The abbreviations BPI or PIT (for Boolean algebras) are sometimes used to refer to this additional axiom.
joins), as do the posets within this article, then this is equivalently characterized as a non-empty lower set I that is closed for binary suprema (that is,
The above statement led to various generalized prime ideal theorems, each of which exists in a weak and in a strong form.
Note that the last property is in fact self-dual—only the prior assumption that I is an ideal gives the full characterization.
Thus the following (strong) maximal ideal theorem (MIT) for Boolean algebras is equivalent to BPI: Note that one requires "global" maximality, not just maximality with respect to being disjoint from F. Yet, this variation yields another equivalent characterization of BPI: The fact that this statement is equivalent to BPI is easily established by noting the following theorem: For any distributive lattice L, if an ideal I is maximal among all ideals of L that are disjoint to a given filter F, then I is a prime ideal.
The proof for this statement (which can again be carried out in ZF set theory) is included in the article on ideals.
Hence, when taking the equivalent duals of all former statements, one ends up with a number of theorems that equally apply to Boolean algebras, but where every occurrence of ideal is replaced by filter[citation needed].
It is worth noting that for the special case where the Boolean algebra under consideration is a powerset with the subset ordering, the "maximal filter theorem" is called the ultrafilter lemma.
It is known that all of these statements are consequences of the Axiom of Choice, AC, (the easy proof makes use of Zorn's lemma), but cannot be proven in ZF (Zermelo-Fraenkel set theory without AC), if ZF is consistent.
Yet, the BPI is strictly weaker than the axiom of choice, though the proof of this statement, due to J. D. Halpern and Azriel Lévy is rather non-trivial.
Indeed, it turns out that the MITs for distributive lattices and even for Heyting algebras are equivalent to the axiom of choice.
Furthermore, observe that Heyting algebras are not self dual, and thus using filters in place of ideals yields different theorems in this setting.
Finally, prime ideal theorems do also exist for other (not order-theoretical) abstract algebras.
This situation requires to replace the order-theoretic term "filter" by other concepts—for rings a "multiplicatively closed subset" is appropriate.
This is of practical importance for proving Stone's representation theorem for Boolean algebras, a special case of Stone duality, in which one equips the set of all prime ideals with a certain topology and can indeed regain the original Boolean algebra (up to isomorphism) from this data.
Furthermore, it turns out that in applications one can freely choose either to work with prime ideals or with prime filters, because every ideal uniquely determines a filter: the set of all Boolean complements of its elements.
Many other theorems of general topology that are often said to rely on the axiom of choice are in fact equivalent to BPI.
In linear algebra, the Boolean prime ideal theorem can be used to prove that any two bases of a given vector space have the same cardinality.