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Given a set $S$ we can define the filter consisting of all complements of finite sets, which is usually called Fréchet filter or cofinite filter.

For any $a\in S$ the set $\{A\subseteq S; a\in A\}$ is the principal ultrafilter defined by $a$.

What are "standard" (common, frequently used) notations for these two filters? What notation would you recommend?

I will be using this notation mostly in connection with $\mathcal F$-limits (see e.g. wikipedia or Hindman-Strauss, p.63) and Stone-Čech compactification.


For the Fréchet filter I was able to find


For principal ultrafilters I found:

  • $\mathcal F_a$, e.g. here

  • $\pi_a$, e.g. this paper

  • $e(a)$, e.g. Hindman-Strauss (I think the authors chose the notation $e$ to indicate that this gives the embedding of discrete space on $A$ into the Stone-Čech compactification.)

  • IIRC I have seen $a^*$ (in the context of Stone-Čech compactification), but I cannot find an example right now.

Since points of $A$ and principal ultrafilters are usually identified (i.e. $A$ is identified with the corresponding subspace of $\beta A$) maybe it would make sense in some situations to denote the principal ultrafilter given by $a$ again as $a$, but I think this would be too confusing.

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    P.S. I did not $e$ven ping you - I see that you're both on this site a lot, so you'll probably notice that the question was bumped, in the case you want to look at it again.2011-12-07

2 Answers 2

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Summarizing the above comments by Brian M. Scott and t.b.: It seems that there is no generally accepted notation.

Notation for principal ultrafilters:

  • In general it seems common to use $\mathscr{U}_a$ when $\mathscr{U}$ is used for a generic ultrafilter, $p_a$ when $p$ is used for a generic ultrafilter, etc.

  • From the notations mentioned in the post, at least in some situations, $e(a)$ and $a^*$ are not advisable. (The letter $e$ is chosen rather arbitrarily. The notation $a^*$ might be confused $A^*$, which is often used for $\operatorname{cl}_{\beta D}(A)\setminus A$.)

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A principal filter corresponding to a set $A$ is sometimes denoted as $\uparrow A$ (or $\uparrow^U A$ to mean a filter on the set $U$).