How do hyperoperations like tetration exist if operations are seperate relations and not repeatitions of each other.

I've run into a bit of a conflict in my fundamental understanding of concepts in math. I've always known the arithmetic operation to be extensions of each other. Multiplication is repeated addition, exponentiation is repeated multiplication, tetration is repeated exponentiation, etc. The issue is, I've also run into the idea that each of these operations is a its own separate relation (generally a 3-tuple one where two of the members are operands and one is an output). I'm a bit confused on how these two concepts don't go against each other.

• Why would they go against each other? – ASKASK Mar 30 '16 at 6:11
• I suggest you to analize carefully the meaning of "Something IS repetead Something". If you try to think about it in a more rigorous way you may find that it is compatible to the $3$-uple definition of an operation. In other words it makes sense for a "$3$-ary relation TO BE the ITERATION of another $3$-ary relation" as long as you have a way to define what ITERATION is (repetition) – MphLee Mar 31 '16 at 7:02
• I also suggest you to pay more attention to the terms you use because "to be an EXTENSION" can have a very precise meaning and the usual and most natural one doesn't apply in this case: Multiplication IS NOT an extension of Addition because $+$ is not a subset of $\times$ (if we see them as $3$-ary relation over the naturals $\Bbb N$) – MphLee Mar 31 '16 at 7:08