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In page Ultrafilter:

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Moreover, ultrafilters on a Boolean algebra can be related to maximal ideals and homomorphisms to the 2-element Boolean algebra {true, false} (also known as 2-valued morphisms) as follows:

  • Given a maximal ideal of a Boolean algebra, its complement is an ultrafilter, and there is a unique homomorphism onto {true, false} taking the maximal ideal to "false".
  • Given an ultrafilter on a Boolean algebra, its complement is a maximal ideal, and there is a unique homomorphism onto {true, false} taking the ultrafilter to "true".[citation needed]