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I understand that the notion of a filter is in some sense dual to the notion of an ideal, at least in the context of Boolean algebras1.

Let $f:{\mathbf A} \to {\mathbf B}$ be a Boolean algebra homomorphism. It's easy to verify that the the set $${\mathbf I} = f^{-1}(\{0\})$$ is an ideal in ${\mathbf A}$. The set ${\mathbf I}$ thus defined is commonly known as the kernel of $f$.

Similarly, it is easy to verify that the set $${\mathbf F} = f^{-1}(\{1\})$$ is a filter in ${\mathbf A}$.

My question is: Is there a standard name (a counterpart/dual of "kernel") for the set ${\mathbf F}$ thus defined?

(I've seen the term cokernel in other contexts, but I believe in this case the cokernel of $f$ would be the quotient ${\mathbf B}/\text{Im}(f)$, and it's not even obvious to me that such quotient would exist, let alone that an isomorphism between it and ${\mathbf F}$ would exist.)


1 An ideal in a Boolean algebra ${\mathbf A}$ is defined as a subset ${\mathbf I} \subseteq {\mathbf A}$ such that

  1. $0 \in {\mathbf I}$;
  2. if $p\in {\mathbf I}$ and $q\in {\mathbf I}$, then $p\vee q \in {\mathbf I}$;
  3. if $p\in {\mathbf I}$ and $q\in {\mathbf A}$, then $p\wedge q \in {\mathbf I}$.

The definition of a filter in ${\mathbf A}$ is just the dual of the definition above. Namely, a filter is defined as a subset ${\mathbf F} \subseteq {\mathbf A}$ such that

  1. $1 \in {\mathbf F}$;
  2. if $p\in {\mathbf F}$ and $q\in {\mathbf F}$, then $p\wedge q \in {\mathbf F}$;
  3. if $p\in {\mathbf F}$ and $q\in {\mathbf A}$, then $p\vee q \in {\mathbf F}$.
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1 Answer 1

At least in Introduction to Boolean Algebras by Steve Givant and Paul Halmos it seems that the term cokernel is in fact used. For instance, see page 162 problem 31. Not sure if this is standard though.

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Surely not standard in module theory, where the cokernel of a morphism $f\colon X\to Y$ is the quotient $Y/f(X)$. –  egreg Sep 18 '13 at 17:32
    
No, agreed. Virtually always cokernel means what you have described. But then again, I can think of at least a half-dozen things called "normal" so it might be a case of the same word for different things. –  Alexander Sep 18 '13 at 17:34
    
Maybe “antikernel”, in this case. –  egreg Sep 18 '13 at 17:39

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