As I understand it so far, it's any operation which takes two elements of a given set and outputs another member of that same set. f:S×S→S

I understand that with the natural numbers ℕ, addition and multiplication are allowed but subtraction and division are not because they allow for negative and rational numbers which are not members of ℕ.

I'm a little more iffy with integers ℤ because I was told that only addition and multiplication were allowed for the integers. I know why division is not allowed, but what about subtraction? For any two members of ℤ, there's no subtraction that would return elements that are not in ℤ.

I just want to get a better understanding of this.