Basic concepts question.

I am used to the Cartesian product of two sets: $A \times B = \{(a,b) \mid a \in A, b \in B\}$.

Is there an operator that produces sets instead of tuples? We might call it set-product and define $A \otimes B := \{a \cup b \mid a\in A, b\in B\}$.

What I am actually trying to do is "factorize" a given set-family $\mathcal M \subseteq 2^V$ where $V$ is the base set. That is, I would like to write $\mathcal M = M_1 \otimes M_2 \otimes M_3$ with $M_i \subseteq 2^V$.

example:

• $\mathcal M = \{\{1,2\}, \{1,3\}, \{1,6\}\}$ can be factorized into $\{\{1\}\} \otimes \{\{2\},\{3\},\{6\}\}$

My questions are:

• do people use this operator (if so, what is it called? I don't want to reinvent the wheel)?
• do you know of results relating to the factorizability of sets in this way?

My goal is to connect what I am doing to existing work.

EDIT:

• fixed errors in question and definition of $\otimes$