Arithmetic operations on sets

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.)

• See math.stackexchange.com/questions/659128/… for addition of sets. – NickC Nov 6 '14 at 22:31
• What you mean by adding two sets? If you think that $2\mathbb{Z} = \mathbb{Z} + \mathbb{Z}$, you're wrong. – brick Nov 6 '14 at 22:32
• @brick I didn't expect $2\mathbb{Z}$ to necessarily be $\mathbb{Z} + \mathbb{Z}$, but was wondering if there was addition where it was. I see that "Minkowski sums" seem to be the standard definition for adding sets? In that case, $\mathbb{R} + \mathbb{R} = \mathbb{R}^2$, right? – porglezomp Nov 6 '14 at 23:21
• With the notation suggested in my answer below, $\mathbb Z + \mathbb Z = \mathbb Z$, since every integer is the sum of two integers. – Emanuele Paolini Nov 7 '14 at 13:57

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 $$X + Y = \{ x+y \colon x \in X,\ y \in Y\}.$$