qemist

Lets look at your group-like structure then with distinct left and right identities, and associativity. I feel it's a bit of an abuse of notation to use (-) for both the right and left inverses, because with two distinct identities a can have distinct left and right inverses. So to make things simpler to write out, I'll use juxtaposition as the operator and call R the right identity and L left identity. Then from what you used to define the structure: La = a = aR aa-R = R a-La = L So far this is all the same as what you wrote. We can't know what RL is from the properties we have here, it could be R or L or any other element since neither are acting as an identity in this case. (We could define it to be R as you did, but that's redundant because the rest of the proof shows that RL = R = L) So lets look at LR. LR = (a-La)R = a-L(aR) = a-La = L LR = L(aa-R) = (La)a-R = aa-R = R Therefore L = R, which is a contradiction. So a group-like structure (I'm assuming you mean any structure with a set and a closed binary operation on that set) with inverses, associativity, a left identity, and a right identity (doesn't matter whether the left is the same as the right) as described above by you is actually a group, and a group has a unique identity.

0L + 0R = 0R is just as natural, which implies that 0L = 0R. Therefore the identity element 0 = 0L = 0R is two-sided and unique. This is true in all the algebraic structures I've ever studied. The definition of an identity is an element e of a set with an operation * such that either e*a = a or a*e = a for all elements a belonging to the set. This means that an identity would have act as an identity on any other identities in the set. If it didn't then it couldn't be a proper identity in the first place.

After doing a bit of reading it seems like the closest thing to having more than one zero is in some semigroups where there can be multiple left neutral elements or multiple right neutral elements (I feel like there should be a more general word for these, maybe multiple same-sided neutral elements?), but if there is at least one of each (like in K^2's "proof", with 0L and 0R) then they all end up being equal and the neutral element is unique.

There's no meaningful difference at all. Adding a negative sign in front of a number like that signifies it's additive inverse. Basically any number x has a unique number that we call -x such that x + (-x) = 0. (This is analogous to the idea of a multiplicative inverse, where say 1/3 * 3 = 1). 0 is special because its additive inverse happens to be itself, it's the only number you can add to 0 to make 0.