When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific values of the number system from which a system is constructed?
As an example, if I want to define addition for a number system constructed from nonnegative reals:
(a,b) + (c,d) = [((a+c)/2), ((b+d)/3)]
Is it permissible to include digits that already belong to the reals? The operation of division? What if I wanted to use a signum function?
I recognize that there may be prohibitions against such things when it comes to operation definitions; if they are off-limits, I would appreciate direction to resources or explanations that indicate the basis for such prohibitions.