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In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem states that, in any finite partially ordered set, the maximum size of an antichain of incomparable elements equals the minimum number of chains needed to cover all elements. This number is called the width of the partial order.
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 partial order of dimension 4 (shown as a Hasse diagram) and four total orderings that form a realizer for this partial order.. In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order.
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.)
A complete lattice is a lattice in which every subset of elements of L has an infimum and supremum; this generalizes the analogous properties of the real numbers. An order-embedding is a function that maps distinct elements of S to distinct elements of L such that each pair of elements in S has the same ordering in L as they do in S.
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.
The first diagram makes clear that the power set is a graded poset.The second diagram has the same graded structure, but by making some edges longer than others, it emphasizes that the 4-dimensional cube is a combinatorial union of two 3-dimensional cubes, and that a tetrahedron (abstract 3-polytope) likewise merges two triangles (abstract 2-polytopes).
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.}