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A totally ordered set is a partially ordered set in which any two elements are comparable. The Szpilrajn extension theorem states that every partial order is contained in a total order. Intuitively, the theorem says that any method of comparing elements that leaves some pairs incomparable can be extended in such a way that every pair becomes ...
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 ...
If the number 1 is excluded, while keeping divisibility as ordering on the elements greater than 1, then the resulting poset does not have a least element, but any prime number is a minimal element for it. 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 ...
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.)
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 ...
An element x of S embeds into the completion as its principal ideal, the set ↓ x of elements less than or equal to x. Then (↓ x) u is the set of elements greater than or equal to x, and ((↓ x) u) l = ↓ x, showing that ↓ x is indeed a member of the completion. The mapping from x to ↓ x is an order-embedding. [7]
Now there are also elements of a poset that are special with respect to some subset of the order. This leads to the definition of upper bounds. Given a subset S of some poset P, an upper bound of S is an element b of P that is above all elements of S. Formally, this means that s ≤ b, for all s in S. Lower bounds again are defined by inverting ...
In a directed set, every pair of elements (particularly pairs of incomparable elements) has a common upper bound within the set. If a directed set has a maximal element, it is also its greatest element, [proof 7] and hence its only maximal element. For a directed set without maximal or greatest elements, see examples 1 and 2 above.