Boolean Algebra (structure) - Ideals and Filters

Ideals and Filters

An ideal of the Boolean algebra A is a subset I such that for all x, y in I we have x ∨ y in I and for all a in A we have a ∧ x in I. This notion of ideal coincides with the notion of ring ideal in the Boolean ring A. An ideal I of A is called prime if I ≠ A and if a ∧ b in I always implies a in I or b in I. Furthermore, for every a ∈ A we have that a ∧ -a = 0 ∈ I and then a ∈ I or -a ∈ I for every a ∈ A, if I is prime. An ideal I of A is called maximal if I ≠ A and if the only ideal properly containing I is A itself. For an ideal I, if a ∉ I and -a ∉ I, then I ∪ {a} or I ∪ {-a} is properly contained in another ideal J. Hence, that an I is not maximal and therefore the notions of prime ideal and maximal ideal are equivalent in Boolean algebras. Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A.

The dual of an ideal is a filter. A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x ∧ y in p and for all a in A we have a ∨ x in p. The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and can not be proved in ZF, if ZF is consistent. Within ZF, it is strictly weaker than the axiom of choice. The Ultrafilter Theorem has many equivalent formulations: every Boolean algebra has an ultrafilter, every ideal in a Boolean algebra can be extended to a prime ideal, etc.