Need an operator on sets similar to Cartesian product

Question

Suppose we have two sets, $A = \{S_1, S_2\}, B = \{T_1,T_2\}$ where $S_1, S_2$ and $T_1,T_2$ are also sets. Is there an operator (already widely used) such that $$\operatorname{Operator}(A, B) = \{\{S_1\cup T_1\},\{S_1\cup T_2\},\{S_2\cup T_1\},\{S_2\cup T_2\}\}?$$

Answer 1

There is no widely-used symbol or name attached to the operation you describe.

Feel free to come up with a descriptive name that pleases you.