Mixins do not have to implement anything. Again, you're artificially restricting what's possible. It's sometimes convenient to use mixins to convey relationships among types. I have a mother and a father. My mother has a mother and a father. So does my father. That means my maternal-side cousin and I have grandparents in common and my paternal-side cousin and I also have grandparents in common, but a different set.
In more concrete terms, an add expression is a binary operator. So is a subtract expression. Both are also associative, but only one is commutative. That's a useful thing to express in class hierarchies and using mixins is a great way to do it.