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As another example, consider the positive integers, ordered by divisibility: 1 is a least element, as it divides all other elements; on the other hand this poset does not have a greatest element. This partially ordered set does not even have any maximal elements, since any g divides for instance 2g, which is distinct from it, so g is not
An up-down poset Q(a,b) is a generalization of a zigzag poset in which there are a downward orientations for every upward one and b total elements. [5] For instance, Q(2,9) has the elements and relations > > < > > < > >. In this notation, a fence is a partially ordered set of the form Q(1,n).
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.)
Hasse diagram of the natural numbers, partially ordered by "x≤y if x divides y".The numbers 4 and 6 are incomparable, since neither divides the other. In mathematics, two elements x and y of a set P are said to be comparable with respect to a binary relation ≤ if at least one of x ≤ y or y ≤ x is true.
The usual numeric orderings on the integers or real numbers satisfy these properties; however, unlike the orderings on the numbers, a partial order may have two elements that are incomparable: neither x ≤ y nor y ≤ x holds. Another familiar example of a partial ordering is the inclusion ordering ⊆ on pairs of sets. [2]
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.
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 ...
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).