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An exemple of an ultrafilter

  1. Dec 9, 2007 #1

    quasar987

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    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!
     
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  3. Dec 9, 2007 #2

    Hurkyl

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    I'm not sure what you you mean by that question.
     
  4. Dec 9, 2007 #3

    quasar987

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    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?
     
  5. Dec 9, 2007 #4

    Hurkyl

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    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?
     
  6. Dec 9, 2007 #5

    quasar987

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    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.
     
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