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The monoid is the set of all sets of integers (but reals or complex numbers could work too). Addition between two elements is defined as $a+b = \{\ x+y\ |\ x \in a,\ y \in b\ \}$. As far as I can tell, the only category of algebraic structures this fits into is monoids, since it's not a group.

Is this structure known or used anywhere, and does it have a name?

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This operation is called Minkowski addition. I'm not aware of any special name for the algebraic structure of $\mathcal{P}(\mathbb{Z})$ (or whatever) equipped with Minkowski addition though.

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Another name for the operation is sumset, and it's studied in Arithmetic combinatronics.

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More generally, if $M$ is a monoid, the set $\mathcal{P}(M)$ of subsets of $M$ is a monoid for the following operation: given $X, Y \in \mathcal{P}(M)$, $$ XY = \{xy \mid x \in X, y \in Y \} $$ The monoid $\mathcal{P}(M)$ is sometimes called the power monoid of $M$.

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