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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 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.
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 lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra.It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet).
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]
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 ]
More formally, a countable poset = (,) is an interval order if and only if there exists a bijection from to a set of real intervals, so (,), such that for any , we have < in exactly when <. Such posets may be equivalently characterized as those with no induced subposet isomorphic to the pair of two-element chains , in other words as the ( 2 + 2 ...
Also note that the empty set usually has upper bounds (if the poset is non-empty) and thus a bounded-complete poset has a least element. One may also consider the subsets of a poset which are totally ordered, i.e. the chains. If all chains have a supremum, the order is called chain complete. Again, this concept is rarely needed in the dual form.