About: Boolean prime ideal theorem

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In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory.

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rdfs:label	Boolescher Primidealsatz (de) Boolean prime ideal theorem (en) Théorème de l'idéal premier dans une algèbre de Boole (fr) Twierdzenie o ideale pierwszym (pl) Teorema do ideal primo booliano (pt) Теорема про булеві прості ідеали (uk) 布尔素理想定理 (zh)
rdfs:comment	Twierdzenie o ideale pierwszym – twierdzenie teorii krat rozdzielnych. (pl) 素理想定理（prime ideal theorem）即保证在给定的抽象代数中特定类型之子集的存在性之數學定理。常见的例子就是布尔素理想定理（Boolean prime ideal theorem），它声称在布尔代数中的理想可以被扩展成素理想。这个陈述对于在集合上的滤子的变体叫做叫做。通过考虑不同的带有适当的理想概念的数学结构可获得其他定理，例如環和（环论的）素理想，和分配格和（序理论的）的极大理想。本文关注序理论的素理想定理。尽管各种素理想定理可能看起来简单且直觉，它们一般不能从策梅洛-弗蘭克爾集合論（ZF）的公理推导出来。反而某些陈述等价于选择公理（AC），而其他的如布尔素理想定理，体现了严格弱于AC的性质。由于这个在ZF和ZF+AC (ZFC)之间的中介状态，布尔素理想定理经常被接受为集合论的公理。经常用缩写BPI（对布尔代数）或PIT提及这个额外公理。 (zh) Der boolesche Primidealsatz sagt aus, dass jede boolesche Algebra ein Primideal enthält. Der Beweis dieses Satzes kann nicht ohne transfinite Methoden geführt werden, das bedeutet, dass er nicht aus den Axiomen der Mengenlehre ohne Auswahlaxiom beweisbar ist. Umgekehrt ist das Auswahlaxiom nicht aus dem booleschen Primidealsatz beweisbar, dieser Satz ist also schwächer als das Auswahlaxiom. Außerdem ist der Satz (relativ zu den Axiomen der Zermelo-Fraenkel-Mengenlehre) äquivalent zu einigen anderen Sätzen wie zum Beispiel Gödels Vollständigkeitssatz. (Das bedeutet, dass man aus den Axiomen der Mengenlehre plus dem booleschen Primidealsatz dieselben Sätze beweisen kann wie aus den Axiomen der Mengenlehre plus dem gödelschen Vollständigkeitssatz.) (de) In mathematics, the Boolean prime ideal theorem states that ideals in a Boolean algebra can be extended to prime ideals. A variation of this statement for filters on sets is known as the ultrafilter lemma. Other theorems are obtained by considering different mathematical structures with appropriate notions of ideals, for example, rings and prime ideals (of ring theory), or distributive lattices and maximal ideals (of order theory). This article focuses on prime ideal theorems from order theory. (en) En mathématiques, un théorème de l'idéal premier garantit l'existence de certains types de sous-ensembles dans une algèbre. Un exemple courant est le théorème de l'idéal premier dans une algèbre de Boole, qui énonce que tout idéal d'une algèbre de Boole est inclus dans un idéal premier. Une variante de cet énoncé pour filtres sur des ensembles est connue comme le . D'autres théorèmes sont obtenus en considérant les différentes structures mathématiques avec les notions d'idéal appropriées, par exemple, les anneaux et leurs idéaux premiers (en théorie des anneaux), ou les treillis distributifs et leurs idéaux maximaux (en théorie des ordres). Cet article se concentre sur le théorème de l'idéal premier en théorie des ordres. (fr) Em matemática, um teorema do ideal primo garante a existência de certos tipos de subconjuntos numa álgebra dada. Um exemplo comum é o teorema do ideal primo booleano, o qual afirma que ideais em uma álgebra booleana podem ser estendidos para ideais primos. Uma variação dessa afirmação para filtros em conjuntos é conhecida como o . Outros teoremas são obtidos considerando diferentes estruturas matemáticas com noções apropriadas de ideais, por exemplo, anéis e ideais primos (da teoria dos anéis), ou e ideais maximais (de ). Esse artigo foca nos teoremas do ideal primo da . (pt) Теорема про Булеві прості ідеали в теорії порядку стверджує, що ідеали в булевій алгебрі можуть бути розширені до простих ідеалів. Так як в теорії порядку більшість понять є , і двоїстим до ідеала є фільтр, то аналогічне твердження для фільтрів називається — . Існують аналогічні формулювання і для інших алгебраїчних структур, наприклад, для кілець та їх простих ідеалів (в теорії кілець). Всі ці формулювання не можуть бути доведені в рамках аксіом теорії множин Цермело-Френцеля (ZF). (uk)
rdfs:seeAlso	Set-theoretic topology
dcterms:subject	Axiom of choice Order theory Boolean algebra Theorems in lattice theory
Wikipage page ID	314919 (xsd:integer)
Wikipage revision ID	1087342940 (xsd:integer)
Link from a Wikipage to another Wikipage	Axiom of choice Basis (linear algebra) Vector space De Bruijn–Erdős theorem (graph theory) Order theory Consistent Mathematics Order theory Graph coloring Stone's representation theorem for Boolean algebras Zermelo–Fraenkel set theory Axiom of Choice Distributive lattice Duality (order theory) Join and meet Lattice (order) List of Boolean algebra topics Non-measurable set Stone duality Filter (mathematics) Filter (set theory) Filters in topology Hasse diagram Cardinality Directed set Graph theory Isomorphism Ultrafilter Ring (mathematics) Boolean algebra Hahn-Banach theorem Hausdorff spaces Heyting algebra Theorems in lattice theory Supremum Zorn's lemma Axiom of choice Azriel Levy Boolean algebra (structure) Ideal (order theory) Tychonoff's theorem Up to Journal of the London Mathematical Society Subset Vitali set Lower set Powerset Disjoint set Alexander subbase theorem Stone's representation theorem Ultrafilter lemma
sameAs	Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem Boolean prime ideal theorem
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