**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.

Read more about this topic: Boolean Algebra (structure)

### Famous quotes containing the words ideals and/or filters:

“A philistine is a full-grown person whose interests are of a material and commonplace nature, and whose mentality is formed of the stock ideas and conventional *ideals* of his or her group and time.”

—Vladimir Nabokov (1899–1977)

“Raise a million *filters* and the rain will not be clean, until the longing for it be refined in deep confession. And still we hear, If only this nation had a soul, or, Let us change the way we trade, or, Let us be proud of our region.”

—Leonard Cohen (b. 1934)