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
$\mathrm{Fr}$, $\mathrm{Fr}(S)$ or $\mathfrak{Fr}$ , e.g. this thesis or this paper.
$\operatorname{Cof}(S)$, e.g. this paper
This question asks about the notation for the dual ideal: Notation for the set of all finite subsets of $\mathbb{N}$
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.