$\mathcal N (A):=\mathcal P(A)\setminus\{\varnothing\}$ notation

Question

Define $\mathcal N$

$\mathcal N (A):=\mathcal P(A)\setminus\{\varnothing\}$

Does $\mathcal N$ has a special name and standard notation?

Answer 1

Reading wikipedia I found this interesting notation. The nice thing is that its seems to be a natural choiche even if wikipedia does not give references for this.

The set of subsets of $S$ of cardinality less than $\kappa$ is denoted by $\mathcal{P}_{\kappa}(S)$ or $\mathcal{P}_{<\kappa}(S) $,

Similarly, the set of non-empty subsets of $S$ might be denoted by $\mathcal{P}_{\geq 1}(S)$ .

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Answer 2

I think there is no particular name. However, you can introduce your set as follows:

$$\mathcal P_0 (A):=\mathcal P(A) \setminus\{\varnothing\}$$

recalling that, mutatis mutandis, often $\mathbb{N}_0$ is used in place of $\mathbb{N}$ when one wants to represent the natural number without the $0$:

$$\mathbb{N}_0 = \mathbb{N} \setminus \{0\}$$

or to impose that $0$ is inside the set:

$$\mathbb{N} = \mathbb{N}_0 \cup \{0\}$$