For any set $S$, $\mathcal{P}(S)$ denotes the power set of $S$ and $\emptyset \in \mathcal{P}(S)$ always holds. Essentially, I want to denote the set that equals the power set (of some $S$) but excluding the empty set. I was thinking about writing $\mathcal{P}^+$ and defining that (as $\mathcal{P}^+(S) := \mathcal{P}(S) - \emptyset = \mathcal{P}(S)\setminus \{\emptyset\}$), but this could be a common enough thing that someone already established a notation for it.

Wikipedia et al. don't mention anything, but maybe there is something nevertheless. I would prefer to use an established notation if there is one (while still defining what I mean).

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    $\begingroup$ I don’t know of any standard notation. The only notation that I’ve seen more than once (that I can recall) is $\wp^*(S)$. $\endgroup$ – Brian M. Scott Oct 26 '12 at 15:02

I am not aware of any such notation (and in the business of choice functions, one runs a lot into $\mathcal P(S)\setminus\{\varnothing\}$).

It is fine to make your own, but be sure to be consistent about it, and to define it at the beginning of your work.

There is a risk of having too many notations, it may burden the reader. Sometimes just writing it explicitly works just as well. If you're tired of doing that, write a LaTeX macro.

  • $\begingroup$ I already wrote a LaTeX macro with a provisional $P^+$, and I am aware that too much definitions could burden the reader, but thanks for the advice, anyway :) $\endgroup$ – bitmask Oct 26 '12 at 15:01

I would suggest using something close to standard notation, e.g. $\mathcal{P}_{\lt \omega}(X)$ or $[X]^{\lt \omega}$ is used for the set of finite subsets of $X$, so why not $\mathcal{P}_{\geq 1}(X)$ or $[X]^{\geq 1}$?

  • $\begingroup$ Actually, $\mathcal P_\omega(X)$ is the set of all finite subsets. See the common notation $\mathcal P_\kappa(\lambda)$ in the context of supercompact measures. $\endgroup$ – Asaf Karagila Oct 26 '12 at 15:03
  • $\begingroup$ I tend to write $S_{fin}$ for finite sets. That is an established notation in this field. $\endgroup$ – bitmask Oct 26 '12 at 15:06

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