Can a Larger Proper Filter Contain an Ultrafilter in a Boolean Algebra?

[quasar987]

Let B be a boolean algebra with smaller element 0, and let b different from 0 be in B. Apparently, {x in B: x>=b} is an ultrafilter of B.

I don't understand why a priori there could not exists a larger proper filter containing this one!

 

[Hurkyl]

I'm not sure what you you mean by that question.

 

[quasar987]

Well, an ultrafilter in a boolean algebra (B,u,n,C,0,1) is as far as I understand, a proper filter that is maximal, in the sense that there are no filter containing it other than B itself.

Why is the set {x in B: x>=b} from post #1 an ultrafilter. Why can't there be a filter containing it other than B?

 

[Hurkyl]

Have you tried constructing a counterexample?



Anyways, are you sure you're not missing a condition? e.g. is b supposed to be an atom?

 

[quasar987]

I had vaguely found a counterexample, that's why I asked the question. Now it's clear that a condition such as "b is an atom" is missing. Thanks.