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In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra. A common example is the Boolean prime ideal theorem, which 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 . 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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  • Boolean prime ideal theorem
  • Boolescher Primidealsatz
  • Twierdzenie o ideale pierwszym
  • Teorema do ideal primo booliano
  • 布尔素理想定理
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  • Twierdzenie o ideale pierwszym – twierdzenie teorii krat rozdzielnych.
  • 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,例如环和(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。
  • In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra. A common example is the Boolean prime ideal theorem, which 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 . 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.
  • 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.)
  • 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 reticulado distributivo e ideais maximais (de Teoria da Ordem). Esse artigo foca nos teoremas do ideal primo da teoria da ordem.
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  • In mathematics, a prime ideal theorem guarantees the existence of certain types of subsets in a given algebra. A common example is the Boolean prime ideal theorem, which 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 . 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. Although the various prime ideal theorems may appear simple and intuitive, they cannot be derived in general from the axioms of Zermelo–Fraenkel set theory without the axiom of choice (abbreviated ZF). Instead, some of the statements turn out to be equivalent to the axiom of choice (AC), while others—the Boolean prime ideal theorem, for instance—represent a property that is strictly weaker than AC. It is due to this intermediate status between ZF and ZF + AC (ZFC) that the Boolean prime ideal theorem is often taken as an axiom of set theory. The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
  • 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.) Ersetzt man die boolesche Algebra durch ihre duale boolesche Algebra, so wird der boolesche Primidealsatz zum Ultrafilterlemma.
  • Twierdzenie o ideale pierwszym – twierdzenie teorii krat rozdzielnych.
  • 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 reticulado distributivo e ideais maximais (de Teoria da Ordem). Esse artigo foca nos teoremas do ideal primo da teoria da ordem. Embora os vários teoremas do ideal primo possam parecer simples e intuitivos, eles geralmente não podem ser derivados dos axiomas da teoria dos conjuntos de Zermelo-Fraenkel sem o axioma da escolha (abreviado ZF). Em vez disso, algumas das afirmações acabam sendo equivalentes ao axioma da escolha (AC = Axiom of choice), enquanto outros – o teorema do ideal primo booleano, por exemplo - representam uma propriedade que é estritamente mais fraca que AC. Devido a este estado intermediário entre ZF e ZF + AC (ZFC) que o teorema do ideal primo booleano é frequentemente considerado um axioma da teoria dos conjuntos. As abreviações BPI (Boolean Prime Ideal, em português ideal primo booleano, IPB) ou PIT (Prime Ideal Teorem, teorema do ideal primo em português, TIP) (para álgebras booleanas) são por vezes usadas para se referir à esse axioma adicional.
  • 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,例如环和(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。
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