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...

unions and subsets Semigroup ideal Boolean prime ideal theorem – Ideals in a Boolean algebra can be extended to prime ideals Taylor (1999), p. 141: "A directed...

In mathematics, the prime ideal theorem may be the Boolean prime ideal theorem the Landau prime ideal theorem on number fields This disambiguation page...

(Boolean algebra) Conjunctive normal form Disjunctive normal form Formal system And-inverter graph Logic gate Boolean analysis Boolean prime ideal theorem...

the theorem is equivalent to the Boolean prime ideal theorem, a weakened choice principle that states that every Boolean algebra has a prime ideal. An...

Boolean formula Boolean prime ideal theorem, a theorem which states that ideals in a Boolean algebra can be extended to prime ideals Binary (disambiguation)...

However, the theory of Boolean rings has an inherent asymmetry between the two operators, while the axioms and theorems of Boolean algebra express the symmetry...

ideal theorem (BPI), which is equivalent to the Banach–Alaoglu theorem. Conversely, the Krein–Milman theorem KM together with the Boolean prime ideal...

two-sided ideal coincide, and the term ideal is used alone. Modular arithmetic Noether isomorphism theorem Boolean prime ideal theorem Ideal theory Ideal (order...

choice). Stone's representation theorem for Boolean algebras needs the Boolean prime ideal theorem. The Nielsen–Schreier theorem, that every subgroup of a free...

some form of the axiom of choice; the Boolean prime ideal theorem is known to be sufficient. The proof of the theorem uses geometric features of Hilbert...

continuity Lindenbaum algebra Zorn's lemma Hausdorff maximality theorem Boolean prime ideal theorem Ultrafilter Ultrafilter lemma Tree (set theory) Tree (descriptive...

be proven optimal provided that the heuristic they use is monotonic. In Boolean algebra, a monotonic function is one such that for all ai and bi in {0...

The Boolean prime ideal theorem (BPIT). Stone's representation theorem for Boolean algebras. Any product of Boolean spaces is a Boolean space. Boolean Prime...

In mathematics, Kruskal's tree theorem states that the set of finite trees over a well-quasi-ordered set of labels is itself well-quasi-ordered under...

compactness theorem and to the Boolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact...

of perfect graphs can provide an alternative proof of Dilworth's theorem. The Boolean lattice Bn is the power set of an n-element set X—essentially {1...

theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem...

topological approach to the ultrafilter lemma (equivalently, the Boolean prime ideal theorem), which is a weak form of the axiom of choice. In some ways,...

Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem (BPI), a well-known intermediate point between the axioms of...

compactness theorem) rely on the axiom of choice, and is in fact equivalent over Zermelo–Fraenkel set theory without choice to the Boolean prime ideal theorem. Other...

of sets. However, the proofs of both statements require the Boolean prime ideal theorem, a weak form of the axiom of choice. The free distributive lattice...

Zeilberger–Bressoud theorem (combinatorics) Birkhoff's representation theorem (lattice theory) Boolean prime ideal theorem (mathematical logic) Bourbaki–Witt theorem (order...

Many theorems require the Hahn–Banach theorem, usually proved using the axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices...

ultrafilter does not imply AC. Indeed, it is equivalent to the Boolean prime ideal theorem (BPIT), a well-known intermediate point between the axioms of...

axiom of choice). The principle is also called the Hausdorff maximality theorem or the Kuratowski lemma (Kelley 1955:33). The Hausdorff maximal principle...

interior operator and closure operator to be modal operators on the power set Boolean algebra of an Alexandroff-discrete space, their construction is a special...

Boolean algebras are the same thing. This result depends on the Boolean prime ideal theorem, a choice principle slightly weaker than the axiom of choice...

the Boolean algebra of regular open sets in the Stone space of prime ideals of A. Each element x of A corresponds to the open set of prime ideals not...

p} . Every prime ideal is a maximal ideal in a Boolean ring, i.e., a ring consisting of only idempotent elements. In fact, every prime ideal is maximal...