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- Greatest_element abstract "In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually.Formally, given a partially ordered set (P, ≤), then an element g of a subset S of P is the greatest element of S if s ≤ g, for all elements s of S.Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique. By using ≥ instead of ≤ in the above definition, one defines the least element of S.Like upper bounds, greatest elements may fail to exist. Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either. Similar conclusions hold for least elements. A finite chain always has a greatest and a least element.A greatest element of a partially ordered subset must not be confused with maximal elements of the set which are elements that are not smaller than any other of its elements. A set can have several maximal elements without having a greatest element. However, if it has a greatest element, it can't have any other maximal element.In a totally ordered set both terms coincide; it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. The dual terms are minimum and absolute minimum. Together they are called the absolute extrema.The least and greatest element of the whole partially ordered set plays a special role and is also called bottom and top, or zero (0) and unit (1), or ⊥ and ⊤, respectively. If both exists, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is even a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order.Further introductory information is found in the article on order theory.".
- Greatest_element wikiPageID "663041".
- Greatest_element wikiPageRevisionID "599615468".
- Greatest_element hasPhotoCollection Greatest_element.
- Greatest_element subject Category:Order_theory.
- Greatest_element comment "In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set (poset) is an element of S which is greater than or equal to any other element of S. The term least element is defined dually.Formally, given a partially ordered set (P, ≤), then an element g of a subset S of P is the greatest element of S if s ≤ g, for all elements s of S.Hence, the greatest element of S is an upper bound of S that is contained within this subset. It is necessarily unique.".
- Greatest_element label "Elemento mayor y menor".
- Greatest_element label "Elementy najmniejszy i największy".
- Greatest_element label "Greatest element".
- Greatest_element label "Grootste en kleinste element".
- Greatest_element label "Größtes und kleinstes Element".
- Greatest_element label "最大元".
- Greatest_element sameAs Nejmenší_a_největší_prvek.
- Greatest_element sameAs Größtes_und_kleinstes_Element.
- Greatest_element sameAs Elemento_mayor_y_menor.
- Greatest_element sameAs Elementu_maximo_eta_minimo.
- Greatest_element sameAs 최대_및_최소원소.
- Greatest_element sameAs Grootste_en_kleinste_element.
- Greatest_element sameAs Elementy_najmniejszy_i_największy.
- Greatest_element sameAs m.030tfl.
- Greatest_element sameAs Q1196892.
- Greatest_element sameAs Q1196892.
- Greatest_element wasDerivedFrom Greatest_element?oldid=599615468.
- Greatest_element isPrimaryTopicOf Greatest_element.