Beyond + and -

[Benny]

considering that and we want 3 signs , it seems we should stop trying to extend the normal + and - and start with 3 totally new signs !!

the addition remains the same but i propose the multiplication table is
that they act like the 3 roots of unity.

That sounds like a cool idea. Maybe the problem isn't §, but the existing definitions for + and -. They are coherent on their own, but not neatly extendable.