Is there a dot product of sets?

Question

Let's say I have sets $\{1,2,3\}$ and $\{x,y,z\}$, I'd like the output to be $\{1x, 2y, 3z\}$, is there such an operation?

Let's say I have sets $\{1,2,3\}$ and $\{x,y,z\}$, I'd like the output to be $\{1x,1y,1z,2x,2y,2z,3x,3y,3z\}$, is there such an operation?

The specific case I'm trying to deal with is a set of tuples $\{(s_1, c_1), (s_2, c_2), ...\}$ where $s_i$ is a set, and $c_i$ is a constant. Given 2 of such sets, I'd like to find an operation that allows me to do the following:

$$A \,(something) \, B = \{ \;(s_i \cap s_j, c_i * c_j) \; | \;(s_i, c_i) \in A, (s_j, c_j) \in B \; \}$$

Thanks!

Answer 1

No, there is no dot product of sets.

Your first example is problematic, because

$$\{1,2,3\} = \{2,1,3\}$$

so the product could equally be $\{2x, y, 3z\}$. So what you are describing is not well defined