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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 wikiPageLength "14327".
- Boolean_prime_ideal_theorem wikiPageOutDegree "58".
- Boolean_prime_ideal_theorem wikiPageRevisionID "676829629".
- Boolean_prime_ideal_theorem wikiPageWikiLink Algebra_over_a_field.
- Boolean_prime_ideal_theorem wikiPageWikiLink Axiom_of_choice.
- Boolean_prime_ideal_theorem wikiPageWikiLink Azriel_Lévy.
- Boolean_prime_ideal_theorem wikiPageWikiLink Basis_(linear_algebra).
- Boolean_prime_ideal_theorem wikiPageWikiLink Boolean_algebra_(structure).
- Boolean_prime_ideal_theorem wikiPageWikiLink Cardinality.
- Boolean_prime_ideal_theorem wikiPageWikiLink Category:Axiom_of_choice.
- Boolean_prime_ideal_theorem wikiPageWikiLink Category:Boolean_algebra.
- Boolean_prime_ideal_theorem wikiPageWikiLink Category:Order_theory.
- Boolean_prime_ideal_theorem wikiPageWikiLink Category:Theorems_in_algebra.
- Boolean_prime_ideal_theorem wikiPageWikiLink Consistency.
- Boolean_prime_ideal_theorem wikiPageWikiLink Directed_set.
- Boolean_prime_ideal_theorem wikiPageWikiLink Disjoint_sets.
- Boolean_prime_ideal_theorem wikiPageWikiLink Distributive_lattice.
- Boolean_prime_ideal_theorem wikiPageWikiLink Duality_(order_theory).
- Boolean_prime_ideal_theorem wikiPageWikiLink Filter_(mathematics).
- Boolean_prime_ideal_theorem wikiPageWikiLink Hausdorff_space.
- Boolean_prime_ideal_theorem wikiPageWikiLink Heyting_algebra.
- Boolean_prime_ideal_theorem wikiPageWikiLink Ideal_(order_theory).
- Boolean_prime_ideal_theorem wikiPageWikiLink Infimum_and_supremum.
- Boolean_prime_ideal_theorem wikiPageWikiLink Isomorphism.
- Boolean_prime_ideal_theorem wikiPageWikiLink Join_and_meet.
- Boolean_prime_ideal_theorem wikiPageWikiLink Lattice_(order).
- Boolean_prime_ideal_theorem wikiPageWikiLink List_of_Boolean_algebra_topics.
- Boolean_prime_ideal_theorem wikiPageWikiLink London_Mathematical_Society.
- Boolean_prime_ideal_theorem wikiPageWikiLink Mathematics.
- Boolean_prime_ideal_theorem wikiPageWikiLink Non-measurable_set.
- Boolean_prime_ideal_theorem wikiPageWikiLink Order_theory.
- Boolean_prime_ideal_theorem wikiPageWikiLink Power_set.
- Boolean_prime_ideal_theorem wikiPageWikiLink Prime_ideal.
- Boolean_prime_ideal_theorem wikiPageWikiLink Ring_(mathematics).
- Boolean_prime_ideal_theorem wikiPageWikiLink Stone_duality.
- Boolean_prime_ideal_theorem wikiPageWikiLink Stones_representation_theorem_for_Boolean_algebras.
- Boolean_prime_ideal_theorem wikiPageWikiLink Subset.
- Boolean_prime_ideal_theorem wikiPageWikiLink Topology.
- Boolean_prime_ideal_theorem wikiPageWikiLink Tychonoffs_theorem.
- Boolean_prime_ideal_theorem wikiPageWikiLink Ultrafilter.
- Boolean_prime_ideal_theorem wikiPageWikiLink Up_to.
- Boolean_prime_ideal_theorem wikiPageWikiLink Upper_set.
- Boolean_prime_ideal_theorem wikiPageWikiLink Vector_space.
- Boolean_prime_ideal_theorem wikiPageWikiLink Vitali_set.
- Boolean_prime_ideal_theorem wikiPageWikiLink Zermelo–Fraenkel_set_theory.
- Boolean_prime_ideal_theorem wikiPageWikiLink Zorns_lemma.
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "Boolean prime ideal theorem".
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "Boolean_prime_ideal_theorem#The_ultrafilter_lemma".
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "Ultrafilter Theorem".
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "prime ideal theorem".
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "the ultrafilter lemma".
- Boolean_prime_ideal_theorem wikiPageWikiLinkText "ultrafilter principle".
- Boolean_prime_ideal_theorem wikiPageUsesTemplate Template:Citation.
- Boolean_prime_ideal_theorem wikiPageUsesTemplate Template:Reflist.
- 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 Field.
- Boolean_prime_ideal_theorem type Theorem.
- 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 sameAs Q872088.
- 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 Теорема_про_булеві_прості_ідеали.
- Boolean_prime_ideal_theorem sameAs Q872088.
- Boolean_prime_ideal_theorem sameAs 布尔素理想定理.
- Boolean_prime_ideal_theorem wasDerivedFrom Boolean_prime_ideal_theorem?oldid=676829629.
- Boolean_prime_ideal_theorem isPrimaryTopicOf Boolean_prime_ideal_theorem.