mathematics, a ranked poset is a partially ordered set in which one of the following (non-equivalent) conditions hold: it is a graded poset, or a poset with the...

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. Graded posets play...

Eulerian poset is a graded poset in which every nontrivial interval has the same number of elements of even rank as of odd rank. An Eulerian poset which...

is called graded, sometimes ranked (but see Ranked poset for an alternative meaning), if it can be equipped with a rank function r : L → N {\displaystyle...

poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. Since every rank...

with several meanings Graded poset, a partially ordered set equipped with a rank function, sometimes called a ranked poset Graded vector space, a vector...

In mathematics, a differential poset is a partially ordered set (or poset for short) satisfying certain local properties. (The formal definition is given...

that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P = ( X , ≤ ) {\displaystyle P=(X,\leq )}...

h-vector, which is defined for an arbitrary ranked poset, and proved that for the class of Eulerian posets, the Dehn–Sommerville equations continue to...

square. Smaller posets, and polytopes in particular, are often best visualized in a Hasse diagram, as shown. By convention, faces of equal rank are placed...

the corresponding Weyl group from this poset. The Hasse graph is a visualization of the ordering of the root poset. ADE classification Affine root system...

geometric or matroid lattice comes from a matroid in this way. A lattice is a poset in which any two elements x {\displaystyle x} and y {\displaystyle y} have...

Predicate functional variable propositional variable Proof Quantifier ∃ ! ∀ rank Sentence atomic spectrum Signature String Substitution Symbol function logical/constant...

Stanley defined a Peck poset to be a graded partially ordered set that is rank symmetric, rank unimodal, and strongly Sperner. The posets in the original paper...

root system, the resulting poset is the corresponding Weyl group with the weak order. In general, the poset of regions is ranked by the number of separating...

linear time, if such a diagram exists. In particular, if the input poset is a graded poset, it is possible to determine in linear time whether there is a...

general form, a hierarchy is a partially ordered set or poset. The system in this case is the entire poset, which is constituted of elements. Within this system...

must be equal but may be greater than one) the graph of a differential poset. The transitive closure of the Young–Fibonacci graph is a partial order...

algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal...

combinatorial properties; notably, it is the motivating example of a differential poset. There is a deep theory of random partitions chosen according to the uniform...

cardinals * An operation that takes a forcing poset and a name for a forcing poset and produces a new forcing poset. ∞ The class of all ordinals, or at least...

contains the poset. Then there do exist filters on the poset that are generic over M; that is, that meet all dense open subsets of the poset that happen...

for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic...

partially ordered set (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not...

greedoids have many equivalent definitions in terms of set system, language, poset, simplicial complex, and so on. The following description takes the traditional...

algebraic poset. Since C is also a lattice, it is often referred to as an algebraic lattice in this context. Conversely, if C is an algebraic poset, then...

the reduced homology of a bounded finite poset is another generalization, important in combinatorics. A poset is "bounded" if it has smallest and largest...

∈ X ∧ a ≠ b {\displaystyle a\in X\land b\in X\land a\neq b} cases; the poset definition implies them because elements of X {\displaystyle X} are incomparable...

theorem for subsets of P ( E ) , {\displaystyle {\mathcal {P}}(E),} the poset of all subsets of E. A chain is a subfamily { S 0 , S 1 , … , S r } ⊆ P...

ordered set (poset) for inclusion. Closure operators allow generalizing the concept of closure to any partially ordered set. Given a poset S whose partial...