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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.

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I am not sure I understand your question, but I will try to answer what I think you are asking.

The operation you have defined that maps two ordered pairs of real numbers (not necessarily positive) to another pair of real numbers is well defined. That fact that it mentions $2$ and $3$ is not a problem.

I see nothing in your example that addresses representatives of equivalence classes.

I hope you chose that just as an example. It's probably not particularly nice, or useful.

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