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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]
A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels, [1] or, equivalently, the poset has a maximum k-family consisting of k rank levels. [2] A strict Sperner poset is a graded poset in which all maximum antichains are rank levels. [2]
In this poset, 60 is an upper bound (though not a least upper bound) of the subset {,,,}, which does not have any lower bound (since 1 is not in the poset); on the other hand 2 is a lower bound of the subset of powers of 2, which does not have any upper bound. If the number 0 is included, this will be the greatest element, since this is a ...
A graded poset is a poset with a rank function: compatible with the ordering (i.e. () < <) such that covers = + Index of articles associated with the same name This set index article includes a list of related items that share the same name (or similar names).
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 poset Y is graded: the minimal element is ∅, the unique partition of zero, and the partitions of n have rank n. This means that given two partitions that are comparable in the lattice, their ranks are ordered in the same sense as the partitions, and there is at least one intermediate partition of each intermediate rank. The poset Y is
When a graded poset has a bottom element, one may assume, without loss of generality, that its rank is zero. In this case, the atoms are the elements with rank one. A graded lattice is semimodular if, for every x {\displaystyle x} and y {\displaystyle y} , its rank function obeys the identity [ 1 ]
Algebraic poset. A poset is algebraic if it has a base of compact elements. Antichain. An antichain is a poset in which no two elements are comparable, i.e., there are no two distinct elements x and y such that x ≤ y. In other words, the order relation of an antichain is just the identity relation. Approximates relation. See way-below relation.