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added two examples of "building blocks" commonly used in the classical theory of factorization
Salvo Tringali
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As far as I understand, you start with a "base set" $V$ and consider the monoid obtained by endowing the power set of the power set of $V$ with the binary operation that sends a pair $(A, B)$ of families of subsets of $V$ to $\{X \cup Y \colon X \in A,\, Y \in B\}$, which is still a family of subsets of $V$. If my understanding is correct (is it?), then you may want to read about power monoids, especially if you are interested in questions relating to factorization (as it seems that you are).

Given a (multiplicatively written) monoid $H$, the full power monoid of $H$ is the monoid $\mathcal P(H)$ obtained by equipping the power set of $H$ with the operation of setwise multiplication induced by $H$, so that $AB := \{ab: a \in A, \, b \in B\}$ for all $A, B \in \mathcal P(H)$. The OP is about the special case where $H$ is the monoid obtained by endowing the power set of a certain set $V$ with the binary operation $(X, Y) \mapsto X \cup Y$. So the answer to the question:

do you know of results relating to the factorizability of sets in this way?

is yes in the sense that this is a special case of a broader class of monoids where people (including myself) have studied, and are studying, various properties that can be labelled with the "Factorization" tag. However, you may want to consider that, before even starting to tackle the basic question of the existence of a factorization for certain elements of the monoid you're implicitly considering (and, more generally, of any monoid), you should somehow decide which "building blocks" are allowed in the factorization process, which is a (delicate) question of independent interest.

From the perspective of the classical theory (of factorization), the "canonical choice" would be to use as building blocks either the atoms or the irreducibles of the monoid $M$ under consideration, where an atom is a non-unit $a \in M$ such that $a \ne xy$ for all non-units $x, y \in M$ (this definition is commonly attributed to P.M. Cohn), and an irreducible is a non-unit $a \in M$ such that $a \ne xy$ for all $x, y \in M$ such that $MyM \ne MaM \ne MxM$ (in the commutative case, this definition is commonly attributed to D.D. Anderson & S. Valdes-Leon). However, none of these choices is fit for your case, unless $V$ is a finite set.

By the way, product-sets in all sorts of monoids (and, in particular, sumsets in additively written groups) are one of the main objects of study in arithmetic combinatorics. So, stitching the pieces together, I would say that also the answer to the question:

do people use this operator (if so, what is it called? I don't want to reinvent the wheel)?

is yes (and you are reinventing the wheel).

Salvo Tringali
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