Mathematicians love abstraction, and the field of abstract algebra is no exception. I'm bringing it out here to make it easier to find than in The Math Thread.

Consider some properties of addition:

- It is commutative: a + b = b + a
- It is associative: (a + b) + c = a + (b + c)
- It has an identity: a + 0 = a
- It has inverses: a + (-a) = 0

Take those properties and make some binary (two-argument) operator have them or else have only some of them. What possible operators can there be?

Try adding a unary operator or another binary operator, especially one that is intertwined with the original binary operator in some way. What further possible operators can there be?

But why start with a binary operator? Why not a unary (one-argument) or a ternary (three-argument) one? Unary ones are almost too simple, while ternary ones has not been researched very much.