a given rank value.[1][2]
Graded posets play an important role in combinatorics and can be visualized by means of a Hasse diagram.
^Stanley, Richard (1984), "Quotients of Peck posets", Order, 1 (1): 29–34, doi:10.1007/BF00396271, MR 0745587, S2CID 14857863.
^Butler, Lynne M. (1994), Subgroup Lattices and Symmetric Functions, Memoirs of the American Mathematical Society, vol. 539, American Mathematical Society, p. 151, ISBN 9780821826003.
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differential poset, and in particular to be r-differential (where r is a positive integer), if it satisfies the following conditions: P is graded and locally...
atomistic if every element is the supremum of some set of atoms. A poset is graded when it can be given a rank function r ( x ) {\displaystyle r(x)} mapping...
for u as an initial segment. Indeed, the word length makes this into a gradedposet. The Hasse diagrams corresponding to these orders are objects of study...
distributivity laws of order theory preservation properties of functions between posets. In the following, partial orders will usually just be denoted by their...
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groups act transitively on the set of flags of the polytope. Eulerian posetGradedposet Regular polytope McMullen & Schulte 2002, p. 31 McMullen & Schulte...
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