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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 $$\operatorname{Operator}(A, B) = \{\{S_1\cup T_1\},\{S_1\cup T_2\},\{S_2\cup T_1\},\{S_2\cup T_2\}\}?$$

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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.

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