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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 ...
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 ...
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 ...
A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset. Maximal element. A maximal element of a subset X of a poset P is an element m of X, such that m ≤ x implies m = x, for all x in X. The dual notion is called minimal element. Maximum element. Synonym of greatest element. For a ...
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]
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.
In mathematics, especially in order theory, the greatest element of a subset of a partially ordered set (poset) is an element of that is greater than every other element of . The term least element is defined dually , that is, it is an element of S {\displaystyle S} that is smaller than every other element of S . {\displaystyle S.}
Let n be a positive integer, and let P be the partial order on the elements a i and b i (for 1 ≤ i ≤ n) in which a i ≤ b j whenever i ≠ j, but no other pairs are comparable. In particular, a i and b i are incomparable in P; P can be viewed as an oriented form of a crown graph. The illustration shows an ordering of this type for n = 4.