# Is there a common notation of the power set excluding the empty set?

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).

• I don’t know of any standard notation. The only notation that I’ve seen more than once (that I can recall) is $\wp^*(S)$. – 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 $\mathcal P(S)\setminus\{\varnothing\}$).
• 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 :) – 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 $[X]^{\geq 1}$?
• 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. – Asaf Karagila Oct 26 '12 at 15:03
• I tend to write $S_{fin}$ for finite sets. That is an established notation in this field. – bitmask Oct 26 '12 at 15:06