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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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In the context of $\beta\omega$ it’s not uncommon to denote by $n$ the principal ultrafilter over $n$, though I’m pretty sure that I recently saw $n^*$ in a proof that every fixed ultrafilter is in the centre of $\langle\beta\omega,+\rangle$. 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. – Brian M. Scott Dec 4 '11 at 9:05
Since I tend to use uppercase script letters for filters and ultrafilters, I’ve always used $\mathscr{F}$ for the Fréchet filter. (I corrected your cofinal to cofinite.) – Brian M. Scott Dec 4 '11 at 9:11
I think all the notations are fine, except $e(a)$ (why on earth $e(\cdot)$?) and I don't like the notation $a^\ast$ either: this is because it seems to indicate a contravariant dependence which it isn't. – t.b. Dec 4 '11 at 9:53
@t.b.: I expect that $e(a)$ stems from the common use of $e$ to denote an embedding. My objection to $a^*$ is that I’m accustomed to $A^*$ denoting the set of free ultrafilters containing $A$, i.e., $\operatorname{cl}_{\beta D}(A)\setminus A$. – Brian M. Scott Dec 4 '11 at 10:12
Thanks @Brian, that's an even better reason. Martin edited the explanation "e for embedding" in and this seems to be the right interpretation. The letter $e$ could also stand for "evaluation" if you think of the Stone-Čech compactification in terms of the Gel'fand spectrum of $C_b(X)$. – t.b. Dec 4 '11 at 10:39
up vote 1 down vote accepted

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 $U$).

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