Has this variant on multiplication by a natural number been studied before? Let $X$ denote an additively-denote commutative monoid. Then we get an action $\star$ of $\mathbb{N}$ on the powerset $\mathcal{P}(X)$ as follows: given a natural number $n$ and a set $A \subseteq X$, define $$n \star A = \left\{x \in X : \exists_{I \in \mathbf{FinSet}}\left(|I|=n \wedge \exists_{f:I \rightarrow A}\left(x = \sum_{i \in I} f(i) \right)\right)\right\}.$$
For example:


*

*$2 \star \{x\} = \{2x\}$

*$3 \star \{x\} = \{3x\}$

*$2 \star \{x,y\} = \{2x,x+y,2y\}$

*$3 \star \{x,y\} = \{3x,2x+y,x+2y,3y\}$

*$2 \star \{x,y,z\} = \{2x,2y,2z,x+y,x+z,y+z\}$


So it's basically a variant on multiplication by a natural number scalar in which the thing getting added to itself $n$ times is allowed to vary. Kind of reminds me of multichoose.

Question. Has this variant on scalar multiplication studied before? If so, what is it called, and where can I learn more?


This looks like some sort of convolution on sets. For any two sets $A, B \in P(X)$, define the product
$$ A \cdot B = \{a + b \mid a \in A, b \in B\}$$
For example, we have that
$$ \{x, y\} \cdot \{x, y\} = \{2x, x + y, 2y\} $$
and in general, the star operator $n \star A$ is the $n$-fold product of $A$.
Note that if $e \in X$ is the monoid unit, then $(P(X), \cdot)$ is a monoid with unit $\{e\}$. 
