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- Maximal_element abstract "In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum. The maximum of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S, and the minimum of S is again defined dually. For totally ordered sets, the notions of maximal element and maximum on one hand and minimal element and minimum on the other hand coincide.While a partially ordered set can have at most one each maximum and minimum it may have multiple maximal and minimal elements. Zorn's lemma states that every partially ordered set for which every totally ordered subset has an upper bound contains at least one maximal element. This lemma is equivalent to the well-ordering theorem and the axiom of choice and implies major results in other mathematical areas like the Hahn–Banach theorem and Tychonoff's theorem, the existence of a Hamel basis for every vector space, and the existence of an algebraic closure for every field.As an example, in the collection S = {{d, o}, {d, o, g}, {g, o, a, d}, {o, a, f}}ordered by containment, the element {d, o} is minimal, the element {g, o, a, d} is maximal, the element {d, o, g} is neither, and the element {o, a, f} is both minimal and maximal. By contrast, neither a maximum nor a minimum exists for S.".
- Maximal_element thumbnail Zigzag_poset.svg?width=300.
- Maximal_element wikiPageID "303398".
- Maximal_element wikiPageRevisionID "593677072".
- Maximal_element hasPhotoCollection Maximal_element.
- Maximal_element subject Category:Order_theory.
- Maximal_element comment "In mathematics, especially in order theory, a maximal element of a subset S of some partially ordered set (poset) is an element of S that is not smaller than any other element in S. A minimal element of a subset S of some partially ordered set is defined dually as an element of S that is not greater than any other element in S. The notions of maximal and minimal elements are weaker than those of greatest element and least element which are also known, respectively, as maximum and minimum.".
- Maximal_element label "Elemento maximal y minimal".
- Maximal_element label "Elemento minimal".
- Maximal_element label "Elementy minimalny i maksymalny".
- Maximal_element label "Maximaal en minimaal element".
- Maximal_element label "Maximal element".
- Maximal_element label "Maximales und minimales Element".
- Maximal_element label "Élément maximal".
- Maximal_element label "极大元".
- Maximal_element sameAs Maximální_a_minimální_prvek.
- Maximal_element sameAs Maximales_und_minimales_Element.
- Maximal_element sameAs Elemento_maximal_y_minimal.
- Maximal_element sameAs Élément_maximal.
- Maximal_element sameAs Maximaal_en_minimaal_element.
- Maximal_element sameAs Elementy_minimalny_i_maksymalny.
- Maximal_element sameAs Elemento_minimal.
- Maximal_element sameAs m.01s5j6.
- Maximal_element sameAs Q1475294.
- Maximal_element sameAs Q1475294.
- Maximal_element wasDerivedFrom Maximal_element?oldid=593677072.
- Maximal_element depiction Zigzag_poset.svg.
- Maximal_element isPrimaryTopicOf Maximal_element.
