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 $A \otimes B := \{\{a, b\} \mid a\in A, b\in B\}$.

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 $M_i \subseteq V$.

_example_: 
- $\mathcal M = \{\{1,2\}, \{1,3\}, \{1,6\}\}$
- $\mathcal M = \{1\}\otimes \{3,6\}$