If $A$ is a set and $\mathcal B$ is a set of sets, is there some shorthand for $\left\{A\times B:B\in\mathcal B\right\}$?

Question

Let $A$ be a set and $\mathcal B$ be a set of sets. Suppose we want to define $$M:=\left\{A\times B:B\in\mathcal B\right\}\;.$$ Is there some shorthand for $M$ as we've got for $$X\times Y=\left\{(x,y):x\in X,y\in Y\right\}\;?$$

Answer 1

You could perhaps use $$ M={\huge\cup}_{B\in\mathcal{B}}\{A\times B\} $$

Answer 2

A good thing in maths is the freedom to concoct new notation when needed. I could say, for example, that a natural number $n$ is reluctant if its lesser prime factor $p$ is greater than $\sqrt[3]n$. After that, it is my responsability giving some use to the new name: theorems, theories and the like. This will not the case now, of course.

In algebra it is already common to write $AB$ for the set $\{ab:a\in A,b\in B\}$ when $A$ and $B$ are for example, subsets of the same group. Sadly, we can't write $A\times\mathcal B$ for your set $M$, since this means another thing: a set of sets is... well, a set, and the operator $\times$ applies on sets.

For your situation, if I wanted to create something new, I'd write $\{A\times B\}_{B\in\mathcal B}$.