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Thus, another way of stating Sperner's theorem is that the width of the inclusion order on a power set is (⌊ / ⌋). A graded partially ordered set is said to have the Sperner property when one of its largest antichains is formed by a set of elements that all have the same rank. In this terminology, Sperner's theorem states that the partially ...
Another way to classify incompatible elements is by mass (lanthanide series): light rare-earth elements (LREE) are La, Ce, Pr, Nd, and Sm, and heavy rare-earth elements (HREE) are Eu–Lu. Rocks or magmas that are rich, or only slightly depleted, in light rare-earth elements are referred to as "fertile", and those with strong depletions in LREE ...
An antichain in is a subset of in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in . (However, some authors use the term "antichain" to mean strong antichain , a subset such that there is no element of the poset smaller than two distinct elements of the antichain.)
If a poset has a greatest element, it must be the unique maximal element, but otherwise there can be more than one maximal element, and similarly for least elements and minimal elements. In our running example, {,,} and {} are the maximal and minimal elements. Removing these, there are 3 maximal elements and 3 minimal elements (see Fig. 5).
The transitive reduction or covering graph of the Dedekind–MacNeille completion describes the order relation between its elements in a concise way: each neighbor of a cut must remove an element of the original partial order from either the upper or lower set of the cut, so each vertex has at most n neighbors.
Next it is shown that the poset of partial orders extending , ordered by extension, has a maximal element. The existence of such a maximal element is proved by applying Zorn's lemma to this poset. Zorn's lemma states that a partial order in which every chain has an upper bound has a maximal element. A chain in this poset is a set of relations ...
In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ≤ b. Dropping the requirement of being acyclic, one can also obtain all preorders.
An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains. Dilworth's theorem states that, in any finite partially ordered set, the largest ...