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In 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 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 ...
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]
Junior Commissioned Officers display their rank insignias on their shoulders, Non-Commissioned Officers showcase their rank insignias on mid sleeves, and in combat uniforms, all individuals wear rank insignias on their chest.
An Eulerian poset which is a lattice is an Eulerian lattice. These objects are named after Leonhard Euler . Eulerian lattices generalize face lattices of convex polytopes and much recent research has been devoted to extending known results from polyhedral combinatorics , such as various restrictions on f -vectors of convex simplicial polytopes ...
In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces , such that
Forcing with this poset collapses λ down to κ. Levy collapsing: If κ is regular and λ is inaccessible, then P is the set of functions p on subsets of λ × κ with domain of size less than κ and p(α, ξ) < α for every (α, ξ) in the domain of p. This poset collapses all cardinals less than λ onto κ, but keeps λ as the successor to κ.
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