Has this variant on multiplication by a natural number been studied before?

Question

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?

Answer 1

$n \star A\,$ is called the n-fold iterated sumset on the wikipedia sumset page (which also includes more references):

In additive combinatorics, the sumset (also called the Minkowski sum) of two subsets $A$ and $B$ of an abelian group $G$ (written additively) is defined to be the set of all sums of an element from $A$ with an element from $B$. That is, $$\displaystyle A+B=\{a+b:a\in A,b\in B\}$$ The n-fold iterated sumset of $A$ is $$\displaystyle nA=A+\cdots +A$$ where there are $n$ summands.

Answer 2

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 $\{e\}$.