weak and in a strong form. Weak prime ideal theorems state that every non-trivial algebra of a certain class has at least one prime ideal. In contrast, strong prime ideal theorems require that every ideal that is disjoint from a given filter can be extended to a prime ideal that is still disjoint from that filter. In the case of algebras that are not posets, one uses different substructures instead of filters. Many forms of these theorems are actually known to be equivalent, so that the assertion that "PIT" holds is usually taken as the assertion that the corresponding statement for Boolean algebras (BPI) is valid. Another variation of similar theorems is obtained by replacing each occurrence of prime ideal by maximal ideal. The corresponding maximal ideal theorems (MIT) are often—though not always—stronger than their PIT equivalents. Boolean prime ideal theorem The Boolean prime ideal theorem is the strong prime ideal theorem for Boolean algebras. Thus the formal statement is: Let B be a Boolean algebra, let I be an ideal and let F be a filter of B, such that I and F are disjoint. Then I is contained in some prime ideal of B that is disjoint from F. The weak prime ideal theorem for Boolean algebras simply states: Every Boolean algebra contains a prime ideal. We refer to these statements as the weak and strong BPI. The two are equivalent, as the strong BPI clearly implies the weak BPI, and the reverse implication can be achieved by using the weak BPI to find prime ideals in the appropriate quotient algebra. The BPI can be expressed in various ways. For this purpose, recall the following theorem: For any ideal I of a Boolean algebra B, the following are equivalent: I is a prime ideal. I is a maximal ideal, i.e. for any proper ideal J, if I is contained in J then I = J. For every element a of B, I contains exactly one of {a, ¬a}. This theorem is a well-known fact for Boolean algebras. Its dual establishes the equivalence of prime filters and ultrafilters. Note that the last property is in fact self-dual—only the prior assumption that I is an ideal gives the full characterization. All of the implications within this theorem can be proven in ZF. Thus the following (strong) maximal ideal theorem (MIT) for Boolean algebras is equivalent to BPI: Let B be