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- Boolean_prime_ideal_theorem abstract "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 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.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.".
- Boolean_prime_ideal_theorem wikiPageID "314919".
- Boolean_prime_ideal_theorem wikiPageRevisionID "603349060".
- Boolean_prime_ideal_theorem hasPhotoCollection Boolean_prime_ideal_theorem.
- Boolean_prime_ideal_theorem subject Category:Axiom_of_choice.
- Boolean_prime_ideal_theorem subject Category:Boolean_algebra.
- Boolean_prime_ideal_theorem subject Category:Order_theory.
- Boolean_prime_ideal_theorem subject Category:Theorems_in_algebra.
- Boolean_prime_ideal_theorem type Abstraction100002137.
- Boolean_prime_ideal_theorem type Communication100033020.
- Boolean_prime_ideal_theorem type Message106598915.
- Boolean_prime_ideal_theorem type Proposition106750804.
- Boolean_prime_ideal_theorem type Statement106722453.
- Boolean_prime_ideal_theorem type Theorem106752293.
- Boolean_prime_ideal_theorem type TheoremsInAlgebra.
- Boolean_prime_ideal_theorem comment "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 ultrafilter lemma.".
- Boolean_prime_ideal_theorem label "Boolean prime ideal theorem".
- Boolean_prime_ideal_theorem label "Boolescher Primidealsatz".
- Boolean_prime_ideal_theorem label "Teorema do ideal primo booliano".
- Boolean_prime_ideal_theorem label "Twierdzenie o ideale pierwszym".
- Boolean_prime_ideal_theorem label "布尔素理想定理".
- Boolean_prime_ideal_theorem sameAs Boolescher_Primidealsatz.
- Boolean_prime_ideal_theorem sameAs Twierdzenie_o_ideale_pierwszym.
- Boolean_prime_ideal_theorem sameAs Teorema_do_ideal_primo_booliano.
- Boolean_prime_ideal_theorem sameAs m.01tqbt.
- Boolean_prime_ideal_theorem sameAs Q872088.
- Boolean_prime_ideal_theorem sameAs Q872088.
- Boolean_prime_ideal_theorem sameAs Boolean_prime_ideal_theorem.
- Boolean_prime_ideal_theorem wasDerivedFrom Boolean_prime_ideal_theorem?oldid=603349060.
- Boolean_prime_ideal_theorem isPrimaryTopicOf Boolean_prime_ideal_theorem.