Is there a formal definition for a system where you can perform arithmetic operations on sets, for instance, $2 \cdot \mathbb{Z}$ = the set of even integers?
If there is, what is it called?
What happens when I add two sets?
(Note: I have no idea what to tag this.)
When you have a function $f$, it is customary to abuse the notation to apply $f$ on sets of elements: $$ f(X) = \{ f(x)\colon x \in X\}. $$
So it is not strange to define $$ 2 X = \{2x \colon x \in X\} $$ and $$ X + Y = \{ x+y \colon x \in X,\ y \in Y\}. $$
This is only a notation no special matematical systems are required.