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The reason is that, in general, there are many different possible ways to draw a Hasse diagram for a given poset. The simple technique of just starting with the minimal elements of an order and then drawing greater elements incrementally often produces quite poor results: symmetries and internal structure of the order are easily lost.
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).
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.
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.
The red subset = {1,2,3,4} has two maximal elements, viz. 3 and 4, and one minimal element, viz. 1, which is also its least element. In mathematics , especially in order theory , a maximal element of a subset S {\displaystyle S} of some preordered set is an element of S {\displaystyle S} that is not smaller than any other element in S ...
This poset satisfies the countable chain condition. Forcing with this poset adds ω 2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum hypothesis. More generally, one can replace ω 2 by any cardinal κ so construct a model where the continuum has size at least κ. Here ...
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.
Intuitively, a filter F is a subset of P whose members are elements large enough to satisfy some criterion. [1] For instance, if x ∈ P, then the set of elements above x is a filter, called the principal filter at x. (If x and y are incomparable elements of P, then neither the principal filter at x nor y is contained in the other.)