Matches in DBpedia 2015-10 for { <http://dbpedia.org/resource/Maximal_element> ?p ?o }
- 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. While a partially ordered set can have at most one each maximum and minimum it may have multiple maximal and minimal elements. For totally ordered sets, the notions of maximal element and maximum coincide, and the notions of minimal element and minimum coincide.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.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.".
- Maximal_element thumbnail Zigzag_poset.svg?width=300.
- Maximal_element wikiPageID "303398".
- Maximal_element wikiPageLength "12303".
- Maximal_element wikiPageOutDegree "55".
- Maximal_element wikiPageRevisionID "681925267".
- Maximal_element wikiPageWikiLink Abstract_algebra.
- Maximal_element wikiPageWikiLink Admissible_decision_rule.
- Maximal_element wikiPageWikiLink Algebraic_closure.
- Maximal_element wikiPageWikiLink Axiom_of_choice.
- Maximal_element wikiPageWikiLink Basis_(linear_algebra).
- Maximal_element wikiPageWikiLink Category:Order_theory.
- Maximal_element wikiPageWikiLink Cofinal_(mathematics).
- Maximal_element wikiPageWikiLink Consumer_choice.
- Maximal_element wikiPageWikiLink Consumer_theory.
- Maximal_element wikiPageWikiLink Decision_theory.
- Maximal_element wikiPageWikiLink Directed_set.
- Maximal_element wikiPageWikiLink Dominating_decision_rule.
- Maximal_element wikiPageWikiLink Duality_(order_theory).
- Maximal_element wikiPageWikiLink Efficient_frontier.
- Maximal_element wikiPageWikiLink Family_of_sets.
- Maximal_element wikiPageWikiLink Fence_(mathematics).
- Maximal_element wikiPageWikiLink Field_(mathematics).
- Maximal_element wikiPageWikiLink Finite_set.
- Maximal_element wikiPageWikiLink Greatest_common_divisor.
- Maximal_element wikiPageWikiLink Greatest_element.
- Maximal_element wikiPageWikiLink Hahn–Banach_theorem.
- Maximal_element wikiPageWikiLink Hamel_basis.
- Maximal_element wikiPageWikiLink Inclusion_(set_theory).
- Maximal_element wikiPageWikiLink Inclusion_relation.
- Maximal_element wikiPageWikiLink Lower_set.
- Maximal_element wikiPageWikiLink Mathematical_analysis.
- Maximal_element wikiPageWikiLink Mathematics.
- Maximal_element wikiPageWikiLink Maxima_and_minima.
- Maximal_element wikiPageWikiLink Maximal_common_divisor.
- Maximal_element wikiPageWikiLink Maximum.
- Maximal_element wikiPageWikiLink Modern_portfolio_theory.
- Maximal_element wikiPageWikiLink Order_theory.
- Maximal_element wikiPageWikiLink Pareto_efficiency.
- Maximal_element wikiPageWikiLink Pareto_frontier.
- Maximal_element wikiPageWikiLink Partially_ordered_set.
- Maximal_element wikiPageWikiLink Power_set.
- Maximal_element wikiPageWikiLink Preference.
- Maximal_element wikiPageWikiLink Preferences.
- Maximal_element wikiPageWikiLink Product_order.
- Maximal_element wikiPageWikiLink Rational_choice.
- Maximal_element wikiPageWikiLink Rational_choice_theory.
- Maximal_element wikiPageWikiLink Rational_number.
- Maximal_element wikiPageWikiLink Set_theory.
- Maximal_element wikiPageWikiLink Singleton_(mathematics).
- Maximal_element wikiPageWikiLink Square_root_of_2.
- Maximal_element wikiPageWikiLink Subset.
- Maximal_element wikiPageWikiLink Total_order.
- Maximal_element wikiPageWikiLink Total_preorder.
- Maximal_element wikiPageWikiLink Totally_ordered_set.
- Maximal_element wikiPageWikiLink Tychonoffs_theorem.
- Maximal_element wikiPageWikiLink Upper_and_lower_bounds.
- Maximal_element wikiPageWikiLink Upper_bound.
- Maximal_element wikiPageWikiLink Upper_set.
- Maximal_element wikiPageWikiLink Weak_ordering.
- Maximal_element wikiPageWikiLink Well-ordering_theorem.
- Maximal_element wikiPageWikiLink Zorns_lemma.
- Maximal_element wikiPageWikiLink File:Hasse_diagram_of_powerset_of_3.svg.
- Maximal_element wikiPageWikiLink File:Zigzag_poset.svg.
- Maximal_element wikiPageWikiLinkText "''min'' and ''max''".
- Maximal_element wikiPageWikiLinkText "Maximal element".
- Maximal_element wikiPageWikiLinkText "Maximal elements and minimal elements".
- Maximal_element wikiPageWikiLinkText "maximal and minimal elements".
- Maximal_element wikiPageWikiLinkText "maximal element".
- Maximal_element wikiPageWikiLinkText "maximal or greatest element".
- Maximal_element wikiPageWikiLinkText "maximal".
- Maximal_element wikiPageWikiLinkText "maximality".
- Maximal_element wikiPageWikiLinkText "minimal element".
- Maximal_element wikiPageWikiLinkText "minimal or least element".
- Maximal_element wikiPageWikiLinkText "minimal".
- Maximal_element wikiPageWikiLinkText "nonmaximal".
- Maximal_element hasPhotoCollection Maximal_element.
- Maximal_element wikiPageUsesTemplate Template:Portal.
- Maximal_element wikiPageUsesTemplate Template:Reflist.
- Maximal_element wikiPageUsesTemplate Template:d,_o%7D,_%7Bd,_o,_g%7D,_%7Bg,_o,_a,_d%7D,_%7Bo,_a,_f.
- Maximal_element subject Category:Order_theory.
- Maximal_element hypernym Element.
- Maximal_element type MilitaryUnit.
- Maximal_element type Field.
- 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 "Maximal element".
- Maximal_element sameAs Maximal_i_minimal_(elements).
- 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 극대_원소와_극소_원소.
- 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 Maximal.
- Maximal_element sameAs Максимальные_и_минимальные_элементы.
- Maximal_element sameAs Максималан_и_минималан_елеменат_скупа.
- Maximal_element sameAs Максимальні_та_мінімальні_елементи.